CAT Questions | CAT Algebra Questions: Linear Equations
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Importance of Linear Equations in Management Entrance Exams
Questions based on Linear equations are an important component of the CAT (2-4 questions) and XAT (2-3 questions) and IIFT (2-3 questions) exam. For cracking CAT and other management exams Linear equation is a basic topic in Quantitative aptitude. This is the easiest of all and if practised enough one can get a full score in Linear Equations questions in one attempt.
CAT Quantitative Aptitude questions | CAT Algebra: Linear Equations
Below mentioned are some linear equations problems that were previously asked in CAT:
Q1.) Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if [CAT 2020]
A) k ≠ 2 B) |k| = 2 C) k = 2 D) |k| ≠ 2
Answer
For unique solution,
a1/a2 ≠ b1/b2
k/4 ≠ 1/k
k^2 ≠ 4
|k| ≠ 2
Q.2) 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 _______________ [CAT 2020]
Answer
Given,
x +9 = z or x = z -9
y +1 = z or y = z -1
now x+y < z+5
or z -9 + z – 1 < z+5
or z < 15
so x < 15 -9
or x < 6
y < z -1
y < 14
thus maximum value of 2x + y = 2*5+13 = 23
Q3.) In an examination, Rama’s score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by 6. The revised scores of Anjali, Mohan, and Rama were in the ratio 11:10:3. Then Anjali’s score exceeded Rama’s score by [CAT 2019]
1. 24
2. 26
3. 35
4. 32
Answer
let score of Rama = r, Mohan = m and Anjali = a
as the score was one-twelfth of the sum of the scores of Mohan and Anjali. so 12r = m + a…… (1)
similarly, after the score of each of them increased by 6, a score of scores of Anjali, Mohan, and Rama becomes in the ratio 11: 10 :3 let 11k = r+6, m+6 = 10k and a+6 = 3k, putting values of the 3 in equation 1,
we get a – r = 40 – 8 = 32
Q4) x + 4|y| = 33. How many integer values of (x, y) are possible? [CAT 2019]
1. 6
2. 3
3. 4
4. More than 6
Answer
first, we will reframe the equation:
3x = 33 – 4|y|
since there are two integers x and y, and since |y| is always positive regardless of the sign of y, this means that when you subtract a multiple of 4 from 33, you need to get a multiple of 3.
since 33 is already a multiple of 3, to obtain another multiple of 3, you will have to subtract a multiple of 3 from it. so, y has to be a positive or a negative multiple of 3.
y = 3, -3, 6, -6, 9, -9, 12, -12…etc.
for every value of y, x will have a corresponding integer value.
So there are infinite integer values possible for x and y.
referring to the question back again we can conclude that the answer is “more than 6”
Choice d is the correct answer.
Understanding Concepts of Linear Equations
The best way to understand the linear equations topic is to practice hard and going through the concepts mentioned below:
• An equation whose solution is a straight line and the degree of the equation is equal to 1.
For Example
2 a + 4 = 0 — equation (1)
Solving a particular linear equation means finding a solution to an unknown variable that is usually on but can be several. In the above equation x is the unknown but there can be more than 2 or more variables (unknown) in a linear equation as given below.
6x + 2 y = 24 — (2)
6 x + 4y +10 z =12 — (3)
So, A linear equation is an equation that can be written in the form y= ax + b where x and y are variables & a and b are constants. Note that the exponential power raised to the variable of a linear equation is always 1.
These are examples of linear expressions:
x + 6
2 x + 3
2 x + 5 y
To solve linear equations, we need to care for the following concepts while solving it.
1. A Linear Equation in one variable is defined as ax + b=0 or ax=c, where a, b and c are real numbers. Also, a≠ 0 and x is an unknown variable.
2. The solution of the equation ax + b=0 is x= -b/a. We can also say that -b/a is the root of the linear equation ax + b=0
3. When 2 equations are given, the root of linear equations one can be calculated using either substitution method or elimination method.
4. We can add a number, c, to both sides of the equation without changing the equation i.e. if a=b then by adding constant c we can write the equation as a+c = b+c
5. Similarly, we can subtract a number, c, from both sides of an equation i.e. if a=b then a-c = b-c
6. Moreover, like addition and subtraction, we can also multiply both sides of an equation by a number, c, without changing the equation i.e. if a = b then it can be written as ac = bc for any value of c
7. both sides of an equation can be divided by a non-zero number, c, without changing the equation i.e. if a=b then a/c =b/c
These facts form the basis for solving all kinds of linear equations that we will be looking so it is very important that you know and do not forget these concepts used in linear equations.
Preparation Strategy
Other than Algebra, Linear Equations have considerable application in Arithmetic questions which are generally framed in the Quants section of CAT, XAT, SNAP, NMAT, and other various management entrance exams We will divide the preparation of this topic into three phases following which can help the aspirants to score exceptionally well in the management entrance exams.
Phase I: In Phase, I the candidate can start with remembering linear equations formulas and basic concept understanding. One should try solving the basic questions from the topic Linear Equations. For this aspirant can follow MBAP CAT E Book (Concept Theory) study material and MBAP CAT E Book (Practice Questions). Or alternatively practice from Arun Sharma’s book and videos on Quantitative aptitude and after following the concept can solve the questions under Level of Difficulty 1 from the topic.
Phase II: In Phase II, linear equations solving practice can be done from MBAP CAT Advance E books and Level of difficulty 2 questions from Arun Sharma’s book from the given topic Linear Equation and also practice through the sectional test as time-based practice and analysis of the test is also necessary by this time.
Phase III: In Phase III aspirants can practice the topic from previous year papers of prominent management entrance exams like CAT, XAT, SNAP, NMAT, and various other exams from MBAP Topic wise Previous Year CAT Question. Aspirants can also practice through a mock test and analyse it thoroughly. Also, in this phase aspirants should go through the concepts once again at the same time should try solving advanced problems from the Linear equations.
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