Practice Questions for Algebra - Quadratic Equations

1. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

Two numbers whose sum is 4 and the absolute value of whose difference is 10 are the roots of the equation

(a)
x² + 4x + 21 = 0
(b)
x² – 4x + 21 = 0
(c)
x² + 4x – 21 = 0
(d)
x² – 4x – 21 = 0

2. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

If α and 1/ α are the two roots of the equation ax² + bx + c = 0, then which of the following is always true?

(a)
a = c
(b)
c = b
(c)
b = a
(d)
b = ac

Answer: Option A

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Explanation :

α and 1/ α are the two roots of the equation (i.e., roots are reciprocal of each other)

⇒ Product of the roots = 1

∴ $\frac{c}{a}$ = 1

⇒ c = a

Hence, option (a).

3. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

One root of x² + ax – 27 = 0 is square of the other. Then the value of a is

(a)
3
(b)
6
(c)
-6
(d)
-3

4. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

If the roots, x₁ and x₂, of the quadratic equation x² – 3x + p = 0 also satisfy the equation 5x₂ – 4x₁ = 51, then which of the following is true?

(a)
p = -28
(b)
x₁ = 3, x₂ = 9
(c)
x₁ = 6, x₂ = 3
(d)
None of these

5. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

If α, ß are the roots of the equation 3x² – 4x – 6 = 0, find the equation whose roots are α² + 1 and ß² + 1.

(a)
9x² + 70x + 97 = 0
(b)
9x² - 70x + 97 = 0
(c)
9x² - 70x - 97 = 0
(d)
9x² + 70x - 97 = 0

Answer: Option B

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Explanation :

Since α, ß are roots of the equation 3x² – 4x – 6 = 0

∴ α + ß = $\frac{4}{3}$ and αß = - $\frac{6}{3}$ = - 2

Now QE whose roots are α² + 1 and ß² + 1 can be formed as

x² - (sum of the roots)x + (product of the roots)
∴ x² – (α² + 1 + ß² + 1) + (α² + 1)(ß² + 1)

Now,
⇒ α² + 1 + ß² + 1 = (α + ß)² – 2αß + 2
= $\frac{16}{9}$ + 4 + 2 = $\frac{16}{9}$ + 6 = $\frac{70}{9}$

and (α² + 1)(ß² + 1) = α²ß² + (α² + ß²) + 1 = (αß)² + (α + ß)² - 2αß + 1
= 4 + $\frac{16}{9}$ + 4 + 1 = $\frac{16}{9}$ + 9 = $\frac{97}{9}$

∴ The required equation is
x² - $\frac{70}{9}$ x + $\frac{97}{9}$ = 0

⇒ 9x² - 70x + 97 = 0

Hence, option (b).

6. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

If α and β are the roots of x² + ax + b = 0, then what is the value of $\frac{1}{α^{2}} + \frac{1}{β^{2}}$ ?

(a)
a² - 4b
(b)
(a² - 4b)/2
(c)
(a² - 4b)/b²
(d)
(a² - 2b)/b²

Answer: Option D

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Explanation :

$\frac{1}{α^{2}} + \frac{1}{β^{2}}$ = $\frac{α^{2} + β^{2}}{{(α β)}^{2}}$ = $\frac{{(α + β)}^{2} - 2 α β}{{(α β)}^{2}}$

In the given equation
sum or the roots i.e., α + β = -a
& product of the roots i.e., αβ = b

∴ $\frac{1}{α^{2}} + \frac{1}{β^{2}}$ = $\frac{{(- a)}^{2} - 2 \times b}{{(b)}^{2}}$ = $\frac{a^{2} - 2 b}{b^{2}}$

Hence, option (d).

7. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

Sum of the areas of two squares is 394 m². If the difference of their perimeters is 8 m, find the sides of the two squares?

(a)
9 m, 7 m
(b)
15 m, 13 m
(c)
18 m, 16 m
(d)
10 m, 12 m

8. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

The equation x + $\sqrt{x - 3}$ = 5 has

(a)
two real roots and one imaginary roots
(b)
one real and one imaginary root
(c)
two imaginary roots
(d)
one real root

Answer: Option D

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Explanation :

x + $\sqrt{x - 3}$ = 5

⇒ $\sqrt{x - 3}$ = 5 – x

Squaring on both the sides

⇒ x – 3 = 25 + x² – 10x

⇒ x² – 11x + 28 = 0

⇒ (x – 7)(x – 4) = 0

⇒ x = 7 or 4

Checking these two values in the original equation, only x = 4 satisfies.

Hence, option (d).

9. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

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

(a)
(0, 0)
(b)
(1, -2)
(c)
(1, 2)
(d)
Both (a) and (b)

10. PE 1 - Quadratic Equation | Algebra - Quadratic Equations

For what values of c in the equation 2x² – (a² + 4a + 4)x + a² – 9a = 0 the roots of the equation would be of opposite signs?

(a)
a ∈ (0, 9)
(b)
a ∈ (-9, 0)
(c)
a ∈ [0, 9]
(d)
a ∈ (-9, 9)

Answer: Option A

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Explanation :

For the roots to be opposite in sign, the product of roots should be negative.

∴ $\frac{a^{2} - 9 a}{2}$ < 0

⇒ a(a – 9) < 0

⇒ 0 < a < 9

Hence, option (a).

PE 1 - Quadratic Equation | Algebra - Quadratic Equations

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