Algebra for CAT Quantitative Aptitude: Check Important Formulas and Previous Year Sample Questions
- IIM Ahmedabad Cut Off
- IIM Bangalore Cut Off
- IIM Calcutta Cut Off
- IIM Kozhikode Cut Off
- IIM Trichy Cut Off
- IIM Visakhapatnam Cut Off
- IIM Jammu Cut Off
- IIM Ranchi Cut Off
- IIM Lucknow Cut Off
- IIM Udaipur Cut Off
- IIM Raipur Cut Off
- IIM Bodh Gaya Cut Off
- IIM Amritsar Cut Off
- IIM Kashipur Cut Off
- IIM Sambalpur Cut Off
- IIM Indore Cut Off
- IIM Rohtak Cut Off
- IIM Shillong Cut Off
- IIM Nagpur Cut Off
- IIM Sirmaur Cut Off
- IIT Cut off
CAT Quantitative Aptitude topic algebra includes sub topics like linear equations, quadratic equations, functions, etc. Moreover, the topic is considered one of the most important and of high weightage in CAT Syllabus. To solve the questions asked from Algebra topic easily in CAT Question Paper, candidates are advised to learn the basic formulas and tricks.
CAT 2021 aspiring candidates need to invest time on this topic as the basic concepts as not difficult but to remember various formulas could be time taking and tricky. Read the article to know more about important formulas, sample questions and much more.
Quick Links:
CAT Algebra Preparation: Important Formulas
| Terms | Formulas |
|---|---|
| a2 – b2 | (a – b)(a + b) |
| (a + b)2 | a2 + 2ab + b2 |
| a2 + b2 | (a + b)2 – 2ab |
| (a – b)2 | a2 – 2ab + b2 |
| (a + b + c)2 | a2 + b2 + c2 + 2ab + 2bc + 2ca |
| (a – b – c)2 | a2 + b2 + c2 – 2ab + 2bc – 2ca |
| (a + b)3 | a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b) |
| (a – b)3 | a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b) |
| a3 – b3 | (a – b)(a2 + ab + b2) |
| a3 + b3 | (a + b)(a2 – ab + b2) |
| (a + b)4 | a4 + 4a3b + 6a2b2 + 4ab3 + b4 |
| (a – b)4 | a4 – 4a3b + 6a2b2 – 4ab3 + b4 |
| a4 – b4 | (a – b)(a + b)(a2 + b2) |
| a5 – b5 | (a – b)(a4 + a3b + a2b2 + ab3 + b4) |
| an – bn | (a – b)(an-1 + an-2b+…+ bn-2a + bn-1) |
| (n = 2k), an + bn | (a + b)(an-1 – an-2b +…+ bn-2a – bn-1) |
| (n = 2k + 1), an + bn | (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1) |
| (a + b + c + …)2 | a2 + b2 + c2 + … + 2(ab + ac + bc + ….) |
| (am)(an) | am+n ; (ab)m = ambm ; (am)n = amn |
Must Read:
What is Algebra?
Algebra is a part of mathematics that substitutes letters for numbers. For instance, we express the statement “Rajesh has 3 cars” in algebra as “Rajesh has x cars”. It can be used for finding linear equations, quadratic equations etc.
Algebra Notations
Given below are the major notations in algebra for CAT 2021 -
| Notation | Meaning | Example |
|---|---|---|
| x=y | x is equal to y. | 2=2 |
| x≠y | x is not equal to y. | 2+2≠2 |
| x < y | x is less than y/y is greater than x. | 2<3 |
| x > y | x is greater than y/y is less than x | 5 > 3 |
| x≤y | y is greater than or equal to x or x is less than or equal to y | - |
| x≥y | x is greater than or equal to y or y is less than or equal to x. | - |
| x≈y | x is approximately equal to y. | 5.000001≈5 |
CAT Algebra: Linear Equation
An algebraic equation such as x = 2y + 7 or 3y + 2x − z = 4 in which the greatest degree term in the variable or variables is of the first degree. The graph of this equation will be a straight line if there are two variables in the equation. The general form of a linear equation with two variables( x and y) is -
ax + by + c = 0, a 0, b 0
Here -
- a, b, and c are real numbers.
- a and b are known as coefficients of x and y and c is a constant
The solution for such an equation would result in a pair of values for x and for y, which makes LHS and RHS of the equation equal.
ax + by + c = 0
by = -ac - c
y = -ax - c / b
Here, for every real number x, there exists a real number y that corresponds to x. Thus, we can say that every linear equation in two variables will have infinitely many solutions. All these can be represented on a certain line line on a graph and hence they are called as Linear Equations as the graph of the equation on the x–y Cartesian plane will be a straight line.
Quick Links:
CAT Algebra: Two Linear Equations in Two Unknowns
Two Linear Equations in Two Unknowns has the following form:
a1 x + b1y = c1
a2 x + b2y = c2
where a1, a2, b1, b2, c1, and c2 are arbitrary real numbers. The solution set of the system of linear equations in two unknowns is a pair of real numbers (x0, y0) which satisfies each of the equations of the system.
How to solve Linear Equations?
Graphical Method
- To solve using the graphical method, plot straight lines corresponding to each of the given linear equations in two variables. The point of intersection is the solution for x and y (i.e., two variables) in the system of linear equations.
Elimination Method
- Step 1: Multiply the equations by suitable numbers so that the coefficients of one of the variables become the same in both the equations.
- Step 2: Add or subtract to get one of the variables (which has become the same in Step 1) cancelled. This will lead to obtaining the value of another variable.
- Step 3: Substitute this value in one of the equations to obtain the value of the other variable.
Substitution Method
- Step 1: Express x in terms of y in one of the equations.
- Step 2 :Put the value of x obtained from Step 1 in the 2nd equation. Calculate the value of ‘y’ now.
- Step 3 Substitute the value of ‘y’ in any of the equations to find the value of x.
Factor Theorem
- If g(x) divides f(x), then we say that f(x) is divisible by g(x) or g(x) is a factor of f(x). Factor theorem is a way to find out whether a polynomial g(x) is a factor of a polynomial f(x) or not without actual division.
- Let f(x) be a polynomial of degree greater than or equal to 1 and ‘a’ be a real number such that f(a) = 0. Then, (x − a) is a factor of f(x). Conversely, if (x − a) is a factor of f(x), then f(a) = 0.
Know Best Books
CAT Algebra: Domain and range of a function
| Type of Function | f(x) | Domain of f(x) | Range of f(x) |
|---|---|---|---|
| Constant Function | f(x)=k | Domain of function will be the set | Range of function is constant k |
| Identity Function | I(x)=x | Domain of function is real number | Range of Function = Domain of Function = R |
| Linear Function | f(x)=ax+b | Domain of function is real number | Range of Function is all those values that satisfy the function |
| Quadratic Function | f(x)=ax2 +bx+c | Domain of function is real number | Range would be a parabola |
Quick Links:
- How to Prepare for CAT Seating Arrangement?
- How to Prepare for CAT Logical Reasoning?
- How to Prepare for CAT Venn Diagrams?
CAT Algebra Sample Questions
CAT Algebra Solved Example: Solve the equation 3x − 2y = 7
Solution Let us first find one particular solution and then we will extend that. One solution is (3, 1). Alternatively, we can find it using [x = (7 + 2y)/3] To find the next solution, let us rewrite the equation as 3x = 2y + 7
Using the concepts of remainder, the above equation can be explained as:
LHS = A multiple of 3
RHS = A number that gives remainder 7 when divided by 2, which means a number that gives remainder 1 when divided by 2. Obviously, the next number will be obtained after the LCM (2, 3) = 6. To move ahead by 6, we will cover 2 multiples of 3 (2 × 3) and 3 multiples of 2 (3 × 2). Hence, the next solutions are (5, 4), (7, 7), (9, 10), (11, 13), etc.
Ques. If x is less than 2, then which of the following statements is always true?
- x is negative.
- 2x is greater than or equal to x.
- x2 is greater than or equal to x.
- None of these
Answer: (d) None of these
Question: If x − y = 8, then which of the following must be true?
- Both x and y are positive.
- If x is positive, y must be negative.
- If x is negative, y must be negative.
Options:
- I only
- II only
- I and II
- III only
Answer: (d) III only
Quick Links:
*The article might have information for the previous academic years, please refer the official website of the exam.