Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (More than One Correct Answer)

Let RS be the diameter of the circle $${x^2}\, + \,{y^2} = 1$$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)

A

$$\left( {{1 \over 3}\,,{1 \over {\sqrt 3 }}} \right)$$

B

$$\left( {{1 \over 4}\,,{1 \over 2}} \right)$$

C

$$\left( {{1 \over 3}\,, - {1 \over {\sqrt 3 }}} \right)$$

D

$$\left( {{1 \over 4}\,,-{1 \over 2}} \right)$$

Let, P $$\equiv$$ (cos$$\theta$$, sin$$\theta$$)

$$\therefore$$ equation of tangent and normal at P

x cos$$\theta$$ + y sin$$\theta$$ = 1 ..... (1)

and y = x tan$$\theta$$ ...... (2)

Now, equation of tangent at S : x = 1 ...... (3)

Solving (1) and (3), Q $$\equiv$$ (1, cosec$$\theta$$ $$-$$ cot$$\theta$$)

$$\therefore$$ equation of straight line parallel to RS drawn from Q

y = cosec$$\theta$$ $$-$$ cot$$\theta$$ ..... (4)

Let, E $$\equiv$$ (h, k)

$$\therefore$$ k = h tan$$\theta$$ [from (2)]

or, $$\tan \theta = {k \over h}$$

Again, k = cosec$$\theta$$ $$-$$ cot$$\theta$$ [from (4)]

or, $$k = {{1 - \cos \theta } \over {\sin \theta }}$$

or, $$k = {{1 - {h \over {\sqrt {{h^2} + {k^2}} }}} \over {{k \over {\sqrt {{h^2} + {k^2}} }}}} = {{\sqrt {{h^2} + {k^2}} - h} \over k}$$

or, $${k^2} = \sqrt {{h^2} + {k^2}} - h$$

or, $$h + {k^2} = \sqrt {{h^2} + {k^2}} $$

$$\therefore$$ locus of E $$x + {y^2} = \sqrt {{x^2} + {y^2}} $$

Clearly, points $$\left( {{1 \over 3},{1 \over {\sqrt 3 }}} \right)$$ and $$\left( {{1 \over 3}, - {1 \over {\sqrt 3 }}} \right)$$ are on locus of E.

Therefore, (A) and (C) are the correct options.

2

MCQ (More than One Correct Answer)

A circle S passes through the point (0, 1) and is orthogonal to the circles $${(x - 1)^2}\, + \,{y^2} = 16\,\,and\,\,{x^2}\, + \,{y^2} = 1$$. Then

A

radius of S is 8

B

radius of S is 7

C

centre of S is (- 7, 1)

D

centre of S is (- 8, 1)

Let, the equation of the required circle is

$${x^2} + {y^2} + 2gx + 2fy + c = 0$$ ..... (1)

Circle (I) cuts the circle $${(x - 1)^2} + {y^2} = 16$$

i.e., $${x^2} + {y^2} - 2x = 15$$ orthogonally

$$\therefore$$ $$2( - g + 0) = - 15 + c$$

or, $$ - 2g = - 15 + c$$

The circle (1) also cuts the circle $${x^2} + {y^2} = 1$$ orthogonally.

$$\therefore$$ 0 = $$-$$1 + c or, c = 1

$$\therefore$$ g = 7

Now, the circle (1) passes through the point (0, 1).

$$\therefore$$ $$2f + 1 + c = 0$$ or, $$2f + 1 + 1 = 0$$ or, f = $$-$$1

$$\therefore$$ the equation of the required circle is

$${x^2} + {y^2} + 14x - 2y + 1 = 0$$

whose centre is ($$-$$7, 1) and radius $$ = \sqrt {49 + 1 - 1} = 7$$ units

Therefore, (B) and (C) are the correct option.

Note :

The condition of the circle $${x^2} + {y^2} + 2{g_1}x + 2{f_1}y + {c_1} = 0$$ cuts orthogonally to the circle $${x^2} + {y^2} + 2{g_2}x + 2{f_2}y + {c_2} = 0$$ is $$2{g_1}{g_2} + 2{f_1}{f_2} = {c_1} + {c_2}$$

3

MCQ (More than One Correct Answer)

Circle (s) touching x-axis at a distance 3 from the origin and having an intercept of length $$2\sqrt 7 $$ on y-axis is (are)

A

$${x^2}\, + \,{y^2}\, - \,6x\,\, + 8y\, + 9 = 0$$

B

$${x^2}\, + \,{y^2}\, - \,6x\,\, + 7y\, + 9 = 0$$

C

$${x^2}\, + \,{y^2}\, - \,6x\,\, - 8y\, + 9 = 0$$

D

$${x^2}\, + \,{y^2}\, - \,6x\,\,- 7y\, + 9 = 0$$

4

MCQ (More than One Correct Answer)

The number of common tangents to the circles $${x^2}\, + \,{y^2} = 4$$ and $${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$$ is

A

0

B

1

C

3

D

4

On those following papers in MCQ (Multiple Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

JEE Advanced 2016 Paper 1 Offline (1)

JEE Advanced 2014 Paper 1 Offline (1)

JEE Advanced 2013 Paper 2 Offline (1)

IIT-JEE 1998 (2)

IIT-JEE 1988 (1)

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

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