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Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

If all the words (with or without meaning) having five letters,formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is :

A

$${52^{nd}}$$

B

$${58^{th}}$$

C

$${46^{th}}$$

D

$${59^{th}}$$

Clearly, number of words start with $$A = {{4!} \over {2!}} = 12$$

Number of words start with $$L = 4! = 24$$

Number of words start with $$M = {{4!} \over {2!}} = 12$$

Number of words start with $$SA = {{3!} \over {2!}} = 3$$

Number of words start with $$SL = 3! = 6$$

Note that, next word will be "SMALL"

Hence, the position of word "SMALL" is 58th.

2

MCQ (Single Correct Answer)

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:

A

120

B

72

C

216

D

192

For a four digit number the first place can be filled in 3 ways with 6 or 7 or 8 and the remaining four places in 4! ways i.e., 3 $$\times$$ 4! = 72.

For a five digit number it can be arranged in 5! ways,

$$\therefore$$ total number of integers = (72 + 120) = 192.

3

MCQ (Single Correct Answer)

Let A and B be two sets containing 2 elements and
4 elements respectively. The number of subsets of
A $$ \times $$ B having 3 or more elements is -

A

219

B

211

C

256

D

220

A $$ \times $$ B will have 2 $$ \times $$ 4 = 8 elements.

The number of subsets having atleast 3 elements

=^{8}C_{3} + ^{8}C_{4} + ^{8}C_{5} + ^{8}C_{6} + ^{8}C_{7} + ^{8}C_{8}

= 2^{8} â€“ (^{8}C_{0} + ^{8}C_{1} + ^{8}C_{2}) = 256 â€“ 1 â€“ 8 â€“ 28 = 219

The number of subsets having atleast 3 elements

=

= 2

4

MCQ (Single Correct Answer)

Let $${T_n}$$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $${T_{n + 1}} - {T_n}$$ = 10, then the value of n is :

A

7

B

5

C

10

D

8

Number of possible triangle using n vertices = ^{n}C_{3}

$$ \therefore $$ T_{n} = ^{n}C_{3}

then T_{n + 1} = ^{n + 1}C_{3}

Given, $${T_{n + 1}} - {T_n}$$ = 10

$$ \Rightarrow $$^{n + 1}C_{3} - ^{n}C_{3} = 10

$$ \Rightarrow $$ $${{\left( {n + 1} \right)n\left( {n - 1} \right)} \over 6} - {{n\left( {n - 1} \right)\left( {n - 2} \right)} \over 6}$$ = 10

$$ \Rightarrow $$ 3n(n - 1) = 60

$$ \Rightarrow $$ n(n - 1) = 20

$$ \Rightarrow $$ n^{2} - n - 20 = 0

$$ \Rightarrow $$ (n - 5)(n + 4) = 0

$$ \therefore $$ n = 5

$$ \therefore $$ T

then T

Given, $${T_{n + 1}} - {T_n}$$ = 10

$$ \Rightarrow $$

$$ \Rightarrow $$ $${{\left( {n + 1} \right)n\left( {n - 1} \right)} \over 6} - {{n\left( {n - 1} \right)\left( {n - 2} \right)} \over 6}$$ = 10

$$ \Rightarrow $$ 3n(n - 1) = 60

$$ \Rightarrow $$ n(n - 1) = 20

$$ \Rightarrow $$ n

$$ \Rightarrow $$ (n - 5)(n + 4) = 0

$$ \therefore $$ n = 5

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Complex Numbers

Quadratic Equation and Inequalities

Permutations and Combinations

Mathematical Induction and Binomial Theorem

Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

Probability

Statistics

Mathematical Reasoning

Trigonometric Functions & Equations

Properties of Triangle

Inverse Trigonometric Functions

Straight Lines and Pair of Straight Lines

Circle

Conic Sections

Functions

Limits, Continuity and Differentiability

Differentiation

Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations