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1

JEE Main 2021 (Online) 27th August Evening Shift

MCQ (Single Correct Answer)
Let M and m respectively be the maximum and minimum values of the function
f(x) = tan$$-$$1 (sin x + cos x) in $$\left[ {0,{\pi \over 2}} \right]$$, then the value of tan(M $$-$$ m) is equal to :
A
$$2 + \sqrt 3 $$
B
$$2 - \sqrt 3 $$
C
$$3 + 2\sqrt 2 $$
D
$$3 - 2\sqrt 2 $$

Explanation

Let g(x) = sin x + cos x = $$\sqrt 2 $$ sin$$\left( {x + {\pi \over 4}} \right)$$

g(x)$$\in$$ $$\left[ {1,\sqrt 2 } \right]$$ for x$$\in$$ [0, $$\pi$$/2]

f(x) = tan$$-$$1 (sin x + cos x) $$\in$$ $$\left[ {{\pi \over 4},{{\tan }^{ - 1}}\sqrt 2 } \right]$$

tan$$({\tan ^{ - 1}}\sqrt 2 - {\pi \over 4}) = {{\sqrt 2 - 1} \over {1 + \sqrt 2 }} \times {{\sqrt 2 - 1} \over {\sqrt 2 - 1}} = 3 - 2\sqrt 2 $$
2

JEE Main 2021 (Online) 27th August Morning Shift

MCQ (Single Correct Answer)
If $${({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2} = a$$; 0 < x < 1, a $$\ne$$ 0, then the value of 2x2 $$-$$ 1 is :
A
$$\cos \left( {{{4a} \over \pi }} \right)$$
B
$$\sin \left( {{{2a} \over \pi }} \right)$$
C
$$\cos \left( {{{2a} \over \pi }} \right)$$
D
$$\sin \left( {{{4a} \over \pi }} \right)$$

Explanation

Given $$a = {({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2}$$

$$ = ({\sin ^{ - 1}}x + {\cos ^{ - 1}}x)({\sin ^{ - 1}}x - {\cos ^{ - 1}}x)$$

$$ = {\pi \over 2}\left( {{\pi \over 2} - 2{{\cos }^{ - 1}}x} \right)$$

$$ \Rightarrow 2{\cos ^{ - 1}}x = {\pi \over 2} - {{2a} \over \pi }$$

$$ \Rightarrow {\cos ^{ - 1}}(2{x^2} - 1) = {\pi \over 2} - {{2a} \over \pi }$$

$$ \Rightarrow 2{x^2} - 1 = \cos \left( {{\pi \over 2} - {{2a} \over \pi }} \right)$$
3

JEE Main 2021 (Online) 26th August Evening Shift

MCQ (Single Correct Answer)
If $$\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p} $$, then the value of tan p is :
A
$${{101} \over {102}}$$
B
$${{50} \over {51}}$$
C
100
D
$${{51} \over {50}}$$

Explanation

$$\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}\left( {{2 \over {4{r^2}}}} \right) = \sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}\left( {{{(2r + 1) - (2r - 1)} \over {1 + (2r + 1)(2r - 1)}}} \right)} } $$

= $$\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}(2r + 1) - {{\tan }^{ - 1}}(2r - 1)} $$

= $${\tan ^{ - 1}}(101) - {\tan ^{ - 1}}1 = {\tan ^{ - 1}}{{50} \over {51}}$$
4

JEE Main 2021 (Online) 26th August Morning Shift

MCQ (Single Correct Answer)
Let $$f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$$, 0 < x < 1. Then :
A
$${(1 - x)^2}f'(x) - 2{(f(x))^2} = 0$$
B
$${(1 + x)^2}f'(x) + 2{(f(x))^2} = 0$$
C
$${(1 - x)^2}f'(x) + 2{(f(x))^2} = 0$$
D
$${(1 + x)^2}f'(x) - 2{(f(x))^2} = 0$$

Explanation

$$f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$$

$${\cot ^{ - 1}}\sqrt {{{1 - x} \over x}} = {\sin ^{ - 1}}\sqrt x $$

or $$f(x) = \cos (2{\tan ^{ - 1}}\sqrt x )$$

$$ = \cos {\tan ^{ - 1}}\left( {{{2\sqrt x } \over {1 - x}}} \right)$$

$$f(x) = {{1 - x} \over {1 + x}}$$

Now, $$f'(x) = {{ - 2} \over {{{(1 + x)}^2}}}$$

or $$f'(x){(1 - x)^2} = - 2{\left( {{{1 - x} \over {1 + x}}} \right)^2}$$

or $${(1 - x)^2}f'(x) + 2{(f(x))^2} = 0$$.

Questions Asked from Inverse Trigonometric Functions

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
JEE Main 2022 (Online) 24th June Morning Shift [ Memory Based ] (1)
JEE Main 2021 (Online) 1st September Evening 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 Evening Shift (1)
JEE Main 2021 (Online) 26th August Morning Shift (1)
JEE Main 2021 (Online) 20th July Evening Shift (1)
JEE Main 2021 (Online) 20th July Morning Shift (1)
JEE Main 2021 (Online) 17th March Evening Shift (1)
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 (1)
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 Morning Slot (1)
JEE Main 2020 (Online) 3rd September Morning Slot (1)
JEE Main 2019 (Online) 12th April Morning Slot (1)
JEE Main 2019 (Online) 10th April Evening Slot (1)
JEE Main 2019 (Online) 8th April Morning 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 (1)
JEE Main 2019 (Online) 9th January Evening Slot (1)
JEE Main 2019 (Online) 9th January Morning Slot (1)
JEE Main 2017 (Online) 9th April Morning Slot (1)
JEE Main 2017 (Online) 8th April Morning Slot (1)
JEE Main 2015 (Offline) (1)
JEE Main 2013 (Offline) (1)
AIEEE 2008 (1)
AIEEE 2007 (1)
AIEEE 2005 (1)
AIEEE 2003 (1)
AIEEE 2002 (1)

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