CAT 2022 Slot 2 Question 1-

The number of distinct integer values of n satisfying 4−log^2n/3−log4^n<0, is

Explanation

4−logn23−logn4<04−log^2n/3−log4<0
If logn2=4;n=24=16log2^n=4=2^4=16;
If logn4=3;n=4^3=64log4^n=3=4^3=64;
For any value of n < 16, the numerator is positive. For any value of n > 16, it is negative.
For any value of n < 64, the denominator is positive. For any value of n > 64, it is negative.
For a fraction to be negative, the numerator and the denominator must be of opposite signs.
In this case, n should be between 16 and 64.
The number of values that n can take = 63 – 16 = 47.

CAT 2021 Slot 3 Question 2-

 For a real number a, if log15^a+log32^a/(log15a)(log32a)=

 = 4 then a must lie in the range

A.4 < a < 5

B.3 < a < 4

C.a > 5

D.2 < a < 3

Explanation

 loga15^a+loga32^a/(loga15^a×loga32^a)

  = 4

1/loga32^a+1/loga15^15

= 4

log32^a+log15^a

= 4

log(32×15)^a

 = 4

a^4 = 32 × 15 = 480

We know that, 44 = 256 and 54 = 625

256 < 480 < 625

44 < a4 < 54

4 < a < 5

CAT 2021 Slot 2 Question 3-  If log2^[3 + log3^{4 + log4^(x – 1)}] – 2 = 0 then 4x equals

Explanation

log₂[3+log₃{4+log₄(x-1)}]- 2= 0
log₂[3 + log₃{4 + log₄(x – 1)}] = 2
3 + log₃{4 + log₄(x – 1)} = 4 ( ∵ log_aN = x ⇒ N = a^x )
log₃{4 + log₄(x – 1)} = 1
4 + log₄(x – 1) = 3
log₄(x – 1) = -1
x – 1 = 1414
4x = 5

CAT 2020 Slot 3  Question 4-

 If x^1 = -1 and x^m = x^m + 1 + (m + 1) for every positive integer m, then x^100 equals

A.-5050

B.-5051

C.-5150

D.-5151

Explanation

x_m = x_{m + 1} + (m + 1)
Let’s say k = m + 1
The equation now becomes: x_{k – 1} = x_k + k
x_k = x_{k – 1} – k
x_k = – k + x_{k – 1}

Now the vizualization of this equation becomes far simpler…
x₁ = -1
x₂ = -2 -1
x₃ = -3 -2 -1
x₄ = -4 -3 -2 -1

x₁₀₀ = -100 -99 -98 … -3 -2 -1
x₁₀₀ = -(100 +99 +98 … +3 +2 +1)
x₁₀₀ = -(100(100+1)2100(100+1)2) = -5050

CAT 2020 Slot 3 Question 5-

 Let loga^30 = A, loga^53 = -B and log2^a = 1/3, then log3^a equals

A.2/A+B−3

B.A+B−3/2

C.A+B/2 – 3

D.2/A+B – 3

Explanation

CAT 2020 Slot 3 Question 6-

 2×4×8×16/(log2^4)^2(log4^8)^3(log8^16)^4

 equals

Explanation

 CAT 2020 Slot 3 Question 7-

 If a,b,c are non-zero and 14^a = 36^b = 84^c, then 6b(1/c – 1/a) is equal to

Explanation

CAT 2020 Slot 1 Question 8-  If log4^5 = (log4^y) (log6^ √5), then y equals

Explanation

CAT 2020 Slot 1 Question 9-

 The number of real-valued solutions of the equation 2^x + 2^-x = 2 – (x – 2)^2 is

A.infinite

B.0

C.1

D.2

Explanation

CAT 2020 Slot 1 Question 10-

 If y is a negative number such that 2^y^2log3^5 = 5^log2^3, then y equals

A.log2 (1/3)

B.log2 (1/5)

C.−log2 (1/3)

D.−log2 (1/5)

Explanation

CAT 2019 Slot 2 Question 11-  If 5^x – 3^y = 13438 and 5^x-1 + 3^y+1 = 9686 , then x+y equals

Explanation

CAT 2019 Slot 1 Question 12-

 If (5.55)^x = (0.555)^y = 1000, then the value of 1/x – 1/y is

A.1

B.1/3

C.2/3

D.3

Explanation

CAT 2019 Slot 1 Question 13-

 Let x and y be positive real numbers such that log^5(x + y) + log^5(x – y) = 3, and log^2y – log^2x = 1 – log2^3. Then xy equals

A.25

B.150

C.250

D.100

Explanation

CAT 2018 Slot 2 Question 14-

If p^3 = q^4 = r^5 = s^6, then the value of logs^ (pqr) is equal to

A.24/5

B.1

C.47/10

D.16/5 

Explanation

CAT 2017 Slot 2 Question 15-

 If x is a real number such that log3^5 = log5^(2 + x), then which of the following is true?

A.0 < x < 3

B.23 < x < 30

C.x > 30

D.3 < x < 23

Explanation