CAT - Logarithms

Question:
If log₂X + log₄X = log_0.25and x > 0, then x is

A.  6^-1/6

B.  6^1/6

C.  3^-1/3

D.  6^1/3

Correct Answer: (A)

Explanation:

log₂x + log₄x = log_0.25

log₂x + = log_0.25

log₂x * = log_0.25

log₂x * 3 = 2log_0.25

log₂x³ = log_0.256

log₂x³ = -log₄6

log₂x³ =

log₂x³ =

2log₂x³ = -log₂6

2log₂x³ + log₂6 = 0

log₂6x⁶ = 0

6x⁶ = 1

x⁶ =

x =  Choice (A).

Level of difficulty 2

CAT - Linear Quadratic Equations

Question:

3x + 4|y| = 33. How many integer values of (x, y) are possible?

A.  6

B.  3

C.  4

D.  More than 6

Correct Answer: (D)

Solution:

Let us rearrange the equation:

3x = 33 – 4|y|

Since x and y are integers, and since |y| is always positive regardless of the sign of y, this means that when you subtract a multiple of 4 from 33, you will get a multiple of 3.

Since 33 is already a multiple of 3, in order to obtain another multiple of 3, you will have to subtract a multiple of 3 from it. So, y has to be a positive or a negative multiple of 3.

y = 3, -3, 6, -6, 9, -9, 12, -12...etc.

For every value of y, x will have a corresponding integer value.

So there are infinite integer values possible for x and y.

Level of difficulty 1

Probability Question

Probability Question

Question:

[x] = greatest integer less than equal to x. If x lies between 3 and 5. What is the probability than [x²] = [x]²?

A. Roughly 0.64

B. Roughly 0.5

C. Roughly 0.14

D. Roughly 0.36

Correct Answer: (C)

Explanation:

Let us take a few examples.

[3²] = [3]²

[3.5²] =12             [3.5]² = 9

[4²] = 16               [4]² = 16

For x Î (3, 5). [x]² can only take value 9, 16 and 25.

Let us see when [x²] will be 9, 16 or 25

If [x²] = 9,  x² [9, 10)

=> x[3, )

[x²] = 16 => x² [16, 17)

=> x [4, )

In the given range [x²] = 25 only when x = 5

So [x²] = [x]² when x [3, ] or [4, ) or 5.

Probability =

= = 0.14

Difficulty Level 2