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1

JEE Main 2021 (Online) 27th August Evening Shift

MCQ (Single Correct Answer)
The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl + lm = 0, is :
A
$${\pi \over 2}$$
B
$$\pi - {\cos ^{ - 1}}\left( {{4 \over 9}} \right)$$
C
$${\cos ^{ - 1}}\left( {{8 \over 9}} \right)$$
D
$${\pi \over 3}$$

Explanation

n = 2 (l + m)

lm + n(l + m) = 0

lm + 2(l + m)2 = 0

2l2 + 2m2 + 5ml = 0

$$2{\left( {{l \over m}} \right)^2} + 2 + 5\left( {{l \over m}} \right) = 0$$

2t2 + 5t + 2 = 0

(t + 2)(2t + 1) = 0

$$ \Rightarrow t = - 2; - {1 \over 2}$$

(i) $${l \over m} = - 2$$

$${n \over m} = - 2$$

($$-$$2m, m, $$-$$2m)

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

(ii) $${l \over m} = - {1 \over 2}$$

n = $$-$$2l

(l, $$-$$2l, $$-$$2l)

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

$$\cos \theta = {{ - 2 - 2 + 4} \over {\sqrt 9 \sqrt 9 }} = 0 \Rightarrow 0 = {\pi \over 2}$$
2

JEE Main 2021 (Online) 27th August Morning Shift

MCQ (Single Correct Answer)
Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :
A
3x2 $$-$$ 2y $$-$$ 6 = 0
B
3x2 + 2y $$-$$ 6 = 0
C
2x2 + 3y $$-$$ 9 = 0
D
2x2 $$-$$ 3y + 9 = 0

Explanation

A(0, 6) and B(2t, 0)



Perpendicular bisector of AB is

$$(y - 3) = {t \over 3}(x - t)$$

So, $$C = \left( {0,3 - {{{t^2}} \over 3}} \right)$$

Let P be (h, k)

$$h = {t \over 2};k = \left( {3 - {{{t^2}} \over 6}} \right)$$

$$ \Rightarrow k = 3 - {{4{h^2}} \over 6} \Rightarrow 2{x^2} + 3y - 9 = 0$$
3

JEE Main 2021 (Online) 26th August Morning Shift

MCQ (Single Correct Answer)
Let ABC be a triangle with A($$-$$3, 1) and $$\angle$$ACB = $$\theta$$, 0 < $$\theta$$ < $${\pi \over 2}$$. If the equation of the median through B is 2x + y $$-$$ 3 = 0 and the equation of angle bisector of C is 7x $$-$$ 4y $$-$$ 1 = 0, then tan$$\theta$$ is equal to :
A
$${1 \over 2}$$
B
$${3 \over 4}$$
C
$${4 \over 3}$$
D
2

Explanation



$$\therefore$$ $$M\left( {{{a - 3} \over 2},{{b + 1} \over 2}} \right)$$ lies on 2x + y $$-$$ 3 = 0

$$\Rightarrow$$ 2a + b = 11 ...........(i)

$$\because$$ C lies on 7x $$-$$ 4y = 1

$$\Rightarrow$$ 7a $$-$$ 4b = 1 ......... (ii)

$$\therefore$$ by (i) and (ii) : a = 3, b = 5

$$\Rightarrow$$ C(3, 5)

$$\therefore$$ mAC = 2/3

Also, mCD = 7/4

$$ \Rightarrow \tan {\theta \over 2} = \left| {{{{2 \over 3} - {4 \over 4}} \over {1 + {{14} \over {12}}}}} \right| \Rightarrow \tan {\theta \over 2} = {1 \over 2}$$

$$ \Rightarrow \tan \theta = {{2.{1 \over 2}} \over {1 - {1 \over 4}}} = {4 \over 3}$$
4

JEE Main 2021 (Online) 25th July Evening Shift

MCQ (Single Correct Answer)
Let the equation of the pair of lines, y = px and y = qx, can be written as (y $$-$$ px) (y $$-$$ qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 $$-$$ 4xy $$-$$ 5y2 = 0 is :
A
x2 $$-$$ 3xy + y2 = 0
B
x2 + 4xy $$-$$ y2 = 0
C
x2 + 3xy $$-$$ y2 = 0
D
x2 $$-$$ 3xy $$-$$ y2 = 0

Explanation

Equation of angle bisector of homogeneous
equation of pair of straight line ax2 + 2hxy + by2 is

$${{{x^2} - {y^2}} \over {a - b}} = {{xy} \over h}$$

for x2 – 4xy – 5y2 = 0

a = 1, h = – 2, b = – 5

So, equation of angle bisector is

$${{{x^2} - {y^2}} \over {1 - ( - 5)}} = {{xy} \over { - 2}}$$

$${{{x^2} - {y^2}} \over 6} = {{xy} \over { - 2}}$$

$$ \Rightarrow {x^2} - {y^2} = - 3xy$$

So, combined equation of angle bisector is $$ {x^2} + 3xy - {y^2} = 0$$

Questions Asked from Straight Lines and Pair of Straight Lines

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
JEE Main 2021 (Online) 31st August Evening Shift (1)
JEE Main 2021 (Online) 31st August Morning Shift (1)
JEE Main 2021 (Online) 27th August Evening Shift (1)
JEE Main 2021 (Online) 27th August Morning Shift (1)
JEE Main 2021 (Online) 26th August Morning Shift (1)
JEE Main 2021 (Online) 27th July Evening Shift (2)
JEE Main 2021 (Online) 25th 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 Morning Shift (1)
JEE Main 2021 (Online) 16th March Evening Shift (1)
JEE Main 2021 (Online) 26th February Morning Shift (1)
JEE Main 2021 (Online) 25th February Morning Shift (1)
JEE Main 2021 (Online) 24th February Morning Shift (1)
JEE Main 2020 (Online) 6th September Morning Slot (1)
JEE Main 2020 (Online) 4th September Evening Slot (1)
JEE Main 2020 (Online) 4th September Morning Slot (1)
JEE Main 2020 (Online) 3rd September Evening Slot (1)
JEE Main 2020 (Online) 2nd September Evening Slot (1)
JEE Main 2020 (Online) 9th January Morning Slot (1)
JEE Main 2020 (Online) 8th January Morning Slot (1)
JEE Main 2020 (Online) 7th January Evening Slot (1)
JEE Main 2019 (Online) 12th April Evening Slot (1)
JEE Main 2019 (Online) 10th April Evening Slot (1)
JEE Main 2019 (Online) 10th April Morning Slot (1)
JEE Main 2019 (Online) 9th April Evening Slot (1)
JEE Main 2019 (Online) 9th April Morning Slot (1)
JEE Main 2019 (Online) 8th April Evening Slot (2)
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JEE Main 2019 (Online) 12th January Evening Slot (1)
JEE Main 2019 (Online) 12th January Morning Slot (1)
JEE Main 2019 (Online) 11th January Evening Slot (1)
JEE Main 2019 (Online) 10th January Evening Slot (2)
JEE Main 2019 (Online) 10th January Morning Slot (3)
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