100+ Most Important Algebra Questions for CAT 2025

Let $$x, y,$$ and $$z$$ be real numbers satisfying
$$4(x^{2}+y^{2}+z^{2})=a,$$
$$4(x-y-z)=3+a$$
The a equals

The roots $$\alpha, \beta$$ of the equation $$3x^2 + \lambda x - 1 = 0$$, satisfy $$\cfrac{1}{\alpha^2} + \cfrac{1}{\beta^2} = 15$$.
The value of $$(\alpha^3 + \beta^3)^2$$, is

In the XY-plane, the area, in sq. units, of the region defined by the inequalities
$$y \geq x + 4$$ and $$-4 \leq x^2 + y^2 + 4(x - y) \leq 0$$ 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 x and y are real numbers such that $$4x^2 + 4y^2 - 4xy - 6y + 3 = 0$$, then the value of $$(4x + 5y)$$ is

Consider two sets $$A = \left\{2, 3, 5, 7, 11, 13 \right\}$$ and $$B = \left\{1, 8, 27 \right\}$$. Let f be a function from A to B such that for every element 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

All the values of x satisfying the inequality $$\cfrac{1}{x + 5} \leq \cfrac{1}{2x - 3}$$ are

The number of distinct integer solutions (x, y) of the equation $$\mid x + y \mid + \mid x - y \mid = 2$$, 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

lf 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

If x and y satisfy the equations $$\mid x \mid + x + y = 15$$ and $$x + \mid y \mid - y = 20$$, then $$(x - y)$$ equals

For any non-zero real number x, let $$f(x) + 2f \left(\cfrac{1}{x}\right) = 3x$$. Then, the sum of all possible values of x for which $$f(x) = 3$$, is

The number of distinct real values of x, satisfying the equation $$max \left\{x, 2\right\} - min\left\{x, 2\right\} = \mid x + 2 \mid - \mid x - 2 \mid$$, is

If $$p^{2}+q^{2}-29=2pq-20=52-2pq$$, then the difference between the maximum and minimum possible value of $$(p^{3}-q^{3})$$

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

If a certain amount of money is divided equally among n persons, each one receives Rs 352. However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330. Then, the maximum possible value of n 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 $$\frac{k}{4y}+9y$$ is

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 $$y\neq3$$ and $$\frac{x}{y}<\frac{x+3}{y-3}$$, Will satisfy the condition.

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

A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is

The population of a town in 2020 was 100000. The population decreased by y% from the year 2020 to 2021, and increased by x% from the year 2021 to 2022, where x and y are two natural numbers. If population in 2022 was greater than the population in 2020 and the difference between x and y is 10, then the lowest possible population of the town in 2021 was

A quadratic equation $$x^2 + bx + c = 0$$ has two real roots. If the difference between the reciprocals of the roots is $$\frac{1}{3}$$, and the sum of the reciprocals of the squares of the roots is $$\frac{5}{9}$$, then the largest possible value of $$(b + c)$$ is

Let n be any natural number such that $$5^{n-1} < 3^{n + 1}$$. Then, the least integer value of m that satisfies $$3^{n+1} < 2^{n+m}$$ for each such n, is

For some real numbers a and b, the system of equations $$x + y = 4$$ and $$(a+5)x+(b^2-15)y=8b$$ 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 $$8^n$$, which can be expressed as powers of 2. Then, the smallest possible value of n + m is

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

Let $$\alpha$$ and $$\beta$$ be the two distinct roots of the equation $$2x^{2} - 6x + k = 0$$, such that ( $$\alpha + \beta$$) and $$\alpha \beta$$ are the distinct roots of the equation $$x^{2} + px + p = 0$$. Then, the value of 8(k - p) is

The equation $$x^{3} + (2r + 1)x^{2} + (4r - 1)x + 2 =0$$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

The number of integer solutions of equation $$2|x|(x^{2}+1) = 5x^{2}$$ is

If $$x$$ and $$y$$ are real numbers such that $$x^{2} + (x - 2y - 1)^{2} = -4y(x + y)$$, then the value $$x - 2y$$ 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 ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is

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

Let r be a real number and $$f(x) = \begin{cases}2x -r & ifx \geq r\\ r &ifx; < r\end{cases}$$. Then, the equation $$f(x) = f(f(x))$$ holds for all real values of $$x$$ where

If $$(3+2\sqrt{2})$$ is a root of the equation $$ax^{2}+bx+c=0$$ and $$(4+2\sqrt{3})$$ is a root of the equation $$ay^{2}+my+n=0$$ where a, b, c, m and n are integers, then the value of $$(\frac{b}{m}+\frac{c-2b}{n})$$ is

Suppose k is any integer such that the equation $$2x^{2}+kx+5=0$$ has no real roots and the equation $$x^{2}+(k-5)x+1=0$$ 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 number of distinct integer values of n satisfying $$\frac{4-\log_{2}n}{3-\log_{4}n} < 0$$, 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

Suppose for all integers x, there are two functions f and g such that $$f(x) + f (x - 1) - 1 = 0$$ and $$g(x ) = x^{2}$$. If $$f\left(x^{2} - x \right) = 5$$, then the value of the sum f(g(5)) + g(f(5)) is

Let r and c be real numbers. If r and -r are roots of $$5x^{3} + cx^{2} - 10x + 9 = 0$$, then c equals

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

The number of integer solutions of the equation $$\left(x^{2} - 10\right)^{\left(x^{2}- 3x- 10\right)} = 1$$ 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

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

The largest real value of a for which the equation $$\mid x + a \mid + \mid x - 1 \mid = 2$$ has an infinite number of solutions for x is

Let a and b be natural numbers. If $$a^2 + ab + a = 14$$ and $$b^2 + ab + b = 28$$, then $$(2a + b)$$ equals

Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

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

If $$3x+2\mid y\mid+y=7$$ and $$x+\mid x \mid+3y=1$$ then $$x+2y$$ is:

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

If $$f(x)=x^{2}-7x$$ and $$g(x)=x+3$$, then the minimum value of $$f(g(x))-3x$$ is:

The number of distinct pairs of integers (m,n), satisfying $$\mid1+mn\mid<\mid m+n\mid<5$$ 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

Consider the pair of equations: $$x^{2}-xy-x=22$$ and $$y^{2}-xy+y=34$$. If $$x>y$$, then $$x-y$$ equals

For a real number x the condition $$\mid3x-20\mid+\mid3x-40\mid=20$$ necessarily holds if

For all real values of x, the range of the function $$f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}$$ is:

Suppose one of the roots of the equation $$ax^{2}-bx+c=0$$ is $$2+\sqrt{3}$$, Where a,b and c are rational numbers and $$a\neq0$$. If $$b=c^{3}$$ then $$\mid a\mid$$ equals.

If r is a constant such that $$\mid x^2 - 4x - 13 \mid = r$$ has exactly three distinct real roots, then the value of r is

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

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

The number of integers n that satisfy the inequalities $$\mid n - 60 \mid < \mid n - 100 \mid < \mid n - 20 \mid$$ is

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

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

In how many ways can a pair of integers (x , a) be chosen such that $$x^{2}-2\mid x\mid+\mid a-2\mid=0$$ ?

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

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

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 May, John bought the same amount of rice and the same amount of wheat as he had bought in April, but spent ₹ 150 more due to price increase of rice and wheat by 20% and 12%, respectively. If John had spent ₹ 450 on rice in April, then how much did he spend on wheat in May?

The number of pairs of integers $$(x,y)$$ satisfying $$x\geq y\geq-20$$ and $$2x+5y=99$$

The number of integers that satisfy the equality $$(x^{2}-5x+7)^{x+1}=1$$ is

In a group of 10 students, the mean of the lowest 9 scores is 42 while the mean of the highest 9 scores is 47. For the entire group of 10 students, the maximum possible mean exceeds the minimum possible mean by

Let $$f(x)=x^{2}+ax+b$$ and $$g(x)=f(x+1)-f(x-1)$$. If $$f(x)\geq0$$ for all real x, and $$g(20)=72$$. then the smallest possible value of b is

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

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

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

Dick is thrice as old as Tom and Harry is twice as old as Dick. If Dick's age is 1 year less than the average age of all three, then Harry's age, in years, is

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

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

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

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

Among 100 students, $$x_1$$ have birthdays in January, $$X_2$$ have birthdays in February, and so on. If $$x_0=max(x_1,x_2,....,x_{12})$$, then the smallest possible value of $$x_0$$ is

The area of the region satisfying the inequalities $$\mid x\mid-y\leq1,y\geq0$$ and $$y\leq1$$ is

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

The number of real-valued solutions of the equation $$2^{x}+2^{-x}=2-(x-2)^{2}$$ 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?

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

Let S be the set of all points (x, y) in the x-y plane such that $$\mid x \mid + \mid y \mid \leq 2$$ and $$\mid x \mid \geq 1.$$ Then, the area, in square units, of the region represented by S equals

The number of the real roots of the equation $$2 \cos (x(x + 1)) = 2^x + 2^{-x}$$ 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 8f(m + 1) - f(m) = 2, then m equals

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 (2$$^n$$ - 1) then a is equal to

The product of the distinct roots of $$\mid x^2 - x - 6 \mid = x + 2$$ is

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, b, x, y be real numbers such that $$a^2 + b^2 = 25, x^2 + y^2 = 169$$, and $$ax + by = 65$$. If $$k = ay - bx$$, then

In 2010, a library contained a total of 11500 books in two categories - fiction and nonfiction. In 2015, the library contained a total of 12760 books in these two categories. During this period, there was 10% increase in the fiction category while there was 12% increase in the non-fiction category. How many fiction books were in the library in 2015?

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

If x is a real number, then $$\sqrt{\log_{e}{\frac{4x - x^2}{3}}}$$ is a real number if and only if

The quadratic equation $$x^2 + bx + c = 0$$ has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of $$b^2 + c$$?

Let A be a real number. Then the roots of the equation $$x^2 - 4x - log_{2}{A} = 0$$ are real and distinct if and only if

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

The smallest integer $$n$$ such that $$n^3-11n^2+32n-28>0$$ is

Let f(x)= $$\max(5x, 52-2x^2)$$, where x is any positive real number. Then the minimum possible value of f(x)

If $$U^{2}+(U-2V-1)^{2}$$= −$$4V(U+V)$$ , 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

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

Let $$f(x) =2x-5$$ and $$g(x) =7-2x$$. Then |f(x)+ g(x)| = |f(x)|+ |g(x)| if and only if

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is

How many different pairs(a,b) of positive integers are there such that $$a\geq b$$ and $$\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$$?

The minimum possible value of the sum of the squares of the roots of the equation $$x^2+(a+3)x-(a+5)=0 $$ is

Let $$f(x) = x^{2}$$ and $$g(x) = 2^{x}$$, for all real x. Then the value of f[f(g(x)) + g(f(x))] at x = 1 is

$$f(x) = \frac{5x+2}{3x-5}$$ and $$g(x) = x^2 - 2x - 1$$, then the value of $$g(f(f(3)))$$ is

If $$f_{1}(x)=x^{2}+11x+n$$ and $$f_{2}(x)=x$$, then the largest positive integer n for which the equation $$f_{1}(x)=f_{2}(x)$$ has two distinct real roots is

For how many integers n, will the inequality $$(n - 5) (n - 10) - 3(n - 2)\leq0$$ be satisfied?

The number of solutions $$(x, y, z)$$ to the equation $$x - y - z = 25$$, where x, y, and z are positive integers such that $$x\leq40,y\leq12$$, and $$z\leq12$$ is

If $$x+1=x^{2}$$ and $$x>0$$, then $$2x^{4}$$  is

If a and b are integers of opposite signs such that $$(a + 3)^{2} : b^{2} = 9 : 1$$ and $$(a -1)^{2}:(b - 1)^{2} = 4:1$$, then the ratio $$a^{2} : b^{2}$$ is