Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

Given $${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ for a $$\Delta $$ABC with usual notation.

If $${{\cos A} \over \alpha } = {{\cos B} \over \beta } = {{\cos C} \over \gamma },$$ then the ordered triad ($$\alpha $$, $$\beta $$, $$\gamma $$) has a value

If $${{\cos A} \over \alpha } = {{\cos B} \over \beta } = {{\cos C} \over \gamma },$$ then the ordered triad ($$\alpha $$, $$\beta $$, $$\gamma $$) has a value

A

(19, 7, 25)

B

(7, 19, 25)

C

(5, 12, 13)

D

(3, 4, 5)

b + c = 11$$\lambda $$, c + a = 12$$\lambda $$, a + b = 13$$\lambda $$

$$ \Rightarrow $$ a = 7$$\lambda $$, b = 6$$\lambda $$, c = 5$$\lambda $$

(using cosine formula)

cosA = $${1 \over 5},$$ cosB = $${19 \over 35},$$ cosC = $${5 \over 7},$$

$$\alpha $$ : $$\beta $$ : $$\gamma $$ $$ \Rightarrow $$ 7 : 19 : 25

$$ \Rightarrow $$ a = 7$$\lambda $$, b = 6$$\lambda $$, c = 5$$\lambda $$

(using cosine formula)

cosA = $${1 \over 5},$$ cosB = $${19 \over 35},$$ cosC = $${5 \over 7},$$

$$\alpha $$ : $$\beta $$ : $$\gamma $$ $$ \Rightarrow $$ 7 : 19 : 25

2

MCQ (Single Correct Answer)

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x^{2} – c^{2} = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

A

$${y \over {\sqrt 3 }}$$

B

$${c \over 3}$$

C

$${c \over {\sqrt 3 }}$$

D

$${3 \over 2}$$y

Given a + b = x and ab = y

If x^{2} $$-$$ c^{2} = y $$ \Rightarrow $$ (a + b)^{2} $$-$$ c^{2} = ab

$$ \Rightarrow $$ a^{2} + b^{2} $$-$$ c^{2} = $$-$$ ab

$$ \Rightarrow $$ $${{{a^2} + {b^2} - {c^2}} \over {2ab}} = - {1 \over 2}$$

$$ \Rightarrow \cos C = - {1 \over 2}$$

$$ \Rightarrow \angle C = {{2\pi } \over 3}$$

$$R = {c \over {2\sin C}} = {c \over {\sqrt 3 }}$$

If x

$$ \Rightarrow $$ a

$$ \Rightarrow $$ $${{{a^2} + {b^2} - {c^2}} \over {2ab}} = - {1 \over 2}$$

$$ \Rightarrow \cos C = - {1 \over 2}$$

$$ \Rightarrow \angle C = {{2\pi } \over 3}$$

$$R = {c \over {2\sin C}} = {c \over {\sqrt 3 }}$$

3

MCQ (Single Correct Answer)

With the usual notation, in $$\Delta $$ABC, if $$\angle A + \angle B$$ = 120^{o}, a = $$\sqrt 3 $$ $$+$$ 1, b = $$\sqrt 3 $$ $$-$$ 1 then the ratio $$\angle A:\angle B,$$ is -

A

9 : 7

B

7 : 1

C

5 : 3

D

3 : 1

A + B = 120^{o}

$$\tan {{A - B} \over 2} = {{a - b} \over {a + b}}\cot \left( {{c \over 2}} \right)$$

$$ = {{\sqrt 3 + 1 - \sqrt 3 + 1} \over {2\left( {\sqrt 3 } \right)}}\cot \left( {{{30}^o}} \right) = {1 \over {\sqrt 3 }}.\sqrt 3 = 1$$

$${{A - B} \over 2} = {45^o}$$

$$ \Rightarrow A - B = {90^o}$$

$$ \ A + B = {120^o}$$

$$2A = {210^o}$$

$$A = {105^o}$$

$$B = {15^o}$$

$$ \therefore $$ $$\angle A:\angle B,$$ = 7 : 1

$$\tan {{A - B} \over 2} = {{a - b} \over {a + b}}\cot \left( {{c \over 2}} \right)$$

$$ = {{\sqrt 3 + 1 - \sqrt 3 + 1} \over {2\left( {\sqrt 3 } \right)}}\cot \left( {{{30}^o}} \right) = {1 \over {\sqrt 3 }}.\sqrt 3 = 1$$

$${{A - B} \over 2} = {45^o}$$

$$ \Rightarrow A - B = {90^o}$$

$$ \ A + B = {120^o}$$

$$2A = {210^o}$$

$$A = {105^o}$$

$$B = {15^o}$$

$$ \therefore $$ $$\angle A:\angle B,$$ = 7 : 1

4

MCQ (Single Correct Answer)

Consider a triangular plot ABC with sides AB = 7m, BC = 5m and CA = 6m. A vertical lamp-post at the mid point D of AC subtends an angle 30^{o} at B. The height (in m) of the lamp-post is -

A

$$2\sqrt {21} $$

B

$${3 \over 2}\sqrt {21} $$

C

$$7\sqrt {3} $$

D

$${2 \over 3}\sqrt {21} $$

BD = hcot30

So, 7

$$ \Rightarrow $$ 37 = 3h

$$ \Rightarrow $$ 3h

$$ \Rightarrow $$ h = $$\sqrt {{{28} \over 3}} = {2 \over 3}\sqrt {21} $$

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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

Sets and Relations

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