Algebra - Inequalities & Modulus - Previous Year CAT/MBA Questions

Answer: Option B

Video Explanation

Text Explanation :

A = |x + 3| + |x - 2| - |2x - 8|

The critical points are -3, 2 and 4.

Case 1: x ≥ 4
∴ A = x + 3 + x - 2 - (2x - 8) 
⇒ A = 2x + 1 - 2x + 8
⇒ A = 9

Case 2: 2 ≤ x < 4
∴ A = x + 3 + x - 2 + (2x - 8) 
⇒ A = 2x + 1 + 2x - 8
⇒ A = 4x - 7
∴ A ∈ [1, 9)

Case 3: -3 ≤ x < 2
∴ A = x + 3 - (x - 2) + (2x - 8) 
⇒ A = x + 3 - x + 2 + 2x - 8
⇒ A = 2x - 3
∴ A ∈ [-9, 1)

Case 4: x < -3
∴ A = - (x + 3) - (x - 2) + (2x - 8) 
⇒ A = - x - 3 - x + 2 + 2x - 8
⇒ A = - 9

From the above cases, The maximum value of |A| = 9

Hence, option (b).

Answer: Option E

Text Explanation :

Since 2 ≤ |x – 1| × |y + 3| ≤ 5, the value of |x – 1| × |y + 3| can be 2, 3, 4 or 5. Thus, we have the following cases.

Case 1: |x – 1| × |y + 3| = 2
⇒ |x – 1| = 2 and |y + 3| = 1
⇒ x = -1 and y = -4 or -2

Note: |x - 1| cannot be equal to 1 since x has to be a negative number.

∴ 2 possibilities.

Case 2: |x – 1| × |y + 3| = 3
⇒ |x – 1| = 3 and |y + 3| = 1
⇒ x = -2 and y = -4 or -2

∴ 2 possibilities.

Case 3: |x – 1| × |y + 3| = 4

⇒ |x – 1| = 4 and |y + 3| = 1
⇒ x = -3 and y = -4 or -2

⇒ |x – 1| = 2 and |y + 3| = 2
⇒ x = -1 and y = -1 or -5

∴ 4 possibilities.

Case 3: |x – 1| × |y + 3| = 5

⇒ |x – 1| = 5 and |y + 3| = 1
⇒ x = -4 and y = -4 or -2

∴ 2 possibilities.

∴ Total 2 + 2 + 4 + 2 = 10 possibilities.

Hence, option (e).

Answer: Option C

Text Explanation :

abc – (a + b + c) = a(bc – 1) – b – c .

It is clear that the expression would be maximum when b and c are negative and their modulus are maximum possible.

∴ b = –10 , c = –9.

Now a has to be positive and maximum possible. It has to be 8.

Hence the expression becomes equal to 731.

Hence, option (c).

Answer: Option C

Video Explanation

Text Explanation :

Let a + b + c = X

We need to find the maximum value of:

$\frac{X+d}{X-d}=\frac{(X-d)+2d}{X-d}=1+\frac{2d}{X-d}$

$$\begin{gathered} \frac{}{} \\ \frac{2d}{X-d} \end{gathered}$$

$\frac{\mathrm{is\; maximum\; when\; d\; is\; maximum}}{}$ and X is minimum possible.

∴ d = 25 and X = 26
The highest possible value of the given expression = 1 + (2 × 25) = 51
Hence, option (c).