Core Syllabus Module

Differentiation

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2019

LEVELJEE Advanced

Let $f (x) = \frac{s i n π x}{x ^{2}}, x > 0$ . Let $x_{1} < x_{2} < x_{3} < \dots < x_{n} < \dots$ be all the points of local maximum of $f$ and $y_{1} < y_{2} < y_{3} < \dots < y_{n} < \dots$ be all the points of local minimum of $f$ . Then which of the following options is/are correct ?

* Multiple Correct Options

(A)

∣ x_{n} - y_{n} ∣ > 1

for every

n

(B)

x_{1} < y_{1}

(C)

x_{n} \in (2 n, 2 n + 1/2)

for every

n

(D)

x_{n + 1} - x_{n} > 2

for every

n

JEE Advanced 1994

LEVELJEE Advanced

The circle $x^{2} + y^{2} = 1$ cuts the $x$ -axis at $P$ and $Q$ . Another circle with centre at $Q$ and variable radius intersects the first circle at $R$ above the $x$ -axis and the line segment $P Q$ at $S$ . Find the maximum area of the triangle $QS R$ .

JEE Advanced 2007

LEVELJEE Advanced

Comprehension Passage

If a continuous function

f

defined on the real line

R

, assumes positive and negative values in

R

then the equation

f (x) = 0

has a root in

R

. For example, if it is known that a continuous function

f

R

is positive at some point and its minimum value is negative then the equation

f (x) = 0

has a root in

R

. Consider

f (x) = k e^{x} - x

for all real

x

where

k

is a real constant.

Question 1:

The line $y = x$ meets $y = k e^{x}$ for $k \leq 0$ at

The positive value of $k$ for which $k e^{x} - x = 0$ has only one root is

(A)

1/ e

(B)

(C)

(D)

lo g_{e} 2

Question 3:

For $k > 0$ , the set of all values of $k$ for which $k e^{x} - x = 0$ has two distinct roots is

(A)

(0, 1/ e)

(B)

(1/ e, 1)

(C)

(1/ e, \infty)

(D)

(0, 1)

JEE Advanced 2012

LEVELJEE Advanced

Comprehension Passage

Let

f (x) = (1 - x)^{2} sin^{2} x + x^{2}

for all

x \in R

and let

g (x) = \int_{1}^{x} (\frac{2 ( t - 1 )}{t + 1} - ln t) f (t) d t

for all

x \in (1, \infty)

Question 1:

Consider the statements: $P$ : There exists some $x \in R$ such that $f (x) + 2 x = 2 (1 + x^{2})$ , $Q$ : There exists some $x \in R$ such that $2 f (x) + 1 = 2 x (1 + x)$

(A)

both

P

and

Q

are true

(B)

P is true and Q is false

(C)

P is false and Q is true

(D)

both

P

and

Q

are false

Question 2:

Which of the following is true?

(A)

g

is increasing on

(1, \infty)

(B)

g is decreasing on

(1, \infty)

(C)

g is increasing on

(1, 2)

and decreasing on

(2, \infty)

(D)

g is decreasing on

(1, 2)

and increasing on

(2, \infty)

JEE Advanced 2013

LEVELJEE Advanced

Comprehension Passage

Let

f : [0, 1] \to R

(the set of all real numbers) be a function. Suppose the function

f

is twice differentiable,

f (0) = f (1) = 0

and satisfies

f^{''} (x) - 2 f^{'} (x) + f (x) \geq e^{x}, x \in (0, 1]

Question 1:

Which of the following is true for $0 < x < 1$ ?

(A)

0 < f (x) < \infty

(B)

- 1/2 < f (x) < 1/2

(C)

- 1/4 < f (x) < 1

(D)

- \infty < f (x) < 0

Question 2:

If the function $e^{- x} f (x)$ assumes its minimum in the interval $[0, 1]$ at $x = 1/4$ , which of the following is true?

(A)

f^{'} (x) < f (x), 1/4 < x < 3/4

(B)

f^{'} (x) > f (x), 0 < x < 1/4

(C)

f^{'} (x) < f (x), 0 < x < 1/4

(D)

f^{'} (x) < f (x), 3/4 < x < 1

JEE Advanced 2009

LEVELJEE Advanced

Match the statements/expressions in Column I with the open intervals in Column II.

List-I

(P)

Interval contained in the domain of definition of non-zero solutions of the differential equation

(x - 3)^{2} y^{'} + y = 0

(Q)

Interval containing the value of the integral

\int_{1}^{5} (x - 1) (x - 2) (x - 3) (x - 4) (x - 5) d x

(R)

Interval in which at least one of the points of local maximum of

cos^{2} x + sin x

lies

(S)

Interval in which

tan^{- 1} (sin x + cos x)

is increasing

List-II

(1)

(- \frac{π}{2}, \frac{π}{2})

(2)

(0, \frac{π}{2})

(3)

(\frac{π}{8}, \frac{5 π}{4})

(4)

(0, \frac{π}{8})

(5)

(- π, π)

JEE Main 2022 (29 June Shift 1)

LEVELJEE Advanced

Let $f : R \to R$ be a function defined by : $f (x) = ⎩ ⎨ ⎧ max_{t \leq x} {t^{3} - 3 t} x^{2} + 2 x - 6 [x - 3] + 9 2 x + 1; x \leq 2; 2 < x < 3; 3 \leq x \leq 5; x > 5$ Where $[t]$ is the greatest integer less than or equal to $t$ . Let $m$ be the number of points where $f$ is not differentiable and $I = \int_{- 2}^{2} f (x) d x$ . Then the ordered pair $(m, I)$ is equal to :

(A)

(3, \frac{27}{4})

(B)

(3, \frac{23}{4})

(C)

(4, \frac{27}{4})

(D)

(4, \frac{23}{4})

JEE Main 2021 (17 March Shift 1)

LEVELJEE Advanced

The maximum value of $z$ in the following equation $z = 6 x y + y^{2}$ , where $3 x + 4 y \leq 100$ and $4 x + 3 y \leq 75$ for $x \geq 0$ and $y \geq 0$ is ____ (Round off to the Nearest Integer)

JEE Main 2022 (26 July Shift 2)

LEVELJEE Advanced

Let $P$ and $Q$ be any points on the curves $(x - 1)^{2} + (y + 1)^{2} = 1$ and $y = x^{2}$ , respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval

(A)

(0, \frac{1}{4})

(B)

(\frac{1}{2}, \frac{3}{4})

(C)

(\frac{1}{4}, \frac{1}{2})

(D)

(\frac{3}{4}, 1)

JEE Advanced 2008

LEVELJEE Advanced

Let $f (x)$ be a non-constant twice differentiable function defined on $(- \infty, \infty)$ such that $f (x) = f (1 - x)$ and $f^{'} (\frac{1}{4}) = 0$ . Then,

* Multiple Correct Options

(A)

f^{''} (x)

vanishes at least twice on

[0, 1]

(B)

f^{'} (\frac{1}{2}) = 0

(C)

\int_{- 1/2}^{1/2} f (x + \frac{1}{2}) sin x d x = 0

(D)

\int_{0}^{1/2} f (t) e^{s i n π t} d t = \int_{1/2}^{1} f (1 - t) e^{s i n π t} d t

JEE Advanced 2013

LEVELJEE Advanced

A line $L : y = m x + 3$ meets $y$ -axis at $E (0, 3)$ and the arc of the parabola $y^{2} = 16 x, 0 \leq y \leq 6$ at the point $F (x_{0}, y_{0})$ . The tangent to the parabola at $F (x_{0}, y_{0})$ intersects the $y$ -axis at $G (0, y_{1})$ . The slope $m$ of the line $L$ is chosen such that the area of the triangle $E F G$ has a local maximum. Match List I with List II and select the correct answer using the code given below the lists :

List-I

(P)

m =

(Q)

Maximum area of

Δ E F G

(R)

y_{0} =

(S)

y_{1} =

List-II

Match the statements given in Column-I with the intervals/union of intervals given in Column-II.

List-I

(P)

The set

{Re (\frac{2 i z}{1 - z ^{2}}) : z is a complex number, ∣ z ∣ = 1, z \neq = \pm 1}

(Q)

The domain of the function

f (x) = sin^{- 1} (\frac{8 ( 3 ) ^{x - 2}}{1 - 3 ^{2 (x - 1)}})

(R)

f (θ) = 1 - tan θ - 1 tan θ 1 - tan θ 1 tan θ 1

, then the set

{f (θ) : 0 \leq θ < \frac{π}{2}}

(S)

f (x) = x^{3/2} (3 x - 10), x \geq 0

then

f (x)

is increasing in

List-II

(1)

(- \infty, - 1) \cup (1, \infty)

(2)

(- \infty, - 0) \cup (0, \infty)

(3)

[2, \infty)

(4)

(- \infty, - 1] \cup [1, \infty)

(5)

(0, 0] \cup [2, \infty)

JEE Main 2020 - 9 Jan (Evening)

LEVELJEE Advanced

If $x = 2 sin θ - sin 2 θ$ and $y = 2 cos θ - cos 2 θ$ , $θ \in [0, 2 π]$ , then $\frac{d ^{2} y}{d x ^{2}}$ at $θ = π$ is:

(A)

\frac{3}{8}

(B)

\frac{3}{2}

(C)

\frac{3}{4}

(D)

- \frac{3}{4}

JEE Advanced 1991

LEVELJEE Advanced

Find $\frac{d y}{d x}$ at $x = - 1$ , when $(sin y)^{s i n (\frac{π x}{2})} + \frac{3}{2} sec^{- 1} (2 x) + 2^{x} tan (ln (x + 2)) = 0$ .

JEE(ADVANCED)-201

LEVELJEE Advanced

If $f : R \to R$ is a twice differentiable function such that $f^{''} (x) > 0$ for all $x \in R$ , and $f (\frac{1}{2}) = f (1) = 1$ , then

(A)

f^{'} (1) \leq 0

(B)

0 < f^{'} (1) \leq \frac{1}{2}

(C)

\frac{1}{2} < f^{'} (1) \leq 1

(D)

f^{'} (1) > 1

JEE Advanced 2023

LEVELJEE Advanced

Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d ^{2} f}{d x ^{2}} (x) > 0$ for all $x \in (- 1, 1)$ . For $f \in S$ , let $X_{f}$ be the number of points $x \in (- 1, 1)$ for which $f (x) = x$ . Then which of the following statements is(are) true?

* Multiple Correct Options

(A)

There exists a function

f \in S

such that

X_{f} = 0

(B)

For every function

f \in S

, we have

X_{f} \leq 2

(C)

There exists a function

f \in S

such that

X_{f} = 2

(D)

There does NOT exist any function

f

S

such that

X_{f} = 1

JEE Advanced 2022

LEVELJEE Advanced

Let $α = \sum_{k = 1}^{\infty} sin^{2 k} (\frac{π}{6})$ . Let $g : [0, 1] \to R$ be the function defined by $g (x) = 2^{α x} + 2^{α (1 - x)}$ . Then, which of the following statements is/are TRUE?

* Multiple Correct Options

(A)

The minimum value of

g (x)

2^{7/6}

(B)

The maximum value of

g (x)

1 + 2^{1/3}

(C)

The function

g (x)

attains its maximum at more than one point

(D)

The function

g (x)

attains its minimum at more than one point

JEE Advanced 2021

LEVELJEE Advanced

Let $f : R \to R$ be defined by $f (x) = \frac{x ^{2} - 3 x - 6}{x ^{2} + 2 x + 4}$ . Then which of the following statements is (are) TRUE ?

* Multiple Correct Options

(A)

f

is decreasing in the interval

(- 2, - 1)

(B)

f

is increasing in the interval

(1, 2)

(C)

f

is onto

(D)

Range of

f

[- \frac{3}{2}, 2]

JEE Advanced 2020

LEVELJEE Advanced

Consider all rectangles lying in the region ${(x, y) \in R \times R : 0 \leq x \leq \frac{π}{2} and 0 \leq y \leq 2 sin (2 x)}$ and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

(A)

\frac{3 π}{2}

(B)

π

(C)

\frac{π}{2 3}

(D)

\frac{π 3}{2}

JEE Main 2026 (24 January Shift 1)

LEVELJEE Advanced

Let $(2 α, α)$ be the largest interval in which the function $f (t) = \frac{∣ t + 1∣}{t ^{2}}, t < 0$ , is strictly decreasing. Then the local maximum value of the function $g (x) = 2 lo g_{e} (x - 2) + α x^{2} + 4 x - α, x > 2$ , is .........

Micro-Topic Strategy Matrix

Techniques of Differentiation

Higher Order Derivatives

Tangents, Normals and Rate Measure

Monotonicity

Maxima and Minima

Mean Value Theorems

Chapter Elite-20 Challenge Zone

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The line y=x meets y=kex for k≤0 at

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The positive value of k for which kex−x=0 has only one root is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For k>0, the set of all values of k for which kex−x=0 has two distinct roots is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the statements: P: There exists some x∈R such that f(x)+2x=2(1+x2), Q: There exists some x∈R such that 2f(x)+1=2x(1+x)

Which of the following is true?

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following is true for 0<x<1?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function e−xf(x) assumes its minimum in the interval [0,1] at x=1/4, which of the following is true?

Match the statements/expressions in Column I with the open intervals in Column II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The maximum value of z in the following equation z=6xy+y2, where 3x+4y≤100 and 4x+3y≤75 for x≥0 and y≥0 is ____ (Round off to the Nearest Integer)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P and Q be any points on the curves (x−1)2+(y+1)2=1 and y=x2, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x) be a non-constant twice differentiable function defined on (−∞,∞) such that f(x)=f(1−x) and f′(41​)=0. Then,

List-I

List-II

Match the statements given in Column-I with the intervals/union of intervals given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=2sinθ−sin2θ and y=2cosθ−cos2θ, θ∈[0,2π], then dx2d2y​ at θ=π is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find dxdy​ at x=−1, when (siny)sin(2πx​)+23​​sec−1(2x)+2xtan(ln(x+2))=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f:R→R is a twice differentiable function such that f′′(x)>0 for all x∈R, and f(21​)=f(1)=1, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α=∑k=1∞​sin2k(6π​). Let g:[0,1]→R be the function defined by g(x)=2αx+2α(1−x). Then, which of the following statements is/are TRUE?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be defined by f(x)=x2+2x+4x2−3x−6​. Then which of the following statements is (are) TRUE ?

The line $y = x$ meets $y = k e^{x}$ for $k \leq 0$ at

The positive value of $k$ for which $k e^{x} - x = 0$ has only one root is

For $k > 0$ , the set of all values of $k$ for which $k e^{x} - x = 0$ has two distinct roots is

Consider the statements: $P$ : There exists some $x \in R$ such that $f (x) + 2 x = 2 (1 + x^{2})$ , $Q$ : There exists some $x \in R$ such that $2 f (x) + 1 = 2 x (1 + x)$

Which of the following is true for $0 < x < 1$ ?

If the function $e^{- x} f (x)$ assumes its minimum in the interval $[0, 1]$ at $x = 1/4$ , which of the following is true?

The maximum value of $z$ in the following equation $z = 6 x y + y^{2}$ , where $3 x + 4 y \leq 100$ and $4 x + 3 y \leq 75$ for $x \geq 0$ and $y \geq 0$ is ____ (Round off to the Nearest Integer)

Let $P$ and $Q$ be any points on the curves $(x - 1)^{2} + (y + 1)^{2} = 1$ and $y = x^{2}$ , respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval

Let $f (x)$ be a non-constant twice differentiable function defined on $(- \infty, \infty)$ such that $f (x) = f (1 - x)$ and $f^{'} (\frac{1}{4}) = 0$ . Then,

If $x = 2 sin θ - sin 2 θ$ and $y = 2 cos θ - cos 2 θ$ , $θ \in [0, 2 π]$ , then $\frac{d ^{2} y}{d x ^{2}}$ at $θ = π$ is:

Find $\frac{d y}{d x}$ at $x = - 1$ , when $(sin y)^{s i n (\frac{π x}{2})} + \frac{3}{2} sec^{- 1} (2 x) + 2^{x} tan (ln (x + 2)) = 0$ .

If $f : R \to R$ is a twice differentiable function such that $f^{''} (x) > 0$ for all $x \in R$ , and $f (\frac{1}{2}) = f (1) = 1$ , then

Let $α = \sum_{k = 1}^{\infty} sin^{2 k} (\frac{π}{6})$ . Let $g : [0, 1] \to R$ be the function defined by $g (x) = 2^{α x} + 2^{α (1 - x)}$ . Then, which of the following statements is/are TRUE?

Let $f : R \to R$ be defined by $f (x) = \frac{x ^{2} - 3 x - 6}{x ^{2} + 2 x + 4}$ . Then which of the following statements is (are) TRUE ?