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# RD Sharma solutions for Class 10 Mathematics chapter 4 - Quadratic Equations [Latest edition]

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#### Chapters

Ex. 4.1Ex. 4.2Ex. 4.3Ex. 3.4Ex. 4.4Ex. 4.5Ex. 4.6Ex. 4.7Ex. 4.8Ex. 4.9Others

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.1 [Pages 4 - 5]

Ex. 4.1 | Q 1.01 | Page 4

Check whether the following is quadratic equation or not.

x2 + 6x − 4 = 0

Ex. 4.1 | Q 1.02 | Page 4

Check whether the following is quadratic equation or not.

sqrt(3x^2)-2x+1/2=0

Ex. 4.1 | Q 1.03 | Page 4

Check whether the following is quadratic equation or not.

x^2+1/x^2=5

Ex. 4.1 | Q 1.04 | Page 4

Check whether the following is quadratic equation or not.

x-3/x=x^2

Ex. 4.1 | Q 1.05 | Page 4

Check whether the following is quadratic equation or not.

2x^2-sqrt(3x)+9=0

Ex. 4.1 | Q 1.06 | Page 4

Check whether the following is quadratic equation or not.

x^2 - 2x - sqrtx - 5 = 0

Ex. 4.1 | Q 1.07 | Page 4

Check whether the following is quadratic equation or not.

3x2 - 5x + 9 = x2 - 7x + 3

Ex. 4.1 | Q 1.08 | Page 4

Check whether the following is quadratic equation or not.

x+1/x=1

Ex. 4.1 | Q 1.09 | Page 4

Check whether the following is quadratic equation or not.

x2 - 3x = 0

Ex. 4.1 | Q 1.1 | Page 4

Check whether the following is quadratic equation or not.

(x+1/x)^2=3(1+1/x)+4

Ex. 4.1 | Q 1.11 | Page 4

Check whether the following is quadratic equation or not.

(2𝑥 + 1)(3𝑥 + 2) = 6(𝑥 − 1)(𝑥 − 2)

Ex. 4.1 | Q 1.12 | Page 4

Check whether the following is quadratic equation or not.

x+1/x=x^2, x ≠ 0

Ex. 4.1 | Q 1.13 | Page 4

Check whether the following is quadratic equation or not.

16x2 − 3 = (2x + 5) (5x − 3)

Ex. 4.1 | Q 1.14 | Page 4

Check whether the following is quadratic equation or not.

(x + 2)3 = x3 − 4

Ex. 4.1 | Q 1.15 | Page 4

Check whether the following is quadratic equation or not.

x(x + 1) + 8 = (x + 2) (x - 2)

Ex. 4.1 | Q 2.1 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

x2 - 3x + 2 = 0, x = 2, x = -1

Ex. 4.1 | Q 2.2 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

x2 + x + 1 = 0, x = 0, x = 1

Ex. 4.1 | Q 2.3 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

x^2 - 3sqrt3x+6=0, x=sqrt3, x=-2sqrt3

Ex. 4.1 | Q 2.4 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

x+1/x=13/6, x=5/6, x=4/3

Ex. 4.1 | Q 2.5 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

2x2 - x + 9 = x2 + 4x + 3, x = 2, x =3

Ex. 4.1 | Q 2.6 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

x^2-sqrt2x-4=0, x=-sqrt2, x=-2sqrt2

Ex. 4.1 | Q 2.7 | Page 4

In the following, determine whether the given values are solutions of the given equation or not:

a2x2 - 3abx + 2b2 = 0, x=a/b, x=b/a

Ex. 4.1 | Q 3.1 | Page 4

In the following, find the value of k for which the given value is a solution of the given equation:

7x2 + kx - 3 = 0, x=2/3

Ex. 4.1 | Q 3.2 | Page 4

In the following, find the value of k for which the given value is a solution of the given equation:

x2 - x(a + b) + k = 0, x = a

Ex. 4.1 | Q 3.3 | Page 4

In the following, find the value of k for which the given value is a solution of the given equation:

kx^2+sqrt2x-4=0, x=sqrt2

Ex. 4.1 | Q 3.4 | Page 4

In the following, find the value of k for which the given value is a solution of the given equation:

x2 + 3ax + k = 0, x = -a

Ex. 4.1 | Q 4 | Page 5

Determine if, 3 is a root of the equation given below:

sqrt(x^2-4x+3)+sqrt(x^2-9)=sqrt(4x^2-14x+16)

Ex. 4.1 | Q 5 | Page 5

If x = 2/3 and x = −3 are the roots of the equation ax2 + 7x + b = 0, find the values of aand b.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.2 [Page 8]

Ex. 4.2 | Q 1 | Page 8

The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers, if x denotes the smaller integer.

Ex. 4.2 | Q 2 | Page 8

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 128. Form the quadratic equation to find how many marbles they had to start with, if John had x marbles.

Ex. 4.2 | Q 3 | Page 8

A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of articles produced in a day. On a particular day, the total cost of production was Rs. 750. If x denotes the number of toys produced that day, form the quadratic equation fo find x.

Ex. 4.2 | Q 4 | Page 8

The height of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.

Ex. 4.2 | Q 5 | Page 8

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 1 1 km/hr more than that of the passenger train, form the quadratic equation to find the average speed of express train.

Ex. 4.2 | Q 6 | Page 8

A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Form the quadratic eqiation to find the speed of the train.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.3, 3.4 [Pages 0 - 21]

Ex. 4.3 | Q 1 | Page 19

Solve the following quadratic equations by factorization:

(x − 4) (+ 2) = 0

Ex. 4.3 | Q 2 | Page 19

Solve the following quadratic equations by factorization:

(2x + 3)(3x − 7) = 0

Ex. 4.3 | Q 3 | Page 19

Solve the following quadratic equations by factorization:

3x2 − 14x − 5 = 0

Ex. 4.3 | Q 4 | Page 19

Solve the following quadratic equations by factorization:

9x2 − 3x − 2 = 0

Ex. 4.3 | Q 5 | Page 19

Solve the following quadratic equations by factorization:

1/(x-1)-1/(x+5)=6/7 , x ≠ 1, -5

Ex. 4.3 | Q 6 | Page 19

Solve the following quadratic equations by factorization:

6x2 + 11x + 3 = 0

Ex. 4.3 | Q 7 | Page 19

Solve the following quadratic equations by factorization:

5x2 - 3x - 2 = 0

Ex. 4.3 | Q 8 | Page 19

Solve the following quadratic equations by factorization:

48x2 − 13x − 1 = 0

Ex. 4.3 | Q 9 | Page 19

Solve the following quadratic equations by factorization:

3x2 = -11x - 10

Ex. 4.3 | Q 10 | Page 19

Solve the following quadratic equations by factorization:

25x(x + 1) = -4

Ex. 4.3 | Q 11 | Page 19

Solve the following quadratic equations by factorization:

$16x - \frac{10}{x} = 27$

Ex. 4.3 | Q 12 | Page 19

Solve the following quadratic equations by factorization:

1/x-1/(x-2)=3 , x ≠ 0, 2

Ex. 4.3 | Q 14 | Page 19

Solve the following quadratic equations by factorization:

1/(x+4)-1/(x-7)=11/30 , x ≠ 4, 7

Ex. 4.3 | Q 15 | Page 19

Solve the following quadratic equations by factorization:$\frac{1}{x - 3} + \frac{2}{x - 2} = \frac{8}{x}; x \neq 0, 2, 3$

Ex. 4.3 | Q 16 | Page 19

Solve the following quadratic equations by factorization:

a2x2 - 3abx + 2b2 = 0

Ex. 4.3 | Q 17 | Page 19

Solve the following quadratic equations by factorization:

$9 x^2 - 6 b^2 x - \left( a^4 - b^4 \right) = 0$

Ex. 4.3 | Q 18 | Page 19

Solve the following quadratic equations by factorization:

4x2 + 4bx - (a2 - b2) = 0

Ex. 4.3 | Q 19 | Page 19

Solve the following quadratic equations by factorization:

ax2 + (4a2 - 3b)x - 12ab = 0

Ex. 3.4 | Q 20 | Page 19

Solve the following quadratic equations by factorization: $2 x^2 + ax - a^2 = 0$

Ex. 4.3 | Q 21 | Page 19

Solve the following quadratic equations by factorization: $\frac{16}{x} - 1 = \frac{15}{x + 1}; x \neq 0, - 1$

Ex. 4.3 | Q 22 | Page 20

Solve the following quadratic equations by factorization:

(x+3)/(x+2)=(3x-7)/(2x-3)

Ex. 4.3 | Q 23 | Page 20

Solve the following quadratic equations by factorization:

(2x)/(x-4)+(2x-5)/(x-3)=25/3

Ex. 4.3 | Q 24 | Page 20

Solve the following quadratic equations by factorization:

(x+3)/(x-2)-(1-x)/x=17/4

Ex. 4.3 | Q 25 | Page 20

Solve the following quadratic equations by factorization:

(x-3)/(x+3)-(x+3)/(x-3)=48/7 , x ≠ 3, x ≠ -3

Ex. 4.3 | Q 26 | Page 20

Solve the following quadratic equations by factorization:

1/(x-2)+2/(x-1)=6/x , x ≠ 0

Ex. 4.3 | Q 27 | Page 20

Solve the following quadratic equations by factorization:

(x+1)/(x-1)-(x-1)/(x+1)=5/6 , x ≠ 1, x ≠ -1

Ex. 4.3 | Q 28 | Page 20

Solve the following quadratic equations by factorization:

(x-1)/(2x+1)+(2x+1)/(x-1)=5/2 , x ≠ -1/2, 1

Ex. 4.3 | Q 29 | Page 20

Solve the following quadratic equations by factorization:

$\frac{4}{x} - 3 = \frac{5}{2x + 3}, x \neq 0, - \frac{3}{2}$

Ex. 4.3 | Q 30 | Page 20

Solve the following quadratic equations by factorization: $\frac{x - 4}{x - 5} + \frac{x - 6}{x - 7} = \frac{10}{3}; x \neq 5, 7$

Ex. 4.3 | Q 31 | Page 20

Solve the following quadratic equations by factorization:

$\frac{x - 2}{x - 3} + \frac{x - 4}{x - 5} = \frac{10}{3}; x \neq 3, 5$

Ex. 4.3 | Q 32 | Page 20

Solve the following quadratic equations by factorization: $\frac{5 + x}{5 - x} - \frac{5 - x}{5 + x} = 3\frac{3}{4}; x \neq 5, - 5$

Ex. 4.3 | Q 33 | Page 20

Solve the following quadratic equations by factorization: $\frac{3}{x + 1} - \frac{1}{2} = \frac{2}{3x - 1}, x \neq - 1, \frac{1}{3}$

Ex. 4.3 | Q 34 | Page 20

Solve the following quadratic equations by factorization: $\frac{3}{x + 1} + \frac{4}{x - 1} = \frac{29}{4x - 1}; x \neq 1, - 1, \frac{1}{4}$

Ex. 4.3 | Q 35 | Page 20

Solve the following quadratic equations by factorization: $\frac{2}{x + 1} + \frac{3}{2(x - 2)} = \frac{23}{5x}; x \neq 0, - 1, 2$

Ex. 4.3 | Q 38 | Page 20

Solve the following quadratic equations by factorization: $\sqrt{3} x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0$

Ex. 4.3 | Q 39 | Page 20

Solve the following quadratic equations by factorization:

4sqrt3x^2+5x-2sqrt3=0

Ex. 4.3 | Q 40 | Page 20

Solve the following quadratic equations by factorization:

sqrt2x^2-3x-2sqrt2=0

Ex. 4.3 | Q 42 | Page 20

Solve the following quadratic equations by factorization:

3x^2-2sqrt6x+2=0

Ex. 4.3 | Q 43 | Page 20

Find the roots of the quadratic equation $\sqrt{2} x^2 + 7x + 5\sqrt{2} = 0$.

Ex. 4.3 | Q 44 | Page 20

Solve the following quadratic equations by factorization:

m/nx^2+n/m=1-2x

Ex. 4.3 | Q 45 | Page 20

Solve the following quadratic equations by factorization:

(x-a)/(x-b)+(x-b)/(x-a)=a/b+b/a

Ex. 4.3 | Q 46 | Page 21

Solve the following quadratic equations by factorization:

1/((x-1)(x-2))+1/((x-2)(x-3))+1/((x-3)(x-4))=1/6

Ex. 4.3 | Q 47 | Page 21

Solve the following quadratic equation by factorization: $\frac{a}{x - b} + \frac{b}{x - a} = 2$

Ex. 4.3 | Q 48 | Page 21

Solve the following quadratic equations by factorization: $\frac{x + 1}{x - 1} + \frac{x - 2}{x + 2} = 4 - \frac{2x + 3}{x - 2}; x \neq 1, - 2, 2$

Ex. 4.3 | Q 49 | Page 21

Solve the following quadratic equations by factorization:

a/(x-a)+b/(x-b)=(2c)/(x-c)

Ex. 4.3 | Q 50 | Page 21

Solve the following quadratic equations by factorization:

x2 + 2ab = (2a + b)x

Ex. 4.3 | Q 51 | Page 21

Solve the following quadratic equations by factorization:

(a + b)2x2 - 4abx - (a - b)2 = 0

Ex. 4.3 | Q 52 | Page 21

Solve the following quadratic equations by factorization:

a(x2 + 1) - x(a2 + 1) = 0

Ex. 4.3 | Q 53 | Page 21

Solve the following quadratic equations by factorization:

x2 - x - a(a + 1) = 0

Ex. 4.3 | Q 54 | Page 21

Solve the following quadratic equations by factorization:

x^2+(a+1/a)x+1=0

Ex. 4.3 | Q 55 | Page 21

Solve the following quadratic equations by factorization:

abx2 + (b2 - ac)x - bc = 0

Ex. 4.3 | Q 56

Solve the following quadratic equations by factorization:

a2b2x2 + b2x - a2x - 1 = 0

Ex. 4.3 | Q 57 | Page 21

Solve the following quadratic equations by factorization:

(x-1)/(x-2)+(x-3)/(x-4)=3 1/3, x ≠ 2, 4

Ex. 4.3 | Q 58 | Page 21

Solve the following quadratic equations by factorization: $\frac{1}{2a + b + 2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x}$

Ex. 4.3 | Q 59 | Page 21

Solve the following quadratic equations by factorization:

$3\left( \frac{3x - 1}{2x + 3} \right) - 2\left( \frac{2x + 3}{3x - 1} \right) = 5; x \neq \frac{1}{3}, - \frac{3}{2}$

Ex. 4.3 | Q 60 | Page 21

Solve the following quadratic equations by factorization:

$3\left( \frac{7x + 1}{5x - 3} \right) - 4\left( \frac{5x - 3}{7x + 1} \right) = 11; x \neq \frac{3}{5}, - \frac{1}{7}$

Ex. 4.3 | Q 61 | Page 21

Solve the following quadratic equations by factorization:

(x-5)(x-6)=25/(24)^2

Ex. 4.3 | Q 62 | Page 21

Solve the following quadratic equations by factorization:

7x + 3/x=35 3/5

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.4 [Page 26]

Ex. 4.4 | Q 1 | Page 26

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

x^2-4sqrt2x+6=0

Ex. 4.4 | Q 2 | Page 26

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x2 – 7x + 3  = 0

Ex. 4.4 | Q 3 | Page 26

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

3x2 + 11x + 10 = 0

Ex. 4.4 | Q 4 | Page 26

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x2 + x – 4 =  0

Ex. 4.4 | Q 5 | Page 26

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x2 + x + 4 = 0

Ex. 4.4 | Q 6 | Page 26

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 4x^2 + 4sqrt3x + 3 = 0

Ex. 4.4 | Q 7 | Page 26

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

sqrt2x^2-3x-2sqrt2=0

Ex. 4.4 | Q 8 | Page 26

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

sqrt3x^2+10x+7sqrt3=0

Ex. 4.4 | Q 9 | Page 26

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

x^2-(sqrt2+1)x+sqrt2=0

Ex. 4.4 | Q 10 | Page 26

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

x2 - 4ax + 4a2 - b2 = 0

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.5 [Page 32]

Ex. 4.5 | Q 1.1 | Page 32

Write the discriminant of the following quadratic equations:

2x2 - 5x + 3 = 0

Ex. 4.5 | Q 1.2 | Page 32

Write the discriminant of the following quadratic equations:

x2 + 2x + 4 = 0

Ex. 4.5 | Q 1.3 | Page 32

Write the discriminant of the following quadratic equations:

(x − 1) (2x − 1) = 0

Ex. 4.5 | Q 1.4 | Page 32

Write the discriminant of the following quadratic equations:

x2 - 2x + k = 0, k ∈ R

Ex. 4.5 | Q 1.5 | Page 32

Write the discriminant of the following quadratic equations:

sqrt3x^2+2sqrt2x-2sqrt3=0

Ex. 4.5 | Q 1.6 | Page 32

Write the discriminant of the following quadratic equations:

x2 - x + 1 = 0

Ex. 4.5 | Q 2.01 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

16x2 = 24x + 1

Ex. 4.5 | Q 2.02 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

x2 + x + 2 = 0

Ex. 4.5 | Q 2.03 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

sqrt3x^2+10x-8sqrt3=0

Ex. 4.5 | Q 2.04 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3x2 - 2x + 2 = 0

Ex. 4.5 | Q 2.05 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

2x^2-2sqrt6x+3=0

Ex. 4.5 | Q 2.06 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3a2x2 + 8abx + 4b2 = 0, a ≠ 0

Ex. 4.5 | Q 2.07 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3x^2+2sqrt5x-5=0

Ex. 4.5 | Q 2.08 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

x2 - 2x + 1 = 0

Ex. 4.5 | Q 2.09 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

2x^2+5sqrt3x+6=0

Ex. 4.5 | Q 2.1 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

sqrt2x^2+7x+5sqrt2=0

Ex. 4.5 | Q 2.11 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

2x^2-2sqrt2x+1=0

Ex. 4.5 | Q 2.12 | Page 32

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3x2 - 5x + 2 = 0

Ex. 4.5 | Q 3.1 | Page 32

Solve for x

(x-1)/(x-2)+(x-3)/(x-4)=3 1/3; x ≠ 2, 4

Ex. 4.5 | Q 3.2 | Page 32

Solve for x

1/x - 1/(x-2)=3, x ≠ 0, 2

Ex. 4.5 | Q 3.3 | Page 32

Solve for x

x+1/x=3, x ≠ 0

Ex. 4.5 | Q 3.4 | Page 32

Solve for x: $\frac{16}{x} - 1 = \frac{15}{x + 1}, x \neq 0, - 1$

Ex. 4.5 | Q 3.5 | Page 32

Solve for x: $\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}, x \neq 3, - 5$

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.6 [Pages 41 - 43]

Ex. 4.6 | Q 1.1 | Page 41

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them

2x2 - 3x + 5 = 0

Ex. 4.6 | Q 1.2 | Page 41

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them

2x2 - 6x + 3 = 0

Ex. 4.6 | Q 1.3 | Page 41

Determine the nature of the roots of the following quadratic equation:

3/5x^2-2/3x+1=0

Ex. 4.6 | Q 1.4 | Page 41

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them;

3x^2 - 4sqrt3x + 4 = 0

Ex. 4.6 | Q 1.5 | Page 41

Determine the nature of the roots of the following quadratic equation:

3x^2-2sqrt6x+2=0

Ex. 4.6 | Q 2.01 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

kx2 + 4x + 1 = 0

Ex. 4.6 | Q 2.02 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

kx^2-2sqrt5x+4=0

Ex. 4.6 | Q 2.03 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

3x2 - 5x + 2k = 0

Ex. 4.6 | Q 2.04 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

4x2 + kx + 9 = 0

Ex. 4.6 | Q 2.05 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

2kx2 - 40x + 25 = 0

Ex. 4.6 | Q 2.06 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

9x2 - 24x + k = 0

Ex. 4.6 | Q 2.07 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

4x2 - 3kx + 1 = 0

Ex. 4.6 | Q 2.08 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

x2 - 2(5 + 2k)x + 3(7 + 10k) = 0

Ex. 4.6 | Q 2.09 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

(3k+1)x2 + 2(k + 1)x + k = 0

Ex. 4.6 | Q 2.1 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

kx2 + kx + 1 = -4x2 - x

Ex. 4.6 | Q 2.11 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

(k + 1)x2 + 2(k + 3)x + (k + 8) = 0

Ex. 4.6 | Q 2.12 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

x2 - 2kx + 7k - 12 = 0

Ex. 4.6 | Q 2.13 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

(k + 1)x2 - 2(3k + 1)x + 8k + 1 = 0

Ex. 4.6 | Q 2.14 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

(2k + 1)x2 + 2(k + 3)x + (k + 5) = 0

Ex. 4.6 | Q 2.15 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

4x2 - 2(k + 1)x + (k + 4) = 0

Ex. 4.6 | Q 2.16 | Page 41

Find the values of k for which the roots are real and equal in each of the following equation:

$4 x^2 - 2\left( k + 1 \right)x + \left( k + 1 \right) = 0$

Ex. 4.6 | Q 3.1 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x2 + 3x + k = 0

Ex. 4.6 | Q 3.2 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots: $2 x^2 + x + k = 0$

Ex. 4.6 | Q 3.3 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x2 - 5x - k = 0

Ex. 4.6 | Q 3.4 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

kx2 + 6x + 1 = 0

Ex. 4.6 | Q 3.5 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

3x2 + 2x + k = 0

Ex. 4.6 | Q 4.1 | Page 42

Find the value of k for which the following equations have real and equal roots:

$x^2 - 2\left( k + 1 \right)x + k^2 = 0$

Ex. 4.6 | Q 4.2 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:

k2x2 - 2(2k - 1)x + 4 = 0

Ex. 4.6 | Q 4.3 | Page 42

Find the value of k for which the following equations have real and equal roots:

$\left( k + 1 \right) x^2 - 2\left( k - 1 \right)x + 1 = 0$

Ex. 4.6 | Q 4.4 | Page 42

Find the value of k for which the following equations have real and equal roots:

$x^2 + k\left( 2x + k - 1 \right) + 2 = 0$

Ex. 4.6 | Q 5.1 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:

2x2 + kx + 3 = 0

Ex. 4.6 | Q 5.2 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:

kx(x - 2) + 6 = 0

Ex. 4.6 | Q 5.3 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:

x2 - 4kx + k = 0

Ex. 4.6 | Q 5.4 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:

$kx\left( x - 2\sqrt{5} \right) + 10 = 0$

Ex. 4.6 | Q 5.5 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:$px(x - 3) + 9 = 0$

Ex. 4.6 | Q 5.6 | Page 42

Find the values of k for which the roots are real and equal in each of the following equation:

$4 x^2 + px + 3 = 0$

Ex. 4.6 | Q 6.1 | Page 42

Find the values of k for which the given quadratic equation has real and distinct roots:

kx2 + 2x + 1 = 0

Ex. 4.6 | Q 6.2 | Page 42

Find the values of k for which the given quadratic equation has real and distinct roots:

kx2 + 6x + 1 = 0

Ex. 4.6 | Q 6.3 | Page 42

Find the values of k for which the given quadratic equation has real and distinct roots:

x2 - kx + 9 = 0

Ex. 4.6 | Q 7 | Page 42

For what value of k,  (4 - k)x2 + (2k + 4)x + (8k + 1) = 0, is a perfect square.

Ex. 4.6 | Q 8 | Page 42

Find the least positive value of k for which the equation x2 + kx + 4 = 0 has real roots.

Ex. 4.6 | Q 9 | Page 42

Find the values of k for which the quadratic equation

$\left( 3k + 1 \right) x^2 + 2\left( k + 1 \right)x + 1 = 0$ has equal roots. Also, find the roots.

Ex. 4.6 | Q 10 | Page 42

Find the values of p for which the quadratic equation

$\left( 2p + 1 \right) x^2 - \left( 7p + 2 \right)x + \left( 7p - 3 \right) = 0$ has equal roots. Also, find these roots.
Ex. 4.6 | Q 11 | Page 42

If −5 is a root of the quadratic equation$2 x^2 + px - 15 = 0$ and the quadratic equation $p( x^2 + x) + k = 0$ has equal roots, find the value of k.

Ex. 4.6 | Q 12 | Page 42

If 2 is a root of the quadratic equation $3 x^2 + px - 8 = 0$ and the quadratic equation $4 x^2 - 2px + k = 0$  has equal roots, find the value of k.

Ex. 4.6 | Q 13 | Page 42

If 1 is a root of the quadratic equation $3 x^2 + ax - 2 = 0$ and the quadratic equation $a( x^2 + 6x) - b = 0$ has equal roots, find the value of b.

Ex. 4.6 | Q 14 | Page 42

Find the value of p for which the quadratic equation

$\left( p + 1 \right) x^2 - 6(p + 1)x + 3(p + 9) = 0, p \neq - 1$ has equal roots. Hence, find the roots of the equation.

Disclaimer: There is a misprinting in the given question. In the question 'q' is printed instead of 9.

Ex. 4.6 | Q 15.1 | Page 42

Determine the nature of the roots of the following quadratic equation:

(x - 2a)(x - 2b) = 4ab

Ex. 4.6 | Q 15.2 | Page 42

Determine the nature of the roots of the following quadratic equation:

9a2b2x2 - 24abcdx + 16c2d2 = 0

Ex. 4.6 | Q 15.3 | Page 42

Determine the nature of the roots of the following quadratic equation:

2(a2 + b2)x2 + 2(a + b)x + 1 = 0

Ex. 4.6 | Q 15.4 | Page 42

Determine the nature of the roots of the following quadratic equation:

(b + c)x2 - (a + b + c)x + a = 0

Ex. 4.6 | Q 16.1 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

x2 - kx + 9 = 0

Ex. 4.6 | Q 16.2 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x2 + kx + 2 = 0

Ex. 4.6 | Q 16.3 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

4x2 - 3kx + 1 = 0

Ex. 4.6 | Q 16.4 | Page 42

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x2 + kx - 4 = 0

Ex. 4.6 | Q 17 | Page 42

If the roots of the equation (b - c)x2 + (c - a)x + (a - b) = 0 are equal, then prove that 2b = a + c.

Ex. 4.6 | Q 18 | Page 43

If the roots of the equation (a2 + b2)x2 − 2 (ac + bd)x + (c2 + d2) = 0 are equal, prove that a/b=c/d.

Ex. 4.6 | Q 19 | Page 43

If the roots of the equations ax2 + 2bx + c = 0 and bx^2-2sqrt(ac)x+b = 0 are simultaneously real, then prove that b2 = ac.

Ex. 4.6 | Q 20 | Page 43

If p, q are real and p ≠ q, then show that the roots of the equation (p − q) x2 + 5(p + q) x− 2(p − q) = 0 are real and unequal.

Ex. 4.6 | Q 21 | Page 43

If the roots of the equation (c2 – ab) x2 – 2 (a2 – bc) x + b2 – ac = 0 in x are equal, then show that either a = 0 or a3 + b3 + c3 = 3abc

Ex. 4.6 | Q 22 | Page 43

Show that the equation 2(a2 + b2)x2 + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.

Ex. 4.6 | Q 23 | Page 43

Prove that both the roots of the equation (x - a)(x - b) +(x - b)(x - c)+ (x - c)(x - a) = 0 are real but they are equal only when a = b = c.

Ex. 4.6 | Q 24 | Page 43

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax2 + bx + c = 0 and -ax2 + bx + c = 0 has real roots.

Ex. 4.6 | Q 25 | Page 43

If the equation $\left( 1 + m^2 \right) x^2 + 2 mcx + \left( c^2 - a^2 \right) = 0$ has equal roots, prove that c2 = a2(1 + m2).

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.7, 4.6 [Pages 51 - 53]

Ex. 4.7 | Q 1 | Page 51

Find the consecutive numbers whose squares have the sum 85.

Ex. 4.7 | Q 2 | Page 51

Divide 29 into two parts so that the sum of the squares of the parts is 425.

Ex. 4.7 | Q 3 | Page 51

Two squares have sides x cm and (x + 4)cm. The sum of this areas is 656 cm2. Find the sides of the squares.

Ex. 4.7 | Q 4 | Page 51

The sum of two numbers is 48 and their product is 432. Find the numbers?

Ex. 4.7 | Q 5 | Page 51

If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation.

Ex. 4.7 | Q 6 | Page 51

Find the whole numbers which when decreased by 20 is equal to 69 times the reciprocal of the members.

Ex. 4.7 | Q 7 | Page 51

Find the two consecutive natural numbers whose product is 20.

Ex. 4.7 | Q 8 | Page 51

The sum of the squares of the two consecutive odd positive integers as 394. Find them.

Ex. 4.7 | Q 9 | Page 51

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

Ex. 4.7 | Q 10 | Page 51

The sum of a numbers and its positive square root is 6/25. Find the numbers.

Ex. 4.7 | Q 11 | Page 51

The sum of a number and its square is 63/4. Find the numbers.

Ex. 4.7 | Q 12 | Page 52

There are three consecutive integers such that the square of the first increased by the product of the first increased by the product of the others the two gives 154. What are the integers?

Ex. 4.7 | Q 13 | Page 52

The product of two successive integral multiples of 5 is 300. Determine the multiples.

Ex. 4.7 | Q 14 | Page 52

The sum of the squares of two numbers as 233 and one of the numbers as 3 less than twice the other number find the numbers.

Ex. 4.7 | Q 15 | Page 52

Find the consecutive even integers whose squares have the sum 340.

Ex. 4.7 | Q 16 | Page 52

The difference of two numbers is 4. If the difference of their reciprocals is 4/21. Find the numbers.

Ex. 4.7 | Q 17 | Page 52

Let us find two natural numbers which differ by 3 and whose squares have the sum 117.

Ex. 4.7 | Q 18 | Page 52

The sum of the squares of three consecutive natural numbers as 149. Find the numbers

Ex. 4.7 | Q 19 | Page 52

Sum of two numbers is 16. The sum of their reciprocals is 1/3. Find the numbers.

Ex. 4.7 | Q 20 | Page 52

Determine two consecutive multiples of 3, whose product is 270.

Ex. 4.7 | Q 21 | Page 52

The sum of a number and its reciprocal is 17/4. Find the number.

Ex. 4.7 | Q 22 | Page 52

A two-digit number is such that the products of its digits is 8. When 18 is subtracted from the number, the digits interchange their places. Find the number?

Ex. 4.7 | Q 23 | Page 52

A two digits number is such that the product of the digits is 12. When 36 is added to the number, the digits inter change their places determine the number.

Ex. 4.7 | Q 24 | Page 52

A two digit number is such that the product of the digits is 16. When 54 is subtracted from the number the digits are interchanged. Find the number

Ex. 4.7 | Q 25 | Page 52

Two numbers differ by 3 and their product is 504. Find the number

Ex. 4.7 | Q 26 | Page 52

Two number differ by 4 and their product is 192. Find the numbers?

Ex. 4.7 | Q 27 | Page 52

A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.

Ex. 4.7 | Q 28 | Page 52

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

Ex. 4.7 | Q 29 | Page 52

The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers.

Ex. 4.7 | Q 30 | Page 52

The sum of two number a and b is 15, and the sum of their reciprocals 1/a and 1/b is 3/10. Find the numbers a and b.

Ex. 4.7 | Q 31 | Page 52

The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers.

Ex. 4.7 | Q 32 | Page 52

Three consecutive positive integers are such that the sum of the square of the first and the product of other two is 46, find the integers.

Ex. 4.7 | Q 33 | Page 52

The difference of squares of two number is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

Ex. 4.7 | Q 34 | Page 52

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

Ex. 4.6 | Q 35 | Page 52

Find two consecutive odd positive integers, sum of whose squares is 970.

Ex. 4.7 | Q 36 | Page 52

The difference of two natural numbers is 3 and the difference of their reciprocals is  $\frac{3}{28}$.Find the numbers.

Ex. 4.7 | Q 37 | Page 52

The sum of the squares of two consecutive odd numbers is 394. Find the numbers.

Ex. 4.7 | Q 38 | Page 53

The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples.

Ex. 4.7 | Q 39 | Page 53

The sum of the squares of two consecutive even numbers is 340. Find the numbers.

Ex. 4.7 | Q 40 | Page 53

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is  $\frac{29}{20}$.Find the original fraction.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.8 [Pages 58 - 59]

Ex. 4.8 | Q 1 | Page 58

The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.

Ex. 4.8 | Q 2 | Page 58

A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km/hr from its usual speed. What is the usual speed?

Ex. 4.8 | Q 3 | Page 58

A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speed of the two trains.

Ex. 4.8 | Q 4 | Page 58

A passenger train takes one hour less for a journey of 150 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.

Ex. 4.8 | Q 5 | Page 58

The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/hr more than the speed of going, what was the speed per hour in each direction?

Ex. 4.8 | Q 6 | Page 58

A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane.

Ex. 4.8 | Q 7 | Page 58

An aeroplane take 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed.

Ex. 4.8 | Q 8.1 | Page 58

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.

Ex. 4.8 | Q 8.2 | Page 58

A train travels at a certain average speed for a distance 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than the original speed, If it takes 3 hours to complete total journey, what is its original average speed?

Ex. 4.8 | Q 9 | Page 59

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for a journey. Find the original speed of the train.

Ex. 4.8 | Q 10 | Page 59

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Ex. 4.8 | Q 11 | Page 59

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.

Ex. 4.8 | Q 12 | Page 59

An aeroplane left 50 minutes later than its scheduled time, and in order to reach the destination, 1250 km away, in time, it had to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

Ex. 4.8 | Q 13 | Page 59

While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late by 30 minutes to reach the destination, 1500 km away in time, the pilot increased the speed by 100 km/hr. Find the original speed/hour of the plane.

Ex. 4.8 | Q 14 | Page 59

A motor boat whose speed in still water is 18 km/hr takes 1 hour more to go 24 km up stream that to return down stream to the same spot. Find the speed of the stream.

Ex. 4.8 | Q 15 | Page 59

A car moves a distance of 2592 km with uniform speed. The number of hours taken for the journey is one-half the number representing the speed, in km/hour. Find the time taken to cover the distance.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.9 [Pages 61 - 62]

Ex. 4.9 | Q 1 | Page 61

Ashu is x years old while his mother Mrs Veena is x2 years old. Five years hence Mrs Veena will be three times old as Ashu. Find their present ages.

Ex. 4.9 | Q 2 | Page 61

The sum of ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man's age at the time. Find their present ages.

Ex. 4.9 | Q 3 | Page 61

The product of Shikha's age five years ago and her age 8 years later is 30, her age at both times being given in years. Find her present age.

Ex. 4.9 | Q 4 | Page 61

The product of Ramu's age (in years) five years ago and his age (in years) nice years later is 15. Determine Ramu's present age.

Ex. 4.9 | Q 5 | Page 61

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Ex. 4.9 | Q 6 | Page 61

A girls is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

Ex. 4.9 | Q 7 | Page 62

The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.1 [Page 64]

Ex. 4.1 | Q 1 | Page 64

The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.

Ex. 4.1 | Q 2 | Page 64

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Ex. 4.1 | Q 3 | Page 64

The hypotenuse of a right triangle is 3sqrt10. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be 9sqrt5. How long are the legs of the triangle?

Ex. 4.1 | Q 4 | Page 64

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic EquationsExercise 4.7 [Pages 70 - 71]

Ex. 4.7 | Q 1 | Page 70

The perimeter of a rectangular field is 82 m and its area is 400 m2. Find the breadth of the rectangle.

Ex. 4.7 | Q 2 | Page 70

The length of a hall is 5 m more than its breadth. If the area of the floor of the hall is 84 m2, what are the length and breadth of the hall?

Q 3 | Page 70

Two squares have sides x cm and (x + 4)cm. The sum of this areas is 656 cm2. Find the sides of the squares.

Q 4 | Page 71

The area of a right angled triangle is 165 m2. Determine its base and altitude if the latter exceeds the former by 7 m.

Q 5 | Page 71

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.

Q 6 | Page 71

Is it possible to design a rectangular park of perimeter 80 and area 400 m2? If so find its length and breadth.

Q 7 | Page 71

Sum of the areas of two squares is 640 m2. If the difference of their perimeters is 64 m. Find the sides of the two squares.

Q 8 | Page 71

Sum of the area of two squares is 400 cm2. If the difference of their perimeters is 16 cm, find the sides of two squares.

Q 9 | Page 71

The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one metre more then twice its breadth. Find the length and the breadth of the plot.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic Equations [Pages 73 - 74]

Q 1 | Page 73

A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.

Q 2 | Page 73

If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir?

Q 3 | Page 73

Two water taps together can fill a tank in 9 3/8. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Q 4 | Page 73

Two pipes running together can fill a tank in 11 1/9 minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

Q 5 | Page 74

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic Equations [Pages 80 - 81]

Q 1 | Page 80

A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter costs Rs. one less, the cost would remain unchanged. How long is the piece?

Q 2 | Page 80

Some students planned a picnic. The budget for food was Rs. 480. But eight of these failed to go and thus the cost of food for each member increased by Rs. 10. How many students attended the picnic?

Q 3 | Page 80

A dealer sells an article for Rs. 24 and gains as much percent as the cost price of the article. Find the cost price of the article.

Q 4 | Page 80

Out of a group of swans, 7/2 times the square root of the total number are playing on the share of a pond. The two remaining ones are swinging in water. Find the total number of swans.

Q 5 | Page 80

If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find the original price of the toy.

Q 6 | Page 80

Rs. 9000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got Rs. 160 less. Find the original number of persons.

Q 7 | Page 80

Some students planned a picnic. The budget for food was Rs. 500. But, 5 of them failed to go and thus the cost of food for each member increased by Rs. 5. How many students attended the picnic?

Q 8 | Page 80

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

Q 9 | Page 80

In a class test, the sum of the marks obtained by P in Mathematics and science is 28. Had he got 3 marks more in mathematics and 4 marks less in Science. The product of his marks would have been 180. Find his marks in two subjects.

Q 10 | Page 81

In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects

Q 11 | Page 81

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic Equations [Page 82]

Q 1 | Page 82

Write the value of k for which the quadratic equation x2 − kx + 4 = 0 has equal roots.

Q 2 | Page 82

What is the nature of roots of the quadratic equation 4x2 − 12x − 9 = 0?

Q 3 | Page 82

If $1 + \sqrt{2}$ is a root of a quadratic equation will rational coefficients, write its other root.

Q 4 | Page 82

Write the number of real roots of the equation x2 + 3 |x| + 2 = 0.

Q 5 | Page 82

Write the sum of real roots of the equation x2 + |x| − 6 = 0.

Q 6 | Page 82

Write the set of value of 'a' for which the equation x2 + ax − 1 = 0 has real roots.

Q 7 | Page 82

Is there any real value of 'a' for which the equation x2 + 2x + (a2 + 1) = 0 has real roots?

Q 8 | Page 82

Write the value of λ for which x2 + 4x + λ is a perfect square.

Q 9 | Page 82

Write the condition to be satisfied for which equations ax2 + 2bx + c = 0 and $b x^2 - 2\sqrt{ac}x + b = 0$ have equal roots.

Q 10 | Page 82

Write the set of value of k for which the quadratic equations has 2x2 + kx − 8 = 0 has real roots.

Q 11 | Page 82

Write a quadratic polynomial, sum of whose zeros is $2\sqrt{3}$ and their product is 2.

Q 12 | Page 82

Show that x = −3 is a solution of x2 + 6x + 9 = 0.

Q 13 | Page 82

Show that x = −2 is a solution of 3x2 + 13x + 14 = 0.

Q 14 | Page 82

Find the discriminant of the quadratic equation $3\sqrt{3} x^2 + 10x + \sqrt{3} = 0$.

Q 15 | Page 82

If $x = - \frac{1}{2}$,is a solution of the quadratic equation $3 x^2 + 2kx - 3 = 0$ ,find the value of k.

#### RD Sharma solutions for Class 10 Mathematics Chapter 4 Quadratic Equations [Pages 83 - 85]

Q 1 | Page 83

If the equation x2 + 4x + k = 0 has real and distinct roots, then

• k < 4

• k > 4

• k ≥ 4

• k ≤ 4

Q 2 | Page 83

If the equation x2 − ax + 1 = 0 has two distinct roots, then

• |a| = 2

• |a| < 2

• |a| > 2

• None of these

Q 3 | Page 83

If the equation 9x2 + 6kx + 4 = 0 has equal roots, then the roots are both equal to

• $\pm \frac{2}{3}$

• $\pm \frac{3}{2}$

• 0

• ±3

Q 4 | Page 83

If ax2 + bx + c = 0 has equal roots, then c =

• $\frac{- b}{2a}$

• $\frac{b}{2a}$

• $\frac{- b^2}{4a}$

• $\frac{b^2}{4a}$

Q 5 | Page 83

If the equation ax2 + 2x + a = 0 has two distinct roots, if

• a = ±1

• a = 0

• a = 0, 1

• a = −1, 0

Q 6 | Page 83

The positive value of k for which the equation x2 + kx + 64 = 0 and x2 − 8x + k = 0 will both have real roots, is

• 4

• 8

• 12

• 16

Q 7 | Page 83

The value of $\sqrt{6 + \sqrt{6 + \sqrt{6 +}}} . . . .$ is

• 4

• 3

• -2

• 3.5

Q 8 | Page 83

If 2 is a root of the equation x2 + bx + 12 = 0 and the equation x2 + bx + q = 0 has equal roots, then q =

• 8

• -8

• 16

• -16

Q 9 | Page 83

If the equations $\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0$ has equal roots, then

• ab = cd

• $ad = \sqrt{bc}$

• $ab = \sqrt{cd}$

Q 10 | Page 83

If the roots of the equations $\left( a^2 + b^2 \right) x^2 - 2b\left( a + c \right)x + \left( b^2 + c^2 \right) = 0$ are equal, then

• 2b = a + c

• b2 = ac

• $b = \frac{2ac}{a + c}$

• b = ac

Q 11 | Page 83

If the equation x2 − bx + 1 = 0 does not possess real roots, then

• −3 < b < 3

• −2 < b < 2

• b > 2

• b < −2

Q 12 | Page 83

If x = 1 is a common roots of the equations ax2 + ax + 3 = 0 and x2 + x + b = 0,  then ab =

• 3

• 3.5

• 6

• -3

Q 13 | Page 83

If p and q are the roots of the equation x2 − px + q = 0, then

• p = 1, q = −2

• p = 1, q = −2

• p = −2, q = 0

• p = −2, q = 1

Q 14 | Page 83

If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax2 +bx + 1 = 0 having real roots is

• 10

• 7

• 6

• 12

Q 15 | Page 84

The number of quadratic equations having real roots and which do not change by squaring their roots is

• 4

• 3

• 2

• 1

Q 16 | Page 84

If $\left( a^2 + b^2 \right) x^2 + 2\left( ab + bd \right)x + c^2 + d^2 = 0$ has no real roots, then

• ab = bc

• ab = cd

• ac = bd

Q 17 | Page 84

If the sum of the roots of the equation x2 − x = λ(2x − 1) is zero, then λ =

•  −2

• 2

• $- \frac{1}{2}$

• $\frac{1}{2}$

Q 18 | Page 84

If x = 1 is a common root of ax2 + ax + 2 = 0 and x2 + x + b = 0, then, ab =

• 1

• 2

• 3

• 4

Q 19 | Page 84

The value of c for which the equation ax2 + 2bx + c = 0 has equal roots is

• $\frac{b^2}{a}$

• $\frac{b^2}{4a}$

• $\frac{a^2}{b}$

• $\frac{a^2}{4b}$

Q 20 | Page 84

If $x^2 + k\left( 4x + k - 1 \right) + 2 = 0$ has equal roots, then k =

• $- \frac{2}{3}, 1$

• $\frac{2}{3}, - 1$

• $\frac{3}{2}, \frac{1}{3}$

• $- \frac{3}{2}, - \frac{1}{3}$

Q 21 | Page 84

If the sum and product of the roots of the equation kx2 + 6x + 4k = 0 are real, then k =

• $- \frac{3}{2}$

• $\frac{3}{2}$

• $\frac{2}{3}$

• $- \frac{2}{3}$

Q 22 | Page 84

If sin α and cos α are the roots of the equations ax2 + bx + c = 0, then b2 =

•  a2 − 2ac

•  a2 + 2ac

• a2 − ac

• a2 + ac

Q 23 | Page 84

If 2 is a root of the equation x2 + ax + 12 = 0 and the quadratic equation x2 + ax + q = 0 has equal roots, then q =

• 12

• 8

• 20

• 16

Q 24 | Page 84

If the sum of the roots of the equation $x^2 - \left( k + 6 \right)x + 2\left( 2k - 1 \right) = 0$ is equal to half of their product, then k =

• 6

• 7

• 1

• 5

Q 25 | Page 84

If a and b are roots of the equation x2 + ax + b = 0, then a + b =

• 1

• 2

• -2

• -1

Q 26 | Page 84

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is

• x2 + 4 = 0

• x2 − 4 = 0

•  4x2 − 1 = 0

• x2 − 2 = 0

Q 27 | Page 84

If one of the equation ax2 + bx + c = 0 is three times times the other, then b2 : ac =

• 3 : 1

• 3 : 16

• 16 : 3

• 16 : 1

Q 28 | Page 84

If one root the equation 2x2 + kx + 4 = 0 is 2, then the other root is

• 6

• -6

• -1

• 1

Q 29 | Page 85

If one of the equation x2 + ax + 3 = 0 is 1, then its other root is

• 3

• -3

• 2

• 1

Q 30 | Page 85

If one root of the equation 4x2 − 2x + (λ − 4) = 0 be the reciprocal of the other, then λ =

• 8

• -8

• 4

• -4

Q 31 | Page 85

If y = 1 is a common root of the equations $a y^2 + ay + 3 = 0 \text { and } y^2 + y + b = 0$, then ab equals

• 3

• -7/2

• 6

• -3

Q 32 | Page 85

The values of k for which the quadratic equation  $16 x^2 + 4kx + 9 = 0$  has real and equal roots are

• $6, - \frac{1}{6}$

• 36, −36

•  6, −6

• $\frac{3}{4}, - \frac{3}{4}$

Ex. 4.1Ex. 4.2Ex. 4.3Ex. 3.4Ex. 4.4Ex. 4.5Ex. 4.6Ex. 4.7Ex. 4.8Ex. 4.9Others

## RD Sharma solutions for Class 10 Mathematics chapter 4 - Quadratic Equations

RD Sharma solutions for Class 10 Mathematics chapter 4 (Quadratic Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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