Test NameNo. of QuestionsMarks (of each)TimeTake Test
Test-112340 Minutes
Test-212340 Minutes
Test-313340 Minutes

Question 1.

If x and y are positive real numbers such that logx(x2+12)\log_{x}(x^2+12) = 4 and 3logyx=13\log_{y}x=1, then x + y equals?

A
11
B
68
C
20
D
10

Question 2.

For some positive ral number x, if log3(x)\log_{\sqrt{3}}(x) + logx25logx(0.008)\frac{\log_{x}25}{\log_{x}(0.008)} = 16/ 3 , then the value of log3(3x2)\log_{3}(3x^{2}) is

A
B
C
D

Question 3.

For a real number x, if 1/ 2 , log3(2x9)log34\frac{\log_{3}(2^{x}-9)}{\log_{3}4} , and log5(2x+17/2)log54\frac{\log_{5}(2^{x}+17/2)}{\log_{5}4} are in arithmetic progression, then the common difference is

A
log4 (3/2)
B
log4 (7/2)
C
log4 (7)
D
log4(23/2)

Question 4.

The number of distinct integer values of n satisfying 4log2n3log4n\frac{4-\log_{2}n}{3-\log_{4}n} < 0, is

A
B
C
D

Question 5.

If 5 -log10(1+x)\log_{10}(\sqrt{1+x}) + 4log10(1x)\log_{10}(\sqrt{1-x}) = log10(11x2)\log_{10}(\frac{1}{\sqrt{1-x^{2}}}) , then 100x equals

A
B
C
D

Question 6.

log2 [3 + log3[4+log4(x1)]\log_{3}[{4 + log_{4}(x - 1)}]] - 2 = 0, then 4x equals

A
B
C
D

Question 7.

For a real number a, if log15a+log32alog15a+log32a\frac{\log_{15}a+\log_{32}a}{\log_{15}a+\log_{32}a} = 4, then a must lie in the range

A
4 < a < 5
B
2 < a < 3
C
a > 5
D
3 < a < 4

Question 8.

If y is a negative number such that 2y2log35=5log232^{y^{2}\log_{3}5}=5^{\log_{2}3} , then y equals

A
log₂(1/5)
B
–log₂ (1/3)
C
–log₂ (1/5)
D
log₂ (1/3)

Question 9.

If log₄ 5 = (log₄ y)(log₆ √5), then y equals

A
B
C
D

Question 10.

The value of⁡ loga(ab)+logb(ba)\log_{a}(\frac{a}{b}) + \log_{b}(\frac{b}{a}) , for 1 < a ≤ b cannot be equal to

A
0
B
-1
C
1
D
-0.5

Question 11.

2X4X8X16(log24)2(log48)3(log816)4\frac{2X4X8X16}{(\log_{2}4)^{2}(\log_{4}8)^{3}(\log_{8}16)^{4}} equals .

A
B
C
D

Question 12.

If loga30\log_{a}30 = A, loga5/3\log_{a}5/3 = -B and log2a\log_{2}a = 1/3, thenlog3a\log_{3}a equals

A
(A + B)/2 - 3
B
2/(A + B) - 3
C
2/(A + B - 3)
D
(A + B - 3)/2

Instructions

Let x and y be positive real numbers such that log5(x + y) + log5(x - y) = 3, and log2y - log2x = 1 - log23. Then xy equals

Question 13.

A
250
B
150
C
100
D
25

Question 14.

The real root of the equation 26x2^{6x} + 23x+22^{3x+2}- 21 = 0 is

A
log2(3)3\frac{\log_{2}(3)}{3}
B
log2(9){\log_{2}(9)}
C
log2(7)3\frac{{\log_{2}(7)}}{3}
D
log2(27){\log_{2}(27)}

Question 15.

Let A be a real number. Then the roots of the equation x2x^{2} - 4x - log2A\log_{2}A = 0 are real and distinct if and only if

A
A < 1/16
B
A < 1/8
C
A > 1/8
D
A > 1/16

Instructions

If x is a real number, then loge(4xx2)/3\sqrt{\log_{e}(4x-x^2)/3} is a real number if and only if

Question 16.

A
-3 ≤ x ≤ 3
B
1 ≤ x ≤ 2
C
1 ≤ x ≤ 3
D
-1 ≤ x ≤ 3

Instructions

If x is a positive quantity such that 2x = 3log523^{\log_{5}2} , then x is equal to

Question 17.

A
1+log5(5/3)\log_{5}(5/3)
B
log59\log_{5}9
C
log58\log_{5}8
D
1 +log5(3/5)\log_{5}(3/5)

Instructions

If log1281 = p, then 3((4-p)/(4+p)) is equal to

Question 18.

A
log28\log_{2}8
B
log616\log_{6}16
C
log68\log_{6}8
D
log416\log_{4}16

Instructions

If log2(5 + log3 a) = 3 and log5(4a + 12 + log2 b) = 3, then a + b is equal to

Question 19.

A
40
B
67
C
59
D
32

Instructions

If p3 = q4 = r5 = s6, then the value of logs(pqr) is equal to

Question 20.

A
47/10
B
16/5
C
24/5
D
1

Instructions

The smallest integer n for which 4n > 1719 holds, is closest to

Question 21.

A
33
B
37
C
39
D
35

Instructions

1log2100\frac{1}{\log_{2}100} - 1log4100\frac{1}{\log_{4}100} + 1log5100\frac{1}{\log_{5}100} - 1log10100\frac{1}{\log_{10}100} + 1log20100\frac{1}{\log_{20}100}  - 1log25100\frac{1}{\log_{25}100} + 1log50100\frac{1}{\log_{50}100} ?

Question 22.

A
-4
B
10
C
0
D
1/2

Instructions

Suppose, log3x = log12y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log6G is equal to:

Question 23.

A
√a
B
2a
C
a/2
D
a

Instructions

The value of log0.008√5 + log√381 – 7 is equal to:

Question 24.

A
1/3
B
2/3
C
5/6
D
7/6

Instructions

If x is a real number such that log35 = log5(2 + x), then which of the following is true?

Question 25.

A
0 < x < 3
B
23 < x < 30
C
x > 30
D
3 < x < 23

Instructions

If log (2a × 3b × 5c) is the arithmetic mean of log (22 × 33 × 5), log (26 × 3 × 57), and log (2 × 32 × 54), then a equals

Question 26.

A
B
C
D

Instructions

If logyx = a × logzy = b × logxz = ab, then which of the following pairs of values for (a, b) is not possible?

Question 27.

A
-2 , 1/2
B
1,1
C
0.4, 2.5
D
2, 2

Instructions

If x ≥ y and y > 1, then the value of the expression

logxlog_{x} (x/y) + logylog_{y} (y/x) can never be

Question 28.

A
-1
B
-0.5
C
0
D
1

Instructions

Let u = (logx)2 – 6 log2 x + 12 where x is a real number. Then the equation xu = 256, has

Question 29.

A
no solution for x
B
exactly one solution for x
C
exactly two distinct solutions for x
D
exactly three distinct solutions for x

Instructions

If log3 2, log3 (2x − 5), log3 (2x − 7/2) are in arithmetic progression, then the value of x is equal to

Question 30.

A
5
B
4
C
2
D
3

Question 31.

If 1/3 log3M+3\log_{3} M + 3 \log_{3} N = 1 + log0.008\log_{0.008} 5 , then

A
M9M^{9} = 9/N
B
N9N^{9} = 9/M
C
M3M^{3} = 3/N
D
N9N^{9} = 3/M

Instructions

If log10 x - log10 x^(1/2) = 2 logx 10, then a possible value of x is given by:

Question 32.

A
10
B
1/100
C
1/1000
D
None of these

Instructions

What is the sum of n terms in the series

log m + log (m^2 / n) + log (m^3 / n^2) + log (m^4 / n^3) .......+ log (m^n / n^(n-1)) ?

Question 33.

A
log[3\frac{}{3}]
B
C
D

Instructions

Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.

What is the distance x between two cities A and B in integral number of kilometres?
I. x satisfies the equation log2x\log_{2}x = x\sqrt{x}
II. x ≤ 10 km

Question 34.

A
1
B
2
C
3
D
4

Instructions

If log2 [log7 (x2 - x + 37)] = 1, then what could be the value of ‘x’?

Question 35.

A
3
B
5
C
4
D
None of these

Instructions

If log7 log5 ((x+5)^(1/2) + x^(1/2)) = 0, find the value of x.

Question 36.

A
1
B
0
C
2
D
None of these

Instructions

log6 216 6\sqrt{6} is

Question 37.

A
3
B
3/2
C
7/2
D
None of these

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