Author Archives: Swati

Chapter 8 Co-ordinate Geometry Exercise Ex. 8.1

Question 1

Plot the following points on the graph paper:

(i) (2,5)

(ii) (4,-3)

(iii) (-5,-7)

(iv) (7,-4)

(v) (-3,2)

(vi) (7,0)

(vii) (-4,0)

(viii) (0,7)

(ix) (0,-4)

(x) (0,0)Solution 1

The graph of the given points are:

Question 2

Write the coordinates of each of the following points marked in the graph paper:

Solution 2

The coordinates of the given points are A(3,1), B(6,0), C(0,6), D(-3,0), E(-4,3), F(-2,-4), G(0,-5), H(3,-6), P(7,-3) and Q(7,6)

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

If x = 1 and y = 6 is solution of the equation 8x – ay + a²= 0, find the value of a.Solution 6

Question 7(i)

Write two solutions of the form x = 0, y = a and x = b, y = 0 for the follwoing equation: 5x – 2y = 10Solution 7(i)

Question 7(ii)

Write two solutions of the form x = 0, y = a and x = b, y = 0 for the following equation: -4x + 3y = 12Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Plot the points (3,5) and (-1,3) on a graph paper and verify that the straight line passing through these points also passes through the point (1,4).Solution 4

The given points on the graph:



It is dear from the graph, the straight line passing through these points also passes through the point (1,4).Question 5

From the choices given below, choose the equation whose graph is given in fig.,

(i) y = x

(ii) x + y = 0

(iii) y = 2x

(iv) 2 + 3y = 7x

 Solution 5

Question 6

From the choices given below, choose the equation whose graph is given in fig.,

(i) y = x + 2

(ii) y = x – 2

(iii) y = -x + 2

(iv) x + 2y = 6

 Solution 6

Question 7

Solution 7

Question 8

Draw the graph of the equation 2x + 3y = 12. Find the graph, find the coordinates of the point.

(i) whose y-coordinate is 3.

(ii) whose x-coordinate is -3Solution 8

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

Question 10

Solution 10

Question 11

Solution 11

Question 12

The sum of a two digit number and the number obtained by reversing the order of its digits is 121. If units and ten’s digit of the number are x and y respectively, then write the linear equation representing the above statement.Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Draw the graph of y = |x|.Solution 15

We have,

y = |X|                               …(i)

Putting x = 0, we get y = 0

Putting x = 2, we get y = 2

Putting x = -2, we get y = 2

Thus, we have the following table for the points on graph of |x|.

x	0	2	-2
y	0	2	2

The graph of the equation y = |x|:

Question 16

Draw the graph of y = |x| + 2.Solution 16

We have,

y = |x| + 2                                                  …(i)

Putting x = 0, we get y = 2

Putting x = 1, we get y = 3

Putting x = -1, we get y = 3

Thus, we have the following table for the points on graph of |x| + 2:

x	0	1	-1
y	2	3	3

The graph of the equation y = |x| + 2:

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Ravish tells his daughter Aarushi, “Seven years ago, I was seven times as old as you were then. Also, three years form now, I shall be three times as old as you will be”. If present ages of Aarushi and Ravish are x and y years respectively, represent this situation algebraically as well as graphically.Solution 20

Question 21

Solution 21

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

y + 3 = 0

y = -3

Point A represents -3 on number line.

On Cartesian plane, equation represents all points on x axis for which y = -3Question 1(iii)

Solution 1(iii)

y = 3

Point A represents 3 on number line.

On Cartesian plane, equation represents all points on x axis for which y = 3Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Give the geometrical representation of 2x + 13 = 0 as an equation in

One variableSolution 2(i)

Question 2(ii)

Give the geometrical representation of 2x + 13 = 0 as an equation in

Two variablesSolution 2(ii)

Question 3(i)

Solve the equation 3x + 2 = x – 8, and represent the solution on (i) the number line.Solution 3(i)

Question 3(ii)

Solve the equation 3x + 2 = x – 8, and represent the solution on (ii) the Cartesian plane.Solution 3(ii)

On Cartesian plane, equation represents all points on y axis for which x = -5Question 4

Write the equation of the line that is parallel to x-axis and passing through the point

(i) (0,3)

(ii) (0,-4)

(iii) (2,-5)

(iv) (3,4)Solution 4

(i) The equation of the line that is parallel to x-axis and passing through the point (0,3) is y = 3

(ii) The equation of the line that is parallel to x-axis and passing through the point (0,-4) is y = -4

(iii) The equation of the line that is parallel to x-axis and passing through the point (2,-5) is y = -5

(iv) The equation of the line that is parallel to x-axis and passing through the point (3, 4) is y = 4Question 5

Solution 5

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution 8Degree of a polynomial is the highest power of variable in the polynomial.
Binomial has two terms in it. So binomial of degree 35 can be written as x³⁵ + 7 .  Monomial has only one term in it. So monomial of degree 100 can be written as 7x¹⁰⁰. Concept Insight: Mono, bi and tri means one, two and three respectively. So, monomial is a polynomial having one term similarly for binomials and trinomials. Degree is the highest exponent of variable.  The answer is not unique in such problems . Remember that the terms are always separated by +ve or -ve sign and not with  .  

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.2

Question 1If f(x) = 2x³ – 13x² + 17x + 12, find:

(i) f(2)

(ii) f(-3)

(iii) f(0)Solution 1(i)

f(x) = 2x³ – 13x² + 17x + 12

f(2) = 2(2)³ – 13(2)² + 17(2) + 12

      = 16 – 52 + 34 + 12

      = 10

(ii)

f(-3) = 2(-3)³ – 13(-3)² + 17(-3) + 12

       = -54 – 117 – 51 + 12

       = – 210

(iii)

f(0) = 2(0)³ – 13(0)² + 17(0) + 12

      = 0 – 0 + 0 + 12

      =12Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7Find rational roots of the polynomial f(x) = 2x³ + x² – 7x – 6.Solution 7

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

If the polynomials ax³ + 3x² – 13 and 2x³ -5x + a, when divided by (x-2) leave the same remainder, find the value of a.Solution 10

Question 11Find the remainder when x³ + 3x² + 3x + 1 is divided by 

Solution 11

Question 12

Solution 12

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

What must be added to 3x³ + x² – 22x + 9 so that the result is exactly divisible by 3x² + 7x – 6?Solution 25

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Using factor theorem, factorize: x⁴ – 7x³ + 9x² + 7x -10Solution 4

Question 5

Using factor theorem, factorize: 3x³ – x² – 3x + 1Solution 5

Question 6

Using factor theorem, factorize each of the following polynomials:

x³ – 23x² + 142x – 120Solution 6

Question 7

Using factor theorem, factorize: y³ – 7y + 6Solution 7

Question 8

Using factor theorem, factorize: x³ – 10x² – 53x – 42Solution 8

Question 9

Using factor theorem, factorize: y³ – 2y² – 29y – 42Solution 9

Question 10

Using factor theorem, factorize: 2y³ – 5y² – 19y + 42Solution 10

Question 11

x³ + 13x² + 32x + 20Solution 11

         Let p(x) = x³ + 13x² + 32x + 20
         The factors of 20 are 1,  2,  4,  5 … …
         By hit and trial method
         p(- 1) = (- 1)³ + 13(- 1)² + 32(- 1) + 20
                   = – 1 + 13 – 32 + 20
                   = 33 – 33 = 0
         As p(-1) is zero, so x + 1 is a factor of this polynomial p(x).

         Let us find the quotient while dividing x³ + 13x² + 32x + 20 by (x + 1)
          By long division

                   We know that
         Dividend = Divisor  Quotient + Remainder
         x³ + 13x² + 32x + 20 = (x + 1) (x² + 12x + 20) + 0
                                         = (x + 1) (x² + 10x + 2x + 20)
                                         = (x + 1) [x (x + 10) + 2 (x + 10)]
                                         = (x + 1) (x + 10) (x + 2)
                                         = (x + 1) (x + 2) (x + 10)Question 12

Factorise:

x³ – 3x² – 9x – 5Solution 12

        Let p(x) = x³ – 3x² – 9x – 5
        Factors of 5 are 1,  5.
        By hit and trial method
        p(- 1) = (- 1)³ – 3(- 1)² – 9(- 1) – 5
           = – 1 – 3 + 9 – 5 = 0
        So x + 1 is a factor of this polynomial
        Let us find the quotient while dividing x³ + 3x² – 9x – 5 by x + 1
        By long division

                Now, Dividend = Divisor  Quotient + Remainder
 x³ – 3x² – 9x – 5 = (x + 1) (x² – 4 x – 5) + 0
                                     = (x + 1) (x² – 5 x + x – 5)
                                     = (x + 1) [(x (x – 5) +1 (x – 5)]
                                     = (x + 1) (x – 5) (x + 1)
                                     = (x – 5) (x + 1) (x + 1) Question 13

Factorise:

2y³ + y² – 2y – 1Solution 13

         Let p(y) = 2y³ + y² – 2y – 1         By hit and trial method
         p(1) = 2 ( 1)³ + (1)² – 2( 1) – 1
                = 2 + 1 – 2 – 1= 0
         So, y – 1 is a factor of this polynomial
         By long division method,

                    p(y) = 2y³ + y² – 2y – 1
                 = (y – 1) (2y² +3y + 1)
                 = (y – 1) (2y² +2y + y +1)
                 = (y – 1) [2y (y + 1) + 1 (y + 1)]
                 = (y – 1) (y + 1) (2y + 1)

 Question 14

Using factor theorem, factorize: x³ – 2x² – x + 2Solution 14

Question 15

Solution 15

Question 16

Using factor theorem, factroize : x⁴ – 2x³ – 7x² + 8x + 12Solution 16

Question 17

Using factor theorem, factroize : x⁴ + 10x³ + 35x² + 50x + 24Solution 17

Question 18

Using factor theorem, factorize : 2x⁴ – 7x³ – 13x² + 63x – 45Solution 18

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Give possible expressions for the length and breadth of the rectangle having

35y² + 13y – 12 as its area.Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.4

Question 1

Solution 1

Question 2

Solution 2

Question 3Factorise : 27x³ – y³ – z³ – 9xyzSolution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 17

Solution 17

Chapter 4 Algebraic Identities Exercise Ex. 4.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 11

Solution 11

Question 12

Solution 12

Question 13(i)

Solution 13(i)

Question 13(ii)

Solution 13(ii)

Question 13(iii)

Solution 13(iii)

Question 13(iv)

Solution 13(iv)

Question 14

Solution 14

Chapter 4 Algebraic Identities Exercise Ex. 4.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1 (x)Write the following in the expanded form:

(x + 2y + 4z)²Solution 1 (x)

Question 1 (xi)Write the following in the expanded form:

(2x – y + z)²Solution 1 (xi)

Question 1 (xii)Write the following in the expanded form:

(-2x + 3y + 2z)²Solution 1 (xii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 6(v)

Solution 6(v)

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Chapter 4 Algebraic Identities Exercise Ex. 4.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11(i)

Solution 11(i)

Question 11(ii)

Solution 11(ii)

Question 11(iii)

Solution 11(iii)

Question 11(iv)

Solution 11(iv)

Question 11(v)

Solution 11(v)

Question 11(vi)

Solution 11(vi)

Question 12(i)

Solution 12(i)

Question 12(ii)

Solution 12(ii)

Question 12(iii)

Solution 12(iii)

Question 12(iv)

Solution 12(iv)

Question 13

Solution 13

Question 14(i)

Solution 14(i)

Question 14(ii)

Solution 14(ii)

Question 15

Solution 15

Question 16

Solution 16

Question 17(i)

Solution 17(i)

Question 17(ii)

Solution 17(ii)

Question 17(iii)

Solution 17(iii)

Question 17(iv)

Solution 17(iv)

Question 18

Solution 18

Question 19

Solution 19

Chapter 4 Algebraic Identities Exercise Ex. 4.4

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 1 (ix)

Solution 1 (ix)

Question 1 (x)

Solution 1 (x)

Question 1 (xi)

Solution 1 (xi)

Question 1 (xii)

Solution 1 (xii)

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 2 (iii)

Solution 2 (iii)

Question 2 (iv)

Solution 2 (iv)

Question 2 (v)

Solution 2 (v)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6 (i)

Solution 6 (i)

Question 6 (ii)

Solution 6 (ii)

Question 6 (iii)

Solution 6 (iii)

Chapter 4 Algebraic Identities Exercise Ex. 4.5

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 3 Rationalisation Exercise Ex. 3.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 3 Rationalisation Exercise Ex. 3.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 4(vi)

Solution 4(vi)

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 6(v)

Solution 6(v)

Question 6(vi)

Solution 6(vi)

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 9(i)

Simplify:Solution 9(i)

Question 9(ii)

Simplify:Solution 9(ii)

Chapter 2 Exponents of Real Numbers Exercise Ex. 2.1

Question 1(i)

Simplify:

3(a⁴b³)¹⁰ × 5(a²b²)³Solution 1(i)

Question 1(ii)

Simplify:

(2x^-2y³)³Solution 1(ii)

Question 1(iii)

Simplify:

Solution 1(iii)

Question 1(iv)

Simplify:

Solution 1(iv)

Question 1(v)

Simplify:

Solution 1(v)

Question 1(vi)

Simplify:

Solution 1(vi)

Question 2(i)

If a = 3 and b = -2, find the value of:

a^a + b^bSolution 2(i)

Question 2(ii)

If a = 3 and b = -2, find the value of:

a^b + b^aSolution 2(ii)

Question 2(iii)

If a = 3 and b = -2, find the value of:

(a + b)^abSolution 2(iii)

Question 3(i)

Prove that:

Solution 3(i)

Question 3(ii)

Prove that:

Solution 3(ii)

Question 4(i)

Prove that:

Solution 4(i)

Question 4(ii)

Prove that:

Solution 4(ii)

Question 5(i)

Prove that:

Solution 5(i)

Question 5(ii)

Prove that:

Solution 5(ii)

Question 6

Solution 6

Question 7(i)

Simplify:

Solution 7(i)

Question 7(ii)

Simplify:

Solution 7(ii)

Question 7(iii)

Simplify:

Solution 7(iii)

Question 7(iv)

Simplify:

Solution 7(iv)

Question 8(i)

Solve the equation for x:

7^{2x + 3} = 1Solution 8(i)

Question 8(ii)

Solve the equation for x:

2^x+1 = 4^x-3Solution 8(ii)

Question 8(iii)

Solve the equation for x:

2^{5x + 3} = 8^{x + 3}Solution 8(iii)

Question 8(iv)

Solve the equation for x:

Solution 8(iv)

Question 8(v)

Solve the equation for x:

Solution 8(v)

Question 8(vi)

Solve the equation for x:

2^{3x – 7} = 256Solution 8(vi)

Question 9(i)

Solve the equation for x:

2^2x – 2^x+3 + 2⁴ = 0Solution 9(i)

Question 9(ii)

Solve the equation for x:

3^{2x + 4} + 1 = 2.3^{x + 2}Solution 9(ii)

Question 10

If 49392 = a⁴b²c³, find the values of a, b and c, where a, b and c are different positive primes.Solution 10

Question 11

If 1176 = 2^a × 3^b × 7^c, find a, b and c.Solution 11

Question 12

Given 4725 = 3^a 5^b 7^c , find

1. the integral values of a, b and c
2. the values of 2^-a3^b7^c

Solution 12

Question 13

If a = xy^{p – 1}, b = xy^q-1 and c = xy^r-1, prove that a^q-rb^r-p c^p-q = 1.Solution 13

Chapter 2 Exponents of Real Numbers Exercise Ex. 2.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Assuming that x, y, z are positive real numbers, simplify each of the following:

Solution 1(vii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Simplify:

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 2(vii)

Solution 2(vii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Prove that:

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 4(i)

Show that:

Solution 4(i)

Question 4(ii)

Show that:

Solution 4(ii)

Question 4(iii)

Show that:

Solution 4(iii)

Question 4(iv)

Show that:

Note: Question modifiedSolution 4(iv)

Note: Question modifiedQuestion 4(v)

Show that:

(x^a-b)^a+b(x^b-c)^b+c(x^c-a)^c+a = 1Solution 4(v)

Question 4(vi)

Show that:

Solution 4(vi)

Question 4(vii)

Show that:

Solution 4(vii)

Question 4(viii)

Show that:

Solution 4(viii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 10(iii)

Solution 10(iii)

Question 10(iv)

Solution 10(iv)

Question 10(v)

Solution 10(v)

Question 10(vi)

Find the value of x if:

Solution 10(vi)

Question 10(vii)

Find the value of x if:

5^{2x + 3} = 1Solution 10(vii)

Question 10(viii)

Find the value of x if:

Solution 10(viii)

Question 10(ix)

Find the value of x if:

Solution 10(ix)

Question 11

If x = 2^1/3 + 2^2/3, show that x³ – 6x = 6.Solution 11

Question 12

Determine (8x)^x, if 9^x+2 = 240 + 9^x.Solution 12

Question 13

If 3^x+1 = 9^x-2, find the value of 2^1+x.Solution 13

Question 14

If 3^4x = (81)^-1 and 10^1/y = 0.0001, find the value of  2^-x+4y.Solution 14

Question 15

If 5^3x = 125 and 10^y = 0.001 find x and y.Solution 15

Question 16(i)

Solve the equation:

3^{x + 1} = 27 × 3⁴Solution 16(i)

Question 16(ii)

Solve the equation:

Solution 16(ii)

Question 16(iii)

Solve the equation

3^x-1 × 5^2y-3 = 225Solution 16(iii)

Question 16(iv)

Solve the equation:

Solution 16(iv)

Question 16(v)

Solve the equation:

Solution 16(v)

Question 16(vi)

Solve the equation:

Solution 16(vi)

Question 17

Solution 17

Question 18(i)

If a and b are different positive primes such that

Solution 18(i)

Question 18(ii)

If a and b are different positive primes such that

(a + b)^-1(a^-1 + b^-1) = a^xb^y, find x + y + 2.Solution 18(ii)

Question 19

If 2^x × 3^y × 5^z = 2160, find x, y and z. Hence, compute the value of 3^x × 2^-y × 5^-z.Solution 19

Question 20

If 1176 = 2^a × 3^b × 7^c, find the values of a, b and c. hence, compute the value of 2^a × 3^b × 7^-c as a fraction.Solution 20

Question 21(i)

Simplify :

Solution 21(i)

Question 21(ii)

Simplify:

Solution 21(ii)

Question 22

Show that:

Solution 22

Question 23(i)

If a = x^m+ny^l, b = x^n+ly^m and c = x^l+myⁿ, prove that a^m-nbn-lcl-m = 1.Solution 23(i)

Question 23(ii)

If x = a^m+n, y = a^n+l and z = a^l+m, prove that x^myⁿz^l = zⁿy^lz^m.Solution 23(ii)

Chapter 1 Number Systems Exercise Ex. 1.1

Question 1Is zero a rational number? Can you write it in the form , where p and q are integers and q  0?Solution 1Yes zero is a rational number as it can be represented in the   form,  where p and q are integers and q  0 as  etc.

Concept Insight: Key idea to answer this question is “every integer is a rational number and zero is a non negative integer”.  Also 0 can be expressed in  form in various ways as 0 divided by any number is 0. simplest is   .

Question 2Find five rational numbers between 1 and 2.Solution 2

Question 3Find six rational numbers between 3 and 4.Solution 3There are infinite rational numbers in between 3 and 4.  
3 and 4 can be represented as respectively. Now rational numbers between 3 and 4 are    


Concept Insight:  Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent  form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

Question 4Find five rational numbers between  .Solution 4There are infinite rational numbers between 

                 Now rational numbers between are 

Concept Insight: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent   form by multiplying and dividing by the number at least 1 more than the rational numbers required.

Alternatively for any two rational numbers a and b,    is also a rational number which lies between a and b.

Question 5Are the following statements true or false? Give reasons for you answer.

(i) Every whole number is a natural number.

(ii) Every integer is a rational number.

(iii) Every rational number is an integer.

(iv) Every natural number is a whole number.

(v) Every integer is whole number.

(vi) Every rational number is whole number.Solution 5(i) False

(ii) True

(iii) False

(iv)True

(v) False

(vi) False

Chapter 1 Number Systems Exercise Ex. 1.2

Question 1

Solution 1

Question 2Express the follwoing rational numbers as decimals:

Solution 2

(i)



(ii)



(iii)

(iv)



(v)



(vi)

Question 3

Solution 3

Chapter 1 Number Systems Exercise Ex. 1.3

Question 1

Solution 1

Question 2

Solution 2

(i)



(ii)



(iii)



(iv)



(v)



(vi)



(vii)

Chapter 1 Number Systems Exercise Ex. 1.4

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 3(x)

Solution 3(x)

As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.
Question 3(xi)

Solution 3(xi)

Rational number as it can be represented in   form.
Question 3(xii)Examine whether 0.3796 is rational or irrational.Solution 3(xii)0.3796

As decimal expansion of this number is terminating, so it is a rational number.Question 3(xiii)Examine whether 7.478478… is rational or irrational.Solution 3(xiii)
As decimal expansion of this number is non terminating recurring so it is a rational number.
Question 3(xiv)Examine whether 1.101001000100001… is rational or irrational.Solution 3(xiv)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 4(vi)

Solution 4(vi)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10Find three different irrational numbers between the rational numbers Solution 10

 3 irrational numbers are –
0.73073007300073000073 … … …0.75075007500075000075 … … …
0.79079007900079000079 … … …

Concept Insight: There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them.

Alternatively following result can be used to answerIrrational number between two numbers x and y 

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 1 Number Systems Exercise Ex. 1.5

Question 1Complete the following sentences:

(i) Every point on the number line corresponds to a ___ number which may be either ____ or_____.

(ii) The decimal form of an irrational number is neither ______ nor ______.

(iii) The decimal representation of a rational number is either ____ or _____.

(iv) Every real number is either ______ number or ______ number.Solution 1(i) Real, rational, irrartional.

(ii) terminating, repeating.

(iii) terminating, non-terminating and reccuring.

(iv) rational, an irrational.Question 2

Find whether the following sentences are true or false:

(i) Every real number is either rational or irrational.

(ii)  is an irrational number.

(iii) Irrational numbers cannot be represented by points on the number line.Solution 2

(i) True

(ii) True

(iii) FalseQuestion 3

Solution 3

Question 4

Solution 4

Chapter 1 Number Systems Exercise Ex. 1.6

Question 1

Solution 1

Question 2

Solution 2

Chapter 16 – Surface Areas and Volumes Exercise Ex. 16.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

2.2 cubic dm of brass is to be drawn into a cylindrical wire 0.25 cm in diameter. Find the length of the wire.Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Find the number of metallic circular discs with 1.5 cm base diameter and of height 0.2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.Solution 9

Question 10

How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece with dimensions 66 cm × 42 cm × 21 cm.Solution 10

Question 11

How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm.Solution 11

Question 12

Three cubes of a metal whose edges are in the ratio 3 : 4: 5 are melted and converted into a single cube whose diagonal is  cm. Find the number of cones so formed.Solution 12

Question 13

A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.Solution 13

Question 14

Solution 14

Question 15

An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is 1/4 of the radius of the original ball, how many such balls are made? Compare the surface area, of all the smaller balls combined together with that of the original ball.Solution 15

Question 16

Solution 16

Question 17

A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.Solution 17

Question 18

Solution 18

Question 19

How many coins 1.75 cm in diameter and 2 mm thick must be melted to form a cuboid 11 cm  10 cm  7 cm?Solution 19

Question 20

Solution 20

Question 21

A cylindrical bucket, 32 cm high and with a radius of base 18 cm, is filled with sand. This bucket is emptied out on the ground and a conical heap of sand is formed. If the height of conical heap is 24 cm, find the radius and slant height of the heap.Solution 21

Question 22

A solid metallic sphere of radius 5.6 cm is melted and solid cones each of radius 2.8 cm and height 3.2 cm are made. Find the number of such cones formed.Solution 22

Question 23

A solid cuboid of iron with dimensions 53 cm x 40 cm x 15 cm is melted and recast into a cylindrical pipe. The outer and inner diameters of pipe are 8 cm and 7 cm respectively. Find the length of pipe.Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of the balls are 1.5 cm and 2 cm. Find the diameter of the third ball.Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Rain water, which falls on a flat rectangular surface of length 6 m and breadth 4 m is transferred into a cylindrical vessel of internal radius 20 cm. What will be the height of water in the cylindrical vessel if a rainfall of 1 cm has fallen? [Use  = 22/7]Solution 35

Question 36

The rain water from a roof of dimensions 22 m × 20 m drains into a cylindrical vessel having diameter of base 2 m and height 3.5 m. if the rain water collected from the roof just fills the cylindrical vessel, then find the rain fall in cm.Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

150 spherical marbles, each of diameter 1.4 cm are dropped in cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in water. Find the rise in the level of water in the vesselSolution 46

*Answer given in the book is incorrect.Question 47

Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which 

of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by shushant ?Solution 47

Question 48

16 glass spheres each of radius 2 cm are picked into a cuboidal box of internal dimensions 16 cm × 8 cm × 8 cm and then the box is filled with water. Find the volume of water filled in the box.Solution 48

Question 49

Water flows through a cylindrical pipe, whose inner radius is 1 cm, at the rate of 80 cm/sec in an empty cylindrical tank, the radius of whose base is 40 cm. What is the rise of water level in tank in half an hour?Solution 49

Question 50

Water in a canal 1.5 m wide and 6 m deep is flowing with a speed of 10 km/hr. How much area will it irrigate in 30 minutes if 8 cm of standing water is desired?Solution 50

Question 51

A farmer runs a pipe of internal diameter 20 cm from the canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?Solution 51

Question 52

A cylindrical tank full of water is emptied by a pipe at the rate of 225 liters per minute. How much time will it take to empty half the tank, if the diameter of its base is 3 m and its height is 3.5 m? (π = 22/7)Solution 52

Question 53

Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of the base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe.Solution 53

Question 54

Water flows at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will level of water in the pond rise by 21 cm?Solution 54

Question 55

A canal 300 cm wide and 120 cm deep. The water in the canal is flowing with a speed of 20 km/h. How much area will it irrigate in 20 minutes if 8 cm of standing water is desired?Solution 55

Question 56

The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 cm², find the volume of cylinder.Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

A 5 m wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. find the cost of cloth used at the rate of Rs. 25 per metre. (π = 22/7)Solution 60

Question 61

Solution 61

Question 62

The difference between the outer and inner curved surface areas of a hollow right circular cylinder 14 cm long is 88 cm². If the volume of metal used in making the cylinder is 176 cm³, find the outer and inner diameters of the cylinder. (Use  = 22/7)Solution 62

Question 63

Solution 63

Question 64

Solution 64

Question 65

If the total surface area of a solid hemisphere is 462 cm², find its volume.

(π = 22/7)Solution 65

*Answer given in the book is incorrect.Question 66

Water flows at the rate of 10m/minute through a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm?Solution 66

Question 67

A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylindrical full of water of height 180 cm such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is equal to the radius of the cone.Solution 67

Question 68

A heap of rice in the form of a cone of diameter 9 m and height 3.5 m. Find the volume of rice. How much canvas cloth is required to cover the heap?Solution 68

Question 69

A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.Solution 69

Question 70

A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical shaped each of radius 1.5 cm and height 4 cm. How many bottles are needed to empty the bowl?Solution 70

Question 71

A factory manufactures 120,000 pencils daily The pencils are cylindrical in shape each of length 25 cm and circumference of base as 1.5 cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at Rs. 0.05 per dm².Solution 71

Question 72

πThe  part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.Solution 72

Height of the conical vessel h = 24 cm

Radius of the conical vessel r =5 cm

Let h be the height of the cylindrical vessel which is filled by water of the conical vessel.

Radius of the cylindrical vessel =10 cm

Volume of the cylindrical vessel = volume of water

π(10)²h=150π 

h = 150π¸ 100π 

h = 1.5 cm

Thus, the height of the cylindrical vessel is 1.5 cm.

Chapter 16 – Surface Areas and Volumes Exercise Ex. 16.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

A cylindrical tub of radius 5 cm and length 9.8 cm is full of water. A solid in the form of a right circular cone mounted on a hemisphere is immersed in the tub. If the radius of the hemi-sphere is 3.5 cm and height of the cone outside the hemisphere is 5 cm, find the volume of the water left in the tub. (Take  = 22/7)Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

A cylinderical road roller made of iron is 1 m long. Its internal diameter is 54 cm and the thickness of the iron sheet used in making the roller is 9 cm. Find the mass of the roller, if 1 cm³ of iron has 7.8 gm mass. (Use  = 3.14)Solution 16

Question 17

A vessel in the form of a hollow hemisphere mounted by a hollow cylinder. The dijameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

A right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled in cones of height 12 cm and diameter 6 cm having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

A wooden toy is made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy.

(π = 22/7)Solution 28

Question 29

The largest possible sphere is carved out of a wooden solid cube of side 7 cm. find the volume of wood left.(Use  = 22/7)Solution 29

Question 30

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. (π = 22/7)Solution 30

Question 31

The largest cone is curved out from one face of solid cube of side 21 cm. Find the volume of the remaining solid.Solution 31

Question 32

A solid wooden toy is in the form of a hemisphere surmounted by a cone of same radius. The radius of hemisphere is 3.5 cm and the total wood used in the making of toy is 166. Find the height of the toy. Also, find the cost of painting the 6 hemispherical part of the toy at the rate of Rs. 10 per cm². (Take π = 22/7).Solution 32

Question 33

In Fig. 16.57, from a cuboidal solid metalic block, of dimensions 15 cm × 10 cm × 5 cm, a cylindrical hole of diameter 7 cm is drilled out. Find the surface area of the remaining block. (Take π = 22/7).

Solution 33

Question 34

A building is in the form of a cylinder surmounted by a hemi-spherical vaulted done and contains  of air. If the internal diameter of done is equal to its total height above the floor, find the height of the building?Solution 34

Question 35

A pen stand made of wood is in the shape of a cuboid four conical depressions and a cubical depression to hold the pens and pins, respectively. The dimension of the cuboid are 10 cm × 5 cm × 4 cm. The radius of each of the conical depression is 0.5 cm and the depth is 2.1 cm. The edge of the cubical depression is 3 cm. Find the volume of the wood in the entire stand.Solution 35

Question 36

A building is in the form of a cylinder surmounted by a hemispherical dome. The base diameter of the dome is equal to  of the total height of the building. Find the height of the building, if it contains  of air.Solution 36

Question 37

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of cone is 4 cm and the diameter of the base is 8 cm. Determine the volume of the toy. If a cube circumscribes the toy, then find the difference of the volumes of cube and the toy. Also, find the total surface area of the toy.Solution 37

Question 38

A circus tent is in the shape of a cylinder surmounted by a conical top of same diameter. If their common diameter is 56m, the height of the cylindrical part is 6 m and the total height of the tent above the ground is 27 m, find the area of the canvas used in making the tent.Solution 38

Total area of the canvas = curved surface area of the cone + curved surface area of a cylinder radius = 28 m height (cylinder) = 6 m

height (cone) = 21 m

l = slant height of cone

curved surface area of the cone = πrl 

=π×28×35

=×28×35 = 3080 m²

curved surface area of the cylinder = 2πrh 

=2××28×6

=1056

Total area of the canvas = 3080+1056 =4136 m²

Chapter 16 – Surface Areas and Volumes Exercise Ex. 16.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

A milk container of height 16 cm is made of metal sheet in the form of frustum of a cone with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk at the rate of Rs.44 per litre which the container can hold.Solution 10

Question 11

A bucket is in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the capacity and surface area of the bucket. Also, find the cost of milk which can completely fill the container, at the rate of Rs.25 per litre.Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

A solid cone of base radius 10 cm is cut into two parts through the mid-points of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.Solution 18

Question 19

A bucket open at the top, and made up of a metal sheet is in the form of a frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost of metal sheet used in it at the rate of Rs. 10 per 100 cm². (π = 22/7)Solution 19

Question 20

In Fig. 14.75, from the top of a solid cone of height 12 cm and base radius 6 cm, a cone of height 4 cm is removed by a plane parallel to the base. Find the total surface area of the remaining solid.

).

Solution 20

Question 21

The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of two parts.Solution 21

Let the height of the cone be H and the radius be R. This cone is divided into two equal parts.

AQ=1/2 AP

Also,

QP||PC

Therefore,ΔAQD~ΔAPC.

So,

Question 22

A bucket, made of metal sheet, is in the form of a cone whose height is 35 cm and radii of circular ends are 30 cm and 12 cm. How many liters of milk it contains if it is full to the brim? If the milk is sold at 40 per litre, find the amount received by the person.Solution 22

A bucket, made of metal sheet, is in the form of a cone.

R = 15 cm, r = 6 cm and H=35 cm

Now, using the similarity concept, we can writ

Volume of the frustum is

The rate of milk is Rs. 40 per litre.

So, the cost of 51.48 litres is Rs. 2059.20.Question 23

The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm,

(i) Find the area of the metal sheet used to make the bucket.

(ii) Why we should avoid the bucket made by ordinary plastic? (use π = 3.14)Solution 23

(i)

Given:

Radius of lower end (r₁) = Diameter/2 = 5 cm

Radius of upper end (r₂) = Diameter/2 = 15 cm

Height of the bucket (h) = 24 cm

Area of metal sheet used in making the bucket

= CSA of bucket + Area of smaller circular base

Hence, area of the metal sheet used in making the bucket is 1711.3 cm².

(ii)

We should avoid the bucket made by ordinary plastic because it is less strength than metal bucket and also not ecofriendly.

Chapter 15 – Areas Related to Circles Exercise Ex. 15.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Find the radius of a circle whose circumference is equal to the sum of the circumference of two circles of radii 15 cm and 18 cm.Solution 9

Question 10

The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.Solution 10

Question 11

The radii of two circles are 19 cm and 9 cm respectively. Find the radius and area of the circle which has its circumference equal to the sum of the circumferences of the two circles.Solution 11

Area of a circle = πr² = (22/7) × 28 × 28 = 2464 cm²Question 12

The area of a circular playground is 22176 m². Find the cost of fencing this ground at the rate of Rs. 50 per metre.Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

A park is in the form of a rectangle 120 m x 100 m. At the centre of the park there is a circular lawn. The area of park excluding lawn is 8700 m². Find the radius of the circular lawn. (Use = 22/7).Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

An archery target has three regions formed by three concentric circles as shown in figure. If the diameters of the concentric circles are in the ratio 1 : 2 : 3, then find the ratio of the areas of three regions.

Solution 20

Question 21

The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/hr?Solution 21

Question 22

A circular pond is 17.5 m in diameter. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of Rs.25 per m².Solution 22

Question 23

A circular park is surrounded by a road 21 m wide. If the radius of the park is 105 m, find the area of the road.Solution 23

Question 24

A square of diagonal 8 cm is inscribed in a circle. Find the area of the region lying inside the circle and outside the square.Solution 24

Question 25

Solution 25

Question 26

Find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm. A third concentric circle is drawn outside the 7 cm circle, such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles. Find the radius of the third circle correct to one decimal place.


Solution 26

Question 27

A path of width 3.5 m runs around a semi-circular grassy plot whose perimeter is 72 m. Find the area of the path. (Use π = 22/7)Solution 27

Question 28

A circular pond is of diameter 17.5 m. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of Rs. 25 per square meter (π = 3.14)Solution 28

Question 29

The outer circumference of a circular race-track is 528 m. The track is everywhere 14 m wide. Calculate the cost of levelling the track at the rate of 50 paise per square metre (Use  = 22/7).Solution 29

Question 30

Solution 30

Question 31

Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is  h (2r + h).Solution 31

Chapter 15 – Areas Related to Circles Exercise Ex. 15.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

The area of a sector of a circle of radius 5 cm is 5 cm². Find the angle contained by the sector.Solution 9

Question 10

Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm.Solution 10

Question 11

Solution 11

Question 12

The perimeter of a scetor of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.Solution 12

Question 13

The perimeter of a certain sector of a circle of radius 5.6 m is 27.2 m. Find the area of the sector.Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

A sector of 56^o cut out from a circle contains area 4.4 cm². Find the radius of the circle.Solution 17

Question 18

Area of a sector of central angle 200° of a circle is 770 cm². Find the length of the corresponding arc of this sector.Solution 18

Question 19

The length of minute hand of a clock is 5 cm. Find the area swept by the minute hand during the time period 6:05 am and 6:40 am.Solution 19

Question 20

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.Solution 20

*Answer does not match with textbook answer.Question 21

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find

(i) the length of arc

(ii) area of the sector formed by the arc. (use π = 22/7)Solution 21

Question 22

From a circular piece of cardboard of radius 3 cm two sectors of 90° have been cut off. Find the perimeter of the remaining portion nearest hundredth centimeters. (Take π = 22/7)Solution 22

*Note: Answer given in the book is incorrect.Question 23

The area of a sector is one-twelfth that of the complete circle. Find the angle of the sector.Solution 23

Question 24

AB is a chord of a circle with centre O and radius 4 cm. AB is of length 4 cm. Find the area of the sector of the circle formed by chord AB.Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Chapter 15 – Areas Related to Circles Exercise Ex. 15.3

Question 1

Solution 1

Question 2

A chord PQ of length 12 cm subtends an angle of 120^o at the centre of a circle. Find the area of the minor segment cut off by the chord PQ.Solution 2

Question 3

Solution 3

Question 4

A chord 10 cm long is drawn in a circle whose radius is  cm. Find area of both the segments. (Take  = 3.14).Solution 4

Question 5

Solution 5

Question 6

Find the area of the minor segment of a circle of radius 14 cm, when the angle of the corresponding sector is 60°.Solution 6

Question 7

A chord of a circle of radius 10 cm subtends an angle of 90° at the centre. Find the area of the corresponding major segment of the circle. (Use π = 3.14).Solution 7

Question 8

The radius of a circle with centre O is 5 cm. Two radii OA and OB are drawn at right angles to each other. Find the areas of the segments made by the chord AB. (π = 3.14)

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 15 – Areas Related to Circles Exercise Ex. 15.4

Question 1

A plot is in the form of the form of a rectangle ABCD having semi-circle on BC as shown in Fig., If AB = 60 m and BC = 28 m, find the area of the piot.

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

A rectangular piece is 20 m long and 15 m wide. Form its four corners, quadrants of radii 3.5 m have been cut. Find the area of the remaining part.Solution 4

Question 5

In fig., PQRS is a square of side 4 cm. Find the area of the shaded region.

Solution 5

Question 6

Four cows are tethered at four corners of a square plot of side 50 m, so that they just cannot reach one another. What area will be left ungrazed?

Solution 6

Question 7

A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20m × 16m, find the area of the field in which the cow can graze.Solution 7

Question 8

A calf is tied with a rope of length 6 m at the corner of a square grassy lawn of side 20 m. If the length of the rope is increased by 5.5 m, Find the increase in area of the grassy lawn in which the calf can graze.Solution 8

Question 9

Solution 9

Question 10

A rectangular park is 100 m by 50 m. It is surrounded by semi-circular flower beds all round. Find the cost of levelling the semi-circular flower beds at 60 paise per square metre (Ise  = 3.14).Solution 10

Question 11

The inside perimeter of a running track (shown in Fig.) is 400 m. The length of each of the straight portion is 90 m and the ends are semi-circles. If the track is everywhere 14 m wide, find the area of the track. Also, find the length of the outer running track.

Solution 11

Question 12

Find the area of Fig., in square cm, correct to one place of decimal. (Take π = 22/7).

Solution 12

Question 13

From a rectangular region ABCD with AB = 20 cm, a right angle AED with AE = 9 cm and DE = 12 cm, is cut off. On the other end, taking BC as diameter, a semicircle is added on outside the region. Find the area of the shaded region. (π = 22/7)

Solution 13

Question 14

From each of the two opposite corners of a square of side 8.8 cm, a quadrant of a circle of radius 1.4 cm is cut. Another circle of radius 4.2 cm is also cut from the centre as shown in Fig. Find the area of the remaining (shaded) portion of the square. (Use π = 22/7).Solution 14

Question 15

ABCD is a rectangle with AB = 14 cm and BC = 7 cm. Taking DC, BC and AD as diameters, three semi-circles are drawn as shown in the figure. Find the area of the shaded region.

Solution 15

Question 16

ABCD is rectangle, having AB = 20 cm and BC = 14 cm. Two sectors of 180° have been cut off. Calculate :

(i) the area of the shaded region.     (ii) the length of the boundary of the shaded region.

Solution 16

Question 17

The square ABCD is divided into five equal parts, all having same area. The central part is circular and the lines AE, GC, BF and HD lie along the diagonals AC and BD of the square. If AB = 22 cm, find:

(i) the circumference of the central part.   (ii) the perimeter of the part ABEF.

Solution 17

Question 18

In figure, find the area of the shaded region.

(Use π = 3.14)

Solution 18

Question 19

OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the (i) quadrant OACB (ii) shaded region.

Solution 19

Question 20

A square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 21 cm, find the area of the shaded region.

Solution 20

Question 21

Solution 21

Question 22

OE = 20 cm. In sector OSFT, square OEFG is inscribed. Find the area of the shaded region.

Solution 22

Question 23

Solution 23

Question 24

A circle is inscribed in an equilateral triangle ABC of side 12 cm, touching its sides (fig.,). Find the radius of the inscribed circle and the area of the shaded part.

Solution 24

Question 25

In fig., an equilateral triangle ABC of side 6 cm has been inscribed in a circle. Find the area of the shaded region. (Take  = 3.14).

Solution 25

*Answer is not matching with textbook.Question 26

Solution 26

Question 27

Find the area of a shaded region in the given figure, where a circular arc of radius 7 cm has been drawn with vertex A of an equilateral triangle ABC of side 14 cm as centre.

Solution 27

Question 28

A regular hexagon is inscribed in a circle. If the area of hexagon is , find the area of the circle. (Use π it = 3.14)Solution 28

Consider the following figure:

Question 29

ABCDEF is a regular hexagon with centre O (Fig.,). If the area of triangle OAB is 9 cm², find the area of: (i) the hexagon and (ii) the circle in which the hexagon is inscribed.

Solution 29

(i)

According to the figure in the question, there are 6 triangles.

Area of one triangle is 9 cm².

Area of hexagon = 6 × 9 = 54 cm²

(ii)

Area of the equilateral triangle = 9 cm²

Area of the circle in which the hexagon is inscribed

= 

= 

= 

= 65.26 cm²

NOTE: Answer not matching with back answer.Question 30

Four equal circles, each of radius 5 cm, touch each other as shown in Fig. Find the area included between them (Take π = 3.14)

Solution 30

Question 31

Solution 31

Question 32

A child makes a poster on a chart paper drawing a square ABCD of side 14 cm. She draws four circles with centre A, B, C and D in which she suggests different ways to save energy. The circles are drawn in such a way that each circle touches externally two of the three remaining circles. In the shaded region she write a message ‘Save Energy’. Find the perimeter and area of the shaded region. (Use π = 22/7)

Solution 32

Question 33

The diameter of a coin is 1 cm. If four such coins be placed on a table so that the rim of each touches that of the other two, find the area of the shaded region (Take π = 3.1416)

Solution 33

Question 34

Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions 14 cm × 7 cm. find the area of the remaining card board. (π = 22/7)Solution 34

Question 35

AB and CD are two diameters of a circle perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

Solution 35

Question 36

PSR, RTQ and PAQ are three semi-circles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region.

Solution 36

Question 37

Two circles with centres A and B touch each other at the point C. If AC = 8 cm and AB = 3 cm, find the area of the shaded region.

Solution 37

Question 38

ABCD is a square of side 2a. Find the ratio between

(i) the circumferences

(ii) the areas of the incircle and the circum-circle of the square.

Solution 38

Question 39

There are three semicircles, A, B and C having diameter 3 cm each, and another semicircle E having a circle D with diameter 4.5 cm are shown. Calculate:

(i) the area of the shaded region

(ii) the cost of painting the shaded region at the rate of 25 paise per cm², to the nearest rupee.

Solution 39

Question 40

Solution 40

Question 41

O is the centre of a circular arc and AOB is a straight line. Find the perimeter and the area of the shaded region correct to one decimal place. (Take π = 3.142)

Solution 41

Question 42

The boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, find (i) the length of the boundary   (ii) the area of the shaded region.

Solution 42

Question 43

Ab = 36 cm and M is mid-point of AB. Semi-circles are drawn on AB, AM and MB as diameters. A circle with centre C touches all the three circles. Find the area of the shaded region.

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Shows a kite in which BCD is the shape of a quadrant of a circle of radius 42 cm. ABCD is a square and Δ CEF is an isosceles right angled triangle whose equal sides are 6 cm long. Find the area of the shaded region.

Solution 46

Question 47

ABCD is a trapezium of area 24.5 cm². In it, AD ∥ BC, ∠DAB = 90°, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a circle, find the area of the shaded region.

(π = 22/7)

Solution 47

Question 48

ABCD is a trapezium with AB ∥ DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. Circles of equal radii 7 cm with centres A, B, C and D have been drawn. Then, find the area of the shaded region of the figure. (π = 22/7)

Solution 48

Since the data given in the question seems incomplete and inconsistent with the figure, we make the following assumptions to solve it:

1. ABCD a symmetric trapezium with AD = BC

2. AD = BC = 14 cm (the distance between AB and CD is not 14 cm)

Draw perpendiculars to CD from A and B to divide the trapezium into one rectangle and two congruent right angled triangles.

The base of the right angled triangle=(CD – AB) ÷ 2

=(32 – 18) ÷ 2=7 cm

cos∠D = base ÷ hypotenuse = 7 ÷ 14 =1/2

m∠D = 60° 

Hence, m∠A = 120° 

*Answer is not matching with textbook answer.Question 49

Solution 49

Question 50

Solution 50

Question 51

Sides of a triangular field are 15 m, 16 m and 17 m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7 m each to graze in the field. Find the area of the field which cannot be grazed by three animals.Solution 51

Question 52

In the given Fig., the side of square is 28 cm, and radius of each circle is half of the length of the side of the square where O and O’ are centres of the circles. Find the area of shaded region.

Solution 52

According to the question,

Side of a square is 28 cm.

Radius of a circle is 14 cm.

Required area = Area of the square + Area of the two circles – Area of two quadrants …(i)

Area of the square = 28² = 784 cm²

Area of the two circles = 2πr²

= 

= 1232 cm²

Area of two quadrants = 

= 

= 308 cm²

Required area = 784 + 1232 – 308 = 1708 cm²

NOTE: Answer not matching with back answer.Question 53

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park?Solution 53

According to the question,

For a cylindrical tank

d = 2 m, r = 1 m, h = 5 m

Volume of the tank = πr²h

  = 

 = 

After recycling, this water is used irrigate a park of a hospital with length 25 m and breadth 20 m.

If the tank is filled completely, then

Volume of cuboidal park = Volume of tank

h = 0.0314 m = 3.14 cm = p cmQuestion 54

In the figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of shaded region (use π = 3.14).

Solution 54

Join OB.

Here,  is a right triangle.

By Pythagoras theorem,

Therefore, radius of the circle (r) 

Area of the square 

Area of the quadrant of a circle 

Area of the shaded region = Area of quadrant – Area of square

= 128.25 cm²Question 55

In the figure, ABCD is a square with side  and inscribed in a circle. Find the area of the shaded region. (Use π = 3.14).

Solution 55

Join AC.

Here,  is a right triangle.

By Pythagoras theorem,

Therefore, diameter of the circle = 4 cm

So, the radius of the circle (r) = 2 cm

Area of the square 

Area of the circle 

Area of the shaded region = Area of the circle – Area of square

= 4.56 cm²

Chapter 15 – Areas Related to Circles Exercise 15.69

Question 1

If the circumference and the area of a circle are numerically equal, then diameter of the circle is

Solution 1

Correct Option :- (D)

Question 2

If the difference between the circumference and radius of a circle is 37 cm., then using π = , the circumference (in cm) of the circle is

(a) 154

(b) 44

(c) 14

(d) 7 Solution 2

According to the question,

Circumference of a circle = 

 = 

 = 44 cm Question 3

A write can be bent in the form of a circle of radius 56 cm. If it is bent in the form of a square, then its area will be

(a) 3520 cm²

(b) 6400 cm²

(c) 7744 cm²

(d) 8800 cm²Solution 3

Correct option (c)

Question 4

Solution 4

correct option – (c)

Question 5

A circular park has a path of uniform width around it. The difference between the outer and inner circumferences of the circular path is 132 m. Its width is

(a) 20 m

(b) 21 m

(c) 22 m

(d) 24 mSolution 5

correct option – (b)

Question 6

The radius of a wheel is 0.25 m. The number of revolutions it will make to travel a distance of 11 km will be

(a) 2800

(b) 4000

(c) 5500

(d) 7000Solution 6

Correct Option: d

Question 7

The ratio of the outer and inner perimeters of a circular path is 23:22. If the path is 5m wide, the diameter of the inner circle is 

(a) 55m

(b) 110 m

(c) 220 m

(d) 230 mSolution 7

Correct Option: (c)

Question 8

Solution 8

Correct option – (c)

Question 9

Solution 9

Correct option (c)

Question 10

Solution 10

Correct Option ( d )

Question 11

Solution 11

Correct option (a)

Question 12

The perimeter of a triangle is 30 cm and the circumference of its incircle is 88 cm. The area of the triangle is

a. 70 cm²

b. 140 cm²

c. 210 cm²

d. 420 cm² Solution 12

Let r be the radius of the circle.

2pr = 88

Perimeter of a triangle = 30 cm

Semi-perimeter = 15 cm

Hence,

Area of a triangle = r × s …(r = incircle radius, s =semi perimeter)

= 14 × 15

= 210 cm² Question 13

Solution 13

Correct option – (c)

Chapter 15 – Areas Related to Circles Exercise 15.70

Question 14

If the circumference of a circle increases from 4π to 8π, then its area is

(a) halved

(b) doubled

(c) tripled

(d) quadrupledSolution 14

Question 15

If the radius of a circle is diminished by 10%, then its area is diminished by

(a) 10%

(b) 19%

(c) 20%

(d) 36%Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

If the perimeter of a semi-circular protractor is 36 cm, then its diameter is

(a) 10 cm

(b) 12 cm

(c) 14 cm

(d) 16 cmSolution 20

Question 21

Solution 21

Question 22

If the perimeter of a sector of a circle of radius 6.5 cm is 29 cm, then its area is

(a) 58 cm²

(b) 52 cm²

(c) 25 cm²

(d) 56 cm²Solution 22

Question 23

If the area of a sector of a circle bounded by an arc of length 5π cm is equal to 20π cm² , then its radius is

(a) 12 cm

(b) 16 cm

(c) 8 cm

(d) 10 cmSolution 23

Question 24

The area of the circle that can be inscribed in a square of side 10 cm is

(a) 40 π cm²

(b) 30 π cm²

(c) 100 π cm²

(d) 25 π cm²Solution 24

Correct option: (d)

Diameter of circle = side of square

2r = 10

r = 5 cm

Area of circle = πr² = 25 π cm²

Question 25

If the difference between the circumference and radius of a circle is 37 cm, then its area is

(a) 154 cm²

(b) 160 cm²

(c) 200 cm² 

(d) 150 cm²Solution 25

Chapter 15 – Areas Related to Circles Exercise 15.71

Question 26

The area of a circular path of uniform width h surrounding a circular region of radius r is

(a) π (2r + h) r

(b) π (2r + h) h

(c) π (h + r) r

(d) π (h + r) hSolution 26

Correct option: (b)

Inner radius = r

outer radius = r + h

area of shaded region = area of outer circle – area of inner circle

= π (r + h)² – πr²

= π {(r + h)² – r² }

= π (r + h – r) (r + h + r)

= π (2r + h) h

Question 27

Solution 27

Question 28

The area of a circle whose area and circumference are numerically equal, is

(a) 2π sq. units

(b) 4π sq. units

(c) 6π sq. units

(d) 8π sq. unitsSolution 28

Correct option: (b)

area = circumference

πr² = 2πr

r = 2 units

area = πr²

       = 4π sq. unitsQuestion 29

If diameter of a circle is increased by 40%, then its area increases by

(a) 96%

(b) 40%

(c) 80%

(d) 48%Solution 29

Question 30

In figure, the shaded area is

(a) 50 (π – 2) cm²

(b) 25 (π – 2) cm²

(c) 25 (π + 2) cm²

(d) 5 (π – 2) cm²

Solution 30

** img pending

Question 31

Solution 31

Question 32

Solution 32

Chapter 15 – Areas Related to Circles Exercise 15.72

Question 33

If the area of a sector of a circle bounded by an arc of length 5π cm is equal to 20π cm², then the radius of the circle is

(a) 12 cm

(b) 16 cm

(c) 8 cm

(d) 10 cmSolution 33

Question 34

In Figure, the ratio of the areas of two sectors S₁ and S₂ is

(a) 5 : 2

(b) 3 : 5

(c) 5 : 3

(d) 4 : 5Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Chapter 15 – Areas Related to Circles Exercise 15.73

Question 38

In figure, the area of the shaded region is

(a) 3π cm²

(b) 6π cm²

(c) 9π cm²

(d) 7π cm²

Solution 38

Question 39

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is

(a) 13 : 22

(b) 14 : 11

(c) 22 : 13

(d) 11 : 14Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

If a chord of a circle of radius 28 cm makes an angle of 90° at the centre, then the area of the major segment is

(a) 392 cm²

(b) 1456 cm²

(c) 1848 cm²

(d) 2240 cm²Solution 42

Question 43

If area of a circle inscribed in an equilateral triangle is 48π square units, then perimeter of the triangle is

Solution 43

Chapter 13 – Areas Related to Circles Exercise 13.74

Question 44

The hour hand of a clock is 6 cm long. The area swept by it between 11.20 am and 11.55 am is

(a) 2.75 cm²

(b) 5.5 cm²

(c) 11 cm²

(d) 10 cm²Solution 44

Question 45

Solution 45

Question 46

If the area of circle is equal to the sum of the areas of two circles of diameters 10 cm and 24 cm, then diameter of the larger circle (in cm) is

(a) 34

(b) 26

(c) 17

(d) 14Solution 46

Correct option: (b)

radius of Circle = 5 cm

area = π (5)²

       = 25 π

rdius of circle 2 = 12 cm

area = π (12)²

       = 144 π

area of larger circle = 144 π + 25π

                            = 169 π

πr² = 169 π

r² = 169

r = 13

diameter = 2r

              = 26Question 47

If Π is taken as 22/7, the distance (in metres) covered by a wheel of diameter 35 cm, in one revolution, is

(a) 2.2

(b) 1.1

(c) 9.625

(d) 96.25 Solution 47

Question 48

ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is

(a) 5

(b) 4

(c) 3

(d) 25Solution 48

Question 49

Area of the largest triangle that can be inscribed in a semi-circle of a radius r units is

a. r² sq. units

b. 

c. 2r² sq. units

d. Solution 49

Question 50

If the sum of the areas of two circles with radii r₁ and r₂ is equal to the area of a circle of radius r, then

a. r = r₁ + r₂

b. 

c. r₁ + r₂ < r

d. Solution 50

Question 51

If the sum of the circumference of two circles with radii r₁ and r₂ is equal to the circumference of a circle of radius r, then

a. r = r₁ + r₂

b. r₁ + r₂ > r

c. r₁ + r₂ < 2

d. None of theseSolution 51

Question 52

If the circumference of a circle and the perimeter of a square are equal, then

a. Area of the circle = Area of the square

b. Area of the circle < Area of the square

c. Area of the circle > Area of the square

d. Nothing definite can be saidSolution 52

Question 53

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is

a. 22 : 7

b. 14 : 11

c. 7 : 22

d. 11 : 14Solution 53