CAT Algebra questions (2023–2025) — actual Quant questions with solutions

f(x) = (x²+ 3x)(x² + 3x + 2), then the sum of all real roots of the equation √f(x) + 1 = 9701, is

For real values of x, the range of the function f(x) = (2x − 3)/(2x² + 4x − 6)

If 12¹² x × 4²⁴ x + ¹² × 5² y = 8⁴ z × 20¹² x × 243 3x-6 , where x, y and z are natural numbers, then x+y+z equals_____

The set of all real values of x for which (x² − |x + 9| + x) > 0, is

The number of non-negative integer values of k for which the quadratic equation x² – 5x + k = 0 has only integer roots, is

Let 3 ≤ x ≤ 6 and [x²] = ([x])², where [x] is the greatest integer not exceeding x. If S represents all feasible values of x, then a possible subset of S is

A value of c for which the minimum value of f(x) = x² – 4cx + 8c is greater than the maximum value of g(x) = –x² + 3cx – 2c, is

All the values of \( x \) satisfying the inequality \( \frac{1}{x + 5} \leq \frac{1}{2x - 3} \) are:

If \( m \) and \( n \) are natural numbers such that \( n > 1 \), and \( mn = 2^{25} \times 3^{40} \), then \( m - n \) equals:

If x and y are real numbers such that 4x² + 4y² – 4xy – 6y + 3 = 0, then the value of (4x + 5y) is

If x and y satisfy the equations |x| + x + y = 15 and x + |y| – y = 20, then (x – y) equals

The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c, and d, with each digit appearing exactly once in every number, is 153310 + n, where n is a single digit natural number. Then, the value of (a + b + c + d + n) is ______ TITA

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√2)^0.5 − (x − 6√2)^0.5 = 2√2, then x equals:

If 4x² + 4y² − 4xy − 6y + 3 = 0, find value of (4x + 5y).

Let x, y, and z be real numbers satisfying 4(x² + y² + z²) = a, 4(xyz) = 3 + a. Then a equals:

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:

Suppose \( f(x, y) \) is a real-valued function such that \( f(3x + 2y, 2x - 5y) = 19x \) for all real numbers \( x \) and \( y \). The value of \( x \) for which \( f(x, 2x) = 27 \), is:

Let a, b, m and n be natural numbers such that a > 1 and b > 1. If a m b n 144 145 , then the largest possible value of n-m is

The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

For any natural numbers m, n and k, such that k divides both m + 2n and 3m + 4n, k must be a common divisor of

Any non-zero real numbers x, y such that y ≠ 3 and (x/y) < ((x+3)/(y-3)) will satisfy the condition

Let a m × b n = 144. Then the largest possible value of n − m is

The area of the quadrilateral bounded by the Y-axis, the line x = 5, and the lines |x − y| − |5 − x| = 2, is

If x and y are real numbers such that x² + (x − 2y − 1)² = −4y(x + y), then the value of x − 2y is:

Let n be the least positive integer such that 168 is a factor of 1134n. If m is the least positive integer such that 1134n is a factor of 168m, then m + n equals:

If √(5x + 9) + √(5x − 9) = 3(2 + √2), then √(10x + 9) is equal to:

If x and y are positive real numbers such that log_x(x² + 12) = 4 and 3log_y(x) = 1, then x + y equals:

The number of integer solutions of the equation 2|x|(x² + 1) = 5x² is:

If \( n \) is the least positive integer such that \( 168 \) is a factor of \( 1134^n \), and \( m \) is the least positive integer such that \( 1134^m \) is a factor of \( 168^n \), then \( m + n \) equals:

If \( \frac{\sqrt{y} + \sqrt{z}}{\sqrt{x} + \sqrt{z}} = \frac{\sqrt{x} + \sqrt{y}}{\sqrt{y} + \sqrt{z}} \), then which of the following must always be true?