Are you preparing for CAT 2025 and wondering what the weightage of Algebra has been in previous years’ exams? Do you want to know which Algebra topics are most expected in CAT 2025? In this article, we will answer these questions and discuss all the important aspects of Algebra that you must focus on for effective preparation. Stay tuned to know more.
This Story also Contains
Introduction to CAT Algebra in Quantitative Aptitude
Overview of Algebra in the CAT Exam
Type 1 – Linear Equations in CAT Algebra
Type 2 – Quadratic Equations in CAT Algebra
Type 3 – Inequalities and Modulus Equations
Type 4 – Functions & Polynomials in CAT Algebra
Advanced CAT Algebra Problem Types
Algebra Important Topics for CAT 2025 & Beyond
CAT Algebra Preparation Tips
CAT 2025 Preparation Resources by Careers360
Common Mistakes in CAT Algebra Preparation
How to Practice Repeated Algebra Questions in CAT
CAT 2025 Algebra Decoded: 4 Equation Types That Appear Year After Year
Introduction to CAT Algebra in Quantitative Aptitude
Algebra has numerous topics and concepts. It focuses on functions, equations, inequalities, modulus, etc. It helps us to build a strong foundation and carries a significant weightage in the CAT exam. Algebra in CAT will certainly boost your CAT percentile in the quant section.
Why Algebra dominates CAT Quant every year?
Algebra is the most dominant topic in the quantitative aptitude section of CAT and other MBA entrance exams. Why is it so? The answer lies within its consistency over the past few years. Algebra has the highest weightage among the others in CAT. Apart from direct questions from algebra, it acts as an important tool to solve the questions of other domain like Arithmetic, logarithm, surds and exponents, and Modern Mathematics.
Must-know Algebra concepts for CAT candidates
We have analysed the last 5 years of CAT question papers to give you some insight on the concepts that have been asked previously, as listed below:
Topic
Important Concept to understand
Complex numbers
Iota, conjugate of complex number
Linear Equations
Forming and solving linear equations, nature of solutions, graphs
Algebra in the CAT exam forms a key part of the Quantitative Ability section, testing concepts like equations, inequalities, sequences, and functions. Mastery of these topics helps in solving problems quickly and accurately, boosting overall CAT 2025 performance.
Weightage of Algebra in CAT Quant section
As seen in previous years, here is a summary (year-wise and slot-wise) of the number of questions asked from Algebra in CAT
S. No.
Year
Slot
Equations
Functions
Inequalities and modulus
Sequence and Series
Log, surds and indices
Others
Total
1
2020
1
4
1
1
-
2
-
8
2
2020
2
4
1
-
3
1
-
9
3
2020
3
2
1
-
1
3
-
7
4
2021
1
1
1
1
1
1
-
5
5
2021
2
3
1
-
2
1
-
7
6
2021
3
1
1
1
2
1
-
6
7
2022
1
1
2
-
1
-
1
5
8
2022
2
3
1
-
1
1
-
6
9
2022
3
3
1
-
-
-
1
5
10
2023
1
5
-
-
2
1
-
8
11
2023
2
2
-
1
2
1
1
7
12
2023
3
2
1
-
3
-
1
7
13
2024
1
2
1
1
1
1
-
7
14
2024
2
4
1
1
1
2
-
9
15
2024
3
4
1
-
1
1
-
7
Why mastering equations is the game-changer
As we have seen that every year, on average, 3 to 4 questions are asked from equations. Equations can be linear, quadratic, or miscellaneous. Mastering these equations can be the game changer in CAT-2025 as it comprises of healthy weightage in the quantitative aptitude section.
Apart from this, solving equations mainly focuses on finding the value or the range of values of the unknown variable. These questions have a lower possibility of being wrong. Also, practising these questions will help in developing critical and analytical thinking.
Type 1 – Linear Equations in CAT Algebra
Linear equations can be in one dimension, two dimensions or three dimensions. Linear equations are a fundamental topic in CAT Algebra, involving equations of the first degree with one or more variables. These questions test your ability to solve for unknowns efficiently and form the basis for more complex problem-solving.
Example 1: 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: [CAT 2023, slot 3]
Solution:
For the given equations: $x +y = 4$ and $(a+5)x + (b^2 -15)y = 8b$ Condition for infinite many solutions $\frac {1}{a+5} = \frac {1}{ b^2 -15} =\frac {4}{8b}$ Solving $\frac {1}{ b^2 -15} =\frac {4}{8b}$ $⇒8b =4{(b^2 -15)}$ On solving this quadratic equation, we get $b = -3, 5$. Solving $\frac {1}{ a+5} =\frac {4}{8b}$ $⇒8b =4{(a+5)}$ For $b = -3, a = -11$ and hence, $ab = (-3) \times (-11) = 33$ For $b = 5, a = 5$ and hence, $ab = (5) \times (5) = 25$ So, the maximum value of $ab$ is 33.
Example 2: The number of distinct integer solutions $(x, y)$ of the equation $|x+y|+|x-y|=2$, is [CAT 2024, slot 3]
Solution:
We’ll analyze all integer pairs $(x, y)$ such that the equation holds.
Let’s denote: $A = |x + y|$; $B = |x - y|$
We are told: $A + B = 2$ Since both $A$ and $B$ are non-negative and integers, possible values of $(A, B)$ are: $(0, 2)$, $(1, 1)$, and $(2, 0)$
Let’s find all integer solutions $(x, y)$ for each case.
Case 1: $|x + y| = 0$ and $|x - y| = 2$
Then $x + y = 0$ and $x - y = \pm 2$
Solving:
$x + y = 0$, $x - y = 2$ Solving: $x = 1$, $y = -1$
$x + y = 0$, $x - y = -2$ Solving: $x = -1$, $y = 1$
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Quadratic equations in CAT Algebra involve second-degree equations with one or more variables. Questions on this topic assess your skills in factorization, applying the quadratic formula, and analyzing roots for problem-solving.
An equation of the form $ax^2+bx+c=0$ where a, b and c are all real and a is not equal to 0, is a quadratic equation.
Key quadratic identities for CAT
There are a few quadratic identities which are frequently used: 1. Difference of Squares: a² - b² = (a + b)(a - b).
2. Square of a Binomial: (a + b)² = a² + 2ab + b².
3. Square of a Binomial: (a - b)² = a² - 2ab + b².
Approaches to solving quadratic equations
There are a few approaches to solving the quadratic equation:
1. Quadratic Formula (Shridharacharya Formula)
For an equation of the form
$ax^2+bx+c=0$
$x = \frac{-b \pm \sqrt{D}}{2a}$ where $D= b^2-4ac$
Example: Find the roots of $5x^2+8x+3=0$
Solution:
Here
$a=5, b= 8, c =3$
Discriminant, $D=8^2-4 \times 5 \times 3=4$
So, $x= \frac{-8 \pm \sqrt{4}}{10}$ $x=\frac{-3}{5}, -1$
2. Method of splitting the middle term:
Example: Find the roots of $5x^2+8x+3=0$
Solution:
$5x^2+8x+3=0$
⇒ $5x^2+5x+3x+3=0$
⇒ $5x(x+1)+3(x+1)=0$
⇒ $(x+1)(5x+3)=0$
⇒ $x = -1, \frac{-3}{5}$ Common traps and shortcuts in CAT Quant While solving a quadratic equation, a student can make mistakes in
misinterpreting the questions
making calculation errors
While finding discriminants, students often neglect the negative value of the discriminant, which leads to a wrong answer. You must consider each possibility.
Some important points that may help you to solve questions based on quadratic equations effectively:
(i) If D is a perfect square, then the roots are rational and in case it is not a perfect square then the roots are irrational.
(ii) In the case of imaginary roots (D < 0) and if p + iq is one root of the quadratic equation, then the other must be the conjugate p - iq and vice versa (where p and q are real and $i = \sqrt{-1}$
(iii) For an equation of the form
$ax^2+bx+c=0$
If a > 0, D < 0, roots are imaginary.
If a > 0, D = 0, roots are real and identical.
If a > 0, D > 0, roots are real and distinct.
If a < 0, D > 0, roots are imaginary.
If a < 0, D > 0, roots are real and distinct.
If a < 0, D = 0, roots are real and equal.
(iv) For an equation of the form
$ax^2+bx+c=0$
Sum of roots = $\frac{-b}{a}$
Product of roots = $\frac{c}{a}$
Sample questions and solutions
Q.1) If r is a constant such that |x² - 4x - 13| = r has exactly three distinct real roots, then the value of r is:
A) 17
B) 21
C) 15
D) 15
Solution:-
Alternatively,$ \left|x^2-4 x-13\right|=r \text {. } $ This can be represented in two parts: $x^2-4 x-13=r$ if $r$ is positive. $x^2-4 x-13=-r$ if $r$ is negative. Considering the first case$: x^2-4 x-13=r$ The quadratic equation becomes: $x^2-4 x-13-r=0$ The discriminant for this function is : $b^2-4 a c=16-[4 ×(-13-r)]=68+4 r$ Since $r$ is positive, the discriminant is always greater than 0 this must have two distinct roots. For the second case: $x^2-4 x-13+r=0$ the function inside the modulus is negative. The discriminant is $16-(4 ×(r-13))=68-4 r$ In order to have a total of 3 roots, the discriminant must be equal to zero for this quadratic equation to have a total of 3 roots. So, $68-4 r=0$ $r=17$ For $r=17$, we can have exactly 3 roots.
Hence, the correct answer is option (1).
Q.2) Suppose one of the roots of the equation $a x^2-b x+c=0$ is $2+\sqrt{3}$, Where $\mathrm{a}, \mathrm{b}$ and c are rational numbers and $a \neq 0$. If $b=c^3$ then $|a|$ equals
A) 1
B) 2
C) 3
D) 3
Solution:-
Given one root $x = 2 + \sqrt{3}$ and $a,b,c \in \mathbb{Q}, a\neq 0$
Since coefficients are rational, the other root $2 - \sqrt{3}$
Sum of roots: $(2+\sqrt{3}) + (2-\sqrt{3}) = 4$
$= \frac{b}{a} \implies b = 4a$
Product of roots: $(2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1$
Two real numbers or two algebraic expressions related by the symbol > (“Greater Than”) or < (“Less than”) (and by the signs ≥ or ≤) form an inequality.
The inequality consists of two sides, ie. LHS and RHS. LHS and RHS can be algebraic expressions or they can be numbers. The expressions in LHS and RHS have to be considered on the set where LHS and RHS have sense simultaneously. This set is called the set of permissible values of the inequality.
If two or several inequalities contain the same sign (< or >) then they are called inequalities of the same sense. Otherwise, they are called inequalities of the opposite sense.
Now let us consider some basic definitions about inequalities.
For 2 real numbers a and b
The inequality a > b means that the difference a – b is positive.
The inequality a < b means that the difference a – b is negative.
Modulus equations involve an expression with absolute values (e.g., |x|). |always gives a positive value. You require the solutions where the expression inside the modulus is equal to the positive or negative value of the number on the other side of the equation.
For example, |x| = 2 gives x = 2 or – 2. To solve a basic modulus equation like |ax + b| > c, two equations will be formed $ax + b > c$ and $ax + b < -c$, then solve each for x to find the solutions.
The inequality $a \geq b$ means that $a>b$ or $a=b$, that is, a is not less than b.
The inequality $a \leq b$ means that $a<b$ or $a=b$, that is, a is not greater than b.
Notation of Ranges:
1. Ranges where the ends are excluded:
If the value of x is denoted as (1, 2) it means 1 < x < 2 i.e. x is greater than 1 but smaller than 2.
Similarly, if we denote the range of values of $x$ as (-9, 1) U (8, 23), this means that the value of $x$ can be denoted as -9 < x < 1 and 8 < x < 23.
2. Ranges where the Ends are Included
[2, 5] means $2 \leq x \leq 5$
3. Mixed ranges
(3, 21] means $3 < x \leq 21$
Solving Algebra inequalities in less time
You can use following results to solve inequalities in less time:
$a^2 +b^2 \geq 2ab$
$|a+b| \leq |a| + |b|$
$|a-b| \geq |a| - |b|$
Arithmetic mean ≥ Geometric mean.
$\frac{a+b}{2} \geq \sqrt {ab}$
$\frac{a}{b}+\frac{b}{a} \geq 2$ if $a>0$ and $b>0$ or if $a<0$ and $b<0$.
$a^2+b^2+c^2 \geq ab+bc+ca$
$(a+b)(b+c)(c+a) \geq 8abc$ if $a \geq 0, b \geq 0$ and $c \geq 0$, the equation being obtained when $a=b=c$.
If $a+b=2$, then $a^4+b^4 \geq 2$
CAT-level practice questions with solutions
Q.1) The number of integers n that satisfy the inequalities |n – 60| < |n – 100| < |n – 20| is:
A) 21
B) 19
C) 18
D) 18
Solution:-
Given: |n – 60| < |n – 100| < |n – 20|
The distance between these two points can be represented by |x – y| or |y – x|.
Let's take the first part of the inequality |n – 60| < |n – 100| This inequality holds good for the values of n below 80. Hence ‘n’ should be less than 80.
Now Let's check for the later part of the inequality. |n – 100| < |n – 20| This inequality holds good for the values of n above 60. Hence ‘n’ should be greater than 60.
$\therefore$ Values of ‘n’ range from 61 to 79. So, the total possible integers that satisfy this inequality are 19.
Hence, the correct answer is 19.
Q.2) All the values of $x$ satisfying the inequality $\frac{1}{x+5} \leq \frac{1}{2 x-3}$ are
A) $-5<x<\frac{3}{2}$ or $\frac{3}{2}<x \leq 8$
B) $x<-5$ or $x>\frac{3}{2}$
C) $x<-5$ or $\frac{3}{2}<x \leq 8$
D) $x<-5$ or $\frac{3}{2}<x \leq 8$
Solution:-
Given: $\frac{1}{x+5} \leq \frac{1}{2x - 3}$
$⇒2x-3 \leq x+5$
$⇒x \leq 8$
But $x \neq -5$ and $x \neq \frac 32$ as at these values fractions are not defined.
So, the set of solutions becomes
$x<-5$ or $\frac{3}{2}<x \leq 8$
Hence, the correct answer is option 3.
Q.3) If $x$ and $y$ satisfy the equations $|x|+x+y=15$ and $x+|y|-y=20$, then $(x-y)$ equals
A) 15
B) 10
C) 20
D) 20
Solution:-
We are given the equations:
$|x| + x + y = 15 -------\quad \text{(1)}$ $x + |y| - y = 20---------- \quad \text{(2)}$
There are 4 cases:
Case 1: Both $x$ and $y$ are positive Then the equations become $x + x + y = 15 ⇒ 2x +y = 15$ and $x + y - y = 20 ⇒ x=20$ So, $y= 15-40=-25$ which is contradictory. So, this is not possible.
Case 2: $x>0$ and $y<0$ are positive Then the equations become $x + x + y = 15 ⇒ 2x +y = 15$ and $x - y - y = 20 ⇒ x-2y=20$ So, $4x+2y+x-2y=30+20⇒x=10$ and $y=-5$ So, $x-y= 10 - (-5)=15$
Case 3: Both $x$ and $y$ are negative Then the equations become $-x + x + y = 15 ⇒ y = 15$ which is contradictory. So, this is not possible.
Case 4: $x<0$ and $y>0$ Then the equations become $-x + x + y = 15 ⇒ y = 15$ and $x + y - y = 20 ⇒ x=20$ which is contradictory. So, this is not possible.
From Case 2: $x-y=15$
Hence, the correct answer is option 1.
Q.4) The number of distinct real values of $x$, satisfying the equation $\max \{x, 2\}-\min \{x, 2\}=|x+2|-|x-2|$, is
Solution:-
We are given the equation:$\max \{x, 2\}-\min \{x, 2\}=|x+2|-|x-2|$
Left-hand side: $\max \{x, 2\} - \min \{x, 2\} = |x - 2|$ (since difference of max and min of two numbers is just their absolute difference)
So, the equation becomes:
$|x - 2| = |x + 2| - |x - 2|$
Bring all terms to one side:
$2|x - 2| = |x + 2|$
Case 1: $x \geq 2$
Then $|x - 2| = x - 2$, $|x + 2| = x + 2$
The equation becomes:
$2(x - 2) = x + 2 \Rightarrow 2x - 4 = x + 2 \Rightarrow x = 6$
This is valid for $x \geq 2$
Case 2: $-2 \leq x < 2$
Then $|x - 2| = 2 - x$, $|x + 2| = x + 2$
$⇒2(2 - x) = x + 2 \Rightarrow 4 - 2x = x + 2 \Rightarrow 3x = 2 \Rightarrow x = \frac{2}{3}$
But $x = 6$ does not lie in $x < -2$, so discard this solution.
Thus, valid real solutions are:
$x = 6$ and $x = \frac{2}{3}$
There are $2$ distinct real values.
Hence, the correct answer is $2$.
Type 4 – Functions & Polynomials in CAT Algebra
Functions and polynomials in CAT Algebra focus on understanding expressions, relationships, and transformations.
Polynomials:
A polynomial is an expression in x or other variable which may contain one or more terms. Also, the power of the variable should be a whole number.
The degree of a polynomial: It is the highest power of the variable in that polynomial.
It is important to note that the degree of a polynomial can never be negative and it should be a non-negative integer.
For instance, in the polynomial $5x^3 -2x^2 + 3x -7$, the degree is 3, because the highest power of x is 3.
Introduction to Functions:
A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. The set of inputs is called the domain and the set of possible outputs is called the codomain.
Notation:
A function f from set A to set B is denoted by f: A ➔ B.
Understanding functions and graph-based questions
Graph of some important functions:
Function
Description
Graph
Constant Function
f(x)= k
Domain: R
Range: k
Modulus Function
f(x)= x
Where:
x={x, if x>0
-x if x<0
Domain: R
Range: Non-negative real numbers
Signum Function
sgnx={1, if x>0 0, if x=0 -1, if x<0
Domain: R
Range: {- 1, 0, 1}
Identity Function
f(x)= x
Domain: R
Range: R
Greatest Integer Function or step function
f(x)= x
Domain: R
Range: Integer
Reciprocal Function
$f(x)=\frac{1}{x}$
Domain: R – {0}
Range: R – {0}
Function of a Function (Composite Function)
A composite function is the function of another function. If f is a function from A in to B and g is a function from B in to C, then their composite function denoted by (gof) is a function from A in to C defined by
gof (x) = g[f(x)]
Also, composite function fog (read as "f of g") is defined as:
fog (x) = f[g(x)]
Q.1) A function $f$ maps the set of natural numbers to whole numbers, such that $f(x y)=f(x) f(y)+f(x)+f(y)$ for all $x, y$ and $f(p)=1$ for every prime number $p$. Then, the value of $f(160000)$ is
A) 8191
B) 2047
C) 4095
D) 4095
Solution:-
Given: $f(xy) = f(x)f(y) + f(x) + f(y)$ for all $x, y \in \mathbb{N}$ And $f(p) = 1$ for every prime $p$
Sum of all possible values of $ x $ is: $ \frac{-3 + \sqrt{17}}{2} + \frac{-3 - \sqrt{17}}{2} = -3 $
Hence, the correct answer is option 3.
Q.3) Let a, b, and c be non-zero real numbers such that $b^2<4 a c$, and $f(x)=a x^2+b x+c$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be:
A) the set of all positive integers
B) the set of all integers
C) either the empty set or the set of all integers
D) either the empty set or the set of all integers
Solution:-
It is given that $f(x) = ax^2 + bx + c$ and $b^2 < 4ac$ This means that f(x) has imaginary roots and therefore, no real roots at all. If a function has no real roots, the graph of the function can never touch the x-axis, because touching the x-axis means, for some real value of x, the value of f(x) is 0. This means that the function has roots in some real value of x. When we graph this quadratic function f(x), we get a parabola that should never touch the x-axis. Such a parabola should be completely above the x-axis or completely below the x-axis.
If it is completely above the x-axis: ● It means that the value of f(x) is always positive for any value of x. The set of values of x that satisfies the condition that f(x) < 0 is an empty set. ● If it is completely below the x-axis: It means that the value of f(x) is always negative for any value of x. The set of values of x that satisfies the condition that f(x) < 0 is the set of all real numbers. Since Set S contains all the integers ‘m’ that satisfy the above conditions, So, set S is either an empty set or the set of all integers. Hence, the correct answer is option (3).
Q.4) Let $0 \leq a \leq x \leq 100$ and $f(x)=|x-a|+|x-100|+|x-a-50|$. Then the maximum value of f(x) becomes 100 when a is equal to:
A) 25
B) 100
C) 50
D) 50
Solution:-
$\begin{aligned} & x\geq a \text {, so }|x-a|=x-a \\ & x<100 \text {, so }|x-100|=100-x \\ & f(x)=(x-a)+(100-x)+|x-a-50|=100 \\ & ⇒|x-a-50|=a\end{aligned}$ From the graph, we can see that when x = aThen, |x - a - 50| = a ⇒ a = 50 Similarly when x = a + 100 |x – a - 50|= a ⇒ a = 50 So, the value of a is 50 when f(x) is 100.
Hence, the correct answer is option (3).
High-frequency CAT Algebra problems on polynomials
Questions on Polynomials that are asked in CAT mainly based on following concepts:
Sum of zeros and product of zeros
1. Sum of Zeroes (α and β): For a quadratic polynomial $ax^2+bx+c$, the sum of its zeroes (roots) is given by $\frac {-b}{a]$.
For a cubic polynomial (zeros α, β and γ) $ax^3+bx^2+cx+d$, the sum of its zeroes (roots) is given by $\frac {-b}{a]$.
2. Product of Zeroes:
For a quadratic polynomial $ax^2+bx+c$, the product of its zeroes (roots) is given by $\frac {c}{a}$.
For a cubic polynomial $ax^3+bx^2+cx+d$, the product of its zeroes (roots) is given by $\frac {-d}{a]$.
Also, for cubic polynomial (zeros α, β and γ)
αβ +βγ + γα = c/a
Questions based on factor theorem:
Factorization (Factor theorem)
If a polynomial P(x) has a factor (x -a), then P(a) = 0 and conversely, if P(a) = 0 then (x -a) is a factor of P(x). This is the factor theorem.
Q.1) $f(x)=\frac{x^2+2x-15}{x^2-7x-18}$ is negative if and only if
A) $-5<x<-2$ or $3<x<9$
B) $-2<x<3$ or $x>9$
C) $x<-5$ or $-2<x<3$
D) $x<-5$ or $-2<x<3$
Solution:-
$\begin{aligned} & f(x)=\frac{x^2+2 x-15}{x^2-7 x-18}<0 \\ & \frac{(x+5)(x-3)}{(x-9)(x+2)}<0\end{aligned}$ We have four inflection points $-5,-2,3$, and 9 . For $x<-5$, all four terms $(x+5),(x-3),(x-9),(x+2)$ will be negative.
The overall expression will be positive. Similarly, when $x>9$, all four terms will be positive. When $x$ belongs to $(-2,3)$, two terms are negative and two are positive.
The overall expression is positive again. We are left with the range $(-5,-2)$ and $(3,9)$ where the expression will be negative.
Thus, the correct answer is option 1) $-5<x<-2$ or $3<x<9$.
Q.2) The roots $\alpha, \beta$ of the equation $3 x^2+\lambda x-1=0$, satisfy $\frac{1}{\alpha^2}+\frac{1}{\beta^2}=15$. The value of $\left(\alpha^3+\beta^3\right)^2$, is
Q.2) When Rajesh's age was same as the present age of Garima, the ratio of their ages was $3: 2$. When Garima's age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become
A) $5:4$
B) $4:3$
C) $2:1$
D) $2:1$
Solution:-
Let present ages of Rajesh and Garima be $R$ and $G$ respectively. When Rajesh's age was $G$, the ratio of their ages was $3:2$. So, at that time: $R - G$ years ago, Rajesh's age = $G$, Garima's age = $G - (R - G) = 2G - R$
Now, when Garima's age becomes $R$, the time passed = $R - G = \frac{4G}{3} - G = \frac{G}{3}$ At that time, Rajesh's age = $R + \frac{G}{3} = \frac{4G}{3} + \frac{G}{3} = \frac{5G}{3}$
Required ratio = $\frac{5G/3}{R} = \frac{5G/3}{4G/3} = \frac{5}{4}$
Hence, the correct answer is option 1.
Q.3) 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
Solution:-
Let the initial number of fruits be $x$ and apples be $y$.
The stock of mangoes is given to be 40% of x, which shows that the number of mangoes will be $\frac{2x}{5}$.
The total number of sold fruits is given by including the selling of all the fruits- mango, apple and banana.
Now, the total no. of fruits sold
$=\frac{2x}{10}+\frac{4y}{10}+96=\frac{x}{2}$
$⇒\frac{x}{5}+\frac{2y}{5}+96=\frac{x}{2}$
On simplifying, we get,
$2x+4y+960=5x$
$⇒x=\frac{960+4y}{3}$
For $x$ to be a positive integer, let us check for values of $y$.
$4y+960$ has to be divisible by $3$, which means $4y$ will also be divisible by $3$.
For $\frac{4y}{10}$ to be an integer, $y$ has to be divisible by $5$.
We can say, that the smallest value of $y$ has to be multiple of both $3$ and $5$, i.e. $15$.
Now, put the value of $y=15$.
We get, $x=\frac{960+4 \times 15}{3}$
$⇒x=\frac{1020}{3}$
$⇒x=340$
So, the smallest possible number of fruits in stock at the starting of the day will be $340$.
Hence, the correct answer is $340$.
Q.4) For natural numbers x, y, and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is:
A) 34
B) 35
C) 36
D) 36
Solution:-
It is given, y(x + z) = 19 y cannot be 19. If y = 19, then x + z = 1 which is not possible when both x and z are natural numbers. Therefore, y = 1 and x + z = 19 It is given, z(x + y) = 51 z can take values 3 and 17. Case 1: If z = 3, y = 1 and x = 16 xyz = 3 × 1 × 16 = 48 Case 2: If z = 17, y = 1 and x = 2 xyz = 17 × 1 × 2 = 34 So, the minimum value xyz can take is 34. Hence, the correct answer is option (1).
Q.5) The number of integer solutions of the equation $\left(x^2-10\right)^{(x^2-3 x-10)}=1$ is:
A) 4
B) 6
C) 3
D) 3
Solution:-
Case 1: When $x^2-3 x-10=0$ and $x^2-10 \neq 0$
$x^2-3 x-10=0$ $ ⇒(x-5)(x+2)=0$ $\therefore x=5 \text { or }-2 $ Case 2: $x^2-10=1$ $⇒ x^2-11=0 $ $\therefore$ No integer solutions Case 3: $\mathrm{x}^2-10=-1$ and $\mathrm{x}^2-3 \mathrm{x}-10$ is even. $x^2-9=0$ $⇒(x+3)(x-3)=0$ $\therefore\mathrm{x}=-3$ or 3 for $\mathrm{x}=-3$ and $+3$, $\mathrm{x}^2-3 \mathrm{x}-10$ is even.
$\therefore$ In total 4 values of $x$ satisfy the equations.
Hence, the correct answer is option (1).
Algebra Important Topics for CAT 2025 & Beyond
Algebra consists of many topics, as discussed earlier in this article. Based on CAT's previous papers analysis, we have divided these topics into two categories:
High-priority concepts
Lesser Priority concepts
High-priority concepts based on previous CAT papers
Topic
Important Concepts
Function
Questions based on fog and gof, value of a function, graphs
Polynomials
Relation between zeros, factor theorem
Linear Equations
Word problems, Nature of solutions, solving linear pair of equations, equations involving modulus, equations involving sequence and series, equations involving logarithm
Quadratic Equations
Nature of roots, relation between roots, equations involving sequences and series, equations involving logarithms
Linear Inequalities
Inequalities involving modulus
Lesser-seen but tricky Algebra concept
There are a few topics which are less seen but very tricky and important for the CAT exam
Greatest integer function
Questions based on the signum function
Questions involving higher-order polynomials
Harmonic Progressions
Questions involving even and odd functions
CAT Algebra Preparation Tips
In this section, we will focus on a few but important tips to prepare for CAT Algebra.
How to build strong basics in Algebra equations
To build a strong foundation, first learn the basic concepts from the NCERT classes 11 and 12. Start with very basic questions and increase the difficulty level gradually.
After learning the basics, apply the concepts in CAT-level questions.
Shortcuts and time-saving tricks for CAT Quant
After building a strong foundation on the topics, the next step is to learn the shortcuts and time-saving techniques. For this, learn some effective calculation methods with the help of Vedic Maths.
Formula sheet for quick revision
Prepare a CAT formula sheet for linear equation, quadratic equation, polynomials, graphs, etc for quick revision.
Best resources & practice material for CAT Algebra
1. Arun Sharma: A Quantitative Approach for CAT (7th Edition, Pg No: 235 - 242) 2. Quantitative Aptitude for CAT by Nishit K Sinha 3. NCERT class 11 and class 12
CAT 2025 Preparation Resources by Careers360
The candidates can download the various CAT preparation resources designed by Careers360 using the links given below.
eBook Title
Download Links
CAT 2025 Arithmetic Important Concepts and Practice Questions
Common mistakes in CAT Algebra preparation include ignoring fundamentals, skipping practice on tricky problems, mismanaging time, and overlooking shortcuts, which can lower accuracy and speed in the exam.
Common mistakes
How to Rectify?
Ignoring the Basics while preparing
Learn fundamentals from NCERT
Solving the modulus inappropriately
Practice more questions and do not avoid any possible case.
Over-dependencies on formulas
Try to solve conceptually
Ignoring domain and restrictions
Solve the questions within specifies domain
Values within square roots should be non-negative
The square root of negative values is not defined. So, you should better take care of it
Poor time management
Prepare a proper timetable
Do not checking the previous year’s pattern
You should check at least the previous 5 years' papers to know the pattern of the question
How to Practice Repeated Algebra Questions in CAT
Algebra plays a key role in scoring well in the CAT quantitative aptitude section. I will suggest four-step strategies to practice repeated Algebra questions in CAT.
Step 1: Identify patterns in the last 4 to 5 years of CAT papers and categorise the questions concept-wise.
Step 2: Work on basics, memorise key concepts or tricks, and learn shortcuts.
Step 3: Practice with CAT-specific resources.
Step 4: Evaluate, analyse your mistakes, and focus on weaker areas.
Frequently Asked Questions (FAQs)
Q: What kind of questions are asked from Algebra in CAT?
A:
In CAT, questions are generally framed by involving two or more concepts in a single question like:
Sequence and series with log
Inequality with modulus
Equations involving log, surds, and indices etc
Q: How much time should I give to prepare for Algebra?
A:
Algebra is very vast having numerous concepts. It will require a lot of time to practice depending upon your capabilities and previous conceptual knowledge. What I suggest is, prepare a proper time table including all subjects and topics. If you do well in only one section, it will not help you to get into IIMs. So, manage your time and prepare well for all sections.
Q: With which books should I start preparing for Algebra?
A:
If you are a beginner,
Start with NCERT classes 11 and 12. After learning the basics, follow Arun Sharma for Quantitative Aptitude for CAT.
Hey! With an All India Rank (AIR) of 302,821 in NEET and belonging to the BCE category, it is highly unlikely to get a BDS seat in Telangana under the state quota, as the closing ranks for BCE are usually below 50,000. You may consider applying to private colleges under management quota or explore BDS seats in other states, but the chances remain very limited with this rank.
At KIMS Amalapuram, the internship stipend for MBBS students is generally reported to be around 20,000 per month, though some students have mentioned that in certain years no stipend was provided at all, which means it can vary depending on the policies in place at the time of your internship. To get the most accurate and updated information, it is always best to confirm directly with the college administration or recent interns, but on average, you can expect a stipend in the range of 18,000-20,000 per month during the compulsory rotating internship.
Since your payment status shows
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Do not make another payment.
Making a second payment could cause a double debit, which is difficult to get a refund for. You should:
Wait 24-48 hours
for the status to update automatically.
Download and save
a copy of your form and a screenshot of the "S" payment status as proof.
Check your bank statement
to confirm the money has been debited.
If the issue is not fixed after 48 hours,
contact the official CAT helpdesk
immediately with your transaction details.
The CAT 2025 exam is a national-level MBA entrance test for IIMs and top B-schools in India. It will be held on 30th November 2025 in computer-based mode across ~170 cities.
The registration is open from 1st August to 13th September 2025 on
iimcat.ac.in
.
Admit cards will be available from 5th November 2025 onward.
Graduates with at least 50 marks (45 for SC/ST/PwD) are eligible to apply.
The exam tests English, Reasoning, and Quantitative Aptitude in three timed sections.
M/s Deloitte Touche Tohmatsu Limited, one of the top four audit and accounting firms in the world with headquarters at London, UK, and with an operational presence in 153 countries, hires Management Trainees (MT) from all the premier management institutes of India thrice every year, in the months of January, May and September.
Each new group of Management Trainees (MT) have to go through a four month rigorous training schedule, after which they have to pass through a test consisting of a written assessment and a case-analysis. The top hundred ranked Management Trainees (MT) based on the performance in the test are confirmed as Management Executives (ME). The rest are given the opportunity of undergoing the training for four months one more time along with the next batch of Management Trainees (MT) and then passing through the subsequent test consisting of the written assessment and case-analysis. The Management Trainee (MT) who fails to get confirmed as a Management Executive (ME) the second time is fired.
The scatter-graph below depicts the number of Management Trainees (MT) at Deloitte taking the tests from January 2020 till May 2022, and the vis-à-vis hired Management Trainees (MT) at Deloitte who were fired :
It is also known that for the month of September 2019 at Deloitte, 96 hired Management Trainees (MT) failed to be confirmed as a Management Executive (ME) the first time, and that 36 hired Management Trainees (MT) were fired.
Question :
In which test did the minimum number of Management Trainees (MT) get confirmed as a Management Executive (ME) in the second attempt ?
Two friends Moloy and Niloy passed out from the Purulia Institute of Science and Technology with B.Tech degrees in Mechanical Engineering, but even after a year placement was hard to find. So they decided to take the challenge head-on, came down to Kolkata, rented a garage space on Park Street, and having an affinity towards making people enjoy good food, started their firm named 'B.Tech Bread-Omlette Wala'.
They started with three items on the menu. One was the French Toast which could be prepared in 3 minutes. The second was the Egg Tortillas which took 15 minutes to prepare. Any one of Moloy and Niloy could prepare any one of them at a time. The third was the Egg Bhurji with French Fries. This however was prepared on an automated fryer which could prepare 3 servings at a time and took 5 minutes irrespective of the number of servings equal to or below 3. The fryer did not need anyone to attend to it, and the time to put in the raw ingredients could be neglected. So one could tend to the preparation of other items while the Egg Bhurji with French Fries were being prepared.
They wanted to serve the orders as early as possible after the order was given. The individual items in any order were served as and when all the items were ready, and the order was then considered closed. None of the items on the menu were prepared in advance in anticipation of future orders.
On the first day, 3 groups of customers came in and ordered at 6.00 pm, 6.10 pm, and 6.13 pm. The first order was for a plate of Egg Tortillas, two plates of French Toast, and three plates of Egg Bhurji with French Fries. The second order was for a plate of French Toast and two plates of Egg Bhurji with French Fries. The third order was for a plate of Egg Tortilla and a plate of Egg Bhurji with French Fries.
On the backdrop of the above information answer the questions given :
Question:
Assuming that the next customer's order could only be attended to when the previous customer's order was closed, at what time would the first customer's order be considered closed ?
Six sticks of equal lengths were kept in the vertical position in an empty flower-vase, to be arranged at the six corners of a regular hexagon. The two ends of each of the sticks were of different colours.
The top ends of the sticks were one of each of the following colours – Red, Cyan, Pink, Brown, Black and Green. The bottom ends were one of each of the following colours – Blue, Yellow, White, Orange, Purple and Grey. Both the sets of colours mentioned were in no particular order.
It was also known that :
a) The stick with the red colour was opposite to the stick with the blue colour
b) There were exactly two sticks whose both ends had colours whose names started with the same letter
c) The stick with the grey colour was adjacent to the stick with the white colour
d) The stick with the cyan colour was adjacent to both the sticks with the brown colour and the one with the blue colour
e) The stick with the purple colour was adjacent to both the sticks with the grey colour and the one with the green colour
f) The stick with the white colour was opposite to the stick with the green colour
Question :
What was the colour of the bottom end of the stick having brown colour at the top end ?
Two friends Moloy and Niloy passed out from the Purulia Institute of Science and Technology with B.Tech degrees in Mechanical Engineering, but even after a year placement was hard to find. So they decided to take the challenge head-on, came down to Kolkata, rented a garage space on Park Street, and having an affinity towards making people enjoy good food, started their firm named 'B.Tech Bread-Omlette Wala'.
They started with three items on the menu. One was the French Toast which could be prepared in 3 minutes. The second was the Egg Tortillas which took 15 minutes to prepare. Any one of Moloy and Niloy could prepare any one of them at a time. The third was the Egg Bhurji with French Fries. This however was prepared on an automated fryer which could prepare 3 servings at a time and took 5 minutes irrespective of the number of servings equal to or below 3. The fryer did not need anyone to attend to it, and the time to put in the raw ingredients could be neglected. So one could tend to the preparation of other items while the Egg Bhurji with French Fries were being prepared.
They wanted to serve the orders as early as possible after the order was given. The individual items in any order were served as and when all the items were ready, and the order was then considered closed. None of the items on the menu were prepared in advance in anticipation of future orders.
On the first day, 3 groups of customers came in and ordered at 6.00 pm, 6.10 pm, and 6.13 pm. The first order was for a plate of Egg Tortillas, two plates of French Toast, and three plates of Egg Bhurji with French Fries. The second order was for a plate of French Toast and two plates of Egg Bhurji with French Fries. The third order was for a plate of Egg Tortilla and a plate of Egg Bhurji with French Fries.
On the backdrop of the above information answer the questions given :
Question:
Assuming that the next customer's order could only be attended to when the previous customer's order was closed, at what time would the third customer's order be considered closed ?
Two friends Moloy and Niloy passed out from the Purulia Institute of Science and Technology with B.Tech degrees in Mechanical Engineering, but even after a year placement was hard to find. So they decided to take the challenge head-on, came down to Kolkata, rented a garage space on Park Street, and having an affinity towards making people enjoy good food, started their firm named 'B.Tech Bread-Omlette Wala'.
They started with three items on the menu. One was the French Toast which could be prepared in 3 minutes. The second was the Egg Tortillas which took 15 minutes to prepare. Any one of Moloy and Niloy could prepare any one of them at a time. The third was the Egg Bhurji with French Fries. This however was prepared on an automated fryer which could prepare 3 servings at a time and took 5 minutes irrespective of the number of servings equal to or below 3. The fryer did not need anyone to attend to it, and the time to put in the raw ingredients could be neglected. So one could tend to the preparation of other items while the Egg Bhurji with French Fries were being prepared.
They wanted to serve the orders as early as possible after the order was given. The individual items in any order were served as and when all the items were ready, and the order was then considered closed. None of the items on the menu were prepared in advance in anticipation of future orders.
On the first day, 3 groups of customers came in and ordered at 6.00 pm, 6.10 pm, and 6.13 pm. The first order was for a plate of Egg Tortillas, two plates of French Toast, and three plates of Egg Bhurji with French Fries. The second order was for a plate of French Toast and two plates of Egg Bhurji with French Fries. The third order was for a plate of Egg Tortilla and a plate of Egg Bhurji with French Fries.
On the backdrop of the above information answer the questions given :
Question:
Suppose Moloy and Niloy had decided to process multiple orders at the same time, however strictly prioritising a first come first serve basis, when would the second customer's order be considered closed ?
Two friends Moloy and Niloy passed out from the Purulia Institute of Science and Technology with B.Tech degrees in Mechanical Engineering, but even after a year placement was hard to find. So they decided to take the challenge head-on, came down to Kolkata, rented a garage space on Park Street, and having an affinity towards making people enjoy good food, started their firm named 'B.Tech Bread-Omlette Wala'.
They started with three items on the menu. One was the French Toast which could be prepared in 3 minutes. The second was the Egg Tortillas which took 15 minutes to prepare. Any one of Moloy and Niloy could prepare any one of them at a time. The third was the Egg Bhurji with French Fries. This however was prepared on an automated fryer which could prepare 3 servings at a time and took 5 minutes irrespective of the number of servings equal to or below 3. The fryer did not need anyone to attend to it, and the time to put in the raw ingredients could be neglected. So one could tend to the preparation of other items while the Egg Bhurji with French Fries were being prepared.
They wanted to serve the orders as early as possible after the order was given. The individual items in any order were served as and when all the items were ready, and the order was then considered closed. None of the items on the menu were prepared in advance in anticipation of future orders.
On the first day, 3 groups of customers came in and ordered at 6.00 pm, 6.10 pm, and 6.13 pm. The first order was for a plate of Egg Tortillas, two plates of French Toast, and three plates of Egg Bhurji with French Fries. The second order was for a plate of French Toast and two plates of Egg Bhurji with French Fries. The third order was for a plate of Egg Tortilla and a plate of Egg Bhurji with French Fries.
On the backdrop of the above information answer the questions given :
Question:
Suppose Moloy and Niloy had decided to process multiple orders at the same time, however strictly prioritising a first come first serve basis, when would the third customer's order be considered closed ?
Two friends Moloy and Niloy passed out from the Purulia Institute of Science and Technology with B.Tech degrees in Mechanical Engineering, but even after a year placement was hard to find. So they decided to take the challenge head-on, came down to Kolkata, rented a garage space on Park Street, and having an affinity towards making people enjoy good food, started their firm named 'B.Tech Bread-Omlette Wala'.
They started with three items on the menu. One was the French Toast which could be prepared in 3 minutes. The second was the Egg Tortillas which took 15 minutes to prepare. Any one of Moloy and Niloy could prepare any one of them at a time. The third was the Egg Bhurji with French Fries. This however was prepared on an automated fryer which could prepare 3 servings at a time and took 5 minutes irrespective of the number of servings equal to or below 3. The fryer did not need anyone to attend to it, and the time to put in the raw ingredients could be neglected. So one could tend to the preparation of other items while the Egg Bhurji with French Fries were being prepared.
They wanted to serve the orders as early as possible after the order was given. The individual items in any order were served as and when all the items were ready, and the order was then considered closed. None of the items on the menu were prepared in advance in anticipation of future orders.
On the first day, 3 groups of customers came in and ordered at 6.00 pm, 6.10 pm, and 6.13 pm. The first order was for a plate of Egg Tortillas, two plates of French Toast, and three plates of Egg Bhurji with French Fries. The second order was for a plate of French Toast and two plates of Egg Bhurji with French Fries. The third order was for a plate of Egg Tortilla and a plate of Egg Bhurji with French Fries.
On the backdrop of the above information answer the questions given :
Question:
A fourth customer comes in and orders two plates of French Toast at 6.24 pm. Suppose Moloy and Niloy had decided to process multiple orders at the same time, however strictly prioritising a first come first serve basis. For exactly how many minutes would one of the friends be idle from 6.00 pm till serving the last customer, assuming that the four customers were the only ones to have come in within the period being discussed ?
Two friends Moloy and Niloy passed out from the Purulia Institute of Science and Technology with B.Tech degrees in Mechanical Engineering, but even after a year placement was hard to find. So they decided to take the challenge head-on, came down to Kolkata, rented a garage space on Park Street, and having an affinity towards making people enjoy good food, started their firm named 'B.Tech Bread-Omlette Wala'.
They started with three items on the menu. One was the French Toast which could be prepared in 3 minutes. The second was the Egg Tortillas which took 15 minutes to prepare. Any one of Moloy and Niloy could prepare any one of them at a time. The third was the Egg Bhurji with French Fries. This however was prepared on an automated fryer which could prepare 3 servings at a time and took 5 minutes irrespective of the number of servings equal to or below 3. The fryer did not need anyone to attend to it, and the time to put in the raw ingredients could be neglected. So one could tend to the preparation of other items while the Egg Bhurji with French Fries were being prepared.
They wanted to serve the orders as early as possible after the order was given. The individual items in any order were served as and when all the items were ready, and the order was then considered closed. None of the items on the menu were prepared in advance in anticipation of future orders.
On the first day, 3 groups of customers came in and ordered at 6.00 pm, 6.10 pm, and 6.13 pm. The first order was for a plate of Egg Tortillas, two plates of French Toast, and three plates of Egg Bhurji with French Fries. The second order was for a plate of French Toast and two plates of Egg Bhurji with French Fries. The third order was for a plate of Egg Tortilla and a plate of Egg Bhurji with French Fries.
On the backdrop of the above information answer the questions given :
Question:
Had Niloy been absent on that day, and assuming that the next customer's order could only be attended to when the previous customer's order was closed, at what time would the fourth customer's order (refer to the previous question) be considered closed ?
The bar-graph given below shows the foreign exchange reserves of Nepal (in million Rupees) from 2014 to 2021. Answer the following questions based on the graph :
Question:
What was the percentage increase (rounded to the nearest integer, if deemed necessary) in the foreign exchange reserves in 2020 over 2016 ?
The Jadavpur University’s Prince Anwar Shah Road hostel consists of two large separate buildings, one for the ladies and the other for the gents, while having a common kitchen and dining hall. It is the hostel of the CS and the EEC department of engineering students of the university.
In recognition of the growing dissatisfaction and hence complaints among the inmates of the hostel regarding the menu served for dinner, the Dean of the engineering department, Dr Aparesh Sanyal, personally decided to investigate the matter. He set about collecting information about the preference of dinner among the inmates, separately from the gents and the ladies wing of the hostel.
Dr Sanyal was able to gather the following partial information :
Hostel inmates
Menu preference for dinner
Total
Egg Meal
Fish Meal
Chicken Meal
Gents
20
Ladies
64
Total
60
The Warden of the hostel was consulted, who after investigation declared that the following facts were clear :
1. Forty percent of the hostel inmates were ladies
2. One-third of the gentlemen inmates preferred an egg meal for dinner
3. Half the hostel inmates preferred either fish meal or chicken meal
Question:
What proportion of the lady hostel inmates preferred a fish meal for dinner ?
A career as Marketing Director is also known as a marketing expert who is responsible for the overall marketing aspect of the company. He or she oversees plans and develops the company's budget. The marketing Director collaborates with the business team to plan and develop the marketing and branding strategies for the company's products or services.
A Business Development Executive identifies and pursues new business opportunities to drive company growth. They generate leads, build client relationships, develop sales strategies, and analyse market trends. Collaborating with internal teams, they aim to meet sales targets. With experience, they can advance to managerial roles, playing a key role in expanding the company’s market presence and revenue.
Content Marketing Specialists are also known as Content Specialists. They are responsible for crafting content, editing and developing it to meet the requirements of digital marketing campaigns. To ensure that the material created is consistent with the overall aims of a digital marketing campaign, content marketing specialists work closely with SEO and digital marketing professionals.
A Sales Manager leads a sales team to meet targets, formulates strategies, analyses performance, and monitors market trends. They typically hold a degree in management or related fields, with an MBA offering added value. The role often demands over 40 hours a week. Strong leadership, planning, and analytical skills are essential for success in this career.
A marketing manager is a person who oversees a company or product marketing. He or she can be in charge of multiple programmes or goods or can be in charge of one product. He or she is enthusiastic, organised, and very diligent in meeting financial constraints. He or she works with other team members to produce advertising campaigns and decides if a new product or service is marketable.
A Marketing manager plans and executes marketing initiatives to create demand for goods and services and increase consumer awareness of them. A marketing manager prevents unauthorised statements and informs the public that the business is doing everything to investigate and fix the line of products. Students can pursue an MBA in Marketing Management courses to become marketing managers.
An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns.
Digital marketing is growing, diverse, and is covering a wide variety of career paths. Each job function aids in the development of effective digital marketing strategies and techniques. The aims and objectives of the individuals who opt for a career as a digital marketing executive are similar to those of a marketing professional: to build brand awareness, promote company services or products, and increase conversions. Individuals who opt for a career as Digital Marketing Executives, unlike traditional marketing companies, communicate effectively through suitable technology platforms.
Individuals who opt for a career as a business analyst look at how a company operates. He or she conducts research and analyses data to improve his or her knowledge about the company. This is required so that an individual can suggest the company strategies for improving their operations and processes.
In a business analyst job role a lot of analysis is done, things are learned from past mistakes and the successful strategies are enhanced further. A business analyst goes through real-world data in order to provide the most feasible solutions to an organisation. Students can pursue Business Analytics to become Business Analysts.
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From the graph, we can see that when x = aThen,
