Indices And Surds

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

Tips To Solve CAT Indices And Surds

- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.

- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.

- If you are confused, enrolling in CAT online coaching will help you a long way.

- Linear equations can be of one variable or two variables, or three variables.

Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
A general form of two-variable linear equation is ax + by = c.
A general form of three-variable linear equation is ax + by + cz = d.

CAT Indices And Surds PDF

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If $\frac{a}{m} = \frac{b}{n} \neq \frac{c}{p}$ : This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.

If $\frac{a}{m} \neq \frac{b}{n}$ : They have different slopes and hence must intersect at some point, resulting in a unique solution.

If $\frac{a}{m} = \frac{b}{n} = \frac{c}{p}$ : The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.

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 $\sqrt{10x+9}$

3√7

3√5

4√3

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?

x, y and z are in Arithmetic Progression

y, x and z are in Arithmetic Progression

√x, √y and √z are in Arithmetic Progression

√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

5/2

3/2

1/2

Question 5.

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

579

580

289

290

Question 6.

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 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

Question 8.

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

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

Question 10.

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

Infinite

Question 11.

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

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

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

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

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

Question 12.

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 13.

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

2019

2017

2020

2018

Question 14.

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

-16

-12

-24

-20

Question 15.

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

1/3

2/3

Question 16.

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

Question 17.

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

37/4

33/4

35/4

31/4

Question 18.

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 19.

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:

9:4

81:4

1:4

25:4

Question 20.

If 9^2x–1 – 81^x-1 = 1944, then x is

9/4

4/9

1/3

Question 21.

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

3/2

2/5

3/4

4/9

Question 22.

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

2^{1/x}

1/x

1/x^2

2^{x}

Question 23.

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

2^(1/2)

3^(1/3)

4^(1/4)

12^(1/12)

Question 24.

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

x = 2, y = 5

x = 2.5, y = 6

x = 3, y = 5

x = 5, y = 2

Question 25.

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

–8 ≤ x ≤ 1

–1 ≤ x ≤ 8

1 < x < 8

1 ≤ x ≤ 8

Question 26.

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

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

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

\sqrt{13}

Question 27.

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

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

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

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

\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

p + q + r

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

p^{-1}

q^{-1}

r^{-1}

Question 29.

Which of the following is true?

7^{3^{2}}

(7^{3})^{2}

7^{3^{2}}

(7^{3})^{2}

7^{3^{2}}

(7^{3})^{2}

None of these

Question 30.

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?

All of them lie between 0.18 and 0.2

A is twice of C

C is the smallest

B is the smallest

Question 31.

2⁷³ – 2⁷² – 2⁷¹ is the same as

2^{69}

2^{70}

2^{71}

2^{72}

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Tips To Solve CAT Indices And Surds

CAT Indices And Surds PDF

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

Question 5.

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 6.

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 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

Question 8.

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

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