# CAT Algebra Questions with Video Solutions & Formulas

Algebra is one of the important topic in Quant Section of the CAT Exam.  We've compiled a list of the top Algebra Questions with video solutions, tailored to match the difficulty level of the actual CAT exam. Each question comes with both video and text explanations provided by Maruti Sir, ensuring comprehensive understanding. You can check out these Algebra CAT Previous questions.  Practice a good amount of questions in the CAT Algebra, along with Arithmetic and Geometry, forms a crucial part of the Quant section. These are good sources for practice the Algebra questions, Algebra sections is combine of Linear EquationQuadratic EquationInequalitiesFunctions/ProgressionsLogarithms questions with its intricate equations and formulas, is a cornerstone of the CAT.  For CAT aspirants, mastering algebra is not just a choice; it's a necessity. If you're embarking on your CAT preparation journey and looking to conquer the realm of algebra, you've landed in the right place.

Remember, practice is key to mastery. The more you practice, the more confident you'll become. Utilize these exclusive free resources to enhance your skills. Additionally, solving questions from previous CAT papers and taking CAT mock tests regularly will familiarize you with the exam pattern and boost your confidence. These questions aren't mere exercises; they are actual CAT questions, carefully selected to give you a taste of what to expect on the big day. Check out here for Detailed video solutions for Complete Algebra CAT Previous year Questions. Keep practicing and stay consistent!

## CAT Algebra Weightage Over Past 3 Years

 Year Weightage 2023 4 2022 2 2021 2

## CAT Algebra Formulas

### 1. CAT Logarithms Formulas: Properties of logarithm

Important Lograthim formulas for CAT exam are given here. Logarithms questions are frequently asked in the previous CAT papers. In order to ace this topic and solve the CAT questions, aspirants must be well-versed in the basic concepts and formulas. To help the aspirants, we have made a PDF which consists of all the formulas, tips and tricks to solve these questions. Every formula in this PDF is very important. Click on the below link to download the CAT Logarithms, Surds and Indices Formulas PDF.

$$\log_{a}{1} = 0$$$$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$$ $$\log_{a}{b}^{c} = c \log_{a}{b}$$$$${b}^{\log_{b}{x}} = x$$$ $${x}^{\log_{b}{y}} = {y}^{\log_{b}{x}}$$$$${\log_{a}{\sqrt[n]{b}}} = \dfrac{\log_{a}{b}}{n}$$$ $${\log_{a}{b}} = \dfrac{\log_{c}{b}}{\log_{c}{a}}$$$$${\log_{a}{b}}*{\log_{b}{a}}= 1$$$ $$a^m\times\ a^n=a^{m+n}$$$$$\frac{a^m\ \ }{a^n}\ =a^{m-n}$$$ $$\left(a^m\right)^{^n}=a^{m\times\ n}$$$$$\left(a\times\ b\right)^m\ =a^m\times\ b^m$$$ $$a^{-m}=\ \frac{1}{a^m}$$$$$a^{\frac{m}{n}}=\sqrt[\ n]{a^m}$$$

• Logarithms can be used to quickly find the number of digits in an exponent.

### 2. CAT Linear Equations Formulas:

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax+by = c and mx+ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax+by=c and mx+ny=p. Each of these equations represent two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If $$\frac{a}{m}=\frac{b}{n}\neq\frac{c}{p}$$: This means that both the equations have the same slope but different intersect and hence are parallel to each. Hence, there is no point of intersection and no solution.

If $$\frac{a}{m}\neq\frac{b}{n}$$: They have different slopes and hence must intersect at some point. This results in a Unique solution.

$$\frac{a}{m}=\frac{b}{n}=\frac{c}{p}$$: The two lines have the same slope and intercept. Hence they are the same lines. As they have infinite points common between them, there are infinite many solutions possible.

### 3. CAT Quadratic Equations Formulas:

Quadratic equations are an essential topic in the quantitative aptitude section, and it is vital to have a clear understanding of the formulas related to it. To help the aspirants to ace this topic, we have made a PDF containing a comprehensive list of formulas, tips, and tricks that you can use to solve quadratic equation problems with ease and speed. Click on the below link to download CAT Quadratic Equations Formulas PDF.

1. Quadratic Equation - Given Roots.

If roots are given : (x-a)(x-b)=0 => $$x^2 - (a+b)x + ab = 0$$

If sum s and product p of roots are given: $$x^2 - sx + p = 0$$

If roots are reciprocals of roots of equation $$ax^2 + bx + c = 0$$, then equation is $$cx^2 + bx + a = 0$$

If roots are k more than roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y-k)^2 + b(y-k) + c = 0$$

If roots are k times roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y/k)^2 + b(y/k) + c = 0$$

The General Quadratic equation will be in the form of a$$x^{2}$$+b$$x$$+c = 0

The values of ‘x’ satisfying the equation are called the roots of the equation.

The value of roots, p and q = $$\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$

The sum of the roots = p+q = $$\dfrac{-b}{a}$$

Product of roots = p*q = $$\dfrac{c}{a}$$

If c and a are equal then the roots are reciprocal to each other.

If b = 0, then the roots are equal and are opposite in sign.

3. Discriminant Formulas

Let D denote the discriminant $$b^{2}-4ac$$. Hence, depending on the sign and value of D, nature of the roots would be as follows:

D<0 and abs(D) is not a perfect square: Roots are complex and irrational. They can be represented as p+iq and p-iq where p and q are the real and imaginary parts of the complex roots. p is rational and q is irrational.

D < 0 and abs(D) is a perfect square: Roots are complex but rational. They can be represented as p+iq and p-iq where p and q are both rational.

D=0 : Roots are real and equal. X = -b/2a

D>0 and D is not a perfect square: Roots are conjugate surds

D>0 and D is a perfect square: Roots are real, rational and unequal

## CAT 2023 Algebra questions

#### Question 1

If x is a positive real number such that $$x^8 + \left(\frac{1}{x}\right)^8 = 47$$, then the value of $$x^9 + \left(\frac{1}{x}\right)^9$$ is

#### Question 2

Any non-zero real numbers x,y such that $$y\neq3$$ and $$\frac{x}{y}<\frac{x+3}{y-3}$$, Will satisfy the condition.

#### Question 3

If $$x$$ and $$y$$ are positive real numbers such that $$\log_{x}(x^2 + 12) = 4$$ and $$3 \log_{y} x = 1$$, then $$x + y$$ equals

#### Question 4

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

#### Question 5

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

#### Question 6

If $$\sqrt{5x+9} + \sqrt{5x - 9} = 3(2 + \sqrt{2})$$, then $$\sqrt{10x+9}$$ is equal to

#### Question 7

For a real number x, if $$\frac{1}{2}, \frac{\log_3(2^x - 9)}{\log_3 4}$$, and $$\frac{\log_5\left(2^x + \frac{17}{2}\right)}{\log_5 4}$$ are in an arithmetic progression, then the common difference is

#### Question 8

If $$x$$ and $$y$$ are real numbers such that $$x^{2} + (x - 2y - 1)^{2} = -4y(x + y)$$, then the value $$x - 2y$$ is

#### Question 9

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

#### Question 10

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

#### Question 11

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

#### Question 12

For some positive real number x, if $$\log_{\sqrt{3}}{(x)}+\frac{\log_{x}{(25)}}{\log_{x}{(0.008)}}=\frac{16}{3}$$, then the value of $$\log_{3}({3x^{2}})$$ is

#### Question 13

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

#### Question 14

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

#### Question 15

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

#### Question 16

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

#### Question 17

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

#### Question 18

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

#### Question 19

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

#### Question 20

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