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## AP SSC 10th Class Maths Bits 5th Lesson Quadratic Equations with Answers

Question 1.

If x^{2} – px + q = 0(p,q∈R and p ≠ 0, q ≠ 0) has distinct real roots, then write

the condition.

Answer: p^{2} > 4q.

Question 2.

If one root of 2x^{2} + kx – 6 is 2., then find k.

Answer:

k = – 1

Explanation:

2(2)^{2} + k(2) – 6 = 0

⇒ 8 + 2k – 6 = O

⇒ 2k + 2 = 0 ⇒ k = -1

Question 3.

If the equation x^{2} + 5x + k = 0 has real and distinct roots, then find the value of ‘k’.

Answer:

k > 6.25

Explanation:

Real and distinct roots so,

b^{2} – 4ac > 0

⇒ 25 – 4 . 1. k > 0

⇒ 25 > 4k = k > \(\frac{25}{4}\) > 6.25

Question 4.

Frame a quadratic equation, whose roots are 2 + \(\sqrt{3}\) and 2 – \(\sqrt{3}\) ?

Answer:

x^{2} – 4x + 1 = 0

Explanation:

x^{2} – (2 + \(\sqrt{3}\) +2 – \(\sqrt{3}\))x + (2 + \(\sqrt{3}\)) (2 – \(\sqrt{3}\))

⇒ x^{2} – 4x + 1 = 0

Question 5.

In a quadratic equation ax^{2} + bx + c = 0, if b^{2} – 4ac > 0, then write the nature of the roots.

Answer:

Roots are real and distinct.

Question 6.

Create the quadratic equation, whose zeroes are \(\sqrt{2}\) and – \(\sqrt{2}\) ?

Answer:

x^{2} – 2 = 0.

Explanation:

\(x^{2}-(\sqrt{2}-\sqrt{2}) x+(\sqrt{2})(-\sqrt{2})=0\)

⇒ x^{2} – 2 = 0

Question 7.

For which positive value of x the qua-dratic equation 4x\(\sqrt{3}\) -9 = 0 satisfies ?

Answer:

\(\frac{3}{2}\)

Question 8.

If the roots of x^{2} + 6x + 5 = 0 are a and P, then find the value of sum of the roots.

Answer:

-6

Explanation:

α + β = \(\frac{-b}{a}\) = — 6

Question 9.

Write the discriminant of 6x^{2} – 5x + 1 = 0.

Answer:

D = 1

Explanation:

D = b^{2} – 4ac = 25 – 4 . 6 . 1

⇒ 25 – 24 = 1 > 0 D = 1

Question 10.

Write the quadratic polynomial having \(\frac { 1 }{ 3 }\) and \(\frac { 1 }{ 2 }\) as its zeroes.

Answer:

x^{2} – \(\frac{5 x-1}{6}\) = 0 ⇒ 6x^{2} – 5x + 1 = 0

Explanation:

Question 11.

If a number is 132 smaller than its square, then find the number.

Answer:

12

Explanation:

x + 132 = x^{2}

⇒ x^{2} – x – 132 = 0

By solve the equation, ∴ x = 12

Question 12.

Write the general form of a quadratic equation in variable ‘x’.

Answer:

ax^{2} + bx + c = 0 (a ≠ 0).

Question 13.

Make the quadratic polynomial, whose zeroes are 2 and 3.

Answer:

x^{2} – 5x + 6.

Question 14.

If α, β are the roots of x^{2} – 10x + 9 = 0, thep find the value of | α – β |.

Answer:

8

Explanation:

x^{2} – 9x – x + 9 = 0

⇒ x(x – 9) – 1 (x – 9) = 0

⇒ (x-9)(x- 1) = 0

x = 9 and 1, |α – β| = |9- 1| = 8

Question 15.

Write the discriminant of adjacent dia-gram indicates.

Answer:

b^{2} – 4ac > 0.

Question 16.

If the roots of a quadratic equation px^{2} + qx + r = 0 are imaginary, then write the condition of discriminant.

Answer:

q^{2} < 4pr (or) q^{2} – 4pr < 0.

Question 17.

Two angles are complementary. If the large angle is twice the measure of a smaller angle, then find the value of smaller angle.

Answer:

30°

Explanation:

x + y = 90°

⇒ x + 2x = 90°

⇒ 3x = 90° ⇒ x = 30°

Question 18.

Observe the following graphs.

Which as them are the graphs of qua-dratic polynomials ?

Answer:

(i) and (iv).

Question 19.

Write the possible number of roots to a quadratic equation.

Answer:

At a maximum of 2.

Question 20.

If 1 is a common root of ax^{2} + ax + 2 = 0 and x^{2} + x + 6 = 0, then find a-b.

Answer:

2

Question 21.

Find the product of roots of quadratic equation ax^{2} + bx + c = 0.

Answer:

\(\frac{\text { c }}{\text { a }}\)

Question 22.

Write the number of diagonals in a polygon, having ‘n’ sides.

Answer:

\(\frac{n(n-3)}{2}\)

Question 23.

Find the discriminant of quadratic equation 2x^{2} + x – 4 = 0.

Answer:

33

Question 24.

A quadratic equation ax^{2} + bx + c = 0 has two distinct real roots, then write the condition.

Answer:

b^{2} – 4ac >0.

Question 25.

Draw the shape of quadratic equation which having distinct roots ?

Answer:

Question 26.

The sum of a number and its reciprocal is \(\frac { 5 }{ 2 }\) then find the number.

Answer:

2 or \(\frac { 1 }{ 2 }\)

Explanation:

x + \(\frac{1}{x}=\frac{5}{2}\)

⇒ \(\frac{x^{2}+1}{x}=\frac{5}{2}\)

⇒ 2x^{2} + 2 = 5x

⇒ 2x^{2} – 5x + 2 = 0

⇒ 2x^{2} – 4x r x + 2 = 0

⇒ 2x(x – 2) – 1 (x – 2) – 0 ⇒ (x – 2) (2x – 1) = 0 1

∴ x = 2 or 1/2.

Question 27.

Find the roots of the equation 4x^{2} – 4\(\sqrt{3}\) x + 3 = 0.

Answer:

\(-\frac{\sqrt{3}}{2}\)

Question 28.

Find the positive root of \(\sqrt{3 x^{2}+6}=9\)

Answer:

5

Explanation:

3x^{2} + 6 = 81

⇒ 3x^{2} = 81 – 6 = 75

⇒ x^{2} = \(\) = 25 ⇒ x = 5

Question 29.

Find the roots of the quadratic equation (7x – 1) (2x + 3) = 0.

Answer:

\(\frac{1}{7}, \frac{-3}{2}\)

Question 30.

If the sum of the squares of two con-secutive odd numbers is 74, then find the smaller number.

Answer:

5 (or)-7

Explanation:

(2x + 1)^{2} + (2x + 3)^{2} – 74

⇒ 4x^{2} + 4x + 1 + 4x^{2} + 12x + 9 — 74′

⇒ 8x^{2} + 16x + 10 = 74

⇒ 8x^{2} + 16x – 64 = 0

⇒ 8(x^{2} + 2x – 8) = 0

⇒ x^{2} + 4x – 2x – 8 = 0

⇒ x(x + 4) – 2 (x + 4) = 0

⇒ x = – 4, 2

∴ x = – 4, then smaller number

= 2 . (-4) + 1 = -8 + 1 = -7

∴ x = 2, then smaller number

= 2 . (2) + 1 = 4 + 1 = 5

Question 31.

Write the standard form of a cubic polynomial.

Answer:

ax^{3} + bx^{2} + cx + d = 0; (a ≠ 0).

Question 32.

Write the discriminant of 5x^{2}– 3x – 2 = 0.

Answer:

49

Question 33.

Create the quadratic equation whose roots are – 2 and – 3.

Answer:

x^{2} + 5x + 6 = 0

Question 34.

Find the roots of the quadratic equation \(\frac{x^{2}-8}{x^{2}+20}=\frac{1}{2}\)

Answer:

±6

Explanation:

2x^{2} – 16 = x^{2} + 20

⇒ x^{2} – 36 ⇒ x = ±6.

Question 35.

Find the roots of the equation 3x^{2} – 2\(\sqrt{6}\) x + 2 = 0.

Answer:

\(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}\)

Question 36.

Find the roots of the quadratic equa- tion \(\left(x-\frac{1}{3}\right)^{2}\) = 9.

Answer:

\(\frac { 10 }{ 3 }\) (or) \(\frac { -8 }{ 3 }\).

Explanation:

Question 37.

On solving x^{2} + 5 = – 6x, find the value of ‘x’

Answer:

– 1 or – 5.

Question 38.

Simplified form of \(\frac{\mathbf{x}}{\mathbf{x}-\mathbf{y}}-\frac{\mathbf{y}}{\mathbf{x}+\mathbf{y}}\)

Answer:

\(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}\)

Question 39.

Find the sum of roots of bx^{2} + ax + c = 0.

Answer:

\(\frac{-a}{b}\)

Question 40.

Find the roots of 2x^{2} – x + \(\frac { 1 }{ 8 }\) = 0.

Answer:

\(\frac{1}{4}, \frac{1}{4}\)

Question 41.

If x + \(\frac{1}{x}\) = 2, then find \(x^{2}+\frac{1}{x^{2}}\).

Answer:

2

Explanation:

x + \(\frac { 1 }{ x }\) = 2

⇒ \(x^{2}+\frac{1}{x^{2}}\) + 2 = 4 ⇒ x^{2} + \(x^{2}+\frac{1}{x^{2}}\) = 2

Question 42.

If 3y^{2} — 192, then find ‘y’.

Answer:

y = ± 8

Question 43.

How many diagonals has a pentagon?

Answer:

’9′

Question 44.

If α and β are the roots of the quadratic equation x^{2} – 3x + 1 = 0, then find \(\left(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}\right)\)

Answer:

7

Question 45.

If \(\mathbf{a}^{\mathbf{x}^{2}-4 \mathbf{x}+3}\) = 1, then find x (a # 0).

Answer:

1 or 3.

Question 46.

Find discriminant of the quadratic equation x + \(\frac { 1 }{ x }\) = 3.

Answer:

5

Question 47.

Create the quadratic equation with 2 < x < 3.

Answer:

x^{2} – 5x + 6 < 0.

Explanation:

x^{2} – (2 + 3)x + 2 . 3 < 0

⇒ x^{2} – 5x + 6 < 0

Question 48.

p(x) = x^{2} + 2x + 1, then find p(x^{2}).

Answer:

x^{4} + 2x^{2} + 1

Question 49.

x^{2} – 7x – 60 = 0, then find ‘x’.

Answer:

12 and -5.

Question 50.

\(\frac{1}{a+3}+\frac{1}{a-3}+\frac{6}{9-a^{2}}\) is equal to ?

Answer:

\(\frac{2}{a+3}\)

Question 51.

Find the roots of \(\sqrt{2} x^{2}+7 x+5 \sqrt{2}\) = 0.

Answer:

\(\frac{-5}{\sqrt{2}} \text { or }-\sqrt{2}\)

Question 52.

Find the roots of a quadratic equation \((\sqrt{2} x+3)(5 x-\sqrt{3})=0\)

Answer:

\(\frac{-3}{\sqrt{2}}, \frac{\sqrt{3}}{5}\)

Question 53.

4x^{2} + ky – 2 = 0 has no real roots, then find ‘k’.

Answer:

k < – \(\sqrt{32}\)

Question 54.

The sum of a number and its reciprocal is \(\frac { 50 }{ 7 }\), then find the number.

Answer:

7 (or) \(\frac { 1 }{ 7 }\)

Explanation:

x + \(\frac{1}{x}=\frac{50}{7}\)

⇒ \(\frac{x^{2}+1}{x}=\frac{50}{7}\)

⇒ 7x^{2} + 7 — 50x

⇒ 7x^{2} – 50x + 7 — 0

⇒ 7x^{2} – 49x – x + 7 = 0

⇒ 7x (x – 7) – 1 (x – 7) – 0

⇒ (x – 7) (7x – 1) – 0

=+ x = 7 (or) 1/7

Question 55.

Find the roots of the quadratic equation \(\frac{9}{x^{2}-27}=\frac{25}{x^{2}-11}\)

Answer:

±6;

Question 56.

Write the nature of the roots of a qua-dratic equation 4x^{2} – 12x + 9 = 0.

Answer:

Real and equal.

Question 57.

3x^{2} + (- k)x + 8 = 0 has no real roots, then find k’.

Answer:

k < 4\(\sqrt{6}\)

Explanation:

No real roots. So D < 0,

(-k)^{2} – 4 . 3 . 8 < 0

⇒ k^{2} – 96 < 0

⇒ k^{2} < 96

⇒ k < \(\sqrt{96}\)

⇒ k < \(4 \sqrt{6}\)

Question 58.

Find the discriminant of 3x^{2} – 2x = \(\frac{-1}{3}\).

Answer:

D = 0

Question 59.

Find the product of the roots of 1 =x^{2}.

Answer:

-1

Question 60.

x(x + 4) = 12, then find ‘x’.

Answer:

– 6 or 2.

Question 61.

Form a quadratic equation from, x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}.

Answer:

2x^{2} – 13x + 9 = 0.

Explanation:

x^{3} – 4x^{2} – x + 1 = x^{3} – 3 . x^{2} . 2 + 3 . x . 2^{2} – 2^{3}

⇒ x^{3} – 4x^{2} – x + 1 = x^{3} – 6x^{2} + 12x – 8

⇒ x^{3} – 4x^{2} – x + 1 = x^{3} + 6x^{2} – 12x + 8 = 0

⇒ 2x^{2} – 13x + 9 = 0

Question 62.

Find the product of the roots of x^{2} + 7x = 0.

Answer:

0

Question 63.

\(\frac{2 a^{2}+a-1}{a+1}+\frac{3 a^{2}+5 a+2}{3 a+2}+\frac{4-a^{2}}{a+2}\) is equal to ?

Answer:

2 (a +1)

Question 64.

1 and \(\frac { 3 }{ 2 }\) are the roots of which qua-dratic equation ?

Answer:

2x^{2} – 5x + 3 = 0.

Question 65.

If b^{2} < 4ac, then draw the shape of graph.

Answer:

Question 66.

\(\sqrt{\mathbf{k}+\mathbf{1}}\) = 3, then find ‘k’.

Answer:

k = 8

Question 67.

\(\sqrt{x}=\sqrt{2 x-1}\), then find ‘x’.

Answer:

x = 1

Question 68.

If \(\frac{1}{x-2}+\frac{2}{x-1}=\frac{6}{x}\) then find ‘x’

Answer:

3 or \(\frac { 4 }{ 3 }\)

Question 69.

Find the coefficient of ‘x’ in a pine qua-dratic equation.

Answer:

0

Question 70.

Write number of distinct line segments that can be formed out of n – points.

Answer:

\(\frac{n(n-1)}{2}\)

Question 71.

The product of two consecutive positive integers is 306, then find the largest number.

Answer:

18

Explanation:

x(x + 1) = 306 ⇒ x^{2} + x – 306 = 0

by solving thik Q.E., x = 17

∴ Largest number = x + 1

= 17 + 1 = 18.

Question 72.

Write the nature of roots of 3x^{2} + 13x – 2 = 0.

Answer:

Real and unequal.

Question 73.

If α and β are the roots of x^{2} – 2x + 3 = 0, then find α^{2} + β^{2}

Answer:

α^{2} + β^{2} = – 2.

Question 74.

If (2x – 1) (2x + 3) = 0, then find ‘x’.

Answer:

\(\frac { 1 }{ 2 }\) or \(\frac { -3 }{ 2 }\)

Question 75.

Write the quadratic equation whose one root is 2 – \(\sqrt{3}\) .

Answer:

x^{2} – 4x + 1 = 0

Question 76.

If b^{2} – 4ac > 0, then write nature of the roots of the quadratic equation.

Answer:

Real and distinct.

Question 77.

Find product of the roots of ax^{2} + bx + c = 0. c

Answer:

c/a

Question 78.

Write the nature of the roots of a qua-dratic equation 4x^{2} + 5x + 1 = 0.

Answer:

Real and distinct.

Question 79.

Write the quadratic equation whose roots are – 1,6.

Answer:

x^{2} – 5x – 6 = 0.

Question 80.

Create the quadratic equation whose roots are – 3 and – 4.

Answer:

x^{2} + 7x 4- 12 = 0.

Question 81.

Find the roots of the quadratic equation (3x + 4)^{2} – 49 = 0.

Answer:

1, \(\frac{-11}{3}\)

Question 82.

If x^{2} – 2x + 1 = 0, then find x + \(\frac{1}{x}\).

Answer:

2

Question 83.

Write nature of the roots of 5x^{2} – x + 1 = 0.

Answer:

Imaginary roots.

Question 84.

Write the nature of the roots of qua-dratic equation 3x^{2} + x + 8 = 0.

Answer:

Imaginary roots.

Question 85.

Find product of the roots of the qua-dratic equation 3x^{2} – 6x + 11 = 0.

Answer:

\(\frac{11}{3}\)

Question 86.

Form a quadratic equation whose roots are k and 1/k.

Answer:

x^{2} – (\(\mathrm{k}+\frac{1}{\mathrm{k}}\))x + 1 = 0

Question 87.

If k^{2} – 8kx + 16 = 0 has equal roots, then find the value of ‘k’.

Answer:

k = ± 1.

Explanation:

(-8k)^{2} – 4(1) (16) = 0

⇒ 64k^{2} = 64 ⇒ k^{2} = 1 ⇒ k = ±1

Question 88.

If the roots of a quadratic equation ax^{2} + bx + c = 0 are real and equal, then find ‘b^{2}‘.

Answer:

4ac

Question 89.

3(x – 4)^{2} – 5(x – 4) = 12, then find ‘x’.

Answer:

7 (or) 8/3.

Explanation:

3(x – 4)^{2} – 5 (x – 4) = 12

3[x^{2} + 16 – 8x] – 5x + 20 — 12

3x^{2} + 48 – 24x – 5x + 20 – 12 — 0

⇒ 3x^{2} – 29x + 56 = 0

⇒ 3x^{2} – 21x – 8x + 56 — 0

⇒ 3x (x – 7) – 8 (x – 7) = 0

⇒ (x – 7) (3x – 8) = 0

⇒ x = 7 (or) x = \(\frac { 8 }{ 3 }\)

Question 90.

If a and pare the roots of x^{2} + x + 1 = 0, then find α^{2} + β^{2}.

Answer:

α^{2} + β^{2} = – 1.

Question 91.

\(\frac{1-\frac{1}{1+x}}{\frac{1}{1+x}}\) is equal to ?

Answer:

x

Question 92.

Find sum of the roots of a pure quadratic equation.

Answer:

0

Question 93.

\(\frac{\mathbf{x}}{\mathbf{a}-\mathbf{b}}=\frac{\mathbf{a}}{\mathbf{x}-\mathbf{b}}\) , then find ‘x’.

Answer:

b – a (or) – a

Question 94.

\(\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}\) x ≠ -4 x or 7 find x’.

Answer:

2 or 1

Question 95.

(1 – 5x) (9x +1) is equal to ?

Answer:

1 + 4x – 5x^{2}.

Question 96.

From the figure, find ’x’.

Answer:

± 10

Explanation:

By Pythagoras theorem,

x^{2} = 6^{2} + 8^{2} = 64 + 36 = 100

x = \(\sqrt{100}\) = ± 10

Question 97.

Find the sum of the roots of the equation 3x^{2} – 7x + 11 = 0.

Answer:

7/3

Question 98.

Find the roots of the quadratic equation \((\sqrt{5} x-3)(\sqrt{5} x-3)\) – 0.

Answer:

\(\frac{3}{\sqrt{5}}, \frac{3}{\sqrt{5}}\)

Question 99.

Write the nature of the roots of the quadratic equation \(\sqrt{3} x^{2}-2 x-\sqrt{3}\).

Answer:

Real and distinct.

Question 100.

If 5x^{2} – kx + 11 = 0 has root x = 3, then find ’k’.

Answer:

k = \(\frac{56}{3}\)

Explanation:

5(3)^{2} – k(3) + 11 = 0

⇒ 45 + 11 – 3k = 0

⇒ 56 – 3k = 0

⇒ 3k = 56 ⇒ k = \(\frac { 56 }{ 3 }\)

Question 101.

Find the value of ‘p’ for which 4x^{2} – 2px + 7 = 0 has a real roots.

Answer:

p > 2\(\sqrt{7}\)

Question 102.

If one root of a quadratic equation is 7 – 7\(\sqrt{3}\) , then find the quadratic equation.

Answer:

x^{2} – 14x + 46 = 0.

Question 103.

If b^{2} – 4ac = 0, then write nature of the roots of the quadratic equation.

Answer:

Real and equal.

Question 104.

Find sum of the roots of ax^{2} + bx + c = 0.

Answer:

\(-\frac{b}{a}\)

Question 105.

If the equation x^{2} – kx + 1 = 0 has equal roots, then find the value of ‘k’.

Answer:

k = ± 2

Explanation:

b^{2} – 4ac = (- k)^{2} – 4 . 1 . 1 = 0

⇒ k^{2} – 4 = 0

⇒ k^{2} = 4 ⇒ k = \(\sqrt{4}\) = ± 2.

Question 106.

Find (he product of the roots of the qua-dratic equation \(\sqrt{2} \mathrm{x}^{2}-3 \mathrm{x}+5 \sqrt{2}\) = 0.

Answer:

5

Question 107.

Write the nature of roots of 3x^{2} + 6x – 2 = 0.

Answer:

Real and distinct.

Question 108.

If the sum of the roots of the quadratic equation 3x^{2} + (2k + 1)x – (k + 5) = 0 is equal to the product of the roots, then find the value of k.

Answer:

4

Explanation:

Sum of the roots = product of the roots

⇒ \(\frac{-(2 k+1)}{3}=\frac{-(k+5)}{3}\)

⇒ – 2k- 1 = -k – 5 ⇒ k = 4

Question 109.

Find the product of zeroes in the above equation.

Answer:

\(\frac{-11}{5}\)

Question 110.

Find the degree of any quadratic equation.

Answer:

2

Question 111.

In the quadratic equation

x^{2} + x – 2 = 0, find the value of a + b + c.

Answer:

a + b + c = 0.

Question 112.

Find the value of \(\left(x+\frac{1}{x}\right)^{2}-\left(y+\frac{1}{y}\right)^{2}-\left(x y-\frac{1}{x y}\right) \cdot\left(\frac{x}{y}-\frac{y}{x}\right)\)

Answer:

0

Question 113.

Form a quadratic equation from x(2x + 3) = x^{2} + 1.

Answer:

x^{2} + 3x – 1 = 0.

Explanation:

2x^{2} + 3x = x^{2} + 1

⇒ x^{2} + 3x – 1 = 0

Question 114.

(x – α) (x – β) = 0, then find the product.

Answer:

x^{2} – (α + β)x + αβ = 0.

Question 115.

If α and β are die roots of x^{2} – 5x + 6 = 0, then find the value of α – β.

Answer:

± 1.

Question 116.

For what values of m’ are the roots of the equation mx^{2} + (m + 3)x + 4 = 0 are equal ?

Answer:

9 or 1.

Explanation:

(m + 3)^{2} – 4 . m . 4 = 0

⇒ (m + 3)^{2} – 16m = 0

⇒ m^{2} + 9 + 6m- 16m = 0

⇒ m^{2} – 10m + 9 = 0

⇒ m^{2} – 9m – m + 9 = 0

⇒ m(m – 9) – 1 (m – 9) = 0

∴ m = 9 or 1

Question 117.

Find the roots of 2x^{2} + x – 4 = 0.

Answer:

x = \(\frac{-1 \pm \sqrt{33}}{4}\)

Question 118.

If kx (x – 2) + 6 = 0 has equal roots, then find k’.

Answer:

k = 6.

Explanation:

kx^{2} – 2kx + 6 = 0

⇒ (2k)^{2} – 4 . k . 6 = 0

⇒ 4k^{2} – 24k = 0

⇒ 4k (k – 6) = 0 ⇒ k = 6

Question 119.

If ‘2’ is a root of x^{2} + 5x + r = 0, then find ‘r’.

Answer:

r = -14

Question 120.

(α + β)^{2} – 2αβ is sequal to ……………

Answer:

α^{2} + β^{2}

Question 121.

Find the value of \(\sqrt{\mathbf{a}+\sqrt{\mathbf{a}+\sqrt{\mathbf{a + \ldots \ldots \infty}}}}\)

Answer:

\(\frac{1+\sqrt{1+4 a}}{2}\)

Question 122.

If the sum of the roots of kx^{2} – 3x + 1 = 0 is \(\frac{-4}{3}\) then find ‘k’.

Answer:

\(\frac{-9}{4}\)

Explanation:

\(\frac{3}{\mathrm{k}}=\frac{-4}{3} \Rightarrow \frac{3 \times 3}{-4}=\mathrm{k} \Rightarrow \mathrm{k}=\frac{-9}{4}\)

Question 123.

\(\frac{n(n+1)}{2}\) = 55, then find ‘n’

Answer:

10

Explanation:

⇒ n^{2} + n = 110 = 0

⇒ n^{2} + n – 110 = 0

⇒ n = 10

Question 124.

If ‘α’ is β root of ax^{2} + bx + c = 0, then find aα^{2} + bα + c.

Answer:

0

Question 125.

If α and β are the roots of the quadratic equation 2x^{2} + 3x – 7 = 0, then find \(\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}\)

Answer:

\(\frac{-37}{14}\)

Question 126.

Find the sum of the roots of -7x + 3x^{2} – 1 = 0.

Answer:

\(\frac{7}{3}\)

Question 127.

Find the roots of a quadratic equation \(\frac{\mathbf{x}}{\mathbf{p}}=\frac{\mathbf{p}}{\mathbf{x}}\)

Answer:

x = p

Explanation:

x^{2} = p^{2} ⇒ x = p

Question 128.

If (x – 3) (x + 3) = 16, then find the value of ‘x’.

Answer:

± 5.

Question 129.

Write the roots of a quadratic equation ax^{2} + bx + c = 0.

Answer:

x = \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

Question 130.

Find the sum of the roots of the quadratic equation 5x^{2} + 4\(\sqrt{3}\)x – 11 = 0.

Answer:

\(\frac{-4 \sqrt{3}}{5}\)

Question 131.

If one root of x^{2} – (p – 1)x + 10 = 0 is 5, then find ‘p’.

Answer:

7

Explanation:

5^{2} – (p – 1) 5 + 10 = 0

⇒ 25 + 10 – 5p + 5 = 0

⇒ 35 = 5p ⇒ p = 7

Question 132.

If one root of x2 – x + k = 0 is square of other, then find ‘k’.

Answer:

k = cube of one root

Explanation:

α = x, β = x^{2}

Product of roots = αβ = \(\frac{\mathrm{c}}{\mathrm{a}}\)

⇒ x.x^{2} = k ⇒ k = x^{3}

k is cube of the first root.

Question 133.

If α, β are the roots of x2 – px + q = 0, then find α^{3} + β^{3}.

Answer:

p^{3} – 3pq

Question 134.

x^{2} + (x + 2)^{2} = 290, then find ‘x’.,

Answer:

11 or – 13

Question 135.

Find the value of \(\sqrt{\mathbf{a} \sqrt{\mathbf{a} \sqrt{\mathbf{a}} \ldots \ldots \infty}}\)

Answer:

a

Question 136.

If \(\frac{-7}{3}\) is a root of 6x^{2} – 13x – 63 = 0, then find other root.

Answer:

\(\frac{9}{2}\)

Question 137.

If b^{2}2 – 4ac < 0, then write nature of the roots of the quadratic equation.

Answer:

Imaginary roots.

Choose the correct answer satistying the following statements.

Question 138.

Statement (A) : The equation x^{2} + 3x + 1 = (x – 2)^{2} is a quadratic equation.

Statement (B) : Any equation of the form ax2 + bx + c = 0 where a ± 0, is called a quadratic equation.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(iii)

Explanation:

We have, x^{2} + 3x + 1 = (x – 2)^{2}

⇒ x^{2} + 3x + 1 = x^{2} – 4x + 4

⇒ 7x – 3 = 0, it is not of the form ax^{2} + bx + c = 0

So, A is incorrect but B is correct.

Hence (iii) is the correct option.

Question 139.

Statement (A) : The roots of the qua-dratic equation x^{2} + 2x + 2 = 0 are imaginary.

Statement (B) : If discriminant D = b^{2} – 4ac < 0, then the roots of quadratic equation ax^{2} + bx + c = 0 are imaginary.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(i)

Explanation:

x^{2} + 2x + 2 = 0

∴ Discriminant, D = b^{2} – 4ac

= (2)^{2} – 4 x 1 x 2

= 4 – 8 = -4 < 0

∴ Roots are imaginary.

So, both A and B are correct and B explains Answer: Hence, (i) is the correct option.

Question 140.

Statement (A) : The value of k = 2, if one root of the quadratic equation

6x^{2} – x – k = 0 is \(\frac{2}{3}\)

Statement (B) : The quadratic equation ax^{2} + bx + c = 0, a ≠ 0 has two roots.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(i)

Explanation:

As one root is \(\frac{2}{3}\) ⇒ x = \(\frac{2}{3}\)

So, both A and B are correct but B does not explain Answer:

∴ Hence, (i) is the correct option.

Question 141.

Statement (A) : The equation 8x^{2} + 3kx + 2 = 0 has equal roots, then the value of k is ± \(\frac{8}{3}\).

Statement (B) : The equation ax^{2} + bx + c = 0 has equal roots if D = b^{2} – 4ac = 0.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(i)

Explanation:

8x^{2} + 3kx + 2 = 0

∴ Discriminant, D = b^{2} – 4ac

= (3k)^{2} – 4 x 8 x 2

= 9k^{2} – 64

For equal roots, D = 0

⇒ 9k^{2} – 64 = 0

⇒ 9k^{2} = 64

⇒ k^{2} = \(\frac { 64 }{ 9 }\)

⇒ 9k^{2} = ±\(\frac { 8 }{ 3 }\)

So, A and B both correct and B explains Answer: Hence, (i) is the correct option.

Question 142.

Statement (A) : The values of x are \(\frac{-a}{2}\), a for a quadratic equation 2x^{2} + ax – a^{2} = 0.

Statement (B) : For quadratic equation ax^{2} + bx + c = 0.

x = \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(iii)

Explanation:

2x^{2} + ax – a^{2} =0

So, A is incorrect but B is correct. Hence, (iii) is the correct option.

Question 143.

Statement (A) : The equation (x – p) (x – r) + λ(x – q) (x – s) = 0, p < q < r < s, has non-real roots if λ > 0.

Statement (B) : The equation ax^{2} + bx + c = 0, a, b,c ∈ R, has non-real roots if b^{2} – 4ac < 0.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(iii)

Explanation:

Statement (A):

Let f(x) = (x – p) (x – r) + λ(x – q) (x- s)

f(p) = λ(p – q) (p – s)

f(q) = (q – p) (q – r)

f(s) = (s – p) (s – r)

f(r) = λ(r – q) (r – s)

If λ > 0, then f(p) > 0, f(q) < 0, f(r) < 0 and f(s) > 0.

⇒ f(x) = 0 has one real root between p and q and other real root between r and s.

Statement – B is obviously true. Option (iii) is correct.

Question 144.

Statement (A) : If roots of the equation x^{2} – bx + c = 0 are two consecutive integers, then b^{2} – 4c = 1.

Statement (B) : If a, b, c are odd integer, then the roots of the equation 4abc x^{2} + (b^{2} – 4ac)x – b = 0 are real and distinct.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(ii)

Explanation:

Statement (A) : Given equation , x^{2} – bx + c = 0.

Let α, β be two roots such that |α – β| = 1. .

⇒ (α + β)^{2} – 4αβ = 1.

⇒ b^{2} – 4c = 1

Statement (B): Given equation

4abc x^{2} + (b^{2} – 4ac) x – b = 0

D = (b^{2} – 4ac)^{2} + 16 ab^{2} c

D = (b^{2} – 4ac)^{2} > 0

Hence roots are real and unequal. Option (ii) is correct.

Question 145.

Statement (A) : If 1 ≤ a ≤ 2, then \(\sqrt{a+2 \sqrt{a-1}}+\sqrt{a-2 \sqrt{a-1}}=2\)

Statement (B) : If 1 ≤ a ≤ 2, then (a – 1) > 1.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(ii)

Explanation:

If 1 ≤ a ≤ 2 ⇒ 0 ≤ a- 1 ≤ 1

Statement – A is true.

Statement – B is false.

Option – (ii) is correct.

Question 146.

Statement (A): If one root is \(\sqrt{3}-\sqrt{2}\), then the equation of lowest degree with rational coefficients x^{4} – 10x^{2} + 1 = 0.

Statement (B): For a polynomial equa-tion with rational coefficient irrational roots occurs in pairs.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(i)

Explanation:

x = \(\sqrt{3}-\sqrt{2}\), x^{2} = 5 – 2\(\sqrt{6}\)

(x^{2} – 5)^{2} = 24

x^{4} – 10x^{2} + 25 = 24

x^{4} – 10x^{2} + 1 = 0

For polynomial equation with rational coefficients irrational roots occurs in pairs.

Option (i) is correct.

Question 147.

Statement (A): Degree of the polynomial 5x^{2} + 3x + 4 is 2.

Statement (B) : The degree of a poly-nomial of one variable is the highest value of the exponent of the variable.

i) Both A and B are true.

ii) A is true, B is false.

iii) A is false, B is true.

iv) Both A and B are false.

Answer:

(i)

Read the below passages and answer to the following questions.

Let us consider a quadratic equation x^{2} + 3ax + 2a^{2} = 0.

If the above equation has roots α,β and it is given that α^{2} + β^{2} = 5.

Question 148.

Find value of ‘a’.

Answer:

±1.

Explanation:

α + β = – 3a; αβ = 2a^{2}

a^{2} + p^{2} = 5

⇒ (α + β)^{2} – 2αβ = 5

⇒ (- 3a)^{2} – 2(2a^{2}) = 5

⇒ 9a^{2} – 4a^{2} = 5

⇒ 5a^{2} = 5 ⇒ a = ± 1

Question 149.

Find value of ‘D’ for the above qua-dratic equation.

Answer:

D > 0.

Explanation:

D = (3a)^{2} – 4(2a^{2})

= 9a^{2} – 8a^{2} = a^{2} = 1 > 0

Question 150.

Find the product of roots.

Answer:

2

Explanation:

αβ = 2a^{2} = 2(1) = 2

Let us consider a quadratic equation x^{2} + λx + λ + 1.25 = O, where λ is a constant. The value of A such that the above quadratic equation has

Question 151.

Two distinct roots.

Answer:

λ > 5 or λ < – 1.

Explanation:

The equation has two distinct roots if b^{2} – 4ac > 0.

∴ (λ – 5)(λ + 1) > 0

⇒ Either λ – 5 > 0 (or) λ + 1 > 0

⇒ λ > 5 (or) λ > -1

∴ λ > 5

⇒ λ – 5 <0 (or) λ + 1 < 0

⇒ λ < 5 (or) λ < – 1

∴ λ < -1 Hence the given equation has two dis-tinct roots for λ > 5 (or) λ < – 1

Question 152.

Two coincident roots.

Answer:

λ = 5 or λ = -1.

Explanation:

The equation has two coincident roots if b^{2} – 4ac = 0

⇒ (λ – 5) (λ + 1) = 0

⇒ Either λ – 5 = 0 (or) λ = 5

⇒ λ + 1 = 0

⇒ λ = – 1

⇒ λ = 5 or – 1

Hence the given equation has coincident roots for λ = 5 or – 1.

The area of a rectangular plot is 528 m^{2}. The length of the plot is one more than twice its breadth.

Question 153.

Which mathematical concept is used to find area of above plot ?

Answer:

Quadratic equation.

Question 154.

Write the breadth and length of above given plot.

Answer:

Let breadth = x m, length = 2x + 1 m.

Question 155.

Write the equation of area of above given plot.

Answer:

Area = length x breadth

= x(2x + 1) – 2x^{2} + x = 528 m^{2}.

The hypotenuse of a right triangle is 25 cm. We know that the difference in lengthof the other two sides is 5 cm.

Question 156.

Write the lengths of smaller and larger sides.

Answer:

Smaller side = x m

Larger side = (x + 5) cm.

Question 157.

Write the hypotenuse of the triangle.

Answer:

x^{2} + (x + 5)^{2} = (25)^{2}

i.e., x^{2} + 5x – 300 = 0

Question 158.

Which mathematical concept is used to find out the values of dimensions ?

Answer:

Quadratic equations.

Question 159.

Column -II give roots of quadratic equations given in column – I, match them correctly.

Answer:

A – (iv), B – (ii).

Question 160.

Column – II give roots of quadratic equations given in column -1, match them correctly.

Answer:

A – (i), B – (iii).

Question 161.

Write the correct matching.

Answer:

A – (ii), B – (iv).

Question 162.

Write die correct matching.

Answer:

A – (iii), B – (i).

Question 163.

Column – II give pair at two numbers for solution to problems given in column -I. Match them correctly.

Answer:

A – (iv), B – (ii).

Question 164.

Column – II give pair at two numbers for solution to problems given in column -I.

Match them correctly.

Answer:

A – (i), B – (iii).

Question 165.

D is the discriminait of the quadratic equation ax^{2} + bx + e = 0.

Answer:

A – (ii), B – (i).

Question 166.

D Is the discriminant of the quadratic equation ax^{2} + bx + c = O.

Answer:

A – (ii), B – (i).

Question 167.

Write a quadratic equation with roots 3 and 4.

Answer:

x^{2} – 7x + 12 = 0

Question 168.

Draw the rough graph of the quadratic equation ax^{2} + bx + c = 0, when b^{2} – 4ac < 0.

Answer: