# CAT Exam- Quadratic Equation, Algebra Concepts, LR & DI |Day 14

## Part-1: Quantitative Aptitude for CAT 2022

Quadratic Equations are the equations with degree 2. Quadratic Equations are an important part of Algebra as well as for the CAT exam. You can expect 1 or 2 algebra questions for CAT Exam (based on analysis of previous years question papers).

Quadratic Equations are of the form of ax2 + bx + c =0.

Here ‘x’ denotes variable (unknown) and a, b, c are constants and a≠ 0.

To prepare Quadratic Equations, you should invest 4-5 hours.

Quadratic equation problems for CAT can be categorized into the following types:

• Questions that involve calculations of some unknown variable.
• Questions that involve calculations of the nature of roots.
• Formation of the quadratic equation.

### How to solve quadratic Equations and quadratic equation questions for CAT.

There are three methods which are generally used to find the roots (zeros) of a quadratic equation.

• Method of factorization
Example: 2x2 + 5x + 3 = 0

⇒ 2x2 + 2x + 3x + 3 = 0

⇒ 2x (x + 1) + 3 (x + 1) = 0

⇒ (x + 1) (2x + 3) = 0

⇒ x = -1 or x = -3/2.
This method is suitable only if factors of the quadratic polynomial are possible.

For equation ax2 + bx + c =0.

x = [-b ± √D] /2a

where, D is discriminant which is equal to b2 - 4ac.

Example: 2x2 + 5x + 3 = 0

D = 52 - 4 x 2 x 3 = 1

x = [-5 ± √1] /4 = [-5 ± 1] /4

x = -1 or -3/2.

• Method of completing the square:
Convert the given equation in the form of either (A+B)2 or (A-B)2.
Example:
2x2 + 5x + 3 = 0
⇒ x2 + 5x/2 + 3/2 = 0
⇒ x2 + 2 x 5x/4 + 25/16 – 25/16 + 3/2 = 0
⇒ (x + 5/4)2 – 1/16 = 0
⇒ (x + 5/4)2 = 1/16
⇒ (x + 5/4) = ± ¼
⇒ x = ± ¼ - 5/4
⇒ x = -1 or -3/2

### Prerequisite to Solve Quadratic Equation Questions For CAT

For this topic, students must have knowledge of

• Linear equations
• Factors
• Degree of polynomial and zeros of a polynomial.

Importance of Quadratic Equations in Real Life and For CAT Exam:

• Quadratic equations are used in many real-life situations like calculations of the area of an enclosed space.
• In the analysis of complex situations while designing a machine.
• In the calculation of the area of a circle, parabola or ellipse.
• 1-2 questions are asked on this topic every year in CAT.

Tips to Prepare Linear Equations for CAT Exam:

1. Learn all the methods of solving quadratic equations and use those methods according to the situation.

2. If the sum of zeros of a quadratic equation is ‘s’ and the product of zeros is ‘p’. Then the quadratic equation will be
x2 - sx + p = 0

3. The nature of zeros is calculated by just finding the value of D.
If D = 0, zeros/ roots are real and equal.
If D > 0, zeros/ roots are real and unequal.
If D < 0, zeros/ roots are imaginary.

1. Sum of the roots = -b/a; Product of roots = c/a.

2. Always check for feasibility of solutions.

### CAT Previous Year Algebra Questions pdf:

The two Previously Asked Questions from Quadratic Equations in the CAT exam are given below:

Q1. Suppose one of the roots of the equation ax2 - bx + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c3 then |a| equals [CAT question paper - 2021]

1. 2
2. 3
3. 4
4. 1

Solution: (a)

For a quadratic equation, ax2 + b x + c = 0.

Since the roots, a, b, and c are all rational and if one of the roots is irrational, the other root of the equation will also be irrational and a conjugate of the first root.

So, the other root is 2 - √ 3.

Sum of roots = −(-b)/a = b/a = (2 + √ 3) + (2 - √ 3) = 4.

And Product of roots = c/a = (2 + √ 3) (2 - √ 3) = 4 - 3 = 1

But here the quadratic equation is ax2 - b x + c = 0.

Therefore,

b = 4a and c = a.

It is given that b = c3

4a = a3

So, a = 2 or -2.

So, |a| = 2.

Q2. Let m and n be positive integers, If x2 + mx + 2n = 0 and x2 + 2nx + m = 0 have real roots, then the smallest possible value of m + n is [CAT Question Paper 2020]

1. 8
2. 6
3. 5
4. 7

Ans: B

For real roots b2 ≥ 4ac

From first equation: m2 ≥ 4 (2n)

Or, m2 ≥ 8n

From second equation: (2n)2 ≥ 4m

Or, 4n2 ≥ 4m

Or, n2 ≥ m

If n = 1; m2≥ 8; m ≥ 3

If m ≥ 3 at n = 1, n2 ≥ m is not true.

If n = 2; m2 ≥ 16; m ≥ 4

If m ≥ 4 at n = 2, n2 ≥ m which is a valid result, if m = 4 and n = 2

So, the minimum sum of m and n is 4+2 = 6.

References to Study (Theory):

1. How to prepare quantitative aptitude by Arun Sharma (6th edition)
2. 2iim (You-Tube)

References to Study (Practice) Quadratic Equation Problems For CAT:

1. NCERT Class 9-10
3. Previous Years Question Papers

## Part-2: CAT Logical Reasoning and Data Interpretation:

Topic: Cube Part-1 (LR) And CAT Questions On Cubes

Cube is a 3-D figure which consists of 6 faces, 12 edges and 8 vertices.

Questions on cube consist of based on the following concepts:

• Painted surface
• Opposite faces
• Consecutive faces
• Number of cuts to make smaller cubes of equal dimension

Approaches and tips to Solve Cube Questions Logical Reasoning:

1. If all the faces of a cube (n x n x n) painted with the same colour are cut in to (n x n x n) equal parts.

• Number of cubes with 3 faces painted = 8 cubes
• Number of cubes with exactly 2 faces painted = (n-2) x 12
• Number of cubes with exactly 1 face painted = (n – 2)2 x 6
• Number of cubes with 0 faces painted = (n-2)3
1. You can assume your room as a cube to analyze the situation.

2. You can use the approach of the open cube to solve the question.

Six faces can be seen like this

(1,6), (2,4), and (3,5) form opposite pairs.

Adjacent faces to face 3 are 1,2,4 and 6. Similarly, we can find the others.

1. Number of cuts to small cubes of equal size (n) from a larger cuboid of dimension (a x b x c) = (a/2 -1) + (b/2 -1) + (c/2 -1)

Prerequisite to solve Cube And Cuboid Questions:

To Prepare the questions based on Cube, students must have the knowledge of

• This topic requires the basic calculations
• Imagination
• Analytical Approach

Importance of study of Cube:

Learning Cube is important because:

• It improves your analytical thinking which can help in your daily life.
• Puzzles based on cubes are frequently asked in CAT.
• would be able to solve dice questions for CAT,

Questions that can help you to build the concept of Cubes (Based on CAT EXAM):

An Iron cube is coloured black on four adjoining surfaces while the opposite sides are coloured red. The cube is then cut two times vertically and four times horizontally at right angles. Based on the same answer to the next question.

Q1. How many cubes (cuboids) have exactly one face coloured black?

1. 1
2. 2
3. 3
4. 4

Ans: D

One cube/cuboid at each black face has exactly one face black coloured.

Q2. How many cubes/cuboids have only one face painted that too in red?

1. 2
2. 3
3. 4
4. 5

Ans: A

cube/cuboid with each red face has exactly one face red coloured.

Q3. How many cubes/cuboids have more than 2 sides painted?

1. 2
2. 4
3. 6
4. 8

Ans: D

More than two faces painted means 3 faces painted i.e. a number of cubes formed at each corner.

References to Study (Theory):

1. Verbal and Non-Verbal Reasoning by R S Aggarwal. (Basic)

References to Study (Practice):

1. Verbal and Non-Verbal Reasoning by R S Aggarwal. (Basic)
2. LRDI for CAT and GMAT by Nishit K Sinha. (Advance)
3. Previous Years Question Papers

## Part-3: CAT Exam VARC

Topic: How to solve problems of sentence correction based on Tenses.

In previous articles, we have made strategies to prepare tenses. Today we are going to discuss, how to solve problems of sentence correction based on Tenses.

Errors in sentences can be due to any of the following reasons:

• Confusion in present tenses
• Confusion in past tenses
• Confusion in future tenses
• Use of the wrong verb

You have to identify which type of error is there in the sentence and prepare them accordingly.

Importance of Sentence correction for CAT exam and in Real life:

• Error in Tenses can change the sense of what you want to say. So, the correct use of sentences is a must.
• You can expect 4-5 questions from this topic in CAT.

Prerequisite:

1. Basic Knowledge of Nouns, Pronouns and Verbs.
2. Basic knowledge of Tenses.
3. Knowledge of the use of helping verbs.

Practice Questions on Sentence correction that can boost your preparation: [Based on PYQs]

CAT Verbal Ability Questions

Find the correct phrase for underlining.

Q1. The glass of Car had broke while the players were playing football.

1. Broke
2. Was broken
3. No correction is required
4. Was broke

Ans: a

The action was done in past but the time is not known. So, Past indefinite tense is used.

Q2. I will be meeting you in the morning.

1. Meet
2. Meeting
3. No correction is required
4. Met

Ans: (a)

Future indefinite tense should be used.

Q3. Preeti attended dance class every Thursday.

1. Attends
2. No correction required
3. Is attending
4. Will attending

Ans: (a)

Action is in the present time at a regular interval of time. Present indefinite should be used.

Q4. Rani was playing when her father entered the room.

1. Played
2. No correction required
3. Plays
4. Is playing

Ans: (b)

Past continuous tense is to be used as the other sentence is in Past indefinite tense.

Q5. It had stopped raining when I was leaving the office.

1. Leave
2. Was leaving
3. Left
4. Leaves

Ans: (c)

Past perfect tense should be used as the first phrase in the past perfect tense.

Tips to Prepare Error correction in Tenses:

• Practice a lot to improve your grammar.
• Write rules and revise them on regular basis.
• During the examination, check on every possibility of error.
• Attempt Mock-Test papers and solutions after the test to check your mistakes.

References to Study (Theory):

1. Advanced English grammar by Wren and Martin