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Question 1.

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A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is

A
B
C
D

Question 2.

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If 5x+9+5x9\sqrt{5x+9} + \sqrt{5x-9} = 3(2 + √2), then find the value of 10x+9\sqrt{10x+9}

A
3√7
B
3√5
C
4√3
D
7√3

Question 3.

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For some positive and distinct real numbers x, y and z if 1y+z\frac{1}{\sqrt{y}+\sqrt{z}} is the arithmetic mean of 1x+z\frac{1}{\sqrt{x}+\sqrt{z}} and 1x+y\frac{1}{\sqrt{x}+\sqrt{y}} , then the relationship which will always hold true, is?

A
x, y and z are in Arithmetic Progression
B
y, x and z are in Arithmetic Progression
C
√x, √y and √z are in Arithmetic Progression
D
√y, √x and √z are in Arithmetic Progression

Question 4.

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The sum of all possible values of x satisfying the equation 24x222x2+x+16+22x+302^{4x^{2}}-2^{2x^{2}+x+16}+2^{2x+30} = 0, is

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

Question 5.

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Let a, b, m and n be natural numbers such that a > 1 and b > 1. If ambn = 144145, then the largest possible value of n - m is

A
579
B
580
C
289
D
290

Question 6.

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Let n be any natural number such that 5n-1 < 3n+1. Then, the least integer value of m that satisfies 3n+1 < 2n+m for each such n, is?

A
B
C
D

Question 7.

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If (7/5)3x2y(\sqrt{7/5})^{3x-2y} = 875/2401 and (4a/b)6xy(4a/b)^{6x-y} = (2a/b)y6x(2a/b)^{y-6x} ,for all non-zero real values of a and b, then the value of x + y is

A
B
C
D

Question 8.

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For all possible integers n satisfying 2.25 ≤ 2 + 2n+2 ≤ 202, the number of integer values of 3 + 3n+1 is

A
B
C
D

Question 9.

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In ‘n’ is a positive integer such that (107\sqrt[7]{10}) (107)2(\sqrt[7]{10})^{2} (107)2(\sqrt[7]{10})^{2} (107)3(\sqrt[7]{10})^{3} .........(107)n(\sqrt[7]{10})^{n} > 999, then the smallest value of n is

A
B
C
D

Question 10.

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

A
0
B
Infinite
C
2
D
1

Question 11.

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If x = (4096)7+43(4096)^{7+4√3} , then which of the following equals 64?

A
(x)72x43\frac{(x)^{\frac{7}{2}}}{x^{\frac{4}{\sqrt{3}}}}
B
(x)7x23\frac{(x)^{7}}{x^{2\sqrt{3}}}
C
(x)72x23\frac{(x)^{\frac{7}{2}}}{x^{2\sqrt{3}}}
D
(x)7x43\frac{(x)^{7}}{x^{4\sqrt{3}}}

Question 12.

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If a, b, c are non-zero and 14a = 36b = 84c, then 6b(1c\frac{1}{c} - 1b\frac{1}{b}) is equal to

A
B
C
D

Question 13.

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How many pairs (a, b) of positive integers are there such that a ≤ b and ab = 42017?

A
2019
B
2017
C
2020
D
2018

Question 14.

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If m and n are integers such that (√2)19 34 42 9m 8n = 3n 16m (64)1/4, then m is

A
-16
B
-12
C
-24
D
-20

Question 15.

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

A
1/3
B
2/3
C
1
D
3

Question 16.

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If 5x - 3y = 13438 and 5x-1 + 3y+1 = 9686, then x + y equals

A
B
C
D

Question 17.

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Given that x2018y2017 = 1/2 and x2016y2019 = 8, the value of x2 + y3 is

A
37/4
B
33/4
C
35/4
D
31/4

Question 18.

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If N and x are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x, then the largest possible x is

A
B
C
D

Question 19.

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If a and b are integers of opposite signs such that (a + 3)2 : b2 = 9 : 1 and (a - 1)2: (b - 1)2 = 4 : 1, then the ratio a2 : b2 is:

A
9:4
B
81:4
C
1:4
D
25:4

Question 20.

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If 92x–1 – 81x-1 = 1944, then x is

A
3
B
9/4
C
4/9
D
1/3

Question 21.

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If 9x1/29^{x-1/2} - 22x - 2 = 4x - 32x - 3, then x is

A
3/2
B
2/5
C
3/4
D
4/9

Question 22.

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If x = −0.5, then which of the following has the smallest value?

A
21/x2^{1/x}
B
1/x
C
1/x^2
D
2x2^{x}

Question 23.

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Which among 21/2, 31/3, 41/4 , 61/6  and 121/12  is the largest ?

A
2^(1/2)
B
3^(1/3)
C
4^(1/4)
D
12^(1/12)

Question 24.

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What are the values of x and y that satisfy both the equations?

2^(0.7x). 3^(−1. 25y) = 8627\frac{8\sqrt{6}}{27}

40.3x . 90.2y = 8 . (81)1/5  

 

A
x = 2, y = 5
B
x = 2.5, y = 6
C
x = 3, y = 5
D
x = 5, y = 2

Question 25.

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What values of x satisfy x2/3+x1/3x^{2/3}+x^{1/3}-2 \le 0

A
–8 ≤ x ≤ 1
B
–1 ≤ x ≤ 8
C
1 < x < 8
D
1 ≤ x ≤ 8

Question 26.

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Let x = 4+44+4.......\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-....... \infty}}}} Then x equals

A
3
B
1312\frac{\sqrt{13}-1}{2}
C
13+12\frac{\sqrt{13}+1}{2}
D
13\sqrt{13}

Question 27.

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Let y = 12+13+12+13+.....\frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+.....}}}} What is the value of y?

A
13+32\frac{\sqrt{13}+3}{2}
B
1332\frac{\sqrt{13}-3}{2}
C
15+32\frac{\sqrt{15}+3}{2}
D
1532\frac{\sqrt{15}-3}{2}

Question 28.

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If pqr = 1 then 11+p+q1\frac{1}{1+p+q^{-1}} + 11+q+r1\frac{1}{1+q+r^{-1}} + 11+r+p1\frac{1}{1+r+p^{-1}} is equivalent to

A
p + q + r
B
1p+q+r\frac{1}{p+q+r}
C
1
D
p1p^{-1} + q1q^{-1} + r1r^{-1}

Question 29.

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Which of the following is true?

A
7327^{3^{2}} = (73)2(7^{3})^{2}
B
7327^{3^{2}} > (73)2(7^{3})^{2}
C
7327^{3^{2}} < (73)2(7^{3})^{2}
D
None of these

Question 30.

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A, B and C are defined as follows.

A = 2.000004 ÷ [(2.000004)2 + (4.000008)]

B = 3.000003 ÷ [(3.000003)2 + (9.000009)]

C = 4.000002 ÷ [(4.000002)2 + (8.000004)]

Which of the following is true about the values of the above three expressions?

A
All of them lie between 0.18 and 0.2
B
A is twice of C
C
C is the smallest
D
B is the smallest

Question 31.

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273 – 272 – 271 is the same as

A
2692^{69}
B
2702^{70}
C
2712^{71}
D
2722^{72}

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