NEW
New Website Launch
Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...
1

AIEEE 2009

MCQ (Single Correct Answer)
Let $$f\left( x \right) = x\left| x \right|$$ and $$g\left( x \right) = \sin x.$$
Statement-1: gof is differentiable at $$x=0$$ and its derivative is continuous at that point.
Statement-2: gof is twice differentiable at $$x=0$$.
A
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B
Statement-1 is true, Statement-2 is false
C
Statement-1 is false, Statement-2 is true
D
Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-1

Explanation

Given that $$f\left( x \right) = x\left| x \right|\,\,$$ and $$\,\,g\left( x \right) = \sin x$$

So that go

$$f\left( x \right) = g\left( {f\left( x \right)} \right)$$

$$ = g\left( {x\left| x \right|} \right) = \sin x\left| x \right|$$

$$ = \left\{ {\matrix{ {\sin \left( { - {x^2}} \right),} & {if\,\,\,x < 0} \cr {\sin \left( {{x^2}} \right),} & {if\,\,\,x \ge 0} \cr } } \right.$$

$$ = \left\{ {\matrix{ { - \sin \,{x^2},} & {if\,\,\,x < 0} \cr {\sin \,\,{x^2},} & {if\,\,\,x \ge 0} \cr } } \right.$$

$$\therefore$$ $$\left( {go\,f} \right)'\,\,\left( x \right) = \left\{ {\matrix{ { - 2x\,\,\cos \,{x^2},\,\,\,\,if\,\,\,\,x < 0} \cr {2x\,\cos \,{x^2},\,\,\,if\,\,\,\,x \ge 0} \cr } } \right.$$

Here we observe

$$L\left( {gof} \right)'\left( 0 \right) = 0 = R\left( {gof} \right)'\left( 0 \right)$$

$$ \Rightarrow $$ go $$f$$ is differentiable at $$x=0$$

and $$\left( {go\,f} \right)'$$ is continuous at $$x=0$$

Now $$\left( {go\,f} \right)''\left( x \right) = \left\{ {\matrix{ { - 2\cos {x^2} + 4{x^2}\sin {x^2},x < 0} \cr {2\cos {x^2} - 4{x^2}\sin {x^2},x \ge 0} \cr } } \right.$$

Here $$L\left( {gof} \right)''\left( 0 \right) = - 2$$ and $$R\left( {go\,f} \right)''\left( 0 \right) = 2$$

As $$L{\left( {go\,f} \right)^{''}}\left( 0 \right) \ne R\left( {go\,f} \right)''\,\,\left( 0 \right)$$

$$ \Rightarrow go\,f\left( x \right)$$ is not twice differentiable at $$x=0.$$

$$\therefore$$ Statement - $$1$$ is true but statement $$-2$$ is false.
2

AIEEE 2008

MCQ (Single Correct Answer)
How many real solutions does the equation
$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?
A
$$7$$
B
$$1$$
C
$$3$$
D
$$5$$

Explanation

Let $$f\left( x \right) = {x^7} + 14{x^5} + 16{x^3} + 30x - 560$$

$$ \Rightarrow f'\left( x \right) = 7{x^6} + 70{x^4} + 48{x^2} + 30 > 0,\,\forall x \in R$$

$$ \Rightarrow f$$ is an increasing function on $$R$$

Also $$\mathop {\lim }\limits_{x \to \infty } \,\,f\left( x \right) = \infty $$ and $$\mathop {\lim }\limits_{x \to - \infty } \,\,f\left( x \right) = - \infty $$

$$ \Rightarrow $$ The curve $$y = f\left( x \right)$$ crosses $$x$$-axis only once.

$$\therefore$$ $$f\left( x \right) = 0$$ has exactly one real root.
3

AIEEE 2008

MCQ (Single Correct Answer)
Suppose the cubic $${x^3} - px + q$$ has three distinct real roots
where $$p>0$$ and $$q>0$$. Then which one of the following holds?
A
The cubic has minima at $$\sqrt {{p \over 3}} $$ and maxima at $$-\sqrt {{p \over 3}} $$
B
The cubic has minima at $$-\sqrt {{p \over 3}} $$ and maxima at $$\sqrt {{p \over 3}} $$
C
The cubic has minima at both $$\sqrt {{p \over 3}} $$ and $$-\sqrt {{p \over 3}} $$
D
The cubic has maxima at both $$\sqrt {{p \over 3}} $$ and $$-\sqrt {{p \over 3}} $$

Explanation

Let $$y = {x^3} - px + q$$

$$ \Rightarrow {{dy} \over {dx}} = 3{x^2} - p$$

For $${{dy} \over {dx}} = 0 \Rightarrow 3{x^2} - p = 0$$

$$ \Rightarrow x = \pm \sqrt {{p \over 3}} $$

$${{{d^2}y} \over {d{x^2}}} = 6x$$

$${\left. {{{{d^2}y} \over {d{x^2}}}} \right|_{x = \sqrt {{p \over 3}} }} = + ve\,\,\,\,$$ and

$$\,\,\,\,\,\,\,\,\,\,$$ $${\left. {\,\,\,{{{d^2}y} \over {d{x^2}}}} \right|_{x = - \sqrt {{p \over 3}} }} = - ve$$

$$\therefore$$ $$y$$ has ninima at $$x = \sqrt {{p \over 3}} $$

and maxima at $$x = - \sqrt {{p \over 3}} $$
4

AIEEE 2007

MCQ (Single Correct Answer)
If $$p$$ and $$q$$ are positive real numbers such that $${p^2} + {q^2} = 1$$, then the maximum value of $$(p+q)$$ is
A
$${1 \over 2}$$
B
$${1 \over {\sqrt 2 }}$$
C
$${\sqrt 2 }$$
D
$$2$$

Explanation

Given that $${p^2} + {q^2} = 1$$

$$\therefore$$ $$p = \cos \theta $$ and $$q = \sin \theta $$

Then $$p+q$$ $$ = \cos \theta + \sin \theta $$

We know that

$$ - \sqrt {{a^2} + {b^2}} \le a\cos \theta + b\sin \theta \le \sqrt {{a^2} + {b^2}} $$

$$\therefore$$ $$ - \sqrt 2 \le \cos \theta + \sin \theta \le \sqrt 2 $$

Hence max. value of $$p + q$$ is $$\sqrt 2 $$

Questions Asked from Application of Derivatives

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
JEE Main 2022 (Online) 29th July Morning Shift (1)
JEE Main 2022 (Online) 28th July Morning Shift (1)
JEE Main 2022 (Online) 26th July Morning Shift (1)
JEE Main 2022 (Online) 25th July Morning Shift (1)
JEE Main 2022 (Online) 30th June Morning Shift (2)
JEE Main 2022 (Online) 29th June Morning Shift (1)
JEE Main 2022 (Online) 27th June Morning Shift (1)
JEE Main 2022 (Online) 26th June Evening Shift (1)
JEE Main 2022 (Online) 26th June Morning Shift (2)
JEE Main 2022 (Online) 25th June Evening Shift (2)
JEE Main 2022 (Online) 24th June Evening Shift (2)
JEE Main 2022 (Online) 24th June Morning Shift (3)
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) 20th July Morning Shift (1)
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) 25th February Morning Shift (1)
JEE Main 2021 (Online) 24th February Evening Shift (3)
JEE Main 2021 (Online) 24th February Morning Shift (2)
JEE Main 2020 (Online) 6th September Evening Slot (2)
JEE Main 2020 (Online) 6th September Morning Slot (1)
JEE Main 2020 (Online) 5th September Evening Slot (2)
JEE Main 2020 (Online) 5th September Morning Slot (1)
JEE Main 2020 (Online) 4th September Evening Slot (2)
JEE Main 2020 (Online) 3rd September Evening Slot (1)
JEE Main 2020 (Online) 3rd September Morning Slot (1)
JEE Main 2020 (Online) 2nd September Evening Slot (2)
JEE Main 2020 (Online) 2nd September Morning Slot (3)
JEE Main 2020 (Online) 9th January Morning Slot (1)
JEE Main 2019 (Online) 12th April Morning Slot (2)
JEE Main 2019 (Online) 10th April Evening Slot (2)
JEE Main 2019 (Online) 9th April Evening Slot (1)
JEE Main 2019 (Online) 9th April Morning Slot (3)
JEE Main 2019 (Online) 8th April Evening Slot (2)
JEE Main 2019 (Online) 8th April Morning Slot (2)
JEE Main 2019 (Online) 12th January Evening Slot (1)
JEE Main 2019 (Online) 11th January Evening Slot (1)
JEE Main 2019 (Online) 11th January Morning Slot (2)
JEE Main 2019 (Online) 10th January Evening Slot (1)
JEE Main 2019 (Online) 10th January Morning Slot (2)
JEE Main 2018 (Online) 16th April Morning Slot (1)
JEE Main 2018 (Online) 15th April Morning Slot (2)
JEE Main 2017 (Online) 9th April Morning Slot (2)
JEE Main 2017 (Online) 8th April Morning Slot (1)
JEE Main 2016 (Online) 10th April Morning Slot (1)
JEE Main 2016 (Online) 9th April Morning Slot (2)
JEE Main 2016 (Offline) (2)
JEE Main 2015 (Offline) (1)
JEE Main 2014 (Offline) (1)
JEE Main 2013 (Offline) (1)
AIEEE 2012 (3)
AIEEE 2011 (2)
AIEEE 2010 (3)
AIEEE 2009 (2)
AIEEE 2008 (2)
AIEEE 2007 (3)
AIEEE 2006 (2)
AIEEE 2005 (4)
AIEEE 2004 (4)
AIEEE 2003 (1)
AIEEE 2002 (2)

Joint Entrance Examination

JEE Main JEE Advanced WB JEE

Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN

Medical

NEET

CBSE

Class 12