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Test-112340 Minutes
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Test-313340 Minutes

Question 1.

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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.

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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.

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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.

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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.

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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.

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

A
B
C
D

Question 7.

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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.

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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.

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

A
B
C
D

Question 10.

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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.

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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.

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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

Question 13.

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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

A
250
B
150
C
100
D
25

Question 14.

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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.

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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

Question 16.

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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

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

Question 17.

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

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)

Question 18.

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

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

Question 19.

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

A
40
B
67
C
59
D
32

Question 20.

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

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

Question 21.

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

A
33
B
37
C
39
D
35

Question 22.

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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} ?

A
-4
B
10
C
0
D
1/2

Question 23.

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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:

A
√a
B
2a
C
a/2
D
a

Question 24.

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

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

Question 25.

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If x is a real number such that log35 = 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

Question 26.

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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

A
B
C
D

Question 27.

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

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

Question 28.

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

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

A
-1
B
-0.5
C
0
D
1

Question 29.

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

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

Question 30.

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

A
5
B
4
C
2
D
3

Question 31.

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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

Question 32.

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

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

Question 33.

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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)) ?

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

Question 34.

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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

A
1
B
2
C
3
D
4

Question 35.

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

A
3
B
5
C
4
D
None of these

Question 36.

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

A
1
B
0
C
2
D
None of these

Question 37.

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log6 216 6\sqrt{6} is

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

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