05. The Fundamental Unit of Life – Textbook Solutions

Page No. 51

Q1. Who discovered cells and how?
Ans:

  • The cell was first discovered by Robert Hooke in 1665. He examined thin slices of cork under a self-made microscope and saw a multitude of tiny hollow spaces that he remarked looked like the walled compartments of a honeycomb. 
  • He termed these spaces as ‘cell’ meaning ‘small room’ in Latin.
    Microscope and Cork cells

Q2. Why is the cell called the structural and functional unit of life?
Ans:

  • All living organisms are made up of cells. This shows that the cell is the structural unit of life. 
  • Each living cell can perform certain basic functions that are characteristic of all living forms. 
    Example: 
    (i) Phagocytic cells eat or kill unwanted or foreign particles inside the body (e.g., WBCs). 
    (ii) Some cells secrete enzymes and hormones, e.g., pancreatic cells, small intestinal cells, and liver cells.

Page No. 53

Q1. How do substances like COand water move in and out of the cell? Discuss.
Ans: COmoves by diffusion – This cellular waste accumulates in high concentrations in the cell, whereas the concentration of CO2 in the external surroundings is comparatively lower. This difference in the concentration level inside and outside of the cell causes the CO2 to diffuse from a higher (within the cell) region to a lower concentration.
H2O diffuses by osmosis through the cell membrane. It moves from a region of higher concentration to a lower concentrated region through a selectively permeable membrane until equilibrium is reached.


Q2. Why is the plasma membrane called a selectively permeable membrane?
Ans: The plasma membrane allows or permits the entry and exit of some materials in and out of the cell and prevents the movement of some other materials through it. Hence, it is called a selectively permeable membrane.Page No. 55

Q1. Fill in the gaps in the following table illustrating the differences between prokaryotic and eukaryotic cells.

Ans:Page No. 57

Q1. Can you name the two organelles we have studied that contain their genetic material?
Ans: Two organelles that contain their genetic material are Mitochondria and Plastids. Mitochondria help in respiration in the cell, while plastids are responsible for the process of photosynthesis in leaves. 


Q2. If the organization of a cell is destroyed due to some physical or chemical influence, what will happen?
Ans:

  • Cell organelles are responsible for the organization and proper functioning of a cell, as each of them performs some specific functions. 
  • Naturally, if any of these organelles are destroyed, the cell will not be able to perform many basic functions like photosynthesis, respiration, nutrition, etc., and may also result in the stopping of all life activities in the cell. Due to the cell damage, the lysosome bursts, and their enzymes digest such cells.


Q3. Why are lysosomes known as suicide bags?
Ans:

  • Lysosomes are the cell organelles involved in the digestion of any foreign material that enters the cell, as they contain digestive enzymes. 
  • In case any cell is dead or damaged, the lysosome bursts to release the digestive enzymes to digest its cell. 
  • Thus, these are known as ‘suicide bags’.


Q4. Where are proteins synthesized inside the cell?
Ans:

  • Proteins are the building blocks of the body that are made up of various amino acids. They are synthesized inside the ribosomes, are the small structures present in the cytoplasm, or might be attached to the surface of the endoplasmic reticulum. The endoplasmic reticulum is called the rough endoplasmic reticulum due to the presence of ribosomes on its surface. 
  • In simpler words, proteins are the body’s building blocks, made of amino acids. They are created inside ribosomes, which are small structures found in the cytoplasm or attached to the endoplasmic reticulum. When ribosomes are attached to the endoplasmic reticulum, it’s called the rough endoplasmic reticulum because the ribosomes give it a rough surface. 

Page No. 59

Q1. Make a comparison and write down ways in which plant cells are different from animal cells.
Ans:



Q2. How is a prokaryotic cell different from a eukaryotic cell?
Ans:


Q3. What would happen if the plasma membrane ruptures or breaks down?
Ans:

  • The plasma membrane is the selectively permeable membrane that surrounds the cell and allows the entry and exit of selected materials of the cell. 
  • If it ruptures, the contents of the cell will come in direct contact with the surrounding medium, and not only unwanted material will be able to enter freely into the cell, but useful material will also find its way out of the cell easily. 
  • This will seriously disrupt the various metabolic activities of the cell and will result in its imminent death.

Q4. What would happen to the life of a cell if there was no Golgi apparatus?
Ans:

  • If there were no Golgi apparatus, the material synthesized by the endoplasmic reticulum would not be carried to the various parts inside and outside of the cell.
  • Also, as the Golgi apparatus performs the function of storage and modification of the material synthesized in the cell, these materials would not be stored and modified further.
  • Moreover, there will be no production of lysosomes, which will cause the accumulation of waste material, viz., worn out and dead cell organelles within the cell, which will ultimately lead to cell death.


Q5. Which organelle is known as the powerhouse of the cell? Why
Ans:

  • Mitochondria are known as the powerhouse of the cell because these are the sites of cellular respiration.
    Mitochondria
  • The energy that is released by the mitochondria is in the form of ATP molecules and is required for various chemical activities needed for. 
  • The energy stored in ATP is used by the body to make new chemical compounds and to perform mechanical work. 
  • Thus, mitochondria are known as the powerhouse of cells.


Q6. Where do the lipids and proteins constituting the cell membrane get synthesized?
Ans:

  • Lipids and proteins are the essential parts of the plasma membrane, which are synthesized through the endoplasmic reticulum. 
  • The endoplasmic reticulum is found to be of two types based on the substances they synthesize. 
  • The smooth endoplasmic reticulum is responsible for the synthesis of lipids, while the rough endoplasmic reticulum is responsible for the synthesis of proteins. 


Q7. How does an Amoeba obtain its food?
Ans:

  • Amoeba takes in food using temporary finger-like extensions of the cell surface, which fuse over the food particle, forming a food vacuole as shown in the figure. 
  • Inside the food vacuole, complex substances are broken down into simpler ones, which then diffuse into the cytoplasm. 
  • The remaining undigested material is moved to the surface of the cell and thrown out.
    Nutrition in Amoeba

Q8. What is osmosis?
Ans: It is the passage of solvent from a region of high concentration to a region of low concentration through a semipermeable membrane.

Q9. Carry out the following osmosis experiment
Take four peeled potato halves and scoop each one out to make potato cups. One of these potato cups should be made from a boiled potato. Put each potato cup in a trough containing water. Now,
(a) Keep cup A empty.
(b) Put one teaspoon of sugar in cup B
(c) Put one teaspoon salt in cup C
(d) Put one teaspoon of sugar in the boiled potato cup D.
Keep these for two hours. Then observe the four potato cups and answer the following
(i) Explain why water gathers in the hollowed portion of B and C.
(ii) Why is potato A necessary for this experiment?
(iii) Explain why water does not gather in the hollowed-out portions of A and D.

Ans: 
(i) 

  • Water gathers in the hollowed portion of cups B and C because of the process of endosmosis (moving in of the solvent). The potato wall acts as a semi-permeable membrane. 
  • As cups B and C are filled with sugar and salt, respectively, and their outer part is in contact with the water, the concentration of water outside the cups is higher than inside the cups. 
  • So, water moves from its higher concentration towards the lower concentration, i.e., inside the cup. 

(ii) Potato A is necessary for this experiment because:

  • Potato A acts as a control of the experiment. It is very necessary to compare the results of the experiment. 
  • It shows that if the concentration of water is the same on both sides, there will be no movement of water. 

(iii) Water does not gather in the hollowed-out portions of A and D because:

  • Water does not gather in the hollowed-out portions of A as it does not contain a hypertonic solution, so there is no concentration difference and hence no movement of solvent. 
  • Water does not gather in cup D as the cells of the boiled potato are dead, and the potato wall is no longer semi-permeable. Hence, no osmosis occurs.


Q10. Which type of cell division is required for the growth and repair of the body and which type is involved in the formation of gametes?
Ans:

  • For growth and repair, mitotic division (mitosis) is involved as this type of division keeps the chromosome number constant.
  • For gamete formation, meiosis is involved as a reduction of chromosome number is necessary for this case.

04. Structure of the Atom – Textbook Solutions

Page No. 39

Q1. What are canal rays?
Ans: Canal rays are a type of positively charged radiation discovered by E. Goldstein in 1886. They are produced in a gas discharge and are significant for the following reasons:

  • They helped in the identification of protons, a key subatomic particle.
  • Canal rays consist of positive ions that move towards the cathode in a discharge tube.
  • Their discovery contributed to the understanding of atomic structure.

Goldstein’s experimental setup
Q2. If an atom contains one electron and one proton, will it carry any charge or not?
Ans: No, it will not carry any charge because the number of protons (positively charged) is equal to the number of electrons (negatively charged).


Page No. 41
Q1. On the basis of Thomson’s model of an atom, explain how the atom is neutral as a whole.
Ans: According to Thomson’s model, an atom is made up of a sphere of positive charge with electrons embedded within it. This arrangement is designed to create a stable electrostatic balance.

  • The atom is neutral because it contains an equal number of electrons and positively charged particles.
  • The positive charge of the sphere balances the negative charge of the electrons.
  • Thus, the overall charge of the atom is zero, making it electrically neutral.

Thomson’s Model of an atom

Q2. On the basis of Rutherford’s model of an atom, which subatomic particle is present in the nucleus of an atom?
Ans: According to Rutherford’s model of the atom:

  • Protons are present in the nucleus, giving it a positive charge.
  • The nucleus contains most of the atom’s mass.
  • Neutrons are also found in the nucleus, contributing to the overall mass.

Rutherford’s Model of an atom
Q3. Draw a sketch of Bohr’s model of an atom with three shells.
Ans: 

A line diagram showing different energy levels of energies E1, E2 and E3 for the electrons

Q4. What do you think would be the observation if the α-particle scattering experiment is carried out using a foil of a metal other than gold?
Ans: If the metal taken is heavier than gold, then the deviation shown by α-particles will be more, and if the metal taken is lighter than gold, then the deviation shown will be less due to less repulsion between positively α-particles and nucleus.


Q5. Name the three sub-atomic particles of an atom.
Ans: The three sub-atomic particles of an atom are:
(i) Protons
(ii) Electrons
(iii) Neutrons


Q6. Helium atom has an atomic mass of 4 u and two protons in its nucleus. How many neutrons does it have?
Ans: Number of neutrons = Atomic mass – Number of protons 
For a helium atom:

  • Atomic mass = 4 u
  • Number of protons = 2
  • Therefore, number of neutrons = 4 – 2 = 2

Page No. 42

Q1. Write the distribution of electrons in carbon and sodium atoms.
Ans: 

 (a) Electronic distribution in a carbon atom:

  • Atomic number: 6
  • Number of electrons: 6
  • Distribution: K = 2, L = 4

(b) Distribution of electrons in a sodium atom:

  • Atomic number: 11
  • Number of electrons: 11
  • Distribution: K = 2, L = 8, M = 1


Q2. If K and L shell of an atom are full, then what would be the total number of electrons in the atom?
Ans. The maximum number of electrons in an atom with full K and L shells is:

  • K shell: 2 electrons
  • L shell: 8 electrons

Therefore, the total number of electrons in the atom is:

  • Total: 10 electrons

Page No. 44

Q1. How will you find the valency of chlorine, sulphur and magnesium?
Ans: To find the valency of chlorine, sulphur, and magnesium:

Chlorine (Cl):

  • Atomic number: 17
  • Electrons: 2, 8, 7
  • Can gain 1 electron to achieve stability (like argon).
  • Valency: 1

Sulphur (S):

  • Atomic number: 16
  • Electrons: 2, 8, 6
  • Can gain 2 electrons to achieve stability (like argon).
  • Valency: 2

Magnesium (Mg):

  • Atomic number: 12
  • Electrons: 2, 8, 2
  • Can lose 2 electrons to achieve stability (like neon).
  • Valency: 2


Q2. If number of electrons in an atom is 8 and number of protons is also 8, then 
(i) what is the atomic number of the atom and 
(ii) what is the charge on the atom?
Ans:
(i) The atomic number is equal to the number of protons in an atom. In this case:
Atomic number = Number of protons = 8
(ii)  The charge of an atom is determined by the balance between protons and electrons.

  • Since the number of electrons (8) is equal to the number of protons (8), the atom has no overall charge.
  • Thus, the charge on the atom is zero.


Q3. With the help of Table 4.1, find out the mass number of oxygen and sulphur atom.

Ans: 
(a) To find the mass number of Oxygen:
Number of protons = 8
Number of neutrons = 8
Atomic number = 8
Atomic mass number = Number of protons + number of neutrons = 8 + 8 = 16
Therefore, mass number of oxygen = 16
(b) To find the mass number of Sulphur:
Number of protons = 16
Number of neutrons = 16
Atomic number = 16
Atomic mass number = Number of protons + number of neutrons = 16 + 16 = 32

Page No. 45

Q1. For the symbol H, D and T tabulate three sub-atomic particles found in each of them.
Ans: 

Hydrogen (H): The most common isotope, consisting of a single proton and no neutrons.

Deuterium (D): A heavier isotope of hydrogen, with one neutron in addition to the proton.

Tritium (T): A radioactive isotope of hydrogen, containing two neutrons and one proton.


Q2. Write the electronic configuration of any one pair of isotopes and isobars.
Ans: 

Isotopes : Isotopes have the same electronic configuration because they have the same number of electrons and protons, differing only in the number of neutrons.
Example : 12C6 and 14C6 are isotopes, have the same electronic configuration as (2, 4)

Electronic Configuration(2,4) of Carbon atom

Isobars: Isobars have the same mass number but differ in atomic number, leading to different electronic configurations.

Example :  22Ne10 and 22Na11 are isobars. They have different atomic number but mass number is same. 

Page No. 46, 47 & 48

Q1. Compare the properties of electrons, protons and neutrons.

Ans: ElectronProtonNeutron(a) It is negatively charged.(a) It is positively charged.(a) It is neutral.(b) Its absolute mass is equal to 9.1 x 10-31 kg. (b) Its absolute mass is equal to 1.673 x 10-27 kg. (b) Its mass is slightly more than that of protons. Its absolute mass is 1.675 x 10-27 kg.


Q2. What are the limitations of J.J. Thomson’s model of the atom?
Ans: Limitations of JJ Thomson’s model of an atom:
(i) The results of experiments carried out by other scientists could not be explained by this model
(ii) The positive charge is spread all over could not be justified.


Q3. What are the limitations of Rutherford’s model of the atom?
Ans: Limitations of Rutherford’s model of an atom
(i) According to classical electromagnetic theory, a moving charged particle, such as an electron, under the influence of attractive force, loses energy continuously in the form of radiations. As a result of this, the electron should lose energy and, therefore, should move in even smaller orbits, ultimately falling into the nucleus. But the collapse does not occur. There is no explanation for this behaviour.
(ii) According to Rutherford, the electrons revolve around the nucleus in fixed orbits. However, Rutherford did not specify the number of orbits and the number of electrons in each orbit.


Q4. Describe Bohr’s model of the atom.
Ans: Bohr made a bold new suggestion that particles at the atomic level would behave differently from macroscopic (bigger) objects.
According to Bohr’s theory:

  • Electrons travel in specific paths called orbits or energy levels.
  • While in these orbits, electrons do not emit energy.
  • Electrons can move to a higher energy level when they absorb energy.
  • When electrons drop back to a lower energy level, they release energy.

This model helped explain the stability of atoms and their emission spectra.

Bohr’s Model of an atom


Q5. Compare all the proposed models of an atom given in this chapter.
Ans: Thomson modelRutherford modelBohr model(a) The positive charge is spread all over and electrons are embedded in it like seeds in the watermelon. (a) There is a positively charged centre called the nucleus where the mass of the atom is concentrated.(a) Electrons revolve in certain discrete orbits called energy levels.(b ) The positive and negative charges are equal, therefore the atom is neutral.(b) Electrons revolve around the nucleus in well-defined orbits.(b) While revolving in these orbits, electrons do not radiate energy.(c) The nucleus is very small compared to the overall size of the atom.
(c) The size of the nucleus is very small as compared to the size of an atom.(c) These orbits are represented by K, L, M, N or 1, 2, 3, 4.

Q6. Summarize the rules for writing of distribution of electrons in various shells for the first eighteen elements.
Ans: The distribution of electrons in the shells of an atom is known as electronic configuration. To write the electronic configuration for an element, follow these steps:

  • Determine the total number of electrons using the element’s atomic number.
  • Distribute these electrons across the shells in order of increasing energy, following the 2n2 rule.
  • Fill the shell with the lowest energy first, then proceed to the next higher energy shells.
  • List the number of electrons in each shell, starting from the lowest, separated by commas. For example, sodium (atomic number 11) has:

Electronic configuration of Na: K = 2, L = 8, M = 1 

The maximum number of electrons in different shells is:

  • K-shell (1st shell) = 2
  • L-shell (2nd shell) = 8
  • M-shell (3rd shell) = 18
  • N-shell (4th shell) = 32

Key points to remember:

  • The outermost shell can hold a maximum of 8 electrons.
  • Electrons fill shells in a step-wise manner; inner shells must be filled before outer shells.


Q7. Define valency by taking examples of silicon and oxygen.
Ans: The valency of an atom is the number of hydrogen atoms which combine with one atom of an element.
For example, 

  • Silicon (atomic number 14) has the following electronic distribution:
    K = 2, L = 8, M = 4.
    In the outermost shell, there are 4 electrons; hence, one atom of silicon will combine with 4 atoms of hydrogen. So, the valency of silicon is 4.
  • In the case of oxygen (atomic number 8), the electronic distribution in various shells is given below:
    K = 2, L = 6
    There are six valence electrons in the atom of oxygen. So, one atom of oxygen will combine with 2 atoms of hydrogen to make a noble gas configuration. Hence, the valency of oxygen is 2.

Q8. Explain with examples (i) Atomic number, (ii) Mass number, (iii) Isotopes and (iv) Isobars. Give any two uses of isotopes.
Ans: 
(i) Atomic number: The atomic number of an atom is equal to the number of protons in the nucleus of an atom.
Atomic number = Number of protons.
(ii) Mass number: The total mass of the atom is due to the sum of the masses of protons and neutrons present in the nucleus. The mass number of an atom is equal to the sum of the number of protons and neutrons in the nucleus.
Mass number = Number of protons + Number of neutrons.
(iii) Isotopes: Isotopes are the atoms of the same element having the same atomic number but different mass numbers. For example, Hydrogen has three isotopes: protium (1), deuterium (2), and tritium (3).
(iv) Isobars: Isobars are the atoms of the different elements having the same mass number and different atomic numbers. For example, Calcium (atomic number 20) and argon (atomic number 18) both have a mass number of 40.


Uses of isotopes:
(i) Cobalt-60 is a radioactive isotope of cobalt. It is used in radiotherapy for cancer.
(ii) 14C is a radioactive isotope of carbon. It is used in carbon dating.


Q9. Na+  has completely filled K and L shells. Explain.
Ans: Sodium has the atomic number 11. It has 11 electrons in its orbitals, wherein the number of protons is equal to the number of electrons. Hence, its electronic configuration is K-2, L-8, and M-1. The one electron in the M shell is lost, and it obtains a positive charge since it has one more proton than electrons and obtains a positive charge, Na+. Na+ is formed when one electron is lost by a sodium atom, as given below:

The new electronic configuration is K-1; L-8 which is the filled state. Hence it is very difficult to eliminate the electron from a filled state as it is very stable.


Q10. If bromine atom is available in the form of, say, two isotopes  79Br35 (49.7%) and 81Br35 (50.3%) calculate the average atomic mass of bromine atom.
Ans: Average atomic mass of Br
= ( (79 x 49.7) + (81 x 50.3) ) / 100
= (3926.3 + 4074.3) / 100
= 8000.6 / 100
= 80.006 u


Q11. The average atomic mass of a sample of an element X is 16.2 u. What are the percentages of isotopes  in the sample?
Ans: It is given that the average atomic mass of the sample of element X is 16.2 u.
Let the percentage of isotope 18 / 8 X be y%. Thus, the percentage of isotope 16 / 8 X will be (100 – y) %.
Therefore,
18 x (y/100) + 16 x ((100-y)/100) = 16.2
18y/100 + ( 16(100 – y) / 100 ) = 16.2
(18y+1600-16y )/100 = 16.2
18y + 1600 – 16y = 1620
2y + 1600 = 1620
2y = 1620 – 1600
y = 10
Therefore, the percentage of isotope 18 / 8 X is 10%.
And, the percentage of isotope 16 / 8 X is (100 – 10) % = 90%.


Q12. If Z = 3, what would be the valency of the element? Also, name the element.
Ans. Z = 3
Electronic configuration is 2, 1.
Valence electron is equal to 1.
Valency is also equal to 1.
The name of the element is lithium.


Q13. Composition of the nuclei of two atomic species X and Y are given as under

Give the mass numbers of X and Y. What is the relation between the two species?
Ans: Mass number of X = 6 + 6 = 12 
Mass number of Y = 6 + 8 = 14
Atomic number of X = 6 
Atomic number of Y = 6
X and Y are isotopes since both the elements have same atomic number but different mass number.


Q14. For the following statements, write T for ‘True’ and F for ‘False’.
(a) J.J. Thomson proposed that the nucleus of an atom contains only nucleons.
(b) A neutron is formed by an electron and a proton combining together. Therefore, it is neutral.
(c) The mass of an electron is about 1 / 2000 times that of proton.
(d) An isotope of iodine is used for making tincture iodine, which is used as a medicine.

Ans. (a) False (b) False (c) True (d) False

Q15. Rutherford’s alpha-particle scattering experiment was responsible for the discovery of
(a) Atomic nucleus
(b) Electron
(c) Proton
(d) Neutron

Ans: (a) Atomic nucleus
An atomic nucleus is the dense central core of an atom, composed of protons and neutrons held together by the strong nuclear force.


Q16. Isotopes of an element have
(a) the same physical properties
(b) different chemical properties
(c) different number of neutrons
(d) different atomic numbers

Ans: (c) different number of neutrons


Q17. Number of valence electrons in Cl ion are:
(a) 16
(b) 8
(c) 17
(d) 18

Ans: (b) 8
Cl has 8 valence electrons. Cl has 17 + 1 = 18 electrons. Its electronic configuration is 2, 8, 8.


Q18. Which one of the following is a correct electronic configuration of sodium?
(a) 2, 8
(b) 8, 2, 1
(c) 2, 1, 8
(d) 2, 8, 1

Ans: (d)  2 ,8 ,1 
2 ,8 ,1 is the correct electronic configuration of sodium.


Q19. Complete the following table.

Ans: 
Atomic number(Z) =Number of protons
Mass number = Number of neutrons + atomic number
=> Mass number(A) = Number of neutrons + number of neutrons

03. Atoms and Molecules – Textbook Solutions

Page No. 27

Q1. In a reaction, 5.3 g of sodium carbonate reacted with 6 g of ethanoic acid. The products were 2.2 g of carbon dioxide, 0.9 g water and 8.2 g of sodium ethanoate. Show that these observations are in agreement with the law of conservation of mass.
Sodium carbonate ethanoic acid → sodium ethanoate carbon dioxide water
Ans: 

Mass of reactants = 5.3 g + 6 g = 11.3 g
Mass of products = 2.2 g + 0.9 g + 8.2 g = 11.3 g
Mass of reactants = Mass of products
Therefore, the law of conservation of mass is proven.

Q2. Hydrogen and oxygen combine in the ratio of 1 : 8 by mass to form water. What mass of oxygen gas would be required to react completely with 3g of hydrogen gas?
Ans: Since hydrogen and oxygen combine in the ratio of 1:8 by mass, 3g of hydrogen gas will react completely with 24 g of oxygen gas.

Q3. Which postulate of Dalton’s atomic theory is the result of the law of conservation of mass?
Ans: Dalton’s postulate that “atoms can neither be created nor destroyed,” is a result of the law of conservation of mass.

Q4. Which postulate of Dalton’s atomic theory can explain the law of definite proportions?
Ans: Atoms combine in a fixed ratio to form compounds, which can explain the law of definite proportions.Page No. 30

Q1. Define atomic mass unit.
Ans: It is defined as equal to 1/12th of the mass of 1 atom of C-12. It is called unified mass denoted by ‘u’ these days.

Q2. Why is it not possible to see an atom with naked eyes?
Ans: The size of an atom is so small that it is not possible to see it with naked eyes. Also, the atom of an element does not exist independently.Page No. 34

Q1. Write down the formulae of 
(a) sodium oxide
(b) aluminium chloride
(c) sodium sulphide
(d) magnesium hydroxide
Ans:
(a) Formula of Sodium Oxide

(b) Formula of Aluminium Chloride

(c) Formula of Sodium Sulphide

(d) Formula of Magnesium Hydroxide

Q2. Write down the names of compounds represented by the following formulae:
(a) Al2(SO4)3
(b) CaCl2
(c) K2SO4
(d) KNO3
(e) CaCO3
Ans: 
(a) Aluminium sulphate 
(b) Calcium chloride 
(c) Potassium sulphate 
(d) Potassium nitrate 
(e) Calcium carbonate

Q3. What is meant by the term chemical formula?
Ans: The chemical formula of a compound is a symbolic representation of its composition.
Chemical Formula of Water

Q4. How many atoms are present in 
(a) H2S molecule and
(b) PO43- ion?

Ans: 
(i) H2S molecule has 2 atoms of hydrogen and 1 atom of sulphur hence 3 atoms in totality.
(ii) PO43- ion has 1 atom of phosphorus and 4 atoms of oxygen hence 5 atoms in totality.Page No. 35

Q1. Calculate the molecular masses of H2, O2, Cl2, CO2, CH4, C2H6, C2H4, NH3, CH3OH.
Ans: 
Molecular mass of H= 2 × Atomic mass of H
= 2 × 1
= 2 u
Molecular mass of O= 2 × Atomic mass of O
= 2 × 16
= 32 u
Molecular mass of Cl= 2 × Atomic mass of Cl
= 2 × 35.5
= 71 u
Molecular mass of CO= Atomic mass of C2 × Atomic mass of O
= 12+ (2+16) = (12 + 32)u 
= 44 u
Molecular mass of CH4= Atomic mass of C4 × Atomic mass of H
= 12+ (4 x 1)u = (12 + 4)u 
= 16 u
Molecular mass of C2H= 2× Atomic mass of C6× Atomic mass of H
= (2 x 12 + 6 x 1)u = (24 + 6)u 
= 30 u
Molecular mass of C2H= 2 x Atomic mass of C4 × Atomic mass of H
= (2 x 12 + 4 x 1)u = (24 + 4)u 
= 28 u
Molecular mass of NH= Atomic mass of N3 × Atomic mass of H
= (14 + 3 x 1)u = (14 + 3)u 
= 17 u
Molecular mass of CH3OH = Atomic mass of C3 × Atomic mass of H Atomic mass of O Atomic mass of H
= (12 + 3 x 1 + 16 + 1)u = (12 + 3 + 17)u 
= 32 u

Q2. Calculate the formula unit masses of ZnO, Na2O, K2CO3, given atomic masses of Zn = 65 u, Na = 23 u, K = 39 u, C = 12 u, and O = 16 u.
Ans:
(i) Formula unit mass of ZnO 
= 65 + 16 = 81 u 
(ii) Formula unit mass of Na2O
= 2 x 23 + 16 = 46 + 16 = 62 u
(iii) Formula unit mass of K2CO3 
= 2 x 39 + 12 + 3 x 16
= 78 + 12 + 48 = 138 uPage No. 36

Q1. A 0.24 g sample of a compound of oxygen and boron was found by analysis to contain 0.096 g of boron and 0.144 g of oxygen. 
Calculate the percentage composition of the compound by weight.
Ans: Percentage of boron = (mass of boron / mass of the compound) x 100
= (0.096g / 0.24g) x 100  
= 40%
Percentage of oxygen = 100 – percentage of boron
= 100 – 40 
= 60%


Q2. When 3.0 g of carbon is burnt in 8.00 g oxygen, 11.00 g of carbon dioxide is produced. What mass of carbon dioxide will be formed when 3.00 g of carbon is burnt in 50.00 g of oxygen? Which law of chemical combination will govern your answer?
Ans:  When 3.0 g of carbon is burnt in 8.00 g of oxygen, 11.00 g of carbon dioxide is produced.
Given that
3.0 g of carbon combines with 8.0 g of oxygen to give 11.0 of carbon dioxide.
Find out
We need to find out the mass of carbon dioxide that will be formed when 3.00 g of carbon is burnt in 50.00 g of oxygen.
Solution
First, let us write the reaction taking place here.
C + O2 → CO2
As per the given condition, when 3.0 g of carbon is burnt in 8.00 g of oxygen, 11.00 g of carbon dioxide is produced.
3g + 8g →11 g ( from the above reaction)
The total mass of reactants = mass of carbon + mass of oxygen
= 3g+8g
= 11g
The total mass of reactants = Total mass of products
Therefore, the law of conservation of mass is proved.
Then, it also depicts that carbon dioxide contains carbon and oxygen in a fixed ratio by mass, which is 3:8.
Thus, it further proves the law of constant proportions.
3 g of carbon must also combine with 8 g of oxygen only.
This means that (50−8) = 42g of oxygen will remain unreacted.
The remaining 42 g of oxygen will be left un-reactive. In this case, too, only 11 g of carbon dioxide will be formed.
The above answer is governed by the law of constant proportions.


Q3. What are polyatomic ions? Give examples.
Ans: 
Polyatomic ions are ions that contain more than one atom, but they behave as a single unit.
Example: CO32-, H2PO4 


Q4. Write the chemical formulae of the following.
(a) Magnesium chloride
(b) Calcium oxide
(c) Copper nitrate
(d) Aluminium chloride
(e) Calcium carbonate.
Ans: The following are the chemical formula of the above-mentioned list:
(a) Magnesium chloride – MgCl2
(b) Calcium oxide – CaO
(c) Copper nitrate – Cu(NO3)2
(d) Aluminium chloride – AlCl3
(e) Calcium carbonate – CaCO3


Q5. Give the names of the elements present in the following compounds.
(a) Quick lime
(b) Hydrogen bromide
(c) Baking powder
(d) Potassium sulphate.
Ans:  
The following are the names of the elements present in the following compounds:
(a) Quick lime – Calcium and oxygen (CaO)
(b) Hydrogen bromide – Hydrogen and bromine (HBr)
(c) Baking powder – Sodium, Carbon, Hydrogen, Oxygen (NaHCO3)
(d) Potassium sulphate – Sulphur, Oxygen, Potassium (K2SO4)


Q6. Calculate the molar mass of the following substances.
(a) Ethyne, C2H2 
(b) Sulphur molecule, S8 
(c) Phosphorus molecule, P4 (Atomic mass of phosphorus =31)
(d) Hydrochloric acid, HCl
(e) Nitric acid, HNO3
Ans: 
 Listed below is the molar mass of the following substances:
(a) Molar mass of Ethyne C2H2= 2 x Mass of C+2 x Mass of H = (2×12)+(2×1)=24+2=26g
(b) Molar mass of Sulphur molecule S8 = 8 x Mass of S = 8  x 32 = 256g
(c) Molar mass of Phosphorus molecule, P4 = 4 x Mass of P = 4 x 31 = 124g
(d) Molar mass of Hydrochloric acid, HCl = Mass of H+ Mass of Cl = 1+35.5 = 36.5g
(e) Molar mass of Nitric acid, HNO3 =Mass of H+ Mass of Nitrogen + 3 x Mass of O = 1 + 14+
3×16 = 63g

02. Is Matter Around Us Pure? – Textbook Solutions

Page No. 15

Q1. What is meant by a substance?
Ans: A pure substance is one that is made up of only one kind of particle, either atoms or molecules. It has definite composition and distinct properties.  Examples of Pure SubstancesA pure substance consists of only one type of particle, either atoms or molecules. It has a definite composition and distinct properties. Examples include:

  • Oxygen
  • Carbon

In contrast, a mixture contains two or more pure substances. For instance:

  • Sea water is a mixture of salt and water.
  • Soil contains various organic and inorganic materials.

Key points about mixtures:

  • Mixtures can be separated into their components through physical processes.
  • Each component retains its own properties.

Types of mixtures include:

  • Homogeneous mixtures have a uniform composition (e.g., sugar in water).
  • Heterogeneous mixtures have a non-uniform composition (e.g., sand and salt).

In summary, a pure substance has consistent properties, while a mixture contains multiple substances that can vary in composition.

Q2. List the points of differences between homogeneous and heterogeneous mixtures.

Ans:

Page No. 18

Q1. Differentiate between homogeneous and heterogeneous mixtures with examples.
Ans: The following are the differences between heterogeneous and homogenous mixtures.


Q2. How are sol, solution and suspension different from each other?
Ans:


Q3. To make a saturated solution, 36g of sodium chloride is dissolved in 100 g of water at 293 K. Find its concentration at this temperature.

Ans: Mass of solute (NaCl): 36 g

Mass of solvent (H2O): 100 g

Mass of solution: 136 g (NaCl + H2O)

Concentration: Calculated as follows:

  • Concentration = (Mass of solute / Mass of solution) × 100
  • Concentration = (36 g / 136 g) × 100
  • Concentration = 26.47%

Thus, the concentration of the solution is 26.47%.Page No. 19

Q1. Classify the following as chemical or physical changes: 

  • cutting of trees, 
  • melting of butter in a pan, 
  • rusting of almirah, 
  • boiling of water to form steam, 
  • passing of electric current, through water and the water breaking down into hydrogen and oxygen gases, 
  • dissolving common salt in water, 
  • making a fruit salad with raw fruits, and
  • burning of paper and wood.

Ans: The following is the classification into physical and chemical change:


Q2. Try segregating the things around you as pure substances or mixtures.
Ans: Listed below are the classifications based on pure substances and mixtures:
 Page No. 22

Exercises

Q1. Which separation techniques will you apply for the separation of the following? 
(a) Sodium chloride from its solution in water. 
(b) Ammonium chloride from a mixture containing sodium chloride and ammonium chloride. 
(c) Small pieces of metal in the engine oil of a car. 
(d) Different pigments from an extract of flower petals.
(e) Butter from curd.
(f) Oil from water. 
(g) Tea leaves from tea.
(h) Iron pins from sand. 
(i) Wheat grains from husk. 
(j) Fine mud particles suspended in water.

Ans: (a) In water, sodium chloride in its solution can be separated through the process of evaporation (as well as crystallization). 
(b) The sublimation technique is appropriate as ammonium chloride supports sublimation. 
(c) Tiny metal pieces in the engine oil of a car can be filtered manually.
(d) Chromatography can be used to separate different pigments from an extract of flower petals.
(e) The technique of churning can be applied to separate butter from curd. It is based on the concept of difference in density. 
(f) To separate oil from water, which are two immiscible liquids which vary in their densities, using a funnel can be an effective method. 
(g) Tea leaves can be manually separated from tea using simple filtration methods. 
(h) Iron pins can be separated from sand either manually or with the use of magnets as the pins exhibit strong magnetic quality, which can be a key characteristic taken into consideration. 
(i) The differentiating property between husk and wheat is that there is a difference in their mass. If treated with a small amount of wind energy, a remarkable variation in the moving distance is noticed. Hence, to separate them, the sedimentation/winnowing procedure can be applied. 
(j) Due to the property of water, sand or fine mud particles tend to sink in the bottom as it is denser, provided they are undisturbed. Through the process of sedimentation/decantation, water can be separated from fine mud particles, as the technique is established on obtaining clear water by tilting it out.Page No. 23

Q2. Write the steps you would use for making tea. Use the words solution, solvent, solute, dissolve, soluble, insoluble, filtrate and residue.
Ans:  Steps for Making Tea

  • Heat a cup of milk, which acts as the solvent.
  • Add tea powder or leaves, the solute, to the boiling milk.
  • Observe that the tea powder remains insoluble while boiling.
  • Add sugar to the boiling solution and stir.
  • Sugar, being a solute, is soluble in the milk.
  • Continue stirring until the sugar completely dissolves, achieving saturation.
  • Once the raw smell of tea leaves disappears, remove the solution from heat.
  • Filter the mixture to separate the tea powder, which becomes the residue.
  • The liquid that passes through is the filtrate, containing the dissolved sugar and milk.


Q3. Pragya tested the solubility of three different substances at different temperatures and collected the data as given below (results are given in the following table, as grams of substance dissolved in 100 grams of water to form a saturated solution).

(a) What mass of potassium nitrate would be needed to produce a saturated solution of potassium nitrate in 50 grams of water at 313 K? 
(b) Pragya makes a saturated solution of potassium chloride in water at 353 K and leaves the solution to cool at room temperature. What would she observe as the solution cools? Explain. 
(c) Find the solubility of each salt at 293 K. Which salt has the highest solubility at this temperature? 
(d) What is the effect of change of temperature on the solubility of a salt?
Ans: 
(a) Given:
Mass of potassium nitrate required to produce a saturated solution in 100 g of water at 313 K = 62g
To find:
Mass of potassium nitrate required to produce a saturated solution in 50 g of water =?
Required amount = 62 x 50/100 = 31
Hence, 31 g of potassium nitrate is required.
(b) The solubility of potassium chloride in water is decreased when a saturated solution of potassium chloride loses heat at 353 K. Consequently, Pragya would observe crystals of potassium chloride, which would have surpassed its solubility at low temperatures.
(c) As per the given data, that is
Solubility of potassium nitrate at 293K = 32 g
Solubility of sodium chloride at 293K = 36 g
Solubility of potassium chloride at 293K = 35 g
Solubility of ammonium chloride at 293K = 37g
We can observe from this data that ammonium chloride has the highest solubility at 293K.
(d) Effect of change of temperature on the solubility of salts:
The table clearly depicts that the solubility of the salt is dependent upon the temperature and increases with an increase in temperature. With this, we can infer that when a salt arrives at its saturation point at a specific temperature, there is a propensity to dissolve more salt through an increase in the temperature of the solution.


Q4. Explain the following, giving examples. 
(a) Saturated solution 
(b) Pure substance 
(c) Colloid 
(d) Suspension

Ans: (a) Saturated solution: It is the state in a solution at a specific temperature when a solvent is no longer soluble without an increase in temperature. Example: Excess carbon leaves off as bubbles from a carbonated water solution saturated with carbon.
(b) Pure substance: A substance is said to be pure when it comprises only one kind of molecule, atom or compound without adulteration with any other substance or any divergence in the structural arrangement. Examples: Sulphur, diamonds etc.
(c) Colloid: A Colloid is an intermediate between solution and suspension. It has particles of various sizes that range between 2 to 1000 nanometers. Colloids can be distinguished from solutions using the Tyndall effect. Tyndall effect is defined as the scattering of light (light beam) through a colloidal solution. Examples: Milk and gelatin.
(d) Suspension: It is a heterogeneous mixture that comprises solute particles that are insoluble but are suspended in the medium. These particles that are suspended are not microscopic but visible to bare eyes and are large enough (usually larger than a micrometer) to undergo sedimentation.


Q5. Classify each of the following as a homogeneous or heterogeneous mixture. 
soda water, wood, air, soil, vinegar, filtered tea.
Ans: The following is the classification of the given substances into homogenous and heterogenous mixtures.


Q6. How would you confirm that a colorless liquid given to you is pure water?
Ans: 
We can confirm if a colorless liquid is pure by setting it to boil. If it boils at 100°C, it is said to be pure. But if there is a decrease or increase in the boiling point, we infer that water has added impurities and, hence not pure.Page No. 24

Q7. Which of the following materials fall in the category of a “pure substance”? 
(a) Ice 
(b) Milk
(c) Iron 
(d) Hydrochloric acid 
(e) Calcium oxide 
(f) Mercury 
(g) Brick 
(h) Wood 
(i) Air

Ans: The following substances from the above-mentioned list are pure substances:

  • Iron
  • Ice
  • Hydrochloric acid
  • Calcium oxide
  • Mercury


Q8. Identify the solutions among the following mixtures.
(a) Soil
(b) Seawater
(c) Air
(d) Coal
(e) Soda water
Ans: 
The following are the solutions from the above-mentioned list of mixtures:

  • Sea water
  • Air
  • Soda water


Q9. Which of the following will show the “Tyndall effect”?
(a) Salt solution
(b) Milk
(c) Copper sulphate solution
(d) Starch solution
Ans:  
Milk and starch solution demonstrate the Tyndall effect because they are colloidal solutions. In these solutions, light is scattered, making its path visible.

  • The Tyndall effect occurs when light passes through a colloid.
  • Colloidal solutions contain particles that are small enough to scatter light.
  • Examples of colloids include milk and starch solutions.

Q10. Classify the following into elements, compounds, and mixtures. 
(a) Sodium 
(b) Soil 
(c) Sugar solution 
(d) Silver 
(e) Calcium carbonate 
(f) Tin 
(g) Silicon 
(h) Coal 
(i) Air 
(j) Soap 
(k) Methane 
(i) Carbon dioxide 
(m) Blood
Ans:

 


Q11. Which of the following are chemical changes? 
(a) Growth of a plant 
(b) Rusting of iron 
(c) Mixing of iron filings and sand 
(d) Cooking of food 
(e) Digestion of food 
(f) Freezing of water 
(g) Burning of a candle
Ans: Among the options listed, the following are considered chemical changes:

  • Rusting of iron
  • Cooking of food
  • Digestion of food
  • Burning of a candle

The growth of a plant is a complex process involving both chemical and physical changes, while mixing iron filings and sand and freezing water are not chemical changes.

1. Matter in Our Surroundings – Textbook Solutions

Page No. 3

Q1. Which of the following are matter?
Chair, air, love, smell, hate, almonds, thought, cold, lemon water, the smell of perfume.
Ans: The following substances are matter: Chair, Air, Almonds, Lemon water, and the smell of perfume (The smell is caused by volatile substances which are matter, as they occupy space and have mass).

Matter around us

Q2. Give reasons for the following observation.
The smell of hot sizzling food reaches you several meters away, but to get the smell from cold food, you have to go close.
Ans: When the air is heated, the particles in it gain more kinetic energy and move faster. This is why the smell of hot food travels farther, allowing a person to sense it even from several meters away.

Q3. A diver is able to cut through water in a swimming pool. Which property of matter does this observation show?
Ans:
 The particles of every matter have a force of attraction between them. This force keeps the particles together in a matter. In the case of water, the force of attraction between particles is less in comparison to solids. Thus, water molecules flow easily, giving way to a diver.

Showing less intermolecular force between liquid molecules.

Q4. What are the characteristics of the particles of matter?
Ans:  The characteristics of particles of matter are:
(i) Presence of intermolecular spaces between particles.
(ii) Particles are in constant motion.
(iii) They attract each other.Page No. 6

Q1. The mass per unit volume of a substance is called density. 
(density=mass/volume). Arrange the following in the order of increasing density – air, exhaust from the chimneys, honey, water, chalk, cotton and iron.
Ans: The following substances are arranged in increasing density: Air < Exhaust from chimney <  Cotton < Water <  Honey < Chalk < Iron.


Q2. (a) Tabulate the differences in the characteristics of states of matter.
(b) Comment upon the following: rigidity, compressibility, fluidity, filling a gas container, shape, kinetic energy and density.
Ans:  
(a) The differences in the characteristics of the three states of matter solid, liquid and gas are:-

(b)
(i) Rigidity: It is the property of matter that continues to remain in its shape when treated with an external force.
(ii) Compressibility: Particles have the ability to reduce their intermolecular space when an external force is applied, which increases their density. This characteristic is called compressibility.
(iii) Fluidity: It is the ability of a substance to flow or move about freely.
(iv) Filling the gas container: The particles in a container take their shape as they randomly vibrate in all possible directions.
(v) Shape: It is the definite structure of an object within an external boundary.
(vi) Kinetic energy: Motion allows particles to possess energy, which is referred to as kinetic energy. The increasing order of kinetic energy possessed by various states of matter is Solids < Liquids < Gases.
Mathematically, it can be expressed as K.E = 1/2 mv2, where ‘m’ is the mass and ‘v’ is the velocity of the particle.
(vii) Density: It is the mass of a unit volume of a substance. It is expressed as d = M/V, where ‘d’ is the density, ‘M’ is the mass and ‘V’ is the volume of the substance


Q3. Give reasons 
(a) A gas fills completely the vessel in which it is kept. 
(b) A gas exerts pressure on the walls of the container.
(c) A wooden table should be called a solid. 
(d) We can easily move our hand in the air, but to do the same through a solid block of wood, we need a karate expert.
Ans: 

(a) There is a low force of attraction between gas particles. The particles in the filled vessel are free to move about.
(b) Gaseous particles have the weakest attraction force and move randomly in all directions. When a gas particle hits the walls of its container, it applies a force, creating pressure on the walls.
(c) The hardwood table has a clear shape and volume. The wood particles are tightly packed and do not change to fit the shape of a container. This gives the table its solid properties.
(d) The air particles are spread far apart with a lot of space between them, which is why we can move our hands freely through the air. However, in a solid block, the particles are held tightly together by a strong force of attraction, leaving little or no space between them. That’s why breaking a solid block would require the strength of a karate expert.


Q4. Liquids generally have lower density as compared to solids. But you must have observed that ice floats on water. Find out why?

Ans: 

  • The mass per unit volume of a substance is called density (density = mass/volume). 
  • As the volume of a substance increases, its density decreases. In general, solids have higher density than liquids. Water is also a liquid, so it should also have less density than that of a solid, which is ice.
  • Though ice is solid, it has a cage-like structure; hence, there are a large number of empty spaces between its particles. These spaces are larger than the spaces between the particles of water. Thus, for a given mass of water, the volume of ice is greater than that of water.
  • Hence, the density of ice is less than that of water. A substance with a lower density than water can float on water. Therefore, ice floats on water.

Page No. 9

Q1. Convert the following temperature to Celsius scale:
(a) 300 K 
(b) 573 K
Ans: To convert a temperature from the Kelvin scale to the Celsius scale, you simply subtract 273 from the given Kelvin temperature.

(a)  (300 – 273)°C = 27°C
(b) (573 – 273)°C = 300°C


Q2. What is the physical state of water at:
(a) 250°C 
(b) 100°C
Ans: 
(a) At 250°C, the physical state of water is gas as the temperature is beyond its boiling point.
(b) At 100°C, it is in the transition state (both liquid and gaseous states) as the water is at its boiling point. Hence, it would be present in both liquid and gaseous state.


Q3. For any substance, why does the temperature remain constant during the change of state?
Ans: It is due to the latent heat as the heat supplied to increase the temperature of the substance is used up to transform the state of matter of the substance; hence, the temperature stays constant. Latent heat is the heat energy needed to change a substance from one form to another without changing its temperature. For example, when ice melts into water, it absorbs heat (latent heat of fusion) without increasing in temperature. Similarly, when water evaporates into steam, it also absorbs heat (latent heat of vaporization) without a temperature change.

Q4. Suggest a method to liquefy atmospheric gases.
Ans: 
To transform a gas into a liquid, it is necessary to bring its constituent particles or molecules closer. This can be achieved with atmospheric gases by either increasing the pressure or lowering the temperature. Page No. 10

Q1. Why does a desert cooler cool better on a hot dry day?
Ans: A desert cooler works better on hot, dry days because the high temperature and low humidity increase the rate of evaporation. This greater evaporation results in more effective cooling.

Q2. How does the water kept in an earthen pot (matka) become cool during summer?
Ans:
 An earthen pot has tiny pores that allow water to seep through and evaporate from its surface. This evaporation requires energy, which is drawn from the water inside the pot, making the water cooler.

Q3. Why does our palm feel cold when we put some acetone or petrol or perfume on it?
Ans: 
Acetone, petrol, and perfume are highly volatile substances. When applied to our palm, they evaporate quickly, absorbing heat from the skin and making our palm feel cold.

Q4. Why are we able to sip hot tea or milk faster from a saucer rather than a cup?
Ans: A saucer has a larger surface area than a cup, which allows for faster evaporation. This rapid evaporation cools the tea or milk more quickly, enabling us to sip it faster.


Q5. What type of clothes should we wear in summer?
Ans: 
In summer, it is advisable to wear light-colored cotton clothes. Light colors reflect heat, and cotton is breathable, allowing sweat to evaporate and providing a cooling effect on the skin.

Exercises 

Q1. Convert the following temperature to Celsius scale.
(a) 293K 
(b) 470K
Ans: To convert a temperature from the Kelvin scale to the Celsius scale, you simply subtract 273 from the given Kelvin temperature because 0°C=273K.
(a) 293K= (293 – 273)°C = 20°C
(b) 470K= (470 – 273)°C = 197°C


Q2. Convert the following temperatures to the kelvin scale.
(a) 25°C 

(b) 373°C
Ans: 
0°C = 273K
(a) 25°C = (25+273)K = 298K
(b) 373°C = (373+273)K = 646K


Q3. Give reason for the following observations:
(a) Naphthalene balls disappear with time without leaving any solid.
(b) We can get the smell of perfume sitting several metres away.
Ans: 

(a) At room temperature, naphthalene balls undergo sublimation, which means they change directly from a solid to a gaseous state without undergoing the intermediate state, i.e., the liquid state.

Naphthalene Balls

(b) It is because perfumes contain a volatile organic solvent that can easily diffuse through air and, hence, carry the fragrance to people sitting several metres away.


Q4. Arrange the following substances in increasing order of forces of attraction between the particles— water, sugar, oxygen.
Ans: 
Oxygen (gas) < Water (liquid) < Sugar (solid)


Q5. What is the physical state of water at: 
(a) 25°C, (b) 0°C, (c) 100°C?

Ans: The physical state of water at different temperatures is as follows:
(a) At 25°C, water is in a liquid state (typical room temperature).
(b) At 0°C, water is at its freezing point, so both solid (ice) and liquid (water) phases can be observed.
(c) At 100°C, water is at its boiling point, resulting in the presence of both liquid water and gaseous water (water vapor).

Q6. Give two reasons to justify.
(a) Water at room temperature is a liquid.
(b) An iron almirah is a solid at room temperature.
Ans:

(a) Water remains in a liquid state at room temperature for two main reasons:

  1. Its melting or freezing point is lower than room temperature, while its boiling point is higher (100°C).
  2. Water has no fixed shape and flows to take the shape of its container, which indicates that it occupies a fixed volume but does not have a defined shape.

(b) An iron almirah is a solid at room temperature for the following reasons:

  1. Both the melting and boiling points of iron are above room temperature, meaning it remains solid under these conditions.
  2. An iron almirah is rigid and maintains a definite shape, and metals typically have a high density, further confirming that it is solid at room temperature.

Q7. Why is ice at 273 K more effective in cooling than water at the same temperature?
Ans: At 273 K, ice will absorb heat energy or latent heat from the medium during melting to transform into water. As a result, ice has a greater cooling impact than water at the same temperature since water does not absorb the excess heat from the medium.

Q8. What produces more severe burns, boiling water or steam?
Ans:
 Steam produces severe burns. It is because it is an exothermic reaction that releases a high amount of heat, which it consumes during vaporization.

Q9. Name A, B, C, D, E and F in the following diagram showing change in its state.

Ans: Interconversion of three states of matter: Using temperature or pressure, any state of matter can be turned into another.
(A) Solid to Liquid → Melting (or) fusion (or) liquefaction
(B) Liquid to Gas → Evaporation (or) vaporization
(C) Gas to liquid → Condensation
(D) Liquid to Solid → Solidification
(E) Solid to Gas → Sublimation
(F) Gas to Solid → Deposition

12. Statistics – Textbook Solutions (Exercise 12.1)

Q1. A survey conducted by an organisation for the cause of illness and death among the women between the ages 15–44 (in years) worldwide, found the following figures (in %):

(i) Represent the information given above graphically.
(ii) Which condition is the major cause of women’s ill health and death worldwide?
(iii) Try to find out, with the help of your teacher, any two factors which play a major role in the cause in (ii) above being the major cause.
Ans: 
(i) The information given in the question is represented below graphically.

(ii) We can observe from the graph that reproductive health conditions is the major cause of women’s ill health and death worldwide.
(iii) Two factors responsible for cause in (ii) are:

  • Lack of proper care and understanding.
  • Lack of medical facilities.



Q2. The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below.

(i) Represent the above information by a bar graph.
Ans: (i) The information given in the question is represented below graphically.

(ii) From the above graph, we can conclude that the maximum number of girls per thousand boys is present in the section ST. We can also observe that the backward districts and rural areas have more number of girls per thousand boys than non-backward districts and urban areas.

Q3. Given below are the seats won by different political parties in the polling outcome of a state assembly elections:

(i) Draw a bar graph to represent the polling results.
(ii) Which political party won the maximum number of seats?
Ans:
(i) The bar graph representing the polling results is given below:

(ii) From the bar graph it is clear that Party A won the maximum number of seats.

Q4. The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:
 (i) Draw a histogram to represent the given data. [Hint: First make the class intervals continuous]
 (ii) Is there any other suitable graphical representation for the same data?
 (iii) Is it correct to conclude that the maximum number of leaves are 153 mm long? Why?

Ans: (i) The data given in the question is represented in discontinuous class interval. So, we have to make it in continuous class interval. The difference is 1, so taking half of 1, we subtract ½ = 0.5 from lower limit and add 0.5 to the upper limit. Then the table becomes:

(ii) Yes, the data given in the question can also be represented by frequency polygon.
No, we cannot conclude that the maximum number of leaves are 153 mm long because the maximum number of leaves are lying in-between the length of 144.5 – 153.5


Q5. The following table gives the life times of 400 neon lamps:

(i) Represent the given information with the help of a histogram.
(ii) How many lamps have a life time of more than 700 hours?
Ans:
(i) The histogram representation of the given data is given below:

(ii) Number of lamps having life time more than 700 hours = 74 + 62 + 48 = 184.


Q6. The following table gives the distribution of students of two sections according to the marks obtained by them:

Represent the marks of the students of both the sections on the same graph by two frequency polygons. From the two polygons compare the performance of the two sections.
Ans: 
The class-marks = (lower limit + upper limit)/2
For section A,

For section B:

 Representing these data on a graph using two frequency polygon we get,

 From the graph, we can conclude that the students of Section A performed better than Section B.

Q7. The runs scored by two teams A and B on the first 60 balls in a cricket match are given below:

Represent the data of both the teams on the same graph by frequency polygons.
Note: The given class intervals are not continuous. Therefore, we first modify the distribution as continuous.
Ans: 
The data given in the question is represented in discontinuous class interval. So, we have to make it in continuous class interval. The difference is 1, so taking half of 1, we subtract ½ = 0.5 = 0.5 from lower limit and add 0.5 to the upper limit. Then the table becomes:

The data of both the teams are represented on the graph below by frequency polygons.

Q8. A random survey of the number of children of various age groups playing in a park was found as follows:

Draw a histogram to represent the above data.
Ans: 
The width of the class intervals in the given data is varying.
We know that,
The area of rectangle is proportional to the frequencies in the histogram.
Thus, the proportion of the children per year can be calculated as given in the table below.
Now, we draw the histogram taking ages (in years) on the x-axis and corresponding adjusted frequencies on the y-axis as shown below:

Let x-axis = the age of children
y-axis = proportion of children per 1 year interval

Q9. 100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows:

(i) Draw a histogram to depict the given information.
(ii) Write the class interval in which the maximum number of surnames lie.

Ans: 
(i) The width of the class intervals in the given data is varying.
We know that,
The area of rectangle is proportional to the frequencies in the histogram.
Thus, the proportion of the number of surnames per 2 letters interval can be calculated as given in the table below.

(ii) 6-8 is the class interval in which the maximum number of surnames lie.


MEASURES OF CENTRAL TENDENCY

We can make out some important features of given data by considering only certain representatives.

These representatives are called the measures of central tendency or averages. There are three main averages: Mean, Median and Mode.
Mean The mean (or average) of a number of observations is the sum of the values of all the observations divided by the total number of observations. It is denoted by the symbol x and we read it as x-bar.
Thus, the mean

Here, ∑ is a Greek symbol called sigma.
The summation  is read as the sum of the x as i varies from 1 to n.


Mode 

The mode is the value of the observation which occurs most frequently, i.e., an observation with the maximum frequency is called the mode of the data.
Note: In a given data, the value around which there is greatest concentration, is called the mode of the data.

Median After arranging the given data in an ascending or a descending order of magnitude, the value of the middle-most observation is called the median of the data.
Note: For ‘n’ observations (taken in order),

(i) if n is odd, the median = value of observation.
(ii) if n is even, the median = mean of   observations.

11. Surface Areas and Volumes – Textbook Solutions (Exercise 11.1, 11.2, 11.3 and 11.4)

Exercise 11.1

Q1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area (Assume π = 22 / 7)
Ans: Radius of cone r = 10.5/2 = 5.25 cm and Slant height (l) = 10 cm
Curved surface area of cone = πrl
= 22/7 × 5.25 × 10
= 22 × 0.75 × 10 = 165 cm2
Hence, the curved surface area of cone is 165 cm2.


Q2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. (Assume π = 22 / 7)
Ans:
 

Radius (r) = 24/2 m = 12 m
Slant height (l) = 21 m
Total surface area of the cone = πr (l + r) m
= 22/7 × 12 × (21 + 12) m
= (22/7 × 12 × 33) m
= 1244.57 m2


Q3. Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find
(i) radius of the base and
(ii) total surface area of the cone. (Assume π = 22 / 7)

Ans: (i) Curved surface area of cone = 308 cm2 and slant height / = 14 cm
Let, the radius of base of cone = r cm
Curved surface area of cone = πrl
⇒ 308 = 22/7 × r × 14
⇒ 308 = 44r
⇒ r = 308/44 = 7 cm
Hence, the radius of base of cone is 7 cm.
(ii) Total surface area of cone = πr(r + l)
= 22/7 × 7 × (7 + 14)
= 22 × 21 = 462 cm
Hence, the total surface area of cone is 462 cm2.


Q4. A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) slant height of the tent.
(ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is Rs 70. (Assume π = 22 / 7)
Ans:
 (i) Radius of cone r = 24 m and height h = 10 m
Let, the slant height = l m
We know that, l2 = r2 + h2
⇒ l2 =24+102 = 576 + 100 = 676
l = √676 = 26m
(ii) Area of canvas to make the tent = πrl
22/7 × 24 × 26 m2
Cost of 1 m2 canvas = ₹ 70
Therefore, the cost of 22/7 × 24 × 26 m2 canvas = ₹ 70 × 22/7 × 24 × 26 = ₹ 137280
Hence, the cost of canvas to make the tent is ₹ 137280.


Q5. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm. [Use π = 3.14]
Ans: Radius of cone r = 6 m and height h = 8 m
Let, the slant height = l m
We know that, l2 = r2 + h
⇒ l2 = 62 + 82 = 36 + 64 = 100 ⇒ l = √100 = 10 m
Area of tarpaulin to make the tent = πrl
= 3.14 × 6 × 10 = 188.40m
Let, the length of 3 m wide tarpaulin = L, therefore, the area of tarpaulin required = 3 × L
According to question,
3 × L = 188.40 ⇒ L = 188.40/3 = 62.80 m
Extra tarpaulin for stitching margins and wastage = 20 cm = 0.20 m
Therefore, the total length of tarpaulin = 62.80 + 0.20 = 63 m
Hence, the length of 3 m wide tarpaulin is 63 m to make the tent.


Q6. The slant height and base diameter of conical tomb are 25m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs. 210 per 100 m2. (Assume π = 22 / 7).
Ans: Radius of conical tomb r = 14/2 = 7 m and slant height / = 25 m/
Curved surface area of conical tomb = πrl
= 22/7 × 7 × 25 = 550 m
Cost of white washing at the rate of ₹ 210 per 100 m2 = ₹550 × 210/100 = ₹1155
Hence, the cost of white washing curved surface area is ₹ 1155.


Q.7. A joker’s cap is in the form of right circular cone of base radius 7 cm and height 24cm. Find the area of the sheet required to make 10 such caps. (Assume π = 22 / 7)
Ans:
 Radius of conical cap, r = 7 cm
Height of conical cap, h = 24cm We know that, l2 = (r2 + h2)
⇒ l2 = 72 + 242 = 49 + 576 = 625 ⇒ l = √625 = 25 cm
Area of sheet required to make 1 cap = πrl
= 22/7 × 7 × 25 = 550 cm2
Therefore, area of sheet required to make 10 such caps = 10 × 550 = 5500 cm2
Hence, area of sheet to make 10 such caps is 5500 cm2.


Q8. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs. 12 per m2, what will be the cost of painting all these cones? (Use π = 3.14 and take √(1.04) = 1.02).
Ans: 
Radius of cone r = 40/2 = 20 cm = 0.2 m and height h = 1 m
Let the slant height = l m
 We know that l2 = r2 + h
⇒ l2= (0.2)2 + 12 = 0.04 + 1 = 1.04
⇒ l= √1.04 = 1.02 m
Curved surface area of cone = πrl
= 3.14 × 0.2 × 1.02 = 6.4056m
Curved surface area of 50 cones
= 50 × 6.4056
= 32.028 m
Cost of painting at the rate of ₹12 per m
= ₹12 × 32.028
= ₹384.34 (approx.)
Hence, the cost of painting the curved surface of 50 cones is ₹384.34


Exercise 11.2

Q1. Find the surface area of a sphere of radius:
(i) 10.5cm
(ii) 5.6cm
(iii) 14cm
(Assume π = 22 / 7)

Ans: (i) Radius of sphere r = 10.5 cm
Surface area of sphere = 4πr
= 4 × 22/7 × 10.5 × 10.5 = 4 × 22 × 1.5 × 10.5 = 1386.00 cm2
Hence, the surface area of sphere is 1386 cm2.
(ii) Radius of sphere r = 5.6 cm
Surface area of sphere = 4πr
= 4 × 22/7 × 5.6 × 5.6 = 4 × 22 × 0.8 × 5.6 = 394.24 cm
Hence, the surface area of sphere is 394.24 cm2.
(iii) Radius of sphere r = 14 cm
Surface area of sphere = 4πr
= 4 × 22/7 × 14 × 14 = 4 × 22 × 2 × 14 = 2464 cm
Hence, the surface area of sphere is 2464 cm2.


Q2. Find the surface area of a sphere of diameter:
(i) 14 cm
(ii) 21 cm
(iii) 3.5 cm
(Assume π = 22 / 7)

Ans: (i) Radius of sphere r = 14/2 = 7 cm
Surface area of sphere = 4πr
= 4 × 22/7 × 7 × 7 = 4 × 22 × 7 = 616 cm
Hence, the surface area of sphere is 616 cm2.
(ii) Radius of sphere r = 21/2 = 10.5 cm
Surface area of sphere = 4πr
= 4 × 22/7 × 10.5 × 10.5 = 4 × 22 × 1.5 × 10.5 = 1386 cm
Hence, the surface area of sphere is 1386 cm2.
(iii) Radius of sphere r = 3.5/2 = 1.75 cm
Surface area of sphere = 4π
= 4 × 22/7 × 1.75 × 1.75 = 4 × 22 × 0.25 × 1.75
= 38.50 cm
Hence, the surface area of sphere is 38.5 cm2.


Q3. Find the total surface area of a hemisphere of radius 10 cm. [Use π = 3.14]
Ans:
 Radius of hemisphere r = 10 cm
Surface area of hemisphere = 3πr2
= 3 × 3.14 × 10 × 10 = 942 cm2
Hence, the total surface area of hemisphere is 942 cm2.


Q4. The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.
Ans: Let r be the initial radius and R be the increased radius of balloons.
r = 7cm and R = 14cm
Ratio of the surface area =4πr2/4πR2
= r2/R2
= (7 × 7)/(14 × 14) = 1/4
Thus, the ratio of surface areas = 1 : 4


Q5. A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm2. (Assume π = 22 / 7)
Ans: Radius of the bowl (r) = 10.5/2 cm = 5.25 cm
Curved surface area of the hemispherical bowl = 2πr2
= (2 × 22/7 × 5.25 × 5.25) cm2
= 173.25 cm2
Rate of tin – plating is = ₹16 per 100 cm2
Therefor, cost of 1 cm= ₹16/100 
Total cost of tin-plating the hemisphere bowl = 173.25 × 16/100
= ₹27.72

Q6. Find the radius of a sphere whose surface area is 154 cm2. (Assume π = 22/7)
Ans:
 Let r be the radius of the sphere.
Surface area = 154 cm2
⇒ 4πr= 154
⇒ 4 × 22/7 × r= 154
⇒ r= 154/(4 × 22/7)
⇒ r= 49/4
⇒ r = 7/2 = 3.5 cm

Q7. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.
Ans: Let the diameter of earth be r and that of the moon will be r/4
Radius of the earth = r/2
Radius of the moon = r/8
Ratio of their surface area = 4π(r/8)2/4π(r/2)2
= (1/64)/(1/4)
= 4/64 = 1/16
Thus, the ratio of their surface areas is 1:16

Q8. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5cm. Find the outer curved surface of the bowl. (Assume π = 22 / 7)
Ans: 

Internal radius of hemispherical bowl r = 5 cm and thickness = 0.25 cm
Therefore,
The outer radius of hemispherical bowl
= R = 5 + 0.25 = 5.25 cm
Outer curved surface area of hemispherical bowl = 2πR
= 2 × 22/7 × 5.25 × 5.25
= 2 × 22 × 0.75 × 5.25
= 173.25 cm
Hence, the outer curved surface area of hemispherical bowl is 173.25 cm2.


Q9. A right circular cylinder just encloses a sphere of radius r (see Fig). Find
(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in(i) and (ii).

Ans: (i) Radius of sphere = radius of cylinder = r
Hence, the surface area of sphere = 4πr
(ii) Radius of cylinder = r and height h = diameter of sphere = 2r
Hence, the curved surface area of cylinder = 2πrh = 2πr(2r) = 4πr
(iii) Now, Surface area of sphere/Curved surface area of cylinder = 4πr2/4πr= 1/1
Hence, the ratio of surface area of sphere to curved surface area of cylinder is 1 : 1.

Exercise 11.3

Q1. Find the volume of the right circular cone with
(i) radius 6 cm, height 7 cm
(ii) radius 3.5 cm, height 12 cm (Assume π = 22 / 7)
Ans: 
(i) Radius (r) = 6 cm
Height (h) = 7 cm
Volume of the cone = 1/3 πr2h
= (1/3 × 22/7 × 6 × 6 × 7) cm
= 264 cm
(ii) Radius (r) = 3.5 cm
Height (h) = 12 cm
Volume of the cone = 1/3 πr2h
Volume of the cone = 1/3 πr2h
= 154 cm3


Q2. Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm (Assume π = 22 / 7)
Ans: (i) 
Radius (r) = 7 cm
Slant height (l) = 25 cm
Let h be the height of the conical vessel.
∴ h = √l2 – r2
⇒ h = √252 – 7
⇒ h = √576 
⇒ h = 24 cm
Volume of the cone = 1/3 πr2h
= (1/3 × 22/7 × 7 × 7 × 24) cm3
= 1232 cm3  
Capacity of the vessel = (1232/1000) = 1.232 litres
(ii) Height (h) = 12 cm
Slant height (l) = 13 cm
Let r be the radius of the conical vessel.
∴ r = √l2 – h
⇒ r = √132 – 122
⇒ r = √25 
⇒ r = 5 cm
Volume of the cone = 1/3 πr2h
= (1/3 × 22/7 × 5 × 5 × 12) cm
= (2200/7) cm
Capacity of the vessel = (2200/7000) l = 11/35 litres


Q3. The height of a cone is 15cm. If its volume is 1570cm3, find the diameter of its base. (Use π = 3.14)
Ans: 
 Height (h) = 15 cm
Volume = 1570 cm3
Let the radius of the base of cone be r cm
∴ Volume = 1570 cm
⇒ 1/3 πr2h = 1570
⇒ 13 × 3.14 × r2 × 15 = 1570
⇒ r2 = 1570/(3.14×5) = 100
⇒ r = 10 cm 
Diameter of base = 2r = 2 x 10 = 20 cm 


Q4. If the volume of a right circular cone of height 9 cm is 48π cm3, find the diameter of its base.
Ans: 
 Given height of a cone, h = 9 cm
Volume of the cone = 48
(1/3)r2h = 48
(1/3)r× 9 = 48
3r2 = 48
r2 = 48/3 = 16
r = 4
So diameter = 2 × radius
= 2 × 4
= 8 cm
Hence the diameter of the cone is 8 cm.


Q5. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres? (Assume π = 22 / 7)
Ans: 
 Radius of pit r = 3.5/2 = 1.75 m and height h = 12 m.
Volume of pit = 1/3 πr2h
= 1/3 × 22/7 × 1.75 × 1.75 × 12 = 38.5 m3
= 38.5 Kilolitres [∴ 1 m3 = 1 kilolitres]
Hence, the capacity of the pit is 38.5 kilolitres.

Q6. The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28 cm, find
(i) height of the cone
(ii) slant height of the cone
(iii) curved surface area of the cone (Assume π = 22 / 7)

Ans:  (i) Diameter of the base of the cone = 28 cm
Radius (r) = 28/2 cm = 14 cm
Let the height of the cone be h cm
Volume of the cone = 13πr2h = 9856 cm3 
⇒ 1/3 πr2h = 9856
⇒ 1/3 × 22/7 × 14 × 14 × h = 9856
⇒ h = (9856 × 3)/(22/7 × 14 × 14)
⇒ h = 48 cm
(ii) Radius (r) = 14 m
Height (h) = 48 cm
Let l be the slant height of the cone
l2 = h+ r2
⇒ l2 = 48+ 142
⇒ l2 = 2304+196
⇒ l= 2500
⇒ ℓ = √2500 = 50 cm

(iii) Radius (r) = 14 m
Slant height (l) = 50 cm
Curved surface area = πrl
= (22/7 × 14 × 50) cm2
= 2200 cm2

Q7. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.
Ans: 
If the triangle is revolved about 12 cm side, a cone will be formed.

Therefore, the radius of cone r = 5 cm, height h = 12 cm and slant height l =13 cm.
Volume of solid (cone) = 1/3 πr2
= 1/3 × π × 5 × 5 × 12 = 100 π cm
Hence, the volume of solid is 100π cm3.


Q8. If the triangle ABC in the Question 7 is revolved about the side 5cm, then find the volume of the solids so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.
Ans: 

A right-angled ΔABC is revolved about its side 5cm, a cone will be formed of radius as 12 cm, height as 5 cm, and slant height as 13 cm.

Volume of cone = (1/3) πr2h; where r is the radius and h be the height of cone

= (1/3) × π × 12 × 12 × 5

= 240 π

The volume of the cones of formed is 240π cm3.

So, required ratio = (result of question 7) / (result of question 8) = (100π) / (240π) = 5 / 12 = 5 : 12.


Q9. A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas. (Assume π = 22 / 7)
Ans:  
Radius (r) of heap = (10.5 / 2) m = 5.25

Height (h) of heap = 3m

Volume of heap = (1 / 3)πr2h

= (1 / 3) × (22 / 7) × 5.25 × 5.25 × 3

= 86.625 m3

The volume of the heap of wheat is 86.625 m3. Again,

Area of canvas required = CSA of cone = πrl, where l = 

After substituting the values, we have

= (22 / 7) × 5.25 × 6.05

= 99.825

Therefore, the area of the canvas is 99.825 m2.

Exercise 11.4

Q1. Find the volume of a sphere whose radius is
(i) 7 cm
(ii) 0.63 m (Assume π = 22 / 7)
Ans: (i)
 Radius of sphere, r = 7 cm
Using, Volume of sphere = (4 / 3) πr3
= (4 / 3) × (22 / 7) × 73 = 4312 / 3
Hence, volume of the sphere is 
(ii) Radius of sphere, r = 0.63 m
Using, volume of sphere = (4 / 3) πr3
= (4 / 3) × (22 / 7) × 0.633 = 1.0478
Hence, volume of the sphere is 1.05 m3 (approx).

Q2. Find the amount of water displaced by a solid spherical ball of diameter
(i) 28 cm
(ii) 0.21 m (Assume π = 22 / 7)
Ans: (i) 
Radius of spherical ball r = 28/2 = 14 cm
Volume of water displaced by spherical ball = 4/3 πr3
= 4/3 × 22/7 × 14 × 14 × 14
= 4/3 × 22 × 2 × 14 × 14 = 

or =11498.67 cm3

(ii) Radius of spherical ball r = 0.21/2 = 0.105 m

Volume of water displaced by spherical ball = 4/3 πr

= 4/3 × 22/7 × 0.105 × 0.105 × 0.105 = 4 × 22 × 0.005 × 0.63 × 0.63 = 0.004861 m

Hence, the volume of water displaced by spherical ball is 0.004861 m3.


Q3. The diameter of a metallic ball is 4.2cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm3? (Assume π = 22 / 7)
Ans:  
Radius of metallic ball r = 4.2/2 = 2.1 cm
Therefore, the volume of metallic ball = 4/3 πr
= 4/3 × 22/7 × 2.1 × 2.1 × 2.1 
= 4 × 22 × 0.1 × 2.1 × 2.1 = 38.808 cm3
Here, the mass of 1 cm3 = 8.9 g 
So, the mass of 38.808 cm3 = 8.9 × 38.808 = 345.39 g (approx.)
Hence, the mass of the ball is 345.39 gram.

Q4. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
Ans: 
 Let the diameter of earth be “d”. Therefore, the radius of earth will be will be d/2
Diameter of moon will be d / 4 and the radius of moon will be d/8
Find the volume of the moon:
Volume of the moon = (4 / 3) πr3 = (4 / 3) π (d / 8)3 = 4 / 3π(d3 / 512)
Find the volume of the earth:
Volume of the earth = (4 / 3) πr3 = (4 / 3) π (d / 2)3 = 4 / 3π(d3 / 8)
Fraction of the volume of the earth is the volume of the moon

Volume of moon is of the 1 / 64 volume of earth.

Q5. How many litres of milk can a hemispherical bowl of diameter 10.5cm hold? (Assume π = 22 / 7)
Ans: 
 Diameter of hemispherical bowl = 10.5 cm
Radius of hemispherical bowl, r = 10.5 / 2 cm = 5.25 cm
Formula for volume of the hemispherical bowl = (2 / 3) πr3
Volume of the hemispherical bowl = (2 / 3) × (22 / 7) × 5.253 = 303.1875
Volume of the hemispherical bowl is 303.1875 cm3
Capacity of the bowl = (303.1875) / 1000 L = 0.303 litres(approx.)
Therefore, hemispherical bowl can hold 0.303 litres of milk.

Q6. A hemi spherical tank is made up of an iron sheet 1cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank. (Assume π = 22/7)
Ans: 
 Inner Radius of the tank, (r) = 1m
Outer Radius (R) = 1.01m
Volume of the iron used in the tank = (2 / 3) π (R3 – r3)
Put values,
Volume of the iron used in the hemispherical tank = (2 / 3) × (22 / 7) × (1.013 – 13) = 0.06348
So, volume of the iron used in the hemispherical tank is 0.06348 m3.

Q7. Find the volume of a sphere whose surface area is 154 cm2. (Assume π = 22 / 7)
Ans: 
 Surface area of sphere A = 154 cm2
Let, the radius of sphere = r cm
We know that the surface area of sphere = 4πr

Volume of surface 4/3 πr3

Hence, the volume of sphere is .


Q8. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹ 4989.60. If the cost of white-washing is ₹ 20 per square meter, find the
(i) Inside surface area of the dome
(ii) volume of the air inside the dome (Assume π = 22 / 7)
Ans: (i) 
Let the internal radius of dome = r m
Internal surface area of dome = 2πr
Cost of white washing at the rate of ₹2 = 2πr2 × ₹2 = ₹4πr2
⇒ ₹4πr2 = ₹498.96
⇒ 4 × 22/7 × r2 = 498.96

Therefore, the internal surface of dome = 2πr
= 2 × 22/7 × (6.3)2
= 2 × 22/7 × 6.3 × 6.3 
= 2 × 22 × 0.9 × 6.3
= 249.48 m
Hence, the inside surface area of the dome is 249.48 m2.
(ii) Volume of the air inside the dome = 2/3 πr
= 2/3 × 22/7 × (6.3)
= 2/3 × 22/7 × 6.3 × 6.3 × 6.3
= 2 × 22 × 0.3 × 6.3 × 6.3
= 523.9 cm
= Hence, the volume of the air inside the dome is 523.9 cm3.


Q9. Twenty-seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S’. Find the
(i) radius r’ of the new sphere,
(ii) ratio of Sand S’.
Ans:  Volume of the solid sphere = (4 / 3)πr3
Volume of twenty seven solid sphere = 27 × (4 / 3)πr3 = 36 πr3
(i) New solid iron sphere radius = r’
Volume of this new sphere = (4/3)π(r’)3
(4 / 3)π(r’)3 = 36 πr3
(r’)3 = 27r3
r’= 3r
Radius of new sphere will be 3r (thrice the radius of original sphere)
(ii) Surface area of iron sphere of radius r, S = 4πr2
Surface area of iron sphere of radius r’= 4π (r’)2
Now
S / S’ = (4πr2) / ( 4π (r’)2)
S / S’ = r2 / (3r’)2 = 1 / 9
The ratio of S and S’ is 1: 9.


Q10. A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm3) is needed to fill this capsule? (Assume π = 22 / 7)
Ans: 
 Radius of capsule r = 3.5/2 = 1.75 mm
Volume of medicine to fill the capsule = 4/3πr3  
= 4/3 × 22/7 × 1.75 × 1.75 × 1.75
= 4/3 × 22 × 0.25 × 1.75 × 1.75
= 22.46 mm3 (approx.)
Hence, 22.46 mmmedicine is required to fill this capsule.

10. Heron’s Formula – Textbook Solutions (Exercise 10.1)

Q1. A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?
Ans: For an equilateral triangle with side ‘a’, area = (√3/4) a2 

∵ Each side of the triangle = a cm

The perimeter of the signal board will be
∴ a + a + a = 180 cm
3a = 180 cm
a= (180/3) = 60 cm
Now, Semi-perimeter (s)= (180/2) = 90 cm
∵ Area of triangle 
Area of the given triangle

= 30 x 30 x√3 = 900√3 cm2
Thus, the area of the given triangle = 900√3 cm2.


Q2. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see Fig). The advertisements yield an earning of ₹ 5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?
Ans: The sides of the triangular wall are a = 122 m, b = 120 m, c = 22 m.

Now, the perimeter will be 

∵ The area of a triangle is given by

∵ Rent for 1 year (i.e. 12 months) per m2 = Rs 5000
∴ Rent for 3 months per m2 = Rs 5000 x (3/12)
⇒ Rent for 3 months for 1320 m2 = 5000 x (3/12) x 1320 = 5000 x 3 x 110 = Rs 16,50,000.


Q3. There is a slide in a park. One of its side walls has been painted in some colour with the message “KEEP THE PARK GREEN AND CLEAN” (see Fig). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

Ans: Sides of the wall a = 15 m, b = 11 m, c = 6 m.

 The area of the triangular surface of the wall

Thus, the required area painted in colour = 20√2 m2.


Q4. Find the area of a triangle, two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Ans: Let the sides of the triangle be a = 18 cm, b = 10 cm and c =?
∵ Perimeter (2s) = 42 cm
s = (42/2)  = 21 cm
c = 42 – (18 + 10) cm = 14 cm
∵ Area of a triangle = 
∴ Area of the given triangle

Thus, the required area of the triangle = 21√11 cm2.


Q5. Side of a triangle is in the ratio of 12: 17: 25 and its perimeter is 540 cm. Find its area.
Ans: The perimeter of the triangle = 540 cm.
Semi-perimeter of the triangle, s= (540/2) = 270 cm
∵ The sides are in the ratio of 12: 17: 25.
∴ Let, the side be a = 12x cm, b = 17x cm, c = 25x cm.
12x + 17x + 25x = 540
54x = 540
x = (540/54) = 10
∴ a = 12 x 10 = 120 cm, b = 17 x 10 = 170 cm, c = 25 x 10 = 250 cm.
(s – a) = (270 – 120) cm = 150 cm
(s – b) = (270 – 170) cm = 100 cm
(s – c) = (270 – 250) cm = 20 cm
∴ Area of the triangle

= 10 x 10 x 3 x 3 x 5 x 2 cm2 = 9,000 cm2
Thus, the required area of the triangle = 9,000 cm2.

Q6. An isosceles triangle has a perimeter of 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
Ans: Equal sides of the triangle are 12 cm each.
Let the third side = x cm.

∵ Perimeter = 30 cm
∴ 12 cm + 12 cm + x cm = 30 cm
x = 30 – 12 – 12 = 6 cm
Semi-perimeter = (30/2) cm = 15 cm
∴ Area of the triangle

Thus, the required area of the triangle = 9√15 cm2.

09. Circles – Textbook Solutions (Exercise 9.1, 9.2 and 9.3)

Exercise 9.1

Q1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
Ans: To recall, a circle is a collection of points whose every point is equidistant from its centre. So, two circles can be congruent only when the distance of every point of both circles is equal from the centre.
In the second part of the question, it is given that AB = CD, i.e. two equal chords. Now, it is to be proven that angle AOB is equal to angle COD.
Proof:
Consider the triangles ΔAOB and ΔCOD,
OA = OC and OB = OD (Since they are the radii of the circle)
AB = CD (As given in the question)
So, by SSS congruency, ΔAOB ≅ ΔCOD
∴ By CPCT, we get
∠AOB = ∠COD. 
Hence proved.


Q2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Ans: Consider the following diagramHere, it is given that ∠AOB = ∠COD, i.e. they are equal angles. Now, we will have to prove that the line segments AB and CD are equal, i.e. AB = CD.
Proof:
In triangles ΔAOB and ΔCOD,
∠AOB = ∠COD (as given in the question)
OA = OC and OB = OD (these are the radii of the circle)
So, by SAS congruency, ΔAOB ≅ ΔCOD.
∴ By the rule of CPCT, we get AB = CD
Hence proved.

Exercise 9.2

Q1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
Ans: Given parameters are:
OP = 5cm
OS = 4cm and
PS = 3cm
Also, PQ = 2PR
Now, suppose RS = x. The diagram for the same is shown below.
Consider the ΔPOR,
OP2 = OR2 + PR2
⇒ 52 = (4 – x)2 + PR2
⇒ 25 = 16 + x2 – 8x + PR2
∴ PR2 = 9 – x2 + 8x — (i)
Now consider ΔPRS,
PS2 = PR2 + RS2
⇒ 32 = PR2 + x2
∴ PR2 = 9 – x2 — (ii)
By equating equation (i) and equation (ii) we get,
9 – x2 + 8x = 9 – x2
⇒ 8x = 0
⇒ x = 0
Now, put the value of x in equation (i)
PR2 = 9 – 02
⇒ PR = 3cm
∴ The length of the cord i.e. PQ = 2PR
So, PQ = 2 × 3 = 6cm.

Q2. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
Ans:
 Let AB and CD be two equal cords (i.e. AB = CD). In the above question, it is given that AB and CD intersect at a point, say, E.
It is now to be proven that the line segments AE = DE and CE = BE
Construction Steps:
Step 1: From the center of the circle, draw a perpendicular to AB i.e. OM ⊥ AB
Step 2: Similarly, draw ON ⊥ CD.
Step 3: Join OE. Now, the diagram is as follows-
Proof:
From the diagram, it is seen that OM bisects AB and so, OM ⊥ AB
Similarly, ON bisects CD and so, ON ⊥ CD
It is known that AB = CD. So,
AM = ND — (i)
and MB = CN — (ii)
Now, triangles ΔOME and ΔONE are similar by RHS congruency since
∠OME = ∠ONE (They are perpendiculars)
OE = OE (It is the common side)
OM = ON (AB and CD are equal and so, they are equidistant from the centre)
∴ ΔOME≅ ΔONE
ME = EN (by CPCT) — (iii)
Now, from equations (i) and (ii) we get,
AM + ME = ND + EN
So, AE = ED
Now from equations (ii) and (iii) we get,
MB – ME = CN – EN
So, EB = CE (Hence proved).

Q3. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
Ans: From the question we know the following:
AB and CD are 2 chords which are intersecting at point E.
PQ is the diameter of the circle.
AB = CD.
Now, we will have to prove that BEQ = CEQ
For this, the following construction has to be done:
Construction:
Draw two perpendiculars are drawn as OM ⊥ AB and ON ⊥ D. Now, join OE. The constructed diagram will look as follows:
Now, consider the triangles ΔOEM and ΔOEN.
Here,
OM = ON [Since the equal chords are always equidistant from the centre]
OE = OE [It is the common side]
∠OME = ∠ONE [These are the perpendiculars]
So, by RHS congruency criterion, ΔOEM ≅ ΔOEN.
Hence, by CPCT rule, MEO = NEO
∴ BEQ = CEQ (Hence proved).

Q4. If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig).
Ans: The given image is as follows:
First, draw a line segment from O to AD such that OM ⊥ AD.
So, now OM is bisecting AD since OM ⊥ AD.
Therefore, AM = MD — (i)
Also, since OM ⊥ BC, OM bisects BC.
Therefore, BM = MC — (ii)
From equation (i) and equation (ii),
AM – BM = MD – MC
∴ AB = CD.

Q5. Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?
Ans:
Let the positions of Reshma, Salma and Mandip be represented as A, B and C respectively.
From the question, we know that AB = BC = 6cm.
So, the radius of the circle i.e. OA = 5cm
Now, draw a perpendicular BM ⊥ AC.
Since AB = BC, ABC can be considered as an isosceles triangle. M is mid-point of AC. BM is the perpendicular bisector of AC and thus it passes through the centre of the circle.
Now,
let AM = y and
OM = x
So, BM will be = (5 – x).
By applying Pythagorean theorem in ΔOAM we get,
OA2 = OM2 +AM2
⇒ 52 = x2 +y2 — (i)
Again, by applying Pythagorean theorem in ΔAMB,
AB2 = BM2 +AM2
⇒ 62 = (5 – x)2 + y2 — (ii)
Subtracting equation (i) from equation (ii), we get
36 – 25 = (5 – x)2 + y2 – x2 – y2
Now, solving this equation we get the value of x as
x = 7 / 5
Substituting the value of x in equation (i), we get
y2 + (49 / 25) = 25
⇒ y2 = 25 – (49 / 25)
Solving it we get the value of y as
y = 24 / 5
Thus,
AC = 2 × AM
= 2 × y
= 2 × (24 / 5) m
AC = 9.6 m
So, the distance between Reshma and Mandip is 9.6 m.

Q6. A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.
Ans: First, draw a diagram according to the given statements. The diagram will look as follows.
Here the positions of Ankur, Syed and David are represented as A, B and C respectively. Since they are sitting at equal distances, the triangle ABC will form an equilateral triangle.
AD ⊥ BC is drawn. Now, AD is median of ΔABC and it passes through the centre O.
Also, O is the centroid of the ΔABC. OA is the radius of the triangle.
OA = 2 / 3 AD
Let the side of a triangle a metres then BD = a / 2 m.
Applying Pythagoras theorem in ΔABD,
AB2 = BD2 + AD2
⇒ AD2 = AB2 – BD2
⇒ AD2 = a2 – (a / 2)2
⇒ AD2 = 3a2 / 4
⇒ AD = √3a / 2
OA = 2 / 3 AD
20 m = 2 / 3 × √3a / 2
a = 20√3 m
So, the length of the string of the toy is 20√3 m.

Exercise 9.3

Q1. In Fig, A, B and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

Ans: It is given that,
AOC = AOB+BOC
So, AOC = 60°+30°
∴ AOC = 90°
It is known that an angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle.
So,
ADC = (½)AOC
= (½)× 90° = 45°


Q2. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Ans:

Given: In circle C (O, r), OA = AB.
To find: ∠ADB and ∠ACB.

Solution:
In △OAB:
OA = AB
OA = OB
Hence, OA = OB = AB

⇒ ABC is an equilateral triangle.
Therefore, ∠AOB = 60° [Each angle of an equilateral triangle is 60°]

∠AOB = 2∠ADB
⇒ ∠ADB = ½ ∠AOB
⇒ ∠ADB = ½ × 60° = 30°

ACBD is a cyclic quadrilateral.
Therefore, ∠ACB + ∠ADB = 180° [The sum of either pair of opposite angles of a cyclic quadrilateral is 180°]

⇒ ∠ACB + 30° = 180°
⇒ ∠ACB = 180° – 30° = 150°


Q3. In Fig, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

Ans: The angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle.
So, the reflex POR = 2×PQR
We know the values of angle PQR as 100°
So, POR = 2×100° = 200°
∴ POR = 360°-200° = 160°
Now, in ΔOPR,
OP and OR are the radii of the circle
So, OP = OR
Also, OPR = ORP
Now, we know the sum of the angles in a triangle is equal to 180 degrees
So,
POR+OPR+ORP = 180°
OPR+OPR = 180°-160°
As OPR = ORP
2OPR = 20°
Thus, OPR = 10°


Q4. In Fig, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

Ans: We know that angles in the segment of the circle are equal so,
∠BAC = ∠BDC
Now in the in ΔABC, the sum of all the interior angles will be 180°
So, ∠ABC + ∠BAC + ∠ACB = 180°
Now, by putting the values,
∠BAC = 180° – 69° – 31°
So, ∠BAC = 80°
∴ ∠BDC = 80°.


Q5. In Fig, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.

Ans: We know that the angles in the segment of the circle are equal.
So,
∠ BAC = ∠ CDE
Now, by using the exterior angles property of the triangle
In ΔCDE we get,
∠ CEB = ∠ CDE + ∠ DCE
We know that ∠ DCE is equal to 20°
So, ∠ CDE = 110°
∠ BAC and ∠ CDE are equal
∴ ∠ BAC = 110°.


Q6. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.

Ans: Consider the following diagram.
Consider the chord CD,
We know that angles in the same segment are equal.
So, ∠ CBD = ∠ CAD
∴ ∠ CAD = 70°
Now, ∠ BAD will be equal to the sum of angles BAC and CAD.
So, ∠ BAD = ∠ BAC + ∠ CAD
= 30° + 70°
∴ ∠ BAD = 100°
We know that the opposite angles of a cyclic quadrilateral sums up to 180 degrees.
So,
∠ BCD + ∠ BAD = 180°
It is known that ∠ BAD = 100°
So, ∠ BCD = 80°
Now consider the ΔABC.
Here, it is given that AB = BC
Also, ∠ BCA = ∠ CAB (They are the angles opposite to equal sides of a triangle)
∠ BCA = 30°
also, ∠ BCD = 80°
∠ BCA + ∠ ACD = 80°
Thus, ∠ ACD = 50° and ∠ ECD = 50°.


Q7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Ans:  Given:

1. The diagonals of the cyclic quadrilateral are diameters of the circle.

2. The quadrilateral is cyclic (its vertices lie on the circle).

Proof: 

A diameter of a circle subtends a 90 angle on the circumference of the circle (by the property of a semicircle).

Let the cyclic quadrilateral be ABCD, and let the diagonals AC and BD be the diameters of the circle.

Since AC is a diameter:

∠ABC=90∘ and   ∠ADC=90(Both ∠ABC and ∠ADC are subtended by AC on opposite sides of the circle).

Similarly, since BD is a diameter:

∠BAD = 90 and ∠BCD = 90 (Both ∠BAD  and ∠BCD are subtended by BD).

From the above, all four angles of the quadrilateral ABCD are 90. Hence, ABCD is a quadrilateral where:

∠ABC = ∠BCD = ∠CDA = ∠DAB = 90


Q8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Ans: Construction:
Consider a trapezium ABCD with AB ║CD and BC = AD.
Draw AM ⊥ CD and BN ⊥ CD.
In ΔAMD and ΔBNC,
The diagram will look as follows:
In ΔAMD and ΔBNC,
AD = BC (Given)
∠AMD = ∠BCN (By construction, each is 90º)
AM = BN (Perpendicular distance between two parallel lines is same)
ΔAMD ≅ ΔBNC (RHS congruence rule)
∠ADC = ∠BCD (CPCT)…(1)
∠BAD and ∠ADC = 180º….(2) [Sum of the co-interior angles]
∠BAD and ∠BCD = 180º [Using equation (1)]
This equation shows that the opposite angle are supplementary.
Therefore, ABCD is a cyclic quadrilateral.


Q9. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig). Prove that ∠ACP = ∠ QCD.

Ans: Construction:
Join the chords AP and DQ.
For chord AP, we know that angles in the same segment are equal.
So, ∠ PBA = ∠ ACP ……(i)
Similarly for chord DQ,
∠ DBQ = ∠ QCD …..(ii)
It is known that ABD and PBQ are two line segments which are intersecting at B.
At B, the vertically opposite angles will be equal.
∴ ∠ PBA = ∠ DBQ ……(iii)
From equation (i), equation (ii) and equation (iii) we get,
∠ ACP = ∠ QCD.


Q10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Ans: First draw a triangle ABC and then two circles having diameter as AB and AC respectively.
We will have to now prove that D lies on BC and BDC is a straight line.
Proof:
We know that angle in the semi-circle are equal
So, ∠ ADB = ∠ ADC = 90°
Hence, ∠ ADB + ∠ ADC = 180°
∴ ∠ BDC is straight line.
So, it can be said that D lies on the line BC.


Q11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.

Ans: We know that AC is the common hypotenuse and ∠ B = ∠ D = 90°.
Now, it has to be proven that ∠ CAD = ∠ CBD
Since, ∠ ABC and ∠ ADC are 90°, it can be said that They lie in the semi-circle.
So, triangles ABC and ADC are in the semi-circle and the points A, B, C and D are concyclic.
Hence, CD is the chord of the circle with center O.
We know that the angles which are in the same segment of the circle are equal.
∴ ∠ CAD = ∠ CBD


Q12. Prove that a cyclic parallelogram is a rectangle.

Ans: It is given that ABCD is a cyclic parallelogram and we will have to prove that ABCD is a rectangle.
Proof:
Let ABCD be a cyclic parallelogram.
∠A + ∠C = 180º (Opposite angles of a cyclic quadrilateral)….(1)
We know that opposite angles of a parallelogram are equal.
∠A = ∠C and ∠B = ∠D
From equation (1),
∠A + ∠C = 180º
∠A + ∠A = 180º
2∠A = 180º
∠A = 90°
Parallelogram ABCD has one of its interior angles as 90º.
Thus, ABCD is a rectangle.

08. Quadrilaterals – Textbook Solutions

Exercise 8.1

Q1. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Ans: 

Given that,
AC = BD
To Prove:  ABCD is a rectangle if the diagonals of a parallelogram are equal.
To show ABCD is a rectangle, we have to prove that one of its interior angles is right-angled.
Proof,
In ΔABC and ΔBAD,
AB = BA (Common)
BC = AD (Opposite sides of a parallelogram are equal)
AC = BD (Given)
Therefore, ΔABC ≅ ΔBAD [SSS congruency]
∠A = ∠B [Corresponding parts of Congruent Triangles]
also,
∠A+∠B = 180° (Sum of the angles on the same side of the transversal)
⇒ 2∠A = 180°
⇒ ∠A = 90° = ∠B
Therefore, ABCD is a rectangle.
Hence Proved.


Q2. Show that the diagonals of a square are equal and bisect each other at right angles.
Ans: Given: A square is given.
To Prove: The diagonals of a square are the same and bisect each other at 90
Proof: Consider ABCD to be a square.

Consider the diagonals AC and BD intersect each other at a point O.
We have to show that the diagonals of a square are equal and bisect each other at right angles,
AC  =  BD, OA  =  OC, OB  =  OD .
In ΔABC and ΔDCB,
AB = DC (Sides of the square are equal)
∠ABC = ∠DCB (All the interior angles are of the value 90o)
BC = CB (Common side)
∴ ΔABC ≅ ΔDCB (By SAS congruency)
∴ AC= DB (By CPCT)
Hence, the diagonals of a square are equal in length.
In ΔAOB and ΔCOD,
∠AOB = ∠COD (Vertically opposite angles)
∠ABO = ∠CDO (Alternate interior angles)
AB = CD (Sides of a square are always equal)
∴ ΔAOB ≅ ΔCOD (By AAS congruence rule)
∴ AO = CO and OB = OD (By CPCT)
As a result, the diagonals of a square are bisected.
In ΔAOB and ΔCOB,
Because we already established that diagonals intersect each other,
AO = CO
AB = CB (Sides of a square are equal)
BO = BO (Common)
∴ ΔAOB ≅ ΔCOB (By SSS congruency)
∴ ∠AOB = ∠COB (By CPCT)
However, ∠AOB + ∠COB = 180° (Linear pair)
∠AOB +∠AOB = 180°
2 x ∠AOB = 180°
∠AOB = 90°

As a result, the diagonals of a square are at right angles to each other.


Q3. Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig.). Show that

Show that
(i) It is bisecting ∠C also, 
(ii) ABCD is a rhombus

Ans: Given: Diagonal AC of a parallelogram ABCD is bisecting ∠A
(i) ABCD is a parallelogram.
∠DAC = ∠BCA (Alternate interior angles) … (1)
And ∠BAC = ∠DCA (Alternate interior angles) … (2)
However, it is given that AC is bisecting ∠A
∠DAC = ∠BAC … (3)
Putting the values of eq (1) and eq (2) in eq (3)
∠BCA = ∠DCA
Hence, AC is bisecting ∠C
(ii) From Equation (4), we obtain
∠DAC = ∠DCA
DA = DC (Side opposite to equal angles are equal)
However, DA = BC and AB = CD (Opposite sides of a parallelogram)
AB = BC = CD = DA
As a result, ABCD is a rhombus.


Q4. ABCD is a rectangle in which diagonal AC bisects A as well as C . Show that:
(i) ABCD is a square 
(ii) Diagonal BD bisects 
B as well as D.
Ans: Given: ABCD is a rectangle where the diagonal AC bisects ∠A as well as ∠C.

(i) It is given that ABCD is a square.
∠A = ∠C
⇒ 1/2 ∠A = 1/2 ∠C (AC bisects ∠A and ∠C)
⇒ ∠DAC = 1/2 ∠DCA
CD = DA (Sides that are opposite to the equal angles are also equal)
Also, DA = BC and AB = CD (Opposite sides of the rectangle are same)
AB = BC = CD = DA
ABCD is a rectangle with equal sides on all sides.
Hence, ABCD is a square.
(ii) Let us now join BD.
In ΔBCD,
BC = CD (Sides of a square are equal to each other)
∠CDB = ∠CBD (Angles opposite to equal sides are equal)
However, ∠CDB = ∠ABD (Alternate interior angles for AB || CD)
∠CBD = ∠ABD
BD bisects ∠B.
Also, ∠CBD = ∠ADB (Alternate interior angles for BC || AD)
∠CDB = ∠ABD
BD bisects ∠D and ∠B.


Q5. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure).

Show that: 
(i) ΔAPD ≅ ΔCQB 

(ii) AP = CQ 

(iii) ΔAQB ≅ ΔCPD 

(iv) AQ = CP

(v) APCQ is a Parallelogram
Ans: 
(i) In ΔAPD and ΔCQB,
AD = CB (Opposite sides of the parallelogram ABCD)
∠ADP = ∠CBQ (Alternate interior angles for BC || AD)
DP = BQ (Given)
∴ ΔAPD  ΔCQB (Using SAS congruence rule)
(ii) As we had observed that ΔAPD  ΔCQB,
∴ AP= CQ (CPCT)
(iii) In ΔAQB and ΔCPD,
∠ABQ = ∠CDP (Alternate interior angles for AB || CD )
AB = CD (Opposite sides of parallelogram ABCD)
BQ = DP (Given)
∴ ΔAQB  ΔCPD (Using SAS congruence rule)
(iv) Since we had observed that ΔAQB  ΔCPD,
∴ AQ = CP (CPCT)
(v) From the results obtained in (ii) and (iv),
AQ = CP and AP = CQ
APCQ is a parallelogram because the opposite sides of the quadrilateral are equal.


Q6. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (See the given figure).
Show that 
(i) ΔAPB ≅ ΔCQD 
(ii) AP = CQ
Ans: 
(i) In ΔAPB and ΔCQD,
∠APB = ∠CQD (Each 90°)
AB = CD (The opposite sides of a parallelogram ABCD)
∠ABP = ∠CDQ (Alternate interior angles for AB || CD)
∴ ΔAPB  ΔCQD (By AAS congruency)
(ii) By using
∴ ΔAPB  ΔCQD , we obtain
AP = CQ (By CPCT)


Q7. ABCD is a trapezium in which AB || CD and AD = BC (see the given figure).
Show that 
(i) ∠A = ∠B 
(ii) ∠C = ∠D 
(iii) 
ΔABC  ΔBAD 
(iv) diagonal AC = diagonal BD
(Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.)

Ans: Let us extend AB by drawing a line through C, which is parallel to AD, intersecting AE at point E. It is clear that AECD is a parallelogram.
(i) AD = CE (Opposite sides of parallelogram AECD)
However, AD = BC (Given)
Therefore, BC = CE
∠CEB = ∠CBE (Angle opposite to the equal sides are also equal)
Considering parallel lines AD and CE.
AE is the transversal line for them (Angles on a same side of transversal)
(Using the relation ∠CEB = ∠CBE) … (1)
However,  ∠CBE +  ∠CBA  = 180° (Linear pair angles) … (2)
From Equations (1) and (2), we obtain ∠A = ∠B
(ii) AB || CD
Also, ∠C + ∠B = 180° (Angles on a same side of a transversal)
∴ ∠A + ∠D = ∠C + ∠B
However, ∠A = ∠B (Using the result obtained in (i))
∴ ∠C = ∠D
(iii) In ΔABC and ΔBAD,
AB = BA (Common side)
BC = AD (Given)
∠B = ∠A (Proved before)
∴ ΔABC ≅ ΔBAD (SAS congruence rule)
(iv) We had seen that, ΔABC ≅ ΔBAD
∴ AC= BD (By CPCT)

Exercise 8.2

Q1. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is diagonal. Show that:
(i) SR || AC and SR = (1/2) AC
(ii) PQ = SR 
(iii) PQRS is a parallelogram.

Ans: Given: ABCD is a quadrilateral
To prove: (i) SR || AC and SR = 1/2 AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
(i) In ΔADC, S and R are the mid-points of sides AD and CD respectively.
In a triangle, the line segment connecting the midpoints of any two sides is parallel to and half of the third side.
∴ SR || AC and SR = 1/2 AC … (1)
(ii) In ΔABC, P and Q are mid-points of sides AB and BC respectively. Therefore, by using midpoint theorem,
PQ || AC and  PQ = 1/2 AC … (2)
Using Equations (1) and (2), we obtain
PQ || SR and PQ = SR
(iii) From Equation (3), we obtained
PQ || SR and PQ = SR
Clearly, one pair of quadrilateral PQRS opposing sides is parallel and equal. PQRS is thus a parallelogram.


Q2. ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
Ans: 

Given: ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.
To Prove: PQRS is a rectangle.
Construction:
Join AC and BD.
Proof:
In ΔDRS and ΔBPQ,
DS = BQ (Halves of the opposite sides of the rhombus)
∠SDR = ∠QBP (Opposite angles of the rhombus)
DR = BP (Halves of the opposite sides of the rhombus)
ΔDRS ≅ ΔBPQ [SAS congruency]
RS = PQ [CPCT]———————- (i)
In ΔQCR and ΔSAP,
RC = PA (Halves of the opposite sides of the rhombus)
∠RCQ = ∠PAS (Opposite angles of the rhombus)
CQ = AS (Halves of the opposite sides of the rhombus)
ΔQCR ≅ ΔSAP [SAS congruency]
RQ = SP [CPCT]———————- (ii)
Now,
In ΔCDB,
R and Q are the mid points of CD and BC, respectively.
⇒ QR || BD
also,
P and S are the mid points of AD and AB, respectively.
⇒ PS || BD
⇒ QR || PS
PQRS is a parallelogram.
also, ∠PQR = 90°
Now,
In PQRS,
RS = PQ and RQ = SP from (i) and (ii)
∠Q = 90°
PQRS is a rectangle.


Q3. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Ans: 

Given: ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.
To prove: The quadrilateral PQRS is a rhombus.
Proof: Let us join AC and BD.
In ΔABC, P and Q are the mid-points of AB and BC respectively.
∴ PQ || AC and PQ = 1/2 AC (Mid-point theorem) … (1)
Similarly, in ΔADC , SR ||  AR, SR = 1/2 AC (Mid-point theorem) … (2)
Clearly,  PQ || SR and  PQ = SR
It is a parallelogram because one pair of opposing sides of quadrilateral PQRS is equal and parallel to each other.
∴ PS || QR , PS = QR (Opposite sides of parallelogram) … (3)
In ΔBCD, Q and R are the mid-points of side BC and CD respectively.
∴ QR || BD, QR = 1/2 BD (Mid-point theorem) … (4)
Also, the diagonals of a rectangle are equal.
∴ AC = BD … (5)
By using Equations (1), (2), (3), (4), and (5), we obtain
PQ = QR = SR = PS
So, PQRS is a rhombus


Q4. ABCD is a trapezium in which  AB || DC , BD is a diagonal and E is the mid – point of AD. A line is drawn through E parallel to AB intersecting BC at F (see the given figure). Show that F is the mid-point of BC.

Ans: Given: ABCD is a trapezium in which AB || DC , BD is a diagonal and E is the mid – point of AD. A line is drawn through E parallel to AB intersecting BC at F.
To prove: F is the mid-point of BC.
Proof: Let EF intersect DB at G.
We know that a line traced through the mid-point of any side of a triangle and parallel to another side bisects the third side by the reverse of the mid-point theorem.
In  ΔABD, EF || AB and E is the mid-point of AD.
Hence, G will be the mid-point of DB.
As  EF || AB, AB || CD,
∴ EF || CD (Two lines parallel to the same line are parallel)
In  ΔBCD, GF || CD and G is the mid-point of line BD. 
So, by using the converse of mid-point theorem, F is the mid-point of BC.


Q5. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.
Ans: Given: In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively to prove: The line segments AF and EC trisect the diagonal BD.

To Prove: The line segments AF and EC trisect the diagonal BD. 

Proof: ABCD is a parallelogram.
AB || CD
And hence, AE || FC
Again, AB = CD (Opposite sides of parallelogram ABCD)
1/2 AB = 1/2 CD
AE = FC (E and F are mid-points of side AB and CD)
In quadrilateral AECF, one pair of opposite sides (AE and CF) is parallel and the same as each other. So, AECF is a parallelogram.
∴ AF || EC (Opposite sides of a parallelogram)
In  ΔDQC, F is the mid-point of side DC and  FP || CQ (as  AF || EC ). 
So, by using the converse of mid-point theorem, it can be said that P is the mid-point of DQ.
∴ DP= PQ … (1)
Similarly, in ΔAPB , E is the mid-point of side AB and EQ || AP (as  AF || EC ).
As a result, the reverse of the mid-point theorem may be used to say that Q is the mid-point of PB.
∴ PQ = QB … (2)
From Equations (1) and (2),
DP = PQ= BQ
Hence, the line segments AF and EC trisect the diagonal BD.


Q6. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that
(i) D is the mid-point of AC 
(ii) MD ⊥ AC
(iii) CM = MA = 1/2 AB
Ans: Given: ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D.
(i) In ΔABC,
It is given that M is the mid-point of AB and MD || BC.
Therefore, D is the mid-point of AC. (Converse of the mid-point theorem)
(ii) As DM || CB and AC is a transversal line for them, therefore,(Co-interior angles)
(iii) Join MC.
In ΔAMD and  ΔCMD,
AD = CD (D is the mid-point of side AC)
∠ADM = ∠CDM (Each)
DM = DM (Common)
∴ ΔAMD ≅ ΔCMD (By SAS congruence rule)
Therefore,
AM = CM (By CPCT)
However, 
AM = 1/2 AB (M is mid-point of AB)
Therefore, it is said that CM = AM = 1/2 AB.