Text Book Solutions All Class

Multiple Choice Questions

Q1: Find the value of x(a) 98º

(b) 100º

(c) 108º

(d)96º

Ans: (c)

x + 36 + 36 = 180 ⇒ x = 108º

Q2: A pair of angles is called linear pair if the sum of two adjacent angles is?

(a) 180º

(b) 90º

(c) 270º

(d) 360º

Ans: (a)

Q3: Find the value of x, y and z(a) x = 110º , y = 70º, z = 80º(b) x = 70º , y = 110º, z = 60º(c) x = 70º, y = 100º, z = 70º(d) x = 70º, y = 110º, z = 70º

Ans: (d)

Vertically opposite angle theorem and linear pair axiom can be used to find the answer

Q4: An exterior angle of the triangle is 110º. And its two opposite interior angles are in the ratio 5:6. What are the values of those angles?

(a) 50º, 60º

(b) 25º, 30º

(c) 35º, 42º

(d) 40º, 48º

Ans: (a)

Q5: Lines l || k and m || n. Find the value of angle z(a) 45º

(b) 60º

(c) 70º

(d) 50º

Ans: (b)

Q6:Find the value of x(a) 67º(b) 71º(c) 57º(d) None of these

Ans: (a)

Q7: The side BC, AB and AC of the triangle ABC are produced in order forming exterior angles ∠ACD = x, ∠BAE = y and ∠CBF = z then the value of 2x + 2y + 2z is

(a) 180°

(b) 360°

(c) 540°

(d) 720°

Ans: (d)

Let a, b,c be the angles of triangle, then a + b + c = 180Now x = a + b, y = b + c, z = a + cTherefore 2x + 2y + 2z = 4(a + b + c) = 720

Q8: The angles of the triangles are in the ratio 1 : 2 : 3. Then the triangle is

(a) scalene

(b) obtuse angled

(c) acute angled

(d) right angles

Ans: (d)

Let x, 2x, 3x be the angles of triangle, then x + 2x + 3x = 180Therefore x = 30So angles are 30, 60 and 90

True or False

Q1: Pairs of vertically opposite angles is always equal

Ans: True

Q2: The sum of the angles of a triangle is 180º

Ans: True

Q3: If the sum of two adjacent angles is 45º, then two adjacent angles are acute angles

Ans: True

Q4: If a line is perpendicular is one of two parallel lines, then it is also perpendicular to the other

Ans: True

Q5: Two lines are intersected by the transversal, and then the corresponding angles are equal

Ans: False

Q6: Can we have a triangle where all the interior angles are more than 60º

Ans: False

Q7: Sum of two complimentary angles is equal to 90º

Ans: True

Q8: Sum of all the exterior angles of any polygon is always 360º

Ans: True

Fill in the blanks 

Q1: Sum of two supplementary angles is ______.

Ans: 180º

Q2: Two lines parallel to the same line is ____ each other.

Ans: parallel

Q3: An acute angle is always less than _____.

Ans: 90º

Q4: Angles forming a linear pair are ______.

Ans: supplementary

Q5: If one angle of triangle is equal to the sum of other two angles, then the triangle is ______.

Ans: right angle triangle

Q6: If two straight lines intersect, the adjacent angles are ______.

Ans: supplementary

Table Type Question

Ans: Complementary = 90 − xSupplementary = 180 − x

Short Answer Type Questions

Q: Write the type of angles

(i) 

(ii) 

(iii) 

(iv) 

(v) 

(vi) 

(vii) 

Ans:

(i) Acute angle

(ii) Right angle

(iii) Obtuse angle

(iv) Straight angle

(v) Reflex angle

(vi) Vertically opposite angle

(vii) Alternate interior angles

Multiple Choice Questions

Q1: Find the value of x(a) 98º

(b) 100º

(c) 108º

(d) 96º

Q2: A pair of angles is called linear pair if the sum of two adjacent angles is?

(a) 180º

(b) 90º

(c) 270º

d) 360º

Q3: Find the value of x, y and z(a) x = 110º , y = 70º, z = 80º

(b) x = 70º , y = 110º, z = 60º(c) x = 70º, y = 100º, z = 70º(d) x = 70º, y = 110º, z = 70ºQ4: An exterior angle of the triangle is 110º. And its two opposite interior angles are in the ratio 5:6. What are the values of those angles?(a) 50º, 60º(b) 25º, 30º(c) 35º, 42º(d) 40º, 48ºQ5: Lines l || k and m || n. Find the value of angle z(a) 45º

(b) 60º

(c) 70º

(d) 50ºQ6:Find the value of x

(a) 67º

(b) 71º

(c) 57º

(d) None of these

Q7: The side BC, AB and AC of the triangle ABC are produced in order forming exterior angles ∠ACD = x, ∠BAE = y and ∠CBF = z then the value of 2x + 2y + 2z is

(a) 180°

(b) 360°

(c) 540°

(d) 720°

Q8: The angles of the triangles are in the ratio 1 : 2 : 3. Then the triangle is

(a) scalene

(b) obtuse angled

(c) acute angled

(d) right angles

True or False

Q1: Pairs of vertically opposite angles is always equal

Q2: The sum of the angles of a triangle is 180º

Q3: If the sum of two adjacent angles is 45º, then two adjacent angles are acute angles

Q4: If a line is perpendicular is one of two parallel lines, then it is also perpendicular to the other

Q5: Two lines are intersected by the transversal, and then the corresponding angles are equal

Q6: Can we have a triangle where all the interior angles are more than 60º

Q7: Sum of two complimentary angles is equal to 90º

Q8: Sum of all the exterior angles of any polygon is always 360º

Fill in the blanks 

Q1: Sum of two supplementary angles is ______.

Q2: Two lines parallel to the same line is ____ each other.

Q3: An acute angle is always less than _____.

Q4: Angles forming a linear pair are ______.

Q5: If one angle of triangle is equal to the sum of other two angles, then the triangle is ______.

Q6: If two straight lines intersect, the adjacent angles are ______.

Table Type QuestionShort Answer Type Questions

Q: Write the type of angles

(i) 

(ii) 

(iii) 

(iv) 

(v) 

(vi) 

(vii) 

Q1. If A, B and C are three points on a line, and ‘B’ lies between ‘A’ and ‘C’ (as shown in the figure), then prove that: AB + BC = AC

Sol.

In the figure given above, AC coincides with AB+ BC

From Euclid’s Axiom 4: Things which coincide with one another are equal to each other.

So, we write AB+BC=AC

Q2. Prove that an equilateral triangle can be constructed on any given line segment. 

Sol. Equilateral triangle is a angle with all sides are equal

1. Draw a line segment AB of the any length.

2. Take compass put the pointy end at point A & pencile at point B.

3.Draw an arc.

Here we draw an arc of radii AB

4. Now put the pointy end at B and pencil at A.

5. Draw another arc.

Here we draw an arc of radii BA .

6. Mark the intersecting point as C.

7. Join point A to point C by a straight line .

8. Join point B to point C by a straight line

ABC is the triangle.

Q3. Prove that two distinct lines cannot have more than one point in common.

Ans 3. 

Given: Two distinct line l and m

To Prove: Lines l and m have at most one point in common.

Proof: Two distinct lines l and m intersect at a point P.

Let us suppose they will interact at another point, say Q (different from P).

It means two lines l and m passing through two distinct point P and Q.

But it is contrary to the axiom 5.1 which states that “Given two distinct points, there exists one and only one line pass through them”

So our supposition is wrong

Hence, two distinct lines cannot have more than one point in common

Q4. Is the following statement true? “Attempts to prove Euclid’s fifth postulate using  the other postulate and axioms led to the discovery of several other geometries.”

True

Q5. Fill in the blanks to complete the following axioms :
(i) Things, which are equal to the same things, are ………………………….
(ii) If equals are added to equals, the ………………………….
(iii) If equals are subtracted from equals, ………………………….
(iv) Things which coicide with one another are ………………………….
(v) The whole is greater than the ………………………….
(vi) Things which are double of the same things are ………………………….
(vii) Things which are halves of the same things, are ………………………….

Sol (i) equal to one another
(ii) wholes are equal
(iii) the remainders are equal
(iv) equal to one another
(v) part
(vi) equal to one another
(vii) equal to one another

Q6. In the figure, line PQ falls on AB and CD such that (∠1 + ∠2) < 180°. So, lines AB and CD, if produced will intersect on the left of PQ. This is an example of which Postulate of Euclid?

Sol. Fifth postulate

Q7. In the given figure, if AC = BD, then prove that AB = CD

Hint: AC = BC   [given]     …(1)
∴ AC = AB + CD [∵ B lies between A and C]    …(2)
BD = BC + CD [∵ C lies between B and D]      …(3)
From (1), (2) and (3) we have: AB + BC = BC + CD or AB = CD

Sol 

From the figure, it can be observed that

AC = AB + BC

BD = BC + CD

It is given that AC = BD

AB + BC = BC + CD (1)

According to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal.

Subtracting BC from equation (1), we obtain

AB + BC − BC = BC + CD − BC

AB = CD

Q8. Write ‘true’ or ‘false’ for the following statement:
(i) Three lines are concurrent if they have a common point.
(ii) A line separates a plane into three parts, namely the two half-planes and the line itself. (iii) Two distinct lines in a plane cannot have more than one point in common.
(iv) A ray has two endpoints.
(v) A line has indefinite length.

Sol. (i),  (ii), (iii) and  (v) are true.   

Q9. How many lines can pass through:
(i) one point
(ii) two distinct points?

Sol (i) infinite  
(ii) one only

Q10. AB and CD are two distinct lines. In how many points can they at the most intersect? 

Sol. one point only

Q11. Prove that any line segment has one and one mid-point.

Sol. Let AB be a line segment

and let D and E be its two midpoints

now, since D is the midpoints of AB

so, AD=DB

AB=AD+DB=2AD-(1)

Also E is a point of AB

So, AE=EB

AB=AE+EB=2AE-(2)

From eq 1 &2

2AD=2AE

D and E coincide to each other

AB has one and only one mid points

Hence every line segments has one and only one midpoint.

Q12. In the given figure, AB = BC, P is midpoint of AB and Q is midpoint of BC. Show that AP = QC
Hint: Things which are halves of the same thing (or equal things) are equal to one another.

Sol. We have AB = BC…(1) [Given]

Now, A, M. B are the three points on a line, and M lies between A and B, then

AM + MB = AB …(2)

Similarly, BN * NC = BC….(3)

So. we get AM + MB = BN * NC

From (1), (2). (3) and Euclid’s axiom 1

Since M Is the mid-point of AB and N is the mid-point of BC. therefore

2AM = 2NC

Using axiom 6, things which are double of the same thing are equal to one another.

Hence, AM = NC.

Q1. Write each of the following is an equation in two variables:
(i) x = –3
(ii)y = 2
(iii) 2x = 3
(iv) 2y = 5

Sol. 

(i)Given equation, x = -3

The above equation can be written in two variables as,

x + 0.y + 3 = 0

(ii) Given equation, y =2

The above equation can be written in two variables as,

0.x + y – 2 = 0

(iii) Given equation, 2x =3

The above equation can be written in two variables as,

2x + 0.y – 3 = 0

(iv) Given equation, 2y =5

The above equation can be written in two variables as,

2y-5= 0

(0)x + 2y- 5= 0

Q2. Write each of the following equations in the form ax + by + c = 0 and also write the values of a, b and c in each case:
(i) 2x + 3y = 3.47
(ii) x – 9 = √3 y
(iii) 4 = 5x – 8y
(iv) y = 2x

Sol. 

(i) 2x + 3y – 3.47 = 0; a = 2, b = 3 and c = –3.47
(ii) x – √3y – 9 = 0; a = 1, b = – √3 and c = –9
(iii) –5x + 8y + 4 = 0; a = –5, b = 8 and c = 4
(iv) –2x + y + 0 = 0; a = –2, b = 1 and c = 0

Q3. (a) Is (3, 2) a solution of 2x + 3y = 12?
(b) Is (1, 4) a solution of 2x + 3y = 12?
(c) Is  a solution of 2x + 3y = 12?
(d) Is  a solution of 2x + 3y = 12?

Sol.

(a) Yes, 

2(3)+ 3(2)= 6+6 =12

(b) No, 

2(1)+ 3(4)= 2+12 =14

(c) Yes, 

2(-5)+ 3(22/3)= -10+ 22 =12

(d) Yes, 

2(2)+ 3(8/3)= 4+8 =12

Q4. Find four different solutions of the equation x + 2y = 6.

Sol. To find the solutions, substitute different values of y and calculate the corresponding values of x.Hence, four different solutions are:
(6,0), (4,1), (2,2), (0,3)

Q5. Find two solutions for each of the following equations:
(i) 4x + 3y = 12
(ii) 2x + 5y = 0
(iii) 3y + 4 = 0

Sol. 1) 4x+3y=12

for y=4

4x+12=12 x=0

for y=0

4x+0=12 x=3

(0,4) & (3,0) are 2 solution

2) 2x+5y=0

for y=−2

2x−10=0 x=5

for y=−4

2x−20=0 x=10

(5,−2) and (10,−4) are 2 solutions

3) 3y+4=0

y=−4/3 is only solution

Q6. Find the value of k such that x = 2 and y = 1 is a solution of the linear equation 2x – ky + 7 = 8

Sol. We can find the value of k by substituting the values of x and y in the given equation.

By substituting the values of x = 2 and y = 1 in the given equation

2x – ky + 7 = 8

⇒ 2(2) – k(1) + 7 = 8

⇒ 4- k+ 7=8

⇒ -k=8-11

k=3

Therefore, the value of k is 3.

Q7. Draw the graph of  y+x = 4.

Sol. Let x be 0 = (0,4)

Let y be 0 = (4,0)

Q8. Force applied on a body is directly proportional to the acceleration produced in the body.
Write an equation to express this situation and plot the graph of the equation.

Sol. Given that, the force (F) is directly proportional to the acceleration (a).

i.e., F∝a

⇒F=ma [where, ,m=arbitrary constant and take value 6 kg of mass ]

∴                           F=6a

(i) If a=5m/s2, then from Eq. (i), we get

F=6×5=30N

(ii) If a=6m/s2, then from Eq. (i), we get

F=6×6=36N

Here, we find two points A (5, 30) and B (6, 36). So draw the graph by plotting the points and joining the line AB. 

Q9. For each of the graph given in the following figure select the equation whose graph it is from the choices given below:
(i) x + y = 0

(ii) x – y = 0

(iii) 2x = y
(iv) y = 2x + 1 

(i) x + y = 0

(ii) x – y = 0
(iii) y = 2x + 4
(iv) y = x – 4 

(i) x + y = 0

(ii) x – y = 0
(iii) y = 2x + 1
(iv) y = 2x – 4 

(i) x + y = 0
(ii) x – y = 0
(iii) 2x + y = –4
(iv) 2x + y = 4

Sol.
(a) x – y = 0
(b) y = 2x + 4
(c) y = 2x – 4
(d) 2x + y = –4

Q10. Which of the following is not a linear equation in two variables?
(i) px + qy + c = 0
(ii) ax² + bx + c = 0
(iii) 3x + 2y = 5

Sol. 

(ii) ax² + bx + c = 0 

(ii) is not a linear equation because it consists x² in it. Linear equation will not contain any exponent to variables

Q11. One of the solutions of the linear equation 4x – 3y + 6 = 0 is
(i) (3, 2)
(ii) (–3, 2)
(iii) (–3, –2)

Sol.  Option (iii) –3, –2 

Q12. lx + my + c = 0 is a linear equation in x and y. For which of the following, the ordered pair (p, q) satisfies it:
(i) lp + mq + c = 0
(ii) y = 0
(iii) x + y = 0
(iv) x = y

Sol. 

 $lp+mq+c=0$lp+mq+c=0

To check if $(p,q)$(p,q) satisfies the equation: $l\cdot p+m\cdot q+c=0$l⋅p+m⋅q+c=0

This matches the form of the linear equation $lx+my+c=0$lx+my+c=0, so statement (i) is correct.

Q13. What is the equation of the x-axis? 

Sol. The x-axis is the horizontal line where $y=0$y=0.

Equation of the x-axis: y=0.

Q14. What is the equation of the y-axis? 

Sol. The y-axis is the vertical line where $x=0$x=0.

Equation of the y-axis: x=0.

Q15. How many solutions do a linear equation in two variables x and y have?

Sol. 

A linear equation in two variables will have infinite solutions

1. Fill in the blanks:
(i) The abscissa of the origin is ____.
(ii) The ordinate of the point (–5, 3) is ____.
(iii) The coordinates of origin are (____, ____).
(iv) The ordinate of every point on x-axis is ____.
(v) The abscissa of every point on y-axis is ____.
(vi) The axes intersect at a point called ____.

2. Look at the following figure and answer the questions given below.

(i) What are the coordinates of B?
(ii) What are the coordinates of M?
(iii) What are the coordinates of L?
(iv) What are the coordinates of S?

3. Write the coordinates of the points marked on the axes in the following figure.


4. Locate the following points in the Cartesian plane.
(i) (4, 0)
(ii) (0, 4)
(iii) (–3, 0)
(iv) (–3, 3)
(v) (5, –3)
(vi) (–3, 5)
(vii) (–5, 3)
(viii) (2, –7)
(ix) (–7, –7)

ANSWERS
1. (i) 0
(ii) 3
(iii) (0, 0)
(iv) 0
(v) 0
(vi) origin 

2. (i) (4, 3)
(ii) (–3, 4)
(iii) (–4, –4)
(iv) (3, –4) 

3. Coordinates of A are (4, 0); Coordinates of B are (0, 3)
Coordinates of C are (–5, 0); Coordinates of D are (0, –4)
Coordinates of E are (1, 0). 

Multiple Choice Questions

Q1: If x/y + y/x = -1 and (xy, ≠ 0), the value of x³ – y³ is
(a) 1
(b) -1
(c) 0
(d) 2
Ans: (c)
x/y + y/x = -1
x² + y² = − xy
or
x² + y² + xy = 0
Now x³ – y³ = (x − y) (x² + y² + xy) = (x − y) × 0 = 0
Hence (c) is the correct answer

Q2: If p + q + r = 0 ,then the value of 
(a) 1
(b) 3
(c) -1
(d) 0
Ans: (b)
Now we know that p³ + q³ + r³ − 3pqr = (p + q + r) (p² + q² + r² − pq − qr − pr)
as p + q + r = 0
p³ + q³ + r³ − 3pqr = 0 or p³ + q³ + r³ = 3pqr
Now

Hence (b) is the correct answer.

Q3: The product of (x + a) (x + b) is
(a) x² + (a + b)x + ab
(b) x² – (a – b)x + ab
(c) x² + (a – b)x + ab
(d) x² + (a – b)x + ab
Ans: (a)
(x+a) (x+b) = x(x+b) + a(x+b)
$$=x² + bx + ax + ab
$$=x² + (a+b)x + ab

Q4: The value of (x + 2y + 2z)² + (x – 2y – 2z)² is
(a) 2x² + 8y² + 8z²
(b) 2x² + 8y² + 8z² + 8xyz
(c) 2x² +8y² + 8z² – 8yz
(d) 2x² + 8y² + 8z² + 16yz
Ans: (d)

Q5: The value of f(x) = 5x−4×2+3 when x = -1, is: 
(a) 3 
(b) -12 
(c) -6

(d) 6
Ans: (c)

To find the value of $$f(x) = 5x − 4x²+ 3 when $$x = −1, follow these steps:

Substitute x = −1 into the equation:

$$f (−1) = 5(−1) − 4(−1)² + 3

f(−1) = −5 −4 + 3

f(−1) = −6

True or False

Q1: P(x) = x – 1 and g(x) =x² – 2x  + 1 . p(x) is a factor of g(x)
Ans: True, as g(1) = 0

Q2: The factor of 3x² – x – 4 are (x + 1)(3x – 4)
Ans: True, we can get this by split method

Q3: Every linear polynomial has only one zero
Ans: True

Q4: Every real number is the zero’s of zero polynomial
Ans: True

Q5: A binomial may have degree 4
Ans: True, example x⁴ + 1

Q6: 0, 2 are the zeroes of x²– 2x
Ans: True

Q7: The degree of zero polynomial is not defined
Ans: True

Answer the following Questions

Q1: Is 3x^1/2 – 4x + 15 a polynomial of one variable?
Ans: No. it is not a polynomial

Q2: Is ∛x – √2x a polynomial
Ans: No. It is not a polynomial

Q3: What will be the degree of polynomials 30x⁵ – 15x² + 40
Ans: Degree of Polynomial is 5

Q4: Is (y²)^1/2 + 2√3 a polynomial of one variable?
Ans: Yes it is a polynomial of one variable

Q5: What will be the coefficient of x³ in 9x³ – 5x + 20.
Ans: The coefficient of x³ is 9

Q6: Show that x = 1 is a root of the polynomial 3x³ – 4x² + 8x – 7
Ans: On putting x = 1
X = 1 is root of polynomial
3x³ – 4x² + 8x – 7 
3(1)³  – 4(1)² + 8(1) – 7 
3 – 4 + 8 – 7 = 0
X = 1 is root of polynomial

Multiple Choice Questions

Q1: If x/y + y/x = -1 and (xy, ≠ 0), the value of x³ – y³ is
(a) 1
(b) -1
(c) 0
(d) 2

Q2: If p + q + r = 0 ,then the value of 
(a) 1
(b) 3
(c) -1
(d) 0

Q3: The product of (x + a) (x + b) is
(a) x² + (a + b)x + ab
(b) x² – (a – b)x + ab
(c) x² + (a – b)x + ab
(d) x² + (a – b)x + ab

Q4: The value of (x + 2y + 2z)² + (x – 2y – 2z)² is
(a) 2x² + 8y² + 8z²
(b) 2x² + 8y² + 8z² + 8xyz
(c) 2x² +8y² + 8z² – 8yz
(d) 2x² + 8y² + 8z² + 16yz

Q5: The value of f(x) = 5x−4x²+3 when x = -1, is:
(a) 3
(b) -12
(c) -6
(d) 6

True or False

Q1: P(x) = x – 1 and g(x) =x² – 2x  + 1 . p(x) is a factor of g(x)

Q2: The factor of 3x² – x – 4 are (x + 1)(3x – 4)

Q3: Every linear polynomial has only one zero

Q4: Every real number is the zero’s of zero polynomial

Q5: A binomial may have degree 4

Q6: 0, 2 are the zeroes of x²– 2x

Q7: The degree of zero polynomial is not defined

Answer the following Questions

Q1: Is 3x^1/2 – 4x + 15 a polynomial of one variable?

Q2: Is ∛x – √2x a polynomial.

Q3: What will be the degree of polynomials 30x⁵ – 15x² + 40.

Q4: Is (y²)^1/2 + 2√3 a polynomial of one variable?

Q5: What will be the coefficient of x³ in 9x³ – 5x + 20.

Q6: Show that x = 1 is a root of the polynomial 3x³ – 4x² + 8x – 7.

Q.1. What is the value of (23)2?

Solution: (23)2 = 23×2           [∵ (xm)n = xm.n]

= 26 =2 x 2 x 2 x 2 x 2 x 2 = 64

Q.2. What is the value of (625) 1/4 ?Solution:

Q.3. Which of the following are irrational numbers?

√3, √4, √5, 22/7, π, 0 

Solution:  √3, √5 and π are irrational numbers.

√3: Irrational (cannot be expressed as a fraction of two integers)

√4: Rational (equals 2, which is an integer)

√5: Irrational (cannot be expressed as a fraction of two integers)22/7: Rational (a fraction of two integers)

π: Irrational (cannot be expressed as a fraction of two integers)

0: Rational (can be expressed as 0/1)

Q.4. Which of the following are rational?

2√3, 1, 0, 3.14, π, 22/7

Solution: 1, 0, 3.14 and 22/7 are rational numbers.

2√3: Irrational (product of a rational and an irrational number)

1: Rational (an integer, can be expressed as 1/1)

0: Rational (can be expressed as 0/1)

3.14: Rational (a terminating decimal, can be expressed as 314/100)

π: Irrational (cannot be expressed as a fraction of two integers)

22/7: Rational (a fraction of two integers)

Q.5. Which of the following cube roots is not irrational?

3 √5, 3√6, 3√7, 3√8, 3√9

Solution:  3√8 is not an irrational number.

Q.6. Which three integers are equal to their own cube roots?

Solution: –1, 0 and 1

[∵ 3√-1 = -1, 3√1 = 1 and 3√0 = 0 ]

 Q.7. Is the following number rational or irrational?

0.0769230769230769230…

Solution:The given number is rational.  

The number 0.076923076923076923… is a repeating decimal.

The number is rational because it has a repeating pattern and can be expressed as a fraction of two integers.

Q.8. Is 0.666 …a “non-terminating rational” or “non-terminating recurring rational number”?

Solution: Non-terminating recurring rational number.

0.666… (where the 6 repeats indefinitely) is a repeating decimal.

It is a non-terminating recurring rational number because it has a repeating pattern and can be expressed as a fraction (specifically, 2/3)

.Q.9. Is 1.27 27 27 … non-terminating recurring or non-terminating non-recurring number?

Solution: It is a non-terminating recurring rational number.

1.272727… (where 27 repeats indefinitely) is a repeating decimal.

It is a non-terminating recurring rational number because it has a repeating pattern and can be expressed as a fraction.

Q.10. The decimal representation of √3 is 1.73205807… . Is it non-terminating nonrecurring?

Solution: YesThe decimal representation of √3 does not repeat and does not terminate.

Q.11. Is the product of a rational and an irrational number, rational?

Solution: No. It is an irrational number.

Q.12. Write the smallest non-negative integer. 

Solution: The smallest non-negative integer is ‘0’.

Q.13. Write the smallest positive integer.

Solution: The smallest positive integer is ‘1’.

Q.14. Up to how many decimal places, π and 22/7  are same?

Solution: π and 22/7 are same only up to 2 decimal places.

[∵ 22/7  = 3.142857142… and π =3.141592653589…]

Q.15. Which of the following is irrational?

Solution: √7 is irrational.

Q.16. Which of the following is the value of 0.999… expressed as p/q :

 (i) 9/10

(ii)  1

(iii)   999/1000

Solution:(ii) = 1

Q.1. Which of the following is an irrational number?

(a) 

(b) √3(c) 1/2

(d) 

Ans.An irrational number is a number that cannot be expressed as a fraction of two integers and has an infinite non-repeating decimal representation.

Let’s evaluate the options:

(a) √49/64: This is a rational number because both the numerator and denominator are perfect squares, and their square root can be expressed as a fraction of integers. √49/64 = 7/8.

(b) √3: This is an irrational number because the square root of 3 cannot be expressed as a fraction of integers, and its decimal representation goes on infinitely without repeating.

(c) 1/2: This is a rational number because it can be expressed as a fraction of integers.

(d) -√1/4: This is a rational number because √1/4 = 1/2, and the negative sign only changes the sign of the rational number.

So, the irrational number among the given options is: (b) √3

Q.2. The numberin p/q form is

(a) 267/1000

(b) 26/10

(c) 241/900

(d) 241/999

Ans. (c)

Solution:let x be the p/q form, x =

multiply both side by 100,

100 x =  …(i)

multiply both side by 10

1000 x =  ….(ii)

Subtract (ii) – (i)

1000 x – 100 x = 

900 x = 241

⇒ x = 241/900

Hence, option (c) is correct

Try yourself:

Q3: Every point on the number line represents, which of the following numbers?

A.Natural numbers

B.Irrational number

C.Rational number

D.Real numberExplanation

Ans. Every point on the number line represents

a: Real number as the number line represents all real numbers, which includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers, but not imaginary numbers.

Q.4. The decimal representation of a rational number is either:

(a) Terminating or repeating

(b) Non-terminating and non-repeating

(c) Only terminating

(d) Only repeating

Ans: (a) Terminating or repeating

 A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where p and q are integers and $q\ne 0$

A terminating decimal is one that has a finite number of digits after the decimal point. 

A repeating decimal is one where a block of digits repeats infinitely.

Q.5. Insert 3 irrational number between 2.6 and 3.8

Ans. 2.6 and 3.8

Irrational numbers are non repeating non – terminating

2.61010010001…..

2.802002000200002……

3.604004000400004…….

Q.6. What is the decimal form of the following no’s.

(a) 18/11

(b) 3/26

(c) 1/17

(d) 2/13

Ans.(a) 18/11 = 1.63636363…

(b) 3/26 = 0.11538461538

(c) 1/17 = 0.05882352941

(d) 2/13 = 0.15384615384

Q.8. Simplify: 

Ans.

Q.9. Rationalise: 

Ans.

Q.10. Find the value of 

Ans.= 5+4 – 4√5 – 5 – 4 –  4√5  = -8√5

Q.11. If ,find the value of a & b.

Ans.Rationalising LHS ∴ a = 11/7 and b = 6/7

Q.12. Evaluate: 

Ans. 

Q.13. Write the value of 

Ans.= 15

Q.14. Express  in p/q form.

Ans.let x be the p/q form,so, x =  10x = 1000x = 1000x – 10x = –  990x = 15555x= 15555/990= 1037/66

Q.15. Insert five rational no’s between 3/5 and 4/5.

Ans.3/5 and 4/530/50 and 40/50∴ pick any five number between 30 and 4031/50,  32/50,  36/50,  37/50,  39/50