Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

The circle $${x^2} + {y^2} = 4x + 8y + 5$$ intersects the line $$3x - 4y - m$$ at two distinct points if

A

$$ - 35 < m < 15$$

B

$$ 15 < m < 65$$

C

$$ 35 < m < 85$$

D

$$ - 85 < m < -35$$

Circle $${x^2} + {y^2} - 4x - 8y - 5 = 0$$

Center $$=(2,4),$$ Radius $$ = \sqrt {4 + 16 + 5} = 5$$

If circle is intersecting line $$3x-4y=m,$$ at two distinct points.

$$ \Rightarrow $$ length of perpendicular from center to the line $$ < $$ radius

$$ \Rightarrow {{\left| {6 - 16 - m} \right|} \over 5} < 5 \Rightarrow \left| {10 + m} \right| < 25$$

$$ \Rightarrow - 25 < m + 10 < 25 \Rightarrow - 35 < m < 15$$

Center $$=(2,4),$$ Radius $$ = \sqrt {4 + 16 + 5} = 5$$

If circle is intersecting line $$3x-4y=m,$$ at two distinct points.

$$ \Rightarrow $$ length of perpendicular from center to the line $$ < $$ radius

$$ \Rightarrow {{\left| {6 - 16 - m} \right|} \over 5} < 5 \Rightarrow \left| {10 + m} \right| < 25$$

$$ \Rightarrow - 25 < m + 10 < 25 \Rightarrow - 35 < m < 15$$

2

MCQ (Single Correct Answer)

If $$P$$ and $$Q$$ are the points of intersection of the circles

$${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0$$ and $${x^2} + {y^2} + 2x + 2y - {p^2} = 0$$ then there is a circle passing through $$P,Q $$ and $$(1, 1)$$ for:

$${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0$$ and $${x^2} + {y^2} + 2x + 2y - {p^2} = 0$$ then there is a circle passing through $$P,Q $$ and $$(1, 1)$$ for:

A

all except one value of $$p$$

B

all except two values of $$p$$

C

exactly one value of $$p$$

D

all values of $$p$$

The given circles are

$${S_1} \equiv {x^2} + {y^2} + 3x + 7y + 2p - 5 = 0\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

$${S_2} \equiv {x^2} + {y^2} + 2x + 2y - {p^2} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$

$$\therefore$$ Equation of common chord $$PQ$$ is $${S_1} - {S_2} = 0$$

$$ \Rightarrow L \equiv x + 5y + {p^2} + 2p - 5 = 0$$

$$ \Rightarrow $$ Equation of circle passing through $$P$$ and $$Q$$ is

$${S_1} + \lambda \,\,L = 0$$

$$ \Rightarrow \left( {{x^2} + {y^2} + 3x + 7y + 2p - 5} \right) + \lambda $$

$$\left( {x + 5y + {p^2} + 2p - 5} \right) = 0$$

As it passes through $$\left( {1,1} \right),$$ therefore

$$ \Rightarrow \left( {7 + 2p} \right) + \lambda \left( {2p + {p^2} + 1} \right) = 0$$

$$ \Rightarrow \lambda = - {{2p + 7} \over {\left( {p + 1} \right)}},$$

which does not exist for $$p=-1$$

$${S_1} \equiv {x^2} + {y^2} + 3x + 7y + 2p - 5 = 0\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

$${S_2} \equiv {x^2} + {y^2} + 2x + 2y - {p^2} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$

$$\therefore$$ Equation of common chord $$PQ$$ is $${S_1} - {S_2} = 0$$

$$ \Rightarrow L \equiv x + 5y + {p^2} + 2p - 5 = 0$$

$$ \Rightarrow $$ Equation of circle passing through $$P$$ and $$Q$$ is

$${S_1} + \lambda \,\,L = 0$$

$$ \Rightarrow \left( {{x^2} + {y^2} + 3x + 7y + 2p - 5} \right) + \lambda $$

$$\left( {x + 5y + {p^2} + 2p - 5} \right) = 0$$

As it passes through $$\left( {1,1} \right),$$ therefore

$$ \Rightarrow \left( {7 + 2p} \right) + \lambda \left( {2p + {p^2} + 1} \right) = 0$$

$$ \Rightarrow \lambda = - {{2p + 7} \over {\left( {p + 1} \right)}},$$

which does not exist for $$p=-1$$

3

MCQ (Single Correct Answer)

The differential equation of the family of circles with fixed radius $$5$$ units and centre on the line $$y = 2$$ is

A

$$\left( {x - 2} \right){y^2} = 25 - {\left( {y - 2} \right)^2}$$

B

$$\left( {y - 2} \right){y^2} = 25 - {\left( {y - 2} \right)^2}$$

C

$${\left( {y - 2} \right)^2}{y^2} = 25 - {\left( {y - 2} \right)^2}$$

D

$${\left( {x - 2} \right)^2}{y^2} = 25 - {\left( {y - 2} \right)^2}$$

Let the center of the circle be $$(h, 2)$$

$$\therefore$$ Equation of circle is

$${\left( {x - h} \right)^2} + \left( {y - 2} \right){}^2 = 25\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Differentiating with respect to $$x,$$ we get

$$2\left( {x - h} \right) + 2\left( {y - 2} \right){{dy} \over {dx}} = 0$$

$$ \Rightarrow x - h = - \left( {y - 2} \right){{dy} \over {dx}}$$

Substituting in equation $$(1)$$ we get

$${\left( {y - 2} \right)^2}{\left( {{{dy} \over {dx}}} \right)^2} + {\left( {y - 2} \right)^2} = 25$$

$$ \Rightarrow {\left( {y - 2} \right)^2}{\left( {y'} \right)^2} = 25 - {\left( {y - 2} \right)^2}$$

$$\therefore$$ Equation of circle is

$${\left( {x - h} \right)^2} + \left( {y - 2} \right){}^2 = 25\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Differentiating with respect to $$x,$$ we get

$$2\left( {x - h} \right) + 2\left( {y - 2} \right){{dy} \over {dx}} = 0$$

$$ \Rightarrow x - h = - \left( {y - 2} \right){{dy} \over {dx}}$$

Substituting in equation $$(1)$$ we get

$${\left( {y - 2} \right)^2}{\left( {{{dy} \over {dx}}} \right)^2} + {\left( {y - 2} \right)^2} = 25$$

$$ \Rightarrow {\left( {y - 2} \right)^2}{\left( {y'} \right)^2} = 25 - {\left( {y - 2} \right)^2}$$

4

MCQ (Single Correct Answer)

The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is

A

$$(3, -4)$$

B

$$(-3, 4)$$

C

$$(-3, -4)$$

D

$$(3, 4)$$

The given circle is $${x^2} + {y^2} + 2x + 4y - 3 = 0$$

Center $$(-1,-2)$$

Let $$Q$$ $$\left( {\alpha ,\beta } \right)$$ be the point diametrically opposite to the point $$P(1,0),$$

then $${{1 + \alpha } \over 2} = - 1$$ and $${{0 + \beta } \over 2} = - 2$$

$$ \Rightarrow \alpha = - 3,\beta = - 4,$$ So, $$Q$$ is $$\left( { - 3, - 4} \right)$$

Center $$(-1,-2)$$

Let $$Q$$ $$\left( {\alpha ,\beta } \right)$$ be the point diametrically opposite to the point $$P(1,0),$$

then $${{1 + \alpha } \over 2} = - 1$$ and $${{0 + \beta } \over 2} = - 2$$

$$ \Rightarrow \alpha = - 3,\beta = - 4,$$ So, $$Q$$ is $$\left( { - 3, - 4} \right)$$

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

JEE Main 2021 (Online) 27th August Evening Shift (1)

JEE Main 2021 (Online) 26th August Evening Shift (1)

JEE Main 2021 (Online) 26th August Morning Shift (1)

JEE Main 2021 (Online) 27th July Evening Shift (1)

JEE Main 2021 (Online) 27th July Morning Shift (2)

JEE Main 2021 (Online) 22th July Evening Shift (1)

JEE Main 2021 (Online) 20th July Evening Shift (1)

JEE Main 2021 (Online) 18th March Evening Shift (1)

JEE Main 2021 (Online) 18th March Morning Shift (2)

JEE Main 2021 (Online) 17th March Evening Shift (2)

JEE Main 2021 (Online) 17th March Morning Shift (2)

JEE Main 2021 (Online) 16th March Evening Shift (1)

JEE Main 2021 (Online) 26th February Evening Shift (2)

JEE Main 2021 (Online) 26th February Morning Shift (1)

JEE Main 2021 (Online) 25th February Evening Shift (1)

JEE Main 2021 (Online) 24th February Evening Shift (1)

JEE Main 2020 (Online) 5th September Evening Slot (1)

JEE Main 2020 (Online) 4th September Evening Slot (1)

JEE Main 2020 (Online) 9th January Morning Slot (1)

JEE Main 2020 (Online) 8th January Evening Slot (1)

JEE Main 2020 (Online) 7th January Evening Slot (1)

JEE Main 2019 (Online) 12th April Evening Slot (2)

JEE Main 2019 (Online) 12th April Morning Slot (1)

JEE Main 2019 (Online) 10th April Evening Slot (1)

JEE Main 2019 (Online) 10th April Morning Slot (2)

JEE Main 2019 (Online) 9th April Evening Slot (2)

JEE Main 2019 (Online) 9th April Morning Slot (1)

JEE Main 2019 (Online) 8th April Evening Slot (1)

JEE Main 2019 (Online) 8th April Morning Slot (1)

JEE Main 2019 (Online) 12th January Evening Slot (1)

JEE Main 2019 (Online) 12th January Morning Slot (2)

JEE Main 2019 (Online) 11th January Morning Slot (3)

JEE Main 2019 (Online) 10th January Evening Slot (1)

JEE Main 2019 (Online) 10th January Morning Slot (1)

JEE Main 2019 (Online) 9th January Evening Slot (1)

JEE Main 2019 (Online) 9th January Morning Slot (1)

JEE Main 2018 (Online) 16th April Morning Slot (1)

JEE Main 2018 (Offline) (2)

JEE Main 2018 (Online) 15th April Evening Slot (1)

JEE Main 2018 (Online) 15th April Morning Slot (1)

JEE Main 2017 (Online) 9th April Morning Slot (3)

JEE Main 2017 (Online) 8th April Morning Slot (1)

JEE Main 2017 (Offline) (1)

JEE Main 2016 (Online) 10th April Morning Slot (1)

JEE Main 2016 (Online) 9th April Morning Slot (1)

JEE Main 2016 (Offline) (2)

JEE Main 2015 (Offline) (2)

JEE Main 2014 (Offline) (1)

JEE Main 2013 (Offline) (1)

AIEEE 2012 (1)

AIEEE 2011 (1)

AIEEE 2010 (1)

AIEEE 2009 (1)

AIEEE 2008 (2)

AIEEE 2007 (1)

AIEEE 2006 (2)

AIEEE 2005 (4)

AIEEE 2004 (4)

AIEEE 2003 (2)

AIEEE 2002 (4)

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