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JEE Advanced > Calculus

Explore popular questions from Calculus for JEE Advanced. This collection covers Calculus previous year JEE Advanced questions hand picked by experienced teachers.

Q 1.

Correct4

Incorrect-1

If {tex} f ( x ) = \sqrt { \frac { x - \sin x } { x + \cos ^ { 2 } x } } , {/tex} then {tex} \lim _ { x \rightarrow \infty } f ( x ) {/tex} is

A

{tex} 0 {/tex}

B

{tex} \infty {/tex}

{tex} 1 {/tex}

D

none of these

Explanation

Q 2.

Correct4

Incorrect-1

If {tex} G ( x ) = - \sqrt { 25 - x ^ { 2 } } {/tex} then {tex} \lim _ { x \rightarrow 1 } \frac { G ( x ) - G ( 1 ) } { x - 1 } {/tex} has the value

A

{tex} \frac { 1 } { 24 } {/tex}

B

{tex} \frac { 1 } { 5 } {/tex}

C

{tex} - \sqrt { 24 } {/tex}

none of these

Explanation

Q 3.

Correct4

Incorrect-1

If {tex} f ( a ) = 2 , f ^ { \prime } ( a ) = 1 , g ( a ) = - 1 , g ^ { \prime } ( a ) = 2 , {/tex} then the value of {tex} \lim _ { x \rightarrow a } \frac { g ( x ) f ( a ) - g ( a ) f ( x ) } { x - a } {/tex} is

A

{tex} - 5 {/tex}

B

{tex} \frac { 1 } { 5 } {/tex}

5

D

none of these

Explanation

Q 4.

Correct4

Incorrect-1

{tex} \lim _ { n \rightarrow \infty } \left\{ \frac { 1 } { 1 - n ^ { 2 } } + \frac { 2 } { 1 - n ^ { 2 } } + \ldots + \frac { n } { 1 - n ^ { 2 } } \right\} {/tex} is equal to

A

{tex}0{/tex}

{tex} - \frac { 1 } { 2 } {/tex}

C

{tex} \frac { 1 } { 2 } {/tex}

D

none of these

Explanation

Q 5.

Correct4

Incorrect-1

{tex} \begin{aligned} \text { If } f ( x ) & = \frac { \sin [ x ] } { [ x ] } , \quad [ x ] \neq 0 \\ & = 0 ,\quad \quad\ \quad [ x ] = 0 \end{aligned} {/tex}
Where {tex} [ x ] {/tex} denotes the greatest integer less than or equal to {tex} x {/tex}. then {tex} \lim _ { x \rightarrow 0 } f ( x ) {/tex} equals {tex} - {/tex}

A

{tex} 1 {/tex}

B

{tex} 0 {/tex}

C

{tex} - 1 {/tex}

none of these

Explanation


Q 6.

Correct4

Incorrect-1

The function {tex} f ( x ) = \left( x ^ { 2 } - 1 \right) \left| x ^ { 2 } - 3 x + 2 \right| + \cos ( | x | ) {/tex} is NOT differentiable at

A

-1

B

0

C

1

2

Explanation




Q 7.

Correct4

Incorrect-1

For {tex} x \in R , \lim _ { x \rightarrow \infty } \left( \frac { x - 3 } { x + 2 } \right) ^ { x } = {/tex}

A

{tex} e {/tex}

B

{tex} e ^ { - 1 } {/tex}

{tex} e ^ { - 5 } {/tex}

D

{tex} e ^ { 5 } {/tex}

Explanation

Q 8.

Correct4

Incorrect-1

{tex} \lim _ { x \rightarrow 0 } \frac { \sin \left( \pi \cos ^ { 2 } x \right) } { x ^ { 2 } } {/tex} equals

A

{tex} - \pi {/tex}

{tex} \pi {/tex}

C

{tex} \pi / 2 {/tex}

D

{tex} 1{/tex}

Explanation

Q 9.

Correct4

Incorrect-1

The left-hand derivative of {tex} f ( x ) = [ x ] \sin ( \pi x ) {/tex} at {tex} x = k , k {/tex} an integer, is

{tex} ( - 1 ) ^ { k } ( k - 1 ) \pi {/tex}

B

{tex} ( - 1 ) ^ { k - 1 } ( k - 1 ) \pi {/tex}

C

{tex} ( - 1 ) ^ { k } k \pi {/tex}

D

{tex} ( - 1 ) ^ { k - 1 } k \pi {/tex}

Explanation


Q 10.

Correct4

Incorrect-1

Which of the following functions is differentiable at {tex} x = 0 ? {/tex}

A

{tex} \cos ( | x | ) + | x | {/tex}

B

{tex} \cos ( | x | ) - | x | {/tex}

C

{tex} \sin ( | x | ) + | x | {/tex}

{tex} \sin ( | x | ) - | x | {/tex}

Explanation


Q 11.

Correct4

Incorrect-1

The integer n for which {tex} \lim _ { x \rightarrow 0 } \frac { ( \cos x - 1 ) \left( \cos x - e ^ { x } \right) } { x ^ { n } } {/tex} is a finite non-zero number is

A

1

B

2

3

D

4

Explanation


Q 12.

Correct4

Incorrect-1

Let {tex} f: R \rightarrow R {/tex} be such that {tex} f ( 1 ) = 3 {/tex} and {tex} f ^ { \prime } ( 1 ) = 6 . {/tex} Then
{tex} \lim _ { x \rightarrow 0 } \left( \frac { f ( 1 + x ) } { f ( 1 ) } \right) ^ { 1 / x } {/tex} equals

A

{tex}1{/tex}

B

{tex} e ^ { 1 / 2 } {/tex}

{tex} e ^ { 2 } {/tex}

D

{tex} e ^ { 3 } {/tex}

Explanation


Q 13.

Correct4

Incorrect-1

{tex} \lim _ { h \rightarrow 0 } \frac { f \left( 2 h + 2 + h ^ { 2 } \right) - f ( 2 ) } { f \left( h - h ^ { 2 } + 1 \right) - f ( 1 ) } , {/tex} given that {tex} f ^ { \prime } ( 2 ) = 6 {/tex} and {tex} f ^ { \prime } ( 1 ) = 4 {/tex}

A

does not exist

B

is equal to {tex} - 3 / 2 {/tex}

C

is equal to {tex} 3 / 2 {/tex}

is equal to {tex}3 {/tex}

Explanation

Q 14.

Correct4

Incorrect-1

If {tex} ( x ) {/tex} is differentiable and strictly increasing function, then the value of {tex} \lim _ { x \rightarrow 0 } \frac { f \left( x ^ { 2 } \right) - f ( x ) } { f ( x ) - f ( 0 ) } {/tex} is

A

1

B

0

-1

D

2

Explanation

Q 15.

Correct4

Incorrect-1

{tex} \lim _ { x \rightarrow \frac { \pi } { 4 } } \frac { \int _ { 2 } ^ { \sec ^ { 2 } x } f ( t ) d t } { x ^ { 2 } - \frac { \pi ^ { 2 } } { 16 } } {/tex} equals

{tex} \frac { 8 } { \pi } f ( 2 ) {/tex}

B

{tex} \frac { 2 } { \pi } f ( 2 ) {/tex}

C

{tex} \frac { 2 } { \pi } f \left( \frac { 1 } { 2 } \right) {/tex}

D

{tex} 4 f ( 2 ) {/tex}

Explanation


Q 16.

Correct4

Incorrect-1

If {tex} \lim _ { x \rightarrow 0 } \left[ 1 + x \ell n \left( 1 + b ^ { 2 } \right) \right] ^ { 1 / x } = 2 b \sin ^ { 2 } \theta , b > 0 {/tex} and {tex} \theta \in ( - \pi , \pi ] {/tex} then the value of {tex} \theta {/tex} is

A

{tex} \pm \frac { \pi } { 4 } {/tex}

B

{tex} \pm \frac { \pi } { 3 } {/tex}

C

{tex} \pm \frac { \pi } { 6 } {/tex}

{tex} \pm \frac { \pi } { 2 } {/tex}

Explanation


Q 17.

Correct4

Incorrect-1

Let {tex} f ( x ) = \{ \begin{array} { c c } { x ^ { 2 } \left| \cos \frac { \pi } { x } \right| , } & { x \neq 0 } \\ { 0 , } & { x = 0 } \end{array} , x \in R \text { then } f {/tex} is.

A

differentiable both at {tex} x = 0 {/tex} and at {tex} x = 2 {/tex}

differentiable at {tex} x = 0 {/tex} but not differentiable at {tex} x = 2 {/tex}

C

not differentiable at {tex} x = 0 {/tex} but differentiable at {tex} x = 2 {/tex}

D

differentiable neither at {tex} x = 0 {/tex} nor at {tex} x = 2 {/tex}

Explanation





Q 18.

Correct4

Incorrect-1

A solution of the differential equation {tex} \left( \frac { d y } { d x } \right) ^ { 2 } - x \frac { d y } { d x } + y = 0 {/tex} is
{tex} y = \left( C _ { 1 } + C _ { 2 } \right) \cos \left( x + C _ { 3 } \right) - C _ { 4 } e ^ { x + C _ { 5 } } , {/tex} where {tex} C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } {/tex} {tex} C _ { 5 } , {/tex} are arbitrary constants, is

A

{tex} y = 2 {/tex}

B

{tex} y = 2 x {/tex}

{tex} y = 2 x - 4 {/tex}

D

{tex} y = 2 x ^ { 2 } - 4 {/tex}

Explanation

Q 19.

Correct4

Incorrect-1

If {tex} x ^ { 2 } + y ^ { 2 } = 1 , {/tex} then

A

{tex} y y ^ { \prime \prime } - 2 \left( y ^ { \prime } \right) ^ { 2 } + 1 = 0 {/tex}

{tex} y y ^ { \prime \prime } + \left( y ^ { \prime } \right) ^ { 2 } + 1 = 0 {/tex}

C

{tex} y y ^ { \prime \prime } + \left( y ^ { \prime } \right) ^ { 2 } - 1 = 0 {/tex}

D

{tex} y y ^ { \prime \prime } + 2 \left( y ^ { \prime } \right) ^ { 2 } + 1 = 0 {/tex}

Explanation

Q 20.

Correct4

Incorrect-1

If {tex} y ( t ) {/tex} is a solution of {tex} ( 1 + t ) \frac { d y } { d t } - t y = 1 {/tex} and {tex} y ( 0 ) = - 1 , {/tex} then {tex} y ( 1 ) {/tex} is equal to

{tex} - 1 / 2 {/tex}

B

{tex} e + 1 / 2 {/tex}

C

{tex} e - 1 / 2 {/tex}

D

{tex} 1 / 2 {/tex}

Explanation


Q 21.

Correct4

Incorrect-1

If {tex} y = y ( x ) {/tex} and {tex} \frac { 2 + \sin x } { y + 1 } \left( \frac { d y } { d x } \right) = - \cos x , y ( 0 ) = 1 {/tex} then {tex} y \left( \frac { \pi } { 2 } \right) {/tex} equals

{tex} 1 / 3 {/tex}

B

{tex} 2 / 3 {/tex}

C

{tex} - 1 / 3 {/tex}

D

{tex}1{/tex}

Explanation


Q 22.

Correct4

Incorrect-1

If {tex} y = y ( x ) {/tex} and it follows the relation {tex} x \cos y + y \cos x = \pi {/tex} then {tex} y ^ {\prime\prime } ( 0 ) = {/tex}

A

{tex}1{/tex}

B

{tex} - 1 {/tex}

{tex} \pi - 1 {/tex}

D

{tex} - \pi {/tex}

Explanation


Q 23.

Correct4

Incorrect-1

The solution of primitive integral equation {tex} \left( x ^ { 2 } + y ^ { 2 } \right) d y = x y {/tex} {tex} d x {/tex} is {tex} y = y ( x ) {/tex}. If {tex} y ( 1 ) = 1 {/tex} and {tex} \left( x _ { 0 } \right) = e , {/tex} then {tex} x _ { 0 } {/tex} is equal to

A

{tex} \sqrt { 2 \left( e ^ { 2 } - 1 \right) } {/tex}

B

{tex} \sqrt { 2 \left( e ^ { 2 } + 1 \right) } {/tex}

{tex} \sqrt { 3 } e {/tex}

D

{tex} \sqrt { \frac { e ^ { 2 } + 1 } { 2 } } {/tex}

Explanation


Q 24.

Correct4

Incorrect-1

For the primitive integral equation {tex} y d x + y ^ { 2 } d y = x d y {/tex} {tex} x \in R , y > 0 , y = y ( x ) , y ( 1 ) = 1 , {/tex} then {tex} y ( - 3 ) {/tex} is

3

B

2

C

1

D

5

Explanation


Q 25.

Correct4

Incorrect-1

The differential equation {tex} \frac { d y } { d x } = \frac { \sqrt { 1 - y ^ { 2 } } } { y } {/tex} determines a family of circles with

A

variable radii and a fixed centre at {tex} ( 0,1 ) {/tex}

B

variable radii and a fixed centre at {tex} ( 0 , - 1 ) {/tex}

fixed radius {tex}1{/tex} and variable centres along the x-axis.

D

fixed radius {tex}1{/tex} and variable centres along the {tex} y {/tex} -axis.

Explanation