CAT Algebra Previous Year Questions (PYQs)

In an arithmetic progression, if the sum of fourth, seventh and tenth terms is 99, and the sum of the first fourteen terms is 497, then the sum of first five terms is

Let $a_n$ be the $n^{th}$ term of a decreasing infinite geometric progression. If $a_1+a_2+a_3 = 52$ and $a_1a_2+a_2a_3+a_3a_1 = 624$ , then the sum of this geometric progression 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

If $a,b,c$ and $d$ are integers such that their sum is 46, then the minimum possible value of $(a-b)^2 + (a-c)^2 + (a-d)^2$ is

The equations $3x^2 - 5x + p = 0$ and $2x^2 - 2x + q = 0$ have one common root. The sum of the other roots of these two equations is

The set of all real values of $x$ for which $(x^2 - |x + 9| + x) > 0$ , is

Let $f(x) = \dfrac{x}{(2x-1)}$ and $g(x) = \dfrac{x}{(x-1)}$. Then, the domain of the function $h(x) = f(g(x)) + g(f(x))$ is all real numbers except

Let $3 \leq x \leq 6$ and $[x^2] = [x]^2$ , where $[x]$ is the greatest integer not exceeding $x$. If set $S$ represents all feasible values of $x$ , then a possible subset of $S$ is

The number of non-negative integer values of $k$ for which the quadratic equation $x^2 - 5x + k = 0$ has only integer roots, is

In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$ , there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. Then, $k$ equals

A value of $c$ for which the minimum value of $f(x) = x^2 - 4cx + 8c$ is greater than the maximum value of $g(x) = -x^2 + 3cx$ - 2c , is

For any natural number $k$ , let $a_k = 3^k$. The smallest natural number $m$ for which $\{(a_1)^1 \times (a_2)^2 \times \ldots \times (a_{20})^{20}\} < \{a_{21} \times a_{22} \times \ldots \times a_{(20+m)}\}$ , is

If $a - 6b + 6c = 4$ and $6a + 3b - 3c = 50$ , where $a , b$ and $c$ are real numbers, the value of $2a + 3b - 3c$ is

Stocks A, B and C are priced at rupees 120, 90 and 150 per share, respectively. A trader holds a portfolio consisting of 10 shares of stock A, and 20 shares of stocks B and C put together. If the total value of her portfolio is rupees 3300, then the number of shares of stock B that she holds, is

If the equations $x^2 + mx + 9 = 0 , x^2 + nx + 17 = 0$ , and $x^2 + (m + n)x + 35 = 0$ have a common negative root, then the value of $2m + 3n$ is

Consider two sets $A = \{2, 3, 5, 7, 11, 13\}$ and $B = \{1, 8, 27\}$. Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$ , there is at least one element $a$ in $A$ such that $f(a) = b$. Then, the total number of such functions $f$ is

A shop wants to sell a certain quantity (in kg) of grains. It sells half the quantity and an additional 3 kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 3 kg of these grains to the second customer. Finally, when the shop sells half of the remaining quantity and an additional 3 kg of these grains to the third customer, there are no grains left. The initial quantity, in kg, of grains 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

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

Suppose $x_{1}, x_{2}, x_{3}, \ldots, x_{100}$ are in arithmetic progression such that $x_{5}=-4$ and $2 x_{6}+2 x_{9}=x_{11}+x_{13}$ , Then, $x_{100}$ equals

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

A function $f$ maps the set of natural numbers to whole numbers, such that $f (xy) = f (x) f (y) + f (x) + f (y)$ for all $x, y$ and $f (p) = 1$ for every prime number $p$. Then, the value of $f (160000)$ is

The sum of the infinite series is $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^{2}\left(\left(\frac{1}{5}\right)^{2}-\left(\frac{1}{7}\right)^{2}\right)+\left(\frac{1}{5}\right)^{3}\left(\left(\frac{1}{5}\right)^{3}-\left(\frac{1}{7}\right)^{3}\right)+\ldots$ equal to

If x and y are real numbers such that $|x| + x + y = 15$ and $x + |y| - y = 20$ , then $(x - y)$ equals