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# Test Level 1: Inequalities - 1

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## 10 Questions MCQ Test Level-wise Tests for CAT | Test Level 1: Inequalities - 1

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Test Level 1: Inequalities - 1 - Question 1

### N is a negative real number. Which of the following is not true?

Detailed Solution for Test Level 1: Inequalities - 1 - Question 1

The absolute value of any number N is always non-negative.
|N| = N, when N > 0 and |N| = -N, when N < 0.
Since N is a negative number, therefore |N| = N is not true.

Test Level 1: Inequalities - 1 - Question 2

### The number of solutions of the equation |x2 - 4| = 4 + x2 is

Detailed Solution for Test Level 1: Inequalities - 1 - Question 2

|x2 - 4| = 4 + x2
Case I: If x2 - 4 ≥ 0
i.e. x ≤ -2 and x ≥ 2
Then, |x2 - 4| = x2 - 4
⇒ The given equation becomes x2 - 4 = 4 + x2.
⇒ -4 = 4; which is not possible.
Case II: If x2 - 4 < 0
⇒ -2 < x < 2
Then, |x2 - 4| = - (x2 - 4)
Thus, the given equation becomes - (x2 - 4) = 4 + x2.
Or 2x2 = 0
⇒ x = 0
Hence, the given equation has only one solution.

Test Level 1: Inequalities - 1 - Question 3

### Which of the following is not true at all?

Detailed Solution for Test Level 1: Inequalities - 1 - Question 3

Take x = 2, then option (1), (2) will become true and hence those are not the answers.
Take x = – 2, option (4) will be true and hence not the answer.
But for any value of x, 2x > x.
So option (3) is not true at all. It is always false.

Test Level 1: Inequalities - 1 - Question 4

A, B, C and D are four friends and they are having w, x, y and z amount, respectively, such that w - z > 0, x - z < 0 and y - z < 0. Which of the following is necessarily true?

Detailed Solution for Test Level 1: Inequalities - 1 - Question 4

w - z > 0
x - z < 0; this means z - x > 0
y - z < 0; this means z - y > 0
The product of cubes of three positive numbers will also be positive.

Test Level 1: Inequalities - 1 - Question 5

If a < b, then the solution of x2 - (a + b) x + ab < 0 is

Detailed Solution for Test Level 1: Inequalities - 1 - Question 5

x2 - (a + b) x + ab < 0
(x - a) (x - b) < 0
a < x < b

Test Level 1: Inequalities - 1 - Question 6

What values of 'm' satisfy the inequality 3m2 - 21m + 30 < 0?

Detailed Solution for Test Level 1: Inequalities - 1 - Question 6

3m2 - 21m + 30 < 0
m2 - 7m + 10 < 0
⇒ (m - 5)(m - 2) < 0
⇒ 2 < m < 5

Test Level 1: Inequalities - 1 - Question 7

What is the best description of 'x' which satisfies the inequality x2 - 5x + 6 ≤ 0 ?

Detailed Solution for Test Level 1: Inequalities - 1 - Question 7

x2 - 5x + 6 ≤ 0
⇒ (x - 3)(x - 2) ≤ 0
⇒ x ∈ [2, 3] or 2 ≤ x ≤ 3
Therefore, option 4 is the correct answer.

Test Level 1: Inequalities - 1 - Question 8

x – |x| is always

Detailed Solution for Test Level 1: Inequalities - 1 - Question 8

|x| is always positive.
If x is positive, x – |x| = 0.
If x is negative, x – |x| < 0.

Test Level 1: Inequalities - 1 - Question 9

Solve the system of inequalities:
5x + 2 > 3x - 1
3x + 1 > 7x - 4

Detailed Solution for Test Level 1: Inequalities - 1 - Question 9

(i) 5x + 2 > 3x - 1 or 2x > -3

(ii) 3x + 1 > 7x - 4

Test Level 1: Inequalities - 1 - Question 10

3x2 – 7x + 4 ≤ 0

Detailed Solution for Test Level 1: Inequalities - 1 - Question 10

At x = 0, inequality is not satisfied. Thus, option (c) is rejected. Also x = 0 is not a solution of the equation. Since, this is a continuous function, the solution cannot start from 0. Thus options (a) and (b) are not right. Further, we see that the given function is quadratic with real roots. Hence, option (d) is also rejected.

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