Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = 4$$ orthogonally, then the locus of its centre is

A

$$2ax\, - 2by\, - ({a^2}\, + \,{b^2} + 4) = 0$$

B

$$2ax\, + 2by\, - ({a^2}\, + \,{b^2} + 4) = 0$$

C

$$2ax\, - 2by\, + ({a^2}\, + \,{b^2} + 4) = 0$$

D

$$2ax\, + 2by\, + ({a^2}\, + \,{b^2} + 4) = 0$$

Let the variable circle is

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

It passes through $$(a,b)$$

$$\therefore$$ $${a^2} + {b^2} + 2ga + 2fb + c = 0\,\,\,\,\,\,\,...\left( 2 \right)$$

$$(1)$$ cuts $${x^2} + {y^2} = 4$$ orthogonally

$$\therefore$$ $$2\left( {g \times 0 + f \times 0} \right) = c - 4 \Rightarrow c = 4$$

$$\therefore$$ from $$(2)$$ $$\,\,\,{a^2} + {b^2} + 2ga + 2fb + 4 = 0$$

$$\therefore$$ Locus of center $$\left( { - g, - f} \right)$$ is

$${a^2} + {b^2} - 2ax - 2by + 4 = 0$$

or $$2ax + 2by = {a^2} + {b^2} + 4$$

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

It passes through $$(a,b)$$

$$\therefore$$ $${a^2} + {b^2} + 2ga + 2fb + c = 0\,\,\,\,\,\,\,...\left( 2 \right)$$

$$(1)$$ cuts $${x^2} + {y^2} = 4$$ orthogonally

$$\therefore$$ $$2\left( {g \times 0 + f \times 0} \right) = c - 4 \Rightarrow c = 4$$

$$\therefore$$ from $$(2)$$ $$\,\,\,{a^2} + {b^2} + 2ga + 2fb + 4 = 0$$

$$\therefore$$ Locus of center $$\left( { - g, - f} \right)$$ is

$${a^2} + {b^2} - 2ax - 2by + 4 = 0$$

or $$2ax + 2by = {a^2} + {b^2} + 4$$

2

MCQ (Single Correct Answer)

If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $$10\,\pi $$, then the equation of the circle is

A

$${x^2}\, + \,{y^2} + \,2x\, - \,2y - \,23\,\, = 0$$

B

$${x^2}\, + \,{y^2} - \,2x\, - \,2y - \,23\,\, = 0$$

C

$${x^2}\, + \,{y^2} + \,2x\, + \,2y - \,23\,\, = 0$$

D

$${x^2}\, + \,{y^2} - \,2x\, + \,2y - \,23\,\, = 0$$

Two diameters are along

$$2x+3y+1=0$$ and $$3x-y-4=0$$

solving we get center $$(1,-1)$$

circumference $$ = 2\pi r = 10\pi $$

$$\therefore$$ $$r=5$$.

Required circle is, $${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = {5^2}$$

$$ \Rightarrow {x^2} + {y^2} - 2x + 2y - 23 = 0$$

$$2x+3y+1=0$$ and $$3x-y-4=0$$

solving we get center $$(1,-1)$$

circumference $$ = 2\pi r = 10\pi $$

$$\therefore$$ $$r=5$$.

Required circle is, $${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = {5^2}$$

$$ \Rightarrow {x^2} + {y^2} - 2x + 2y - 23 = 0$$

3

MCQ (Single Correct Answer)

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is

A

$${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,62$$

B

$${x^2}\, + \,{y^2} + \,2x\, - \,2y\,\, = \,62$$

C

$${x^2}\, + \,{y^2} + \,2x\, - \,2y\,\, = \,47$$

D

$${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,47$$

$$\pi {r^2} = 154 \Rightarrow r = 7$$

For center on solving equation

$$2x - 3y = 5\& 3x - 4y = 7$$

we get $$x = 1,\,y = - 1$$

$$\therefore$$ center $$=(1,-1)$$

Equation of circle,

$${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = {7^2}$$

$${x^2} + {y^2} - 2x + 2y = 47$$

For center on solving equation

$$2x - 3y = 5\& 3x - 4y = 7$$

we get $$x = 1,\,y = - 1$$

$$\therefore$$ center $$=(1,-1)$$

Equation of circle,

$${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = {7^2}$$

$${x^2} + {y^2} - 2x + 2y = 47$$

4

MCQ (Single Correct Answer)

If the two circles $${(x - 1)^2}\, + \,{(y - 3)^2} = \,{r^2}$$ and $$\,{x^2}\, + \,{y^2} - \,8x\, + \,2y\, + \,\,8\,\, = 0$$ intersect in two distinct point, then

A

$$r > 2$$

B

$$2 < r < 8$$

C

$$r < 2$$

D

$$r = 2.$$

$$\left| {{r_1} - {r_2}} \right| < {C_1}{C_2}$$ for intersection

$$ \Rightarrow r - 3 < 5 \Rightarrow r < 8\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

and $${r_1} + {r_2} > {C_1}{C_2},\,$$

$$r + 3 > 5 \Rightarrow r > 2\,\,\,...\left( 2 \right)$$

From $$\left( 1 \right)$$ and $$\left( 2 \right),$$ $$2 < r < 8.$$

$$ \Rightarrow r - 3 < 5 \Rightarrow r < 8\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

and $${r_1} + {r_2} > {C_1}{C_2},\,$$

$$r + 3 > 5 \Rightarrow r > 2\,\,\,...\left( 2 \right)$$

From $$\left( 1 \right)$$ and $$\left( 2 \right),$$ $$2 < r < 8.$$

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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