Algebra Practice Questions for CAT with Solutions

Question 1:
A test has 20 questions, with 4 marks for a correct answer, –1 mark for a wrong answer, and no marks for an unattempted question. A group of friends took the test. If all of them scored exactly 15 marks, but each of them attempted a different number of questions, what is the maximum number of people who could be in the group?
[1] 3
[2] 4
[3] 5
[4] more than 5
c + w + n = 20

4c – w = 15

Adding the two equations, we get 5c + n = 35

c(max) = 7, when n = 0 & w = 13

c(min) = 4, when n = 15 & w = 1

Maximum number of people who could be in the group = Number of possible values of ‘c’ = 4. Option B

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Question 2:
How many integers x with |x|< 100 can be expressed as $x = \frac{{4 - {y^3}}}{4}$ for some positive integer y?
[1] 0
[2] 3
[3] 6
[4] 4
$x = \frac{{4 - {y^3}}}{4} = 1 - \frac{{{y^3}}}{4}$
y3 = 4(1-x) = 4 – 4x
x = 0 or -3 or -15 or -53
No. of valid values of x = 4. Option D

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Question 3:
The number of roots common between the two equations x3+3x2+4x+5=0 and x3+2x2+7x+3=0 is:
[1] 0
[2] 1
[3] 2
[4] 3
For the roots to be common to the two equation, both the equations must be equal to 0 and hence equal to each other at those values of x

x3+3x2+4x+5 = x3+2x2+7x+3

3x2+4x+5 = 2x2+7x+3
x2 – 3x + 2 = 0
x = 1 or 2
At x = 1 and at x = 2 both the equations become equal to each other
But at x = 1 or at x = 2 none of the original equations become 0.
Number of common roots = 0. Option A

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Question 4:
Let u= ${({\log _2}x)^2} - 6{\log _2}x + 12$ where x is a real number. Then the equation xu=256, has:
[1] no solution for x
[2] exactly one solution for x
[3] exactly two distinct solutions for x
[4] exactly three distinct solutions for x
xu=256
u $lo{g_2}x$ = 8
$lo{g_2}x$ =$\frac{8}{u}$

Putting this in the first equation

u = (8/u)2 – 6×$\frac{8}{u}$ + 12

u3 = 64 – 48u + 12u2
u3 – 12u2 + 48u – 64 =0
(u -4)3 = 0
u = 4
$lo{g_2}x$= $\frac{8}{u}$ = 2
x = 2
We have exactly one solution for x. Option B

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Question 5:
Let a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax2 + bx + c, bx2 + cx + a, and cx2 + ax + b.
[1] 4
[2] 5
[3] 6
[4] 0
For these equations to have real roots

b2 – 4ac ≥ 0

b2 ≥ 4ac

c2 – 4ab ≥ 0

c2 ≥ 4ab

a2 – 4ac ≥ 0

a2 ≥ 4ac

Multiplying the three we get

(abc)2 ≥ 64(abc)2

This is not possible for positive values of a, b & c.

So, there are no real roots for the three given polynomials. Option D

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Algebra Practice Questions for CAT with Solutions
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