Surds and Indices CAT PYQs , 21 Questions with Solutions

For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which

$\left(a_1\right)^1 \times \left(a_2\right)^2 \times \cdots \times \left(a_{20}\right)^{20} < a_{21} \times a_{22} \times \cdots \times a_{(20+m)}$

is

If $9^{x^2 + 2x - 3} - 4\left(3^{x^2 + 2x - 2}\right) + 27 = 0$, then the product of all possible values of $x$ is

The sum of digits of the number $( 625 ) ^ { 65 } \times ( 128 ) ^ { 36 }$, is

If $12^{12x} \times 4^{24x+12} \times 5^{2y} = 8^{4z} \times 20^{12x} \times 243^{3x-6}$, where $x, y$ and $z$ are natural numbers, then $x+y+z$ equals

If $\left(x^2 + \dfrac{1}{x^2}\right) = 25$ and $x > 0$, then the value of $\left(x^7 + \dfrac{1}{x^7}\right)$ is

The sum of all real values of $k$ for which $\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}$ , is

If $(a+b \sqrt{n})$ is the positive square root of $(29-12 \sqrt{5})$, where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is.

If $(x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}$ , then $x$ equals

If $(a+b \sqrt{3})^2=52+30 \sqrt{3}$ , where $a$ and $b$ are natural numbers, then $a+b$ equals

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \frac{4}{10^x} = \frac{81}{2}$, is

If $\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})$ , then $\sqrt{10 x+9}$ is equal to

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 $n$ and $m$ be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$ , which can be expressed as powers of 2 . Then, the smallest possible value of $n+m$ is

If $x$ is a positive real number such that $x^8 + \left(\frac{1}{x}\right)^8 = 47$, then the value of $x^9 + \left(\frac{1}{x}\right)^9$ is

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

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

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

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

The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

Given that $x^{2018} y^{2017} = 1/2$ and $x^{2016} y^{2019} = 8$, the value of $x^2 + y^3$?

If $9^{2x-1} - 81^{x-1} = 1944$, then $x$ is