Indices And Surds

Test Name	No. of Questions	Marks (of each)	Time
Test-1	10	3	40 Minutes
Test-2	10	3	40 Minutes
Test-3	11	3	40 Minutes

Question 1.

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

Question 2.

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

A

3√7

B

3√5

C

4√3

D

7√3

Question 3.

For some positive and distinct real numbers x, y and z if $\frac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\frac{1}{\sqrt{x}+\sqrt{z}}$ and $\frac{1}{\sqrt{x}+\sqrt{y}}$ , 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.

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

A

3

B

5/2

C

3/2

D

1/2

Instructions

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

Question 5.

A

579

B

580

C

289

D

290

Instructions

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

Question 6.

Question 7.

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

Instructions

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

Question 8.

Question 9.

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

Instructions

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

Question 10.

A

0

B

Infinite

C

2

D

1

Question 11.

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

A

\frac{(x)^{\frac{7}{2}}}{x^{\frac{4}{\sqrt{3}}}}

B

\frac{(x)^{7}}{x^{2\sqrt{3}}}

C

\frac{(x)^{\frac{7}{2}}}{x^{2\sqrt{3}}}

D

\frac{(x)^{7}}{x^{4\sqrt{3}}}

Instructions

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

Question 12.

Instructions

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

Question 13.

A

2019

B

2017

C

2020

D

2018

Instructions

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

Question 14.

A

-16

B

-12

C

-24

D

-20

Instructions

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

Question 15.

A

1/3

B

2/3

C

1

D

3

Instructions

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

Question 16.

Instructions

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

Question 17.

A

37/4

B

33/4

C

35/4

D

31/4

Instructions

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

Question 18.

Instructions

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:

Question 19.

A

9:4

B

81:4

C

1:4

D

25:4

Instructions

If 9^2x–1 – 81^x-1 = 1944, then x is

Question 20.

A

3

B

9/4

C

4/9

D

1/3

Instructions

If $9^{x-1/2}$ - 2^{2x - 2} = 4^x - 3^{2x - 3}, then x is

Question 21.

A

3/2

B

2/5

C

3/4

D

4/9

Question 22.

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

A

2^{1/x}

B

1/x

C

1/x^2

D

2^{x}

Instructions

Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

Question 23.

A

2^(1/2)

B

3^(1/3)

C

4^(1/4)

D

12^(1/12)

Instructions

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

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

4^0.3x . 9^0.2y = 8 . (81)^1/5

Question 24.

A

x = 2, y = 5

B

x = 2.5, y = 6

C

x = 3, y = 5

D

x = 5, y = 2

Question 25.

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

A

–8 ≤ x ≤ 1

B

–1 ≤ x ≤ 8

C

1 < x < 8

D

1 ≤ x ≤ 8

Question 26.

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

A

3

B

\frac{\sqrt{13}-1}{2}

C

\frac{\sqrt{13}+1}{2}

D

\sqrt{13}

Question 27.

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

A

\frac{\sqrt{13}+3}{2}

B

\frac{\sqrt{13}-3}{2}

C

\frac{\sqrt{15}+3}{2}

D

\frac{\sqrt{15}-3}{2}

Question 28.

If pqr = 1 then $\frac{1}{1+p+q^{-1}}$ + $\frac{1}{1+q+r^{-1}}$ + $\frac{1}{1+r+p^{-1}}$ is equivalent to

A

p + q + r

B

\frac{1}{p+q+r}

C

1

D

p^{-1}

+

q^{-1}

+

r^{-1}

Question 29.

Which of the following is true?

A

7^{3^{2}}

=

(7^{3})^{2}

B

7^{3^{2}}

>

(7^{3})^{2}

C

7^{3^{2}}

<

(7^{3})^{2}

D

None of these

Instructions

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?

Question 30.

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

Instructions

2⁷³ – 2⁷² – 2⁷¹ is the same as

Question 31.

A

2^{69}

B

2^{70}

C

2^{71}

D

2^{72}

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Question 1.

Question 2.

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

Question 3.

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

Question 4.

The sum of all possible values of x satisfying the equation 24x2−22x2+x+16+22x+302^{4x^{2}}-2^{2x^{2}+x+16}+2^{2x+30}24x2−22x2+x+16+22x+30 = 0, is

Instructions

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

Question 5.

Instructions

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?

Question 6.

Question 7.

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

Instructions

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

Question 8.

Question 9.

In ‘n’ is a positive integer such that (107\sqrt[7]{10}710​) (107)2(\sqrt[7]{10})^{2}(710​)2 (107)2(\sqrt[7]{10})^{2}(710​)2 (107)3(\sqrt[7]{10})^{3}(710​)3 .........(107)n(\sqrt[7]{10})^{n}(710​)n > 999, then the smallest value of n is

Instructions

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

Question 10.

Question 11.

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

Instructions

If a, b, c are non-zero and 14a = 36b = 84c, then 6b(1c\frac{1}{c}c1​ - 1b\frac{1}{b}b1​) is equal to

Question 12.

Instructions

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

Question 13.

Instructions

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

Question 14.

Instructions

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

Question 15.

Instructions

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

Question 16.

Instructions

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

Question 17.

Instructions

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

Question 18.

Instructions

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:

Question 19.

Instructions

If 92x–1 – 81x-1 = 1944, then x is

Question 20.

Instructions

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

Question 21.

Question 22.

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

Instructions

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

Question 23.

Instructions

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}2786​​ 40.3x . 90.2y = 8 . (81)1/5

Question 24.

Question 25.

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

Question 26.

Let x = 4+4−4+4−.......∞\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-....... \infty}}}}4+4−4+4−.......∞​​​​ Then x equals

Question 27.

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

Question 28.

If pqr = 1 then 11+p+q−1\frac{1}{1+p+q^{-1}}1+p+q−11​ + 11+q+r−1\frac{1}{1+q+r^{-1}}1+q+r−11​ + 11+r+p−1\frac{1}{1+r+p^{-1}}1+r+p−11​ is equivalent to

Question 29.

Which of the following is true?

Instructions

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?

Question 30.

Instructions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

If 9^2x–1 – 81^x-1 = 1944, then x is

If $9^{x-1/2}$ - 2^{2x - 2} = 4^x - 3^{2x - 3}, then x is

Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6 and 12^1/12 is the largest ?

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

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

4^0.3x . 9^0.2y = 8 . (81)^1/5

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

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

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

If pqr = 1 then $\frac{1}{1+p+q^{-1}}$ + $\frac{1}{1+q+r^{-1}}$ + $\frac{1}{1+r+p^{-1}}$ is equivalent to

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?

2⁷³ – 2⁷² – 2⁷¹ is the same as