CAT Algebra questions (2020–2025) — 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?

In an examination, there were 75 questions. 3 marks were awarded for each correct answer, 1 mark was deducted for each wrong answer and 1 mark was awarded for each unattempted question. Rayan scored a total of 97 marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

For any real number x, let [x] be the largest Integer less than or equal to x. CAT 2022 Question. TITA _______

Let a, b, c be non-zero real numbers such that b² < 4ac, and f(x) = ax² + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

For any natural number t, suppose the sum of the first t terms of an arithmetic progression is (n + 2n²). If the n^th term of the progression is divisible by 9, then the smallest possible value of n is

The largest real value of a for which the equation |x + a| + |x − 1| = 2 has an infinite number of solutions for x is

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

Let A be the largest positive integer that divides all the numbers of the form 3^k + 4^k + 5^k and B be the largest positive integer that divides all the numbers of the form 4^k + 3(4^k) + 4^k+2, where k is any positive integer. Then (A + B) equals TITA ________

Then the maximum value of f(x) becomes 100 when a is equal to _______ TITA

Consider the arithmetic progression 3, 7, 11, ... and let A_n denote the sum of the first n terms of this progression. Then the value of

If a and b are non-negative real numbers such that a + 2b = 6, then the average of the maximum and minimum possible values of (a + b) is

On day one, there are 100 particles in a laboratory experiment. On day n, where n≥2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals

Consider the pair of equations: x² − xy − x = 22 and y² − xy + y = 34. If x > y, then x − y equals

f(x) = x²+2x−15/x²−7x−18 is negative if and only if

The number of integers n that satisfy the inequalities |n − 60| < |n − 100| < |n − 20| is.

If r is a constant such that |x² − 4 x − 13| = r has exactly three distinct real roots, then the value of r is

If x_0 = 1, x_1 = 2, and x_(n + 2) = 1+xⁿ+1/xⁿ, n = 0, 1, 2, 3,..., then x_(2021) is equal to?

If log₂[3 + log₃{4 + log₄(x − 1)}] − 2 = 0 then 4x equals TITA

Three positive integers x, y and z are in arithmetic progression. If y − x > 2 and xyz = 5(x + y + z), then z − x equals

For all real values of x, the range of the function f(x) = x2+2x+42x2+4x+9 is.

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

For all real numbers x the condition |3x − 20| + |3x − 40| = 20 necessarily holds if.

Suppose one of the roots of the equation a x2 − b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c3 then |a| equals

For a sequence of real numbers x1, x2, ..., xn, if x1 − x2 + x3 − ... + (−1)n + 1xn = n2 + 2n for all natural numbers n, then the sum x49 + x50 equals

If f(x)=x²−7x and g(x)=x+3 , then the minimum value of f(g(x))−3x is

For a real number a, if log15a+log32a/(log15a)(log32a)= = 4 then a must lie in the range

If 3x+2|y|+y=7 and x+|x|+3y=1 , then x+2y is

Consider a sequence of real numbers x1,x2,x3,… such that xn+1=xn+n−1 for all n≥1. If x1=−1 then x100 is equal to

The number of distinct pairs of integers (m, n) satisfying |1+mn| < |m+n| < 5 is

Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if

If f(5 + x) = f(5 − x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

If log4 5 = (log4 y) (log6 √5), then y equals

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?

The number of distinct real roots of the equation (x + 1/x)² − 3(x + 1/x) + 2 = 0 equals

Among 100 students, x1 have birthdays in January, x2 have birthdays in February, and so on. If x0 = max(x1, x2, ..., x12), then the smallest possible value of x0 is

How many distinct positive integer-valued solutions exist to the equation (x² − 7x + 11)(x² − 13x + 42) = 1?

The area of the region satisfying the inequalities |x| − y ≤ 1, y ≥ 0, and y ≤ 1 is.

If y is a negative number such that 2y2log35 = 5log23, then y equals

The number of pairs of integers(x,y) satisfying x ≥ y ≥ −20 and 2x + 5y = 99 is

The value of loga^a/b + logb^b/a, for 1 < a ≤ b cannot be equal to

If x and y are non-negative integers such that x + 9 = z, y + 1 = z and x + y < z + 5, then the maximum possible value of 2x + y equals

Let f(x) = x² + ax + b and g(x) = f(x + 1) − f(x − 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is

The number of integers that satisfy the equality (x2 − 5x + 7)x + 1 = 1 is

In how many ways can a pair of integers (x , a) be chosen such that x² − 2 | x | + | a − 2 | = 0 ?

If x¹ = −1 and x^m = x^m + 1 + (m + 1) for every positive integer m, then x¹⁰⁰ equals

Let N, x and y be positive integers such that N = x + y, 2 25, then how many distinct values are possible for N?

Let loga30 = A, loga5/3= −B and log2a = 1/3, then log3a equals

The area, in sq. units, enclosed by the lines x = 2, y = |x − 2| + 4, the X-axis and the Y-axis is equal to

If f(x+y) = f(x)f(y) and f(5) = 4, then f(10) − f(−10) is equal to

Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m + n is

The points (2 , 1) and (−3 , −4) are opposite vertices of a parellelogram. If the other two vertices lie on the line x + 9y + c = 0, then c is