Algebra Practice Questions for CAT with Solutions

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Question 1:
Given that three roots of f(x) = x4+ax2+bx+c are 2, -3, and 5, what is the value of a+b+c?
[1] -79
[2] 79
[3] -80
[4] 80
We have to find out a + b + c

f(1) is 1 + a + b + c

So, we need to find out f(1) – 1

Let the 4th root be r

Coefficient of x3 is - (Sum of the roots)

0 = - (r + 2 -3 + 5)
r = - 4

So, f(x) = (x – 2) (x + 3) (x + 4)(x – 5)

f(1) = (-1)×4×5×(-4) = 80
a + b + c = f(1) – 1 = 79. Option B

Question 2:
If both a and b belong to the set (1, 2, 3, 4), then the number of equations of the form ax2+bx+1=0 having real roots is
[1] 10
[2] 7
[3] 6
[4] 12
For the equation to have real roots

b2 – 4a ≥ 0

b = 1, No equation exists

b = 2, a = 1. 1 equation exists

b = 3, a = 1 or 2. 2 equations exist

b = 4, a = 1 or 2 or 3 or 4. 4 equations exist

Total equations = 0 + 1 + 2 + 4 = 7. Option B


Question 3:
Rakesh and Manish solve an equation. In solving Rakesh commits a mistake in constant term and finds the root 8 and 2. Manish commits a mistake in the coefficient of x and finds the roots -9 and -1. Find the correct roots.
[1] 9,1
[2] -9,1
[3] -8,-2
[4] None of these
Rakesh’s equation

(x – 8)(x – 2) = 0

x2 – 10x + 16 = 0

Manish’s equation

(x + 9)(x + 1) = 0

x2 + 10x + 9 = 0

Correct equation is x2 – 10x + 9 = 0

(x – 1)(x – 9) = 0
Roots are 9, 1. Option A

Question 4:
The number of quadratic equations which are unchanged by squaring their roots is
[1] 2
[2] 4
[3] 6
[4] None of these.
This would happen if and only if the roots and their squares are the same value.

The roots can be 0 or 1 or a combination of these.

So, valid equations will be formed when

Both roots are 0
Both roots are 1
One root is 0 and the other root is 1

Number of equations = 3. Option D


Question 5:
If the roots of px2+qx+2=0 are reciprocals of each other, then
[1] p = 0
[2] p = -2
[3] p= +2
[4] p = √2
If the roots are reciprocals of each other, product of the roots is 1
2/p = 1
p = 2. Option C

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Algebra Practice Questions for CAT with Solutions
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