Quadratic Equations Test

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Quadratic Equations Test - 1

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following equations is not quadratic?

A
(x + 1)² = 2(x - 3)

B
x² - 2x = -2(3 - x)

C
(x - 2)(x + 1) = (x - 1)(x + 3)

D
None of these

Solution

Option (1) :
(x + 1)² = 2(x - 3)
x² + 2x + 1 = 2x - 6
x² + 7 = 0, a quadratic equation.

Option (2):
x² - 2x = -2(3 - x)
x² - 2x = -6 + 2x
x² - 4x + 6 = 0, a quadratic equation.

Option 3:
(x - 2)(x + 1) = (x - 1)(x + 3)
x² - 2x + x - 2 = x² + 3x - x - 3
-x - 2x + 1 = 0
-3x + 1 = 0, not a quadratic equation, because it is not in the form of ax² + bx + c = 0
Question 2
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Which of the following quadratic equations is in standard form?

A
5x² - 2x + 1 = 5(x² + 1)

B
5x² + 2x + 1 = 0

C
-2x² + 1 - 5x² = 0

D
2x - 5x² - x = 0

Solution

A quadratic equation in the standard form is given in option (2), i.e. 5x² + 2x + 1 = 0, because the standard form of a quadratic equation is ax² + bx + c = 0, where a, b, c ∈R, a ≠0 and a is positive.
Question 3
1 / -0

Which of the following quadratic equations is in standard form?

A
4x + 5x² - 2 = 0

B
x² + 8x - 2 = 0

C
8x + 4x² - 2 = 3

D
4x² - 2 - 2x = 0

Solution

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, c ∈R, a ≠0.
So, the standard form of the given equation is x² + 8x - 2 = 0.
Question 4
1 / -0

Solve the following equation for x:

x² - 7x + 12 = 0

A
3 and 4

B
3 and -4

C
-3 and 4

D
-3 and -4

Solution

x² - 7x + 12 = 0
=> x²- 4x - 3x + 12
=> (x - 3)(x - 4) = 0
So, x = 3 and 4
Question 5
1 / -0

Solve for x:

3x² + 5x - 8 = 0

A
1,

B
1,

C
-1,

D
3, 4

Solution

x =
= -
= 1 or
Question 6
1 / -0

Find x, if 2x² - 8x - 24 = 0.

A
x = 6, -2

B
x = 4, -3

C
x = 1, -1

D
None of these

Solution

2x² - 8x - 24 = 0
x²- 4x - 12 = 0
Or x² - 6x + 2x - 12 = 0
Or x(x - 6) + 2(x - 6) = 0
Or (x - 6)(x + 2) = 0

Thus, x = 6 or x = -2
Question 7
1 / -0

If the equation x² – 5x + 6 = 0 has real roots, then what is the sum and product of the roots of this equation?

A
5, 5

B
6, 5

C
8, 6

D
5, 6

Solution

x² – 5x + 6 = 0
x² – 3x – 2x + 6 = 0
x(x – 3) – 2(x – 3) = 0
(x – 3) (x – 2) = 0
x = 3, 2
Product of roots = 3 x 2
= 6
Sum of roots = 3 + 2
= 5
Question 8
1 / -0

Find the real roots of the quadratic equation x²- 9x + 18 = 0 and for the two roots x and x₁, find a and b such that Log₁₀a = x and Log₁₀ b = x_1.

A
18, 6

B
10³, 10⁹

C
10⁶, 10³

D
None of these

Solution

x² - 9x + 18 = 0
x² - 6x - 3x + 18 = 0
x (x - 6) - 3 (x - 6) = 0
(x - 6) (x - 3) = 0
x = 6, 3
x = 6, x₁ = 3
Log₁₀ a = x Log₁₀ a = 6 a = (10)⁶
Log₁₀ b = x₁ Log₁₀ b = 3 b = (10)³
Question 9
1 / -0

Which of the following equations has no real roots?

A
x² + x - 12 = 0

B
x² – x – 1 = 0

C
2x² + 7x + 5 = 0

D
x² + x + 1 = 0

Solution

In x² + x + 1 = 0, D < 0 (As D < 0, it has no real roots)
Question 10
1 / -0

Find the common root between quadratic equations x² – 5x + 6 = 0 and x² – 7x + 10 = 0.

A
3

B
2

C
5

D
4

Solution

x² – 5x + 6 = 0 x² – 7x + 10 = 0
X² – 3x – 2x + 6 = 0 x² – 5x – 2x + 10 = 0
X (x – 3) – 2 (x – 3) = 0 x (x – 5) – 2 (x – 5) = 0
(x – 3) (x – 2) = 0 (x – 5) (x – 2) = 0
X = 3, (2) x = 5, (2)
Question 11
1 / -0

Find the product of the roots of the following equation:

– 9x² – 8x = 15

A

B

C

D

Solution

– 9 x² – 8 x = 15

First subtract 15 from both sides.

– 9 x² – 8 x – 15 = 0

Or 9 x²+ 8 x + 15 = 0

Comparing the equation with the standard quadratic equation

a x²+ b x + c = 0
a = 9
b = 8
c = 15

Product of roots= c/a = 15/9 = 5/3

Hence, option (4) is correct.
Question 12
1 / -0

What would be the value of p for which the equation 4x² - 5x + p = 0 gives two real equal roots?

A
25

B

C

D
None of these

Solution

Discriminant = b² - 4ac
= (- 5)² - 4 4 p
= 25 - 16p

For equal roots, D = 0
25 - 16p = 0
p =
Question 13
1 / -0

If x = 2 and x = 3 are roots of the equation 3x² – 2mx + 2n = 0, find the values of m and n.

A

B

C

D
None of these

Solution

Sum of roots = 2 + 3 = 5 = 2m/3
This gives us m as 15/2.
Also, product of roots is 2 x 3 = 6 = 2n/3
This gives us n as 9.
Thus, option (3) is correct.
Question 14
1 / -0

Which of the following equations has one of its roots as?

A
x² + 2x + 2 = 0

B
x² + 2x – 2 = 0

C
x² + x + 2 = 0

D
x² – x – 2 = 0

Solution
Question 15
1 / -0

Which of the following quadratic polynomials has sum of its zeros as 1/4 and product of its zeros as -1?

A
4x² + x + 4

B
4x² – x – 4

C
4x³ – x² + 3

D
None of these

Solution

Let the quadratic polynomial be ax² + bx + c and its zeros be and .
Given:

Thus, the quadratic polynomial becomes a{x² - (1/4)x + (-1)}.
(a/4){4x² - x - 4}

Putting a = 4, the quadratic polynomial becomes 4x² – x – 4.