Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :

A

an ellipse

B

a parabola

C

a hyperbola

D

a straight line

Let equation of circle is

x^{2}^{} + y^{2} + 2fx + 2fy + e = 0, it passes through (0, 2b)

$$ \Rightarrow $$ 0 + 4b^{2} + 2g $$ \times $$ 0 + 4f + c = 0

$$ \Rightarrow $$ 4b^{2} + 4f + c = 0 . . . (i)

$$2\sqrt {{g^2} - c} = 4a$$ . . . (ii)

g^{2} $$-$$ c = 4a^{2} $$ \Rightarrow $$ c = $$\left( {{g^2} - 4{a^2}} \right)$$

Putting in equation (1)

$$ \Rightarrow $$ 4b^{2} + 4f + g^{2} $$-$$ 4a^{2} = 0

$$ \Rightarrow $$ x^{2} + 4y + 4(b^{2} $$-$$ a^{2}) = 0, it represent a hyperbola.

x

$$ \Rightarrow $$ 0 + 4b

$$ \Rightarrow $$ 4b

$$2\sqrt {{g^2} - c} = 4a$$ . . . (ii)

g

Putting in equation (1)

$$ \Rightarrow $$ 4b

$$ \Rightarrow $$ x

2

MCQ (Single Correct Answer)

If the area of the triangle whose one vertex is at the vertex of the parabola, y^{2} + 4(x – a^{2}) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is

A

$$5\sqrt 5 $$

B

$${\left( {10} \right)^{2/3}}$$

C

$$5\left( {{2^{1/3}}} \right)$$

D

5

Vertex is (a^{2}, 0)

y^{2} $$=$$ $$-$$(x $$-$$ a^{2}) and x $$=$$ 0 $$ \Rightarrow $$ (0, $$ \pm $$ 2a)

Area of triangle is $$ = {1 \over 2}.$$4a.(a^{2}) = 250

$$ \Rightarrow $$ a^{3} = 125 or a = 5

y

Area of triangle is $$ = {1 \over 2}.$$4a.(a

$$ \Rightarrow $$ a

3

MCQ (Single Correct Answer)

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :

A

$${{13} \over 6}$$

B

2

C

$${{13} \over 12}$$

D

$${{13} \over 8}$$

2b = 5 and 2ae = 13

b^{2} = a^{2}(e^{2} $$-$$ 1) $$ \Rightarrow $$ $${{25} \over 4}$$ = $${{169} \over 4}$$ $$-$$ a^{2}

$$ \Rightarrow $$ a $$=$$ 6 $$ \Rightarrow $$ e $$=$$ $${{13} \over {12}}$$

b

$$ \Rightarrow $$ a $$=$$ 6 $$ \Rightarrow $$ e $$=$$ $${{13} \over {12}}$$

4

MCQ (Single Correct Answer)

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?

A

$$\left( {4\sqrt 2 ,2\sqrt 3 } \right)$$

B

$$\left( {4\sqrt 3 ,2\sqrt 3 } \right)$$

C

$$\left( {4\sqrt 3 ,2\sqrt 2 } \right)$$

D

$$\left( {4\sqrt 2 ,2\sqrt 2 } \right)$$

$${{2{b^2}} \over a} = 8$$ and 2ae $$=$$ 2b

$$ \Rightarrow $$ $${b \over a}$$ = e and 1 $$-$$ e^{2} = e^{2} $$ \Rightarrow $$ e $$=$$ $${1 \over {\sqrt 2 }}$$

$$ \Rightarrow $$ b = 4$$\sqrt 2 $$ and a $$=$$ 8

So equation of ellipse is $${{{x^2}} \over {64}} + {{{y^2}} \over {32}} = 1$$

$$ \Rightarrow $$ $${b \over a}$$ = e and 1 $$-$$ e

$$ \Rightarrow $$ b = 4$$\sqrt 2 $$ and a $$=$$ 8

So equation of ellipse is $${{{x^2}} \over {64}} + {{{y^2}} \over {32}} = 1$$

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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