Algebra — CAT Previous-Year Questions

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

For any natural number n, let a_n be the largest integer not exceeding √n. Then the value of a₁ + a₂ + a₃ + ... + a₅₀ is

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

Let x, y, and z be real numbers satisfying

4(x² + y² + z²) = a,

4(x - y - z) = 3 + a

Then a equals

When 10¹⁰⁰ is divided by 7, the remainder is

If the equations x² + mx + 9 = 0, x² + nx + 17 = 0 and x² + (m + n)x + 35 = 0 have a common negative root, then the value of (2m + 3n) is

Suppose x₁, x₂, x₃, ..., x₁₀₀ are in arithmetic progression such that x₅ = -4 and 2x₆ + 2x₉ = x₁₁ + x₁₃. Then x₁₀₀ equals

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 function f is

The sum of all real values of k for which ${\left(\frac{1}{8}\right)}^k$ × ${\left(\frac{1}{32768}\right)}^{\frac{1}{2}}$ = $\left(\frac{1}{8}\right)$ × ${\left(\frac{1}{32768}\right)}^{\frac{1}{k}}$, is

If x is a positive number such that 4 log₁₀x + 4 log₁₀₀x + 8 log₁₀₀₀x = 13, then the greatest integer not exceeding x, is

The roots of α, β of the equation 3x² + λx - 1 = 0, satisfy $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ = 15.

The value of (α³ + β³)², is

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

If a, b and c are postive real numbers such that a > 10 ≥ b ≥ c and $\frac{{\log }_8(a+b)}{{\log }_{2}c}$ + $\frac{{\log }_{27}(a-b)}{{\log }_{3}c}$ = $\frac{2}{3}$, then the greatest possible integer value of a is

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

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

If m and n are natural numbers such that n > 1, and m = 2²⁵ × 3⁴⁰, then m - n equals

When 3³³³ is divided by 11, the remainder is

The value of x satisfying the inequality $\frac{1}{x+5}$ ≤ $\frac{1}{2x-3}$ are

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

If x and y satisfy the equations |x| + x + y = 15 and x + |y| - y = 20, then (x - y) 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 10⁶⁸ is divided by 13, the remainder is

The number of distinct real values of x, satisfying the equation 

max{x, 2} - min{x, 2} = |x + 2| - |x - 2|, is

For some constant real numbers p, k and a, consider the following system of linear equations in x and y:

px - 4y = 2

3x + ky = a

A necessary condition for the system to have no solution for (x, y), is

(a + b√3)² = 52 + 30√3, where a and b are natural numbers, then a + b equals

For any non-zero real number x, let f(x) + 2f(1/x) = 3x. Then, the sum of all possible values of x for which f(x) = 3, is

The number of distinct integer solutions (x, y) of the equation |x + y| + |x - y| = 2, is

If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8 and 8^f = 9, then the value of the product abcdef is

Consider the sequence t₁ = 1, t₂ = -1 and t_n = $\left(\frac{n-3}{n-1}\right)$t_n-2 for n ≥ 3. Then, the value of the sum $\frac{1}{t_2}$ + $\frac{1}{t_4}$ + $\frac{1}{t_6}$ + ... + $\frac{1}{t_{2022}}$ + $\frac{1}{t_{2024}}$

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

The number of integral solutions of equation 2|x|(x² + 1) = 5x² is?

Let α and β be two distinct root of the equation 2x² – 6x + k = 0, such that (α + β) and αβ are the two roots of the equation x² + px + p = 0. Then the value of 8(k - p)?

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

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 n^th day exceeds one million, then the lowest possible value of n 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 168^m, then m + n equals

The equation x³ + (2r + 1)x² + (4r - 1)x + 2 = 0 has -2 as one of the roots. If the other roots are real, then the minimum possible non-negative integer value of r is?

If $\sqrt{5x+9}$ + $\sqrt{5x-9}$ = 3(2 + √2), then find the value of $\sqrt{10x+9}$?

For some positive and distinct real numbers x, y and z if $\frac{1}{\sqrt{\mathrm{y}}+\sqrt{\mathrm{z}}}$ is the arithmetic mean of $\frac{1}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{z}}}$ and $\frac{1}{\sqrt{\mathrm{x}}+\sqrt{y}}$, then the relationship which will always hold true, 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

The sum of all possible values of x satisfying the equation ${2^{4x}}^2$ - $2^{2x^2+x+16}$ + $2^{2x+30}$ = 0, is

Any non-zero real numbers x, y such that y ≠ 3 and $\frac{\mathrm{x}}{\mathrm{y}}$ < $\frac{\mathrm{x}+3}{\mathrm{y}-3}$, will satisfy the condition

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

Let k be the largest integer such that the equation (x - 1)² + 2kx + 11 = 0 has no real roots. If y is a positive real number, then the least possible value of k/4y + 9y is?

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

For some positive ral number x, if ${\log }_{\sqrt{3}} \left(x\right)$ + $\frac{{\log }_x 25}{{\log }_x (0.008)}$ = $\frac{16}{3}$, then the value of ${\log }_3 \left(3x^2\right)$ is

If p² + q² - 29 = 2pq - 20 = 52 - 2pq, then the difference between the maximum and minimum possible value of (p³ - q³) is

Let both the series a₁, a₂, a₃, ... and b₁, b₂, b₃, ... be in arithmetic progression such that the common differences of both the series are prime numbers. If a₅ = b₉, a₁₉ = b₁₉ and b₂ = 0, then a₁₁ equal?

Let a_n and b_n be two sequences such that a_n = 13 + 6(n - 1) and b_n = 15 + 7(n - 1) for all natural numbers n. Then, the largest three digit integer that is common to both these sequences is

For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b² -15)y = 8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is?

Let n and m be two positive integers such that there are exactly 41 integers greater than 8^m and less than 8ⁿ, 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 x⁸ + ${\left(\frac{1}{\mathrm{x}}\right)}^8$ = 47, then the value of x⁹ + ${\left(\frac{1}{\mathrm{x}}\right)}^9$ is

For a real number x, if $\frac{1}{2}$, $\frac{{\log }_3(2^x-9)}{{\log }_{3}4}$, and $\frac{{\log }_5(2^x+{\displaystyle \frac{17}{2}})}{{\log }_{5}4}$ are in arithmetic progression, then the common difference is

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

A quadratic equation x² + bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is

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?

Let a_n = 46 + 8n and b_n = 98 + 4n be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is

The value of 1 + $\left(1+\frac{1}{3}\right)\frac{1}{4}$ + $\left(1+\frac{1}{3}+\frac{1}{9}\right)\frac{1}{16}$ + $\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right)\frac{1}{64}$ + ..., 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 and b be natural numbers. If a² + ab + a = 14 and b² + ab + b = 28, then (2a + b) equals

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 numbesr of the form 4^k + 3(4^k) + 4^k+2, where k is any positive integer. Then (A + B) equals

Let a, b and 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 natural numbers x, y and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is

For any natural number n, suppose the sum of the first n 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

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum _{n=1}^N\left[\frac{1}{5}+\frac{n}{25}\right]$ = 25, then N is

Let 0 ≤ a ≤ x ≤ 100 and f(x) = |x - a| + |x - 100| + |x - a - 50|. Then the maximum value of f(x) becomes 100 when a is equal to

The number of distinct integer values of n satisfying $\frac{4-{\log }_2\mathrm{n}}{3-{\log }_4\mathrm{n}}$ < 0, is

Suppose for all integers x, there are two functions f and g such that f(x) + f(x - 1) - 1 = 0 and g(x) = x². If f(x² - x) = 5, then the value of the sum f(g(5)) + g(f(5)) is

Consider the arithmetic progressions 3, 7, 11, ... and let An dentoe the sum of the first n terms of this progression. Then the value of $\frac{1}{25}\underset{\mathrm{n}=1}{\overset{25}{\sum {\mathrm{A}}_{\mathrm{n}}}}$

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

For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is:

Let f(x) be a quadratic ploynomial in x such that f(x) ≥ 0 for all real numbers x. If f(2) = 0 and f(4) = 6, then f(-2) is equal to

Let r and c be real numbers. If r and -r are roots of 5x³ + cx² - 10x + 9 = 0, then c equals

Manu earns ₹4000 per month and wants to save an average of ₹550 per month in a year. In the first nine months, his monthly expense was ₹3500, and he foresees that, tenth month onward, his monthly expense will increase to ₹3700. In order to meet his yearly savings target, his monthly earnings, in rupees, from the tenth month onward should be:

Five students, including Amit, appear for an examination in which possible marks are integers between 0 and 50, both inclusive. The average marks for all the students is 38 and exactly three students got more than 32. If no two students got the same marks and Amit got the least marks among the five students, then the difference between the highest and lowest possible marks of Amit is

The number of integral solutions of the equation ${({\mathrm{x}}^2-10)}^{({\mathrm{x}}^2-3\mathrm{x}-10)}$ = 1 is

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:

On day one, there are 100 particles in a laboratory experiment. On day n, where n greater than or 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.

The minimum possibe value of $\frac{x^2-6x+10}{3-x}$, for x < 3, is

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

Suppose k is any integer such that the equation 2x² + kx + 5 = 0 has no real roots and the equation x² + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is

If c = $\frac{16x}{y}$ + $\frac{49y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

Let r be a real number and f(x) = $\{\begin{array}{cc}2\mathrm{x}-\mathrm{r}& \mathrm{if} \mathrm{x}\ge \mathrm{r}\\ \mathrm{r}& \mathrm{if} \mathrm{x}<\mathrm{r}\end{array}.$Then, the equation f(x) = f(f(x)) holds for all real values of x where

If (3 + 2√2) is a root of the equation ax² + bx + c = 0, and (4 + 2√3) is a root of the equation ay² + my + n = 0, where a, b, c, m and n are integers, then the value of $\left(\frac{b}{m}+\frac{c-2b}{n}\right)$ is

A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of â‚¹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is 

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

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

If 5 – ${\log }_{10}\left(\sqrt{1+x}\right)$ + 4${\log }_{10}\left(\sqrt{1-x}\right)$ = ${\log }_{10}\left(\frac{1}{\sqrt{1-x^2}}\right)$, then 100x equals

A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, or a basket of 8 oranges and 7 mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

f(x) = $\frac{x^2+2x-15}{x^2-7x-18}$ is negative if and only if

How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?

If x₀ = 1, x₁ = 2 and x_n+2 = $\frac{1+x_{n+1}}{x_n}$, n = 0, 1, 2, 3, …, then x₂₀₂₁ is equal to

The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15^th group is equal to

A box has 450 balls, each either white or black, there being as many metallic white balls as metallic black balls. If 40% of the white balls and 50% of the black balls are metallic, then the number of non-metallic balls in the box is

For a 4-digit number, the sum of its digits in the thousands, hundreds and tens places is 14, the sum of its digits in the hundreds, tens and units places is 15, and the tens place digit is 4 more than the units place digit. Then the highest possible 4-digit number satisfying the above conditions is

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

log₂ [3 + log₃ {4 + log₄ (x - 1)}] - 2 = 0, then 4x equals

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

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

For a real number x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

Suppose one of the roots of the equation ax² – bx + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c³ then |a| equals

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) = $\frac{x^2+2x+4}{2x^2+4x+9}$ is

For a real number a, if $\frac{lo{g}_{15}a+lo{g}_{32}a}{lo{g}_{15}a\times lo{g}_{32}a}$ = 4, then a must lie in the range

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

The cost of fencing a rectangular plot is ₹ 200 per ft along one side, and ₹ 100 per ft along the three other sides. If the area of the rectangular plot is 60000 sq. ft, then the lowest possible cost of fencing all four sides, in INR, is 

In ‘n’ is a positive integer such that $\left(\sqrt[7]{10}\right)$${\left(\sqrt[7]{10}\right)}^2$...${\left(\sqrt[7]{10}\right)}^n$ > 999, then the smallest value of n is

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 x_n+1 = x_n + n – 1 for all n ≥ 1. If x₁ = -1 then x₁₀₀

A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt. She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is

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

A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of 5 children. How many toffees were there in his stock initially?

How many distinct positive integer-valued solutions exist to the equation ${\left(x^2-7x+11\right)}^{(x^2-13x+42)}$ = 1?

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

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

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

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

The number of distinct real roots of the equation ${\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^2-3\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)+2$ = 0 equals

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

If y is a negative number such that $2^{{\mathrm{y}}^2{\log }_{3}5}=5^{{\log }_{2}3}$, then y equals

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

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

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

Let the m-th and n-th terms of a geometric progression be 3/4 and 12, respectively, where m < n. If the common ratio of the progression is an integer r, then the smallest possible value of r + n - m is

For real x, the maximum possible value of $\frac{x}{\sqrt{1+x^4}}$ is

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

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

If x and y are positive real numbers satisfying x + y = 102, then the minimum possible value of 2601$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)$ is

    The value of ${\log }_a\left(\frac{a}{b}\right)+{\log }_b\left(\frac{b}{a}\right)$, for 1 < a ≤ b cannot be equal to

Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is ₹ 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is

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

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

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

If x₁ = - 1 and x_m = x_m+1 + (m + 1) for every positive integer m, then x₁₀₀ equals

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

Let m and n be natural numbers such that n is even and 0.2 < $\frac{\mathrm{m}}{20},\frac{\mathrm{n}}{\mathrm{m}},\frac{\mathrm{n}}{11}$ < 0..5. Then m – 2n equals

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

Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?

$\frac{2\times 4\times 8\times 16}{\left({{\log }_{2}4)}^2\right({{\log }_{4}8)}^3({{\log }_{8}16)}^4}$ equals

If log_a30 = A, log_a(5/3) = -B and log₂a = 1/3, then log₃a equals

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

How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?

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

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

If a₁ + a₂ + a₃ + … + a_n = 3(2ⁿ⁺¹ – 2), for every n ≥ 1, then a₁₁ equals 

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is

The number of solutions to the equation |x|(6$x^2$ + 1) = 5$x^2$ is

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8 f(m + 1) − f(m) = 2, then m equals

If the population of a town is p in the beginning of any year then it becomes 3 + 2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

If a₁, a₂, ... are in A.P., then, $\frac{1}{\sqrt{a_1}+\sqrt{a_2}}$ + $\frac{1}{\sqrt{a_2}+\sqrt{a_3}}$ + ... + $\frac{1}{\sqrt{a_n}+\sqrt{a_{n+1}}}$ is equal to

The product of the distinct roots of ∣x² − x − 6∣ = x + 2 is

Let x and y be positive real numbers such that log₅(x + y) + log₅(x - y) = 3, and log₂y - log₂x = 1 - log₂3. Then xy equals

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

Consider a function f satisfying  f(x + y) = f(x) f(y) where x, y are positive integers and f(1) = 2. If  f(a + 1) + f(a + 2) +…+ f(a + n) = 16(2ⁿ – 1) then a is equal to

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

Let a, b, x, y be real numbers such that a² + b² = 25 , x² + y² = 169 and ax + by = 65. If k = ay - bx, then

Let a₁ , a₂ be integers such that a₁ - a₂ + a₃ - a₄ + ........ + (-1)^n-1 a_n = n , for n ≥ 1. Then a₅₁ + a₅₂ + ........ + a₁₀₂₃ equals

How many factors of 2⁴ × 3⁵ × 10⁴ are perfect squares which are greater than 1?

What is the largest positive integer such that $\frac{(n^2+7n+12)}{(n^2-n-12)}$ is also positive integer?

Let f be a function such that f(mn) = f(m) × f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals

Let A be a real number. Then the roots of the equation x² - 4x - log₂A = 0 are real and distinct if and only if

If x is a real number, then $\sqrt{{\log }_e\left(\frac{4x-x^2}{3}\right)}$ is a real number if and only if

How many pairs (m,n) of positive integers satisfy the equation m² + 105 = n²?

The quadratic equation x² + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b² + c?

In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is

The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 is

If (2n+1) + (2n+3) + (2n+5) + ... + (2n+47) = 5280 , then what is the value of 1 + 2 + 3 + ... + n?

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

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

If x is a positive quantity such that 2^x = $3^{{\log }_5\left(2\right)}$ , then x is equal to

If log₁₂81 = p, then $3\left(\frac{4-p}{4+p}\right)$ is equal to

If log₂(5 + log₃ a) = 3 and log₅(4a + 12 + log₂ b) = 3, then a + b is equal to

If u² + (u−2v−1)² = −4v(u + v), then what is the value of u + 3v?

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

The number of integers x such that 0.25 ≤ 2^x ≤ 200, and 2^x + 2 is perfectly divisible by either 3 or 4, is 

While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is

Let f(x) = min {2x², 52 − 5x}, where x is any positive real number. Then the maximum possible value of f(x) is

Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is

The arithmetic mean of x, y and z is 80, and that of x, y, z, u and v is 75, where u = (x + y)/2 and v = (y + z)/2. If x ≥ z, then the minimum possible value of x is

If p^₃ = q⁴ = r⁵ = s⁶, then the value of log_s(pqr) is equal to

The smallest integer n for which 4ⁿ > 17¹⁹ holds, is closest to

$\frac{1}{l{\mathrm{og}}_{2}100}-\frac{1}{{\log }_{4}100}+\frac{1}{{\log }_{5}100}-\frac{1}{{\log }_{10}100}+\frac{1}{{\log }_{20}100}-\frac{1}{{\log }_{25}100}+\frac{1}{{\log }_{50}100}?$

If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

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

If a and b are integers such that 2x² − ax + 2 > 0 and x² − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

The smallest integer n such that n³ – 11n² + 32n – 28 > 0 is

Let t₁, t₂,… be real numbers such that t₁ + t₂ + … + t_n = 2n² + 9n + 13, for every positive integer n ≥ 2. If t_k = 103, then k equals

Let f(x) = max {5x, 52 – 2x²}, where x is any positive real number. Then the minimum possible value of f(x) is

The value of the sum 7 × 11 + 11 × 15 + 15 × 19 + ...+ 95 × 99 is

If A = {6²ⁿ - 35n - 1: n = 1, 2, 3, ...} and B = {35(n - 1) : n = 1,2,3,...} then which of the following is true?

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:

Suppose, log₃x = log₁₂y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log₆G is equal to:

If x + 1 = x² and x > 0, then 2x⁴ is:

The value of log_0.008√5 + log_√381 – 7 is equal to:

If 9^2x–1 – 81^x-1 = 1944, then x is

For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied?

If f₁(x) = x² + 11x + n and f₂(x) = x, then the largest positive integer n for which the equation f₁(x) = f₂(x) has two distinct real roots, is :

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)² + (a - c)² + (a - d)² is

The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is

If the square of the 7^th term of an arithmetic progression with positive common difference equals the product of the 3^rd and 17^th terms, then the ratio of the first term to the common difference is:

If f(x) = (5x+2)/(3x-5) and g(x) = x² – 2x – 1, then the value of g(f(f(3))) is:

Let a₁, a₂,.......a_3n be an arithmetic progression with a₁ = 3 and a₂ = 7. If a₁ + a₂ + ......+ a_3n = 1830, then what is the smallest positive integer m such that m(a₁ + a₂ + ..... + a_n) > 1830?

If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

If x is a real number such that log₃5 = log₅(2 + x), then which of the following is true?

Let f(x) = x² and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

The minimum possible value of the squares of the roots of the equation: x² + (a + 3)x – (a + 5) = 0 is