Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

If in a $$\Delta ABC$$, the altitudes from the vertices $$A, B, C$$ on opposite sides are in H.P, then $$\sin A,\sin B,\sin C$$ are in

A

G. P.

B

A. P.

C

A.P-G.P.

D

H. P

$$\Delta = {1 \over 2}{p_1}a = {1 \over 2}{p_2}b = {1 \over 2}{p_3}b$$

$${p_1},{p_2},{p_3},$$ are in $$H.P.$$

$$ \Rightarrow {{2\Delta } \over a},{{2\Delta } \over b},{{2\Delta } \over c}$$ are in $$H.P.$$

$$ \Rightarrow {1 \over a},{1 \over b},{1 \over c},$$ are in $$H.P.$$

$$ \Rightarrow a,b,c$$ are in $$A.P.$$

$$ \Rightarrow $$ $$K\sin A,K\sin B,K\sin C$$ are in $$A.P.$$

$$ \Rightarrow $$ $$\sin A,\sin B,\sin C$$ are in $$A.P.$$

$${p_1},{p_2},{p_3},$$ are in $$H.P.$$

$$ \Rightarrow {{2\Delta } \over a},{{2\Delta } \over b},{{2\Delta } \over c}$$ are in $$H.P.$$

$$ \Rightarrow {1 \over a},{1 \over b},{1 \over c},$$ are in $$H.P.$$

$$ \Rightarrow a,b,c$$ are in $$A.P.$$

$$ \Rightarrow $$ $$K\sin A,K\sin B,K\sin C$$ are in $$A.P.$$

$$ \Rightarrow $$ $$\sin A,\sin B,\sin C$$ are in $$A.P.$$

2

MCQ (Single Correct Answer)

The sides of a triangle are $$\sin \alpha ,\,\cos \alpha $$ and $$\sqrt {1 + \sin \alpha \cos \alpha } $$ for some $$0 < \alpha < {\pi \over 2}$$. Then the greatest angle of the triangle is

A

$${150^ \circ }$$

B

$${90^ \circ }$$

C

$${120^ \circ }$$

D

$${60^ \circ }$$

Let $$a = \sin \alpha ,b = \cos \alpha $$

and $$c = \sqrt {1 + \sin \alpha \cos \alpha } $$

Clearly $$a$$ and $$b < 1$$ but $$c > 1$$

as $$\,\,\,\sin \alpha > 0$$ and $$\cos \alpha > 0$$

$$\therefore$$ $$c$$ is the greatest side and greatest angle is $$C$$

$$\therefore$$ $$\cos C = {{{a^2} + {b^2} - {c^2}} \over {2ab}}$$

$$ = {{{{\sin }^2}\alpha + {{\cos }^2}\alpha - 1 - \sin \alpha \cos \alpha } \over {2\,\sin \alpha \cos \alpha }}$$

$$ = - {1 \over 2}$$

$$\therefore$$ $$C = {120^ \circ }$$

and $$c = \sqrt {1 + \sin \alpha \cos \alpha } $$

Clearly $$a$$ and $$b < 1$$ but $$c > 1$$

as $$\,\,\,\sin \alpha > 0$$ and $$\cos \alpha > 0$$

$$\therefore$$ $$c$$ is the greatest side and greatest angle is $$C$$

$$\therefore$$ $$\cos C = {{{a^2} + {b^2} - {c^2}} \over {2ab}}$$

$$ = {{{{\sin }^2}\alpha + {{\cos }^2}\alpha - 1 - \sin \alpha \cos \alpha } \over {2\,\sin \alpha \cos \alpha }}$$

$$ = - {1 \over 2}$$

$$\therefore$$ $$C = {120^ \circ }$$

3

MCQ (Single Correct Answer)

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $${60^ \circ }$$ and when he retires $$40$$ meters away from the tree the angle of elevation becomes $${30^ \circ }$$. The breadth of the river is

A

$$60\,\,m$$

B

$$30\,\,m$$

C

$$40\,\,m$$

D

$$20\,\,m$$

From the figure

$$\tan {60^ \circ } = {y \over x}$$

$$ \Rightarrow y = \sqrt {3x} .......\left( 1 \right)$$

$$\tan {30^ \circ } = {y \over {x + 40}}$$

$$ \Rightarrow y = {{x + 40} \over {\sqrt 3 }}........\left( 2 \right)$$

From $$(1)$$ and $$(2),$$

$$\sqrt 3 x = {{x + 40} \over {\sqrt 3 }} \Rightarrow x = 20m$$

4

MCQ (Single Correct Answer)

If in a $$\Delta ABC$$ $$a{\cos ^2}\left( {{C \over 2}} \right) + {\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},$$ then the sides $$a, b$$ and $$c$$

A

satisfy $$a+b=c$$

B

are in A.P

C

are in G.P

D

are in H.P

If $$a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2}$$

$$a\left[ {\cos C + 1} \right] + c\left[ {\cos A + 1} \right] = 3b$$

$$\left( {a + c} \right) + \left( {a\cos C + c\cos \,B} \right) = 3b$$

$$a + c + b = 3b$$ or $$a + c = 2b$$

or $$a,b,c$$ are in $$A.P.$$

$$a\left[ {\cos C + 1} \right] + c\left[ {\cos A + 1} \right] = 3b$$

$$\left( {a + c} \right) + \left( {a\cos C + c\cos \,B} \right) = 3b$$

$$a + c + b = 3b$$ or $$a + c = 2b$$

or $$a,b,c$$ are in $$A.P.$$

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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