LFM Pure and Mechanics

isi-entrance 2016 Q29 4 marks Geometric or applied optimisation problem View

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e^{-x}$ is:
(A) $1/e$
(B) 1
(C) $1/2$
(D) $e$

isi-entrance 2016 Q29 4 marks Geometric or applied optimisation problem View

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e ^ { - x }$ is:
(A) $1 / e$
(B) 1
(C) $1 / 2$
(D) $e$

isi-entrance 2016 Q38 4 marks Determine intervals of increase/decrease or monotonicity conditions View

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

isi-entrance 2016 Q38 4 marks Determine intervals of increase/decrease or monotonicity conditions View

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

isi-entrance 2016 Q64 4 marks Determine intervals of increase/decrease or monotonicity conditions View

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

isi-entrance 2016 Q64 4 marks Determine intervals of increase/decrease or monotonicity conditions View

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(A) $f ( x )$ is an increasing function on the real line
(B) $f ( x )$ is a decreasing function on the real line
(C) $f ( x )$ is increasing on $- \infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f ( x )$ is decreasing on $- \infty < x \leq 0$ and increasing on $0 \leq x < \infty$

isi-entrance 2017 Q17 Geometric or applied optimisation problem View

A circular lawn of diameter 20 meters on a horizontal plane is to be illuminated by a light-source placed vertically above the centre of the lawn. It is known that the illuminance at a point $P$ on the lawn is given by the formula $I = \frac{C \sin\theta}{d^2}$ for some constant $C$, where $d$ is the distance of $P$ from the light-source and $\theta$ is the angle between the line joining the centre of the lawn to $P$ and the line joining the light-source to $P$. Then the maximum possible illuminance at a point on the circumference of the lawn is
(A) $\frac{C}{75\sqrt{3}}$
(B) $\frac{C}{100\sqrt{3}}$
(C) $\frac{C}{150\sqrt{3}}$
(D) $\frac{C}{250\sqrt{3}}$.

isi-entrance 2017 Q28 Find absolute extrema on a closed interval or domain View

For a positive real number $\alpha$, let $S_\alpha$ denote the set of points $(x, y)$ satisfying $$|x|^\alpha + |y|^\alpha = 1$$ A positive number $\alpha$ is said to be good if the points in $S_\alpha$ that are closest to the origin lie only on the coordinate axes. Then
(A) all $\alpha$ in $(0,1)$ are good and others are not good.
(B) all $\alpha$ in $(1,2)$ are good and others are not good.
(C) all $\alpha > 2$ are good and others are not good.
(D) all $\alpha > 1$ are good and others are not good.

isi-entrance 2018 Q5 Find critical points and classify extrema of a given function View

Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair?
(A) $( 4,4 )$
(B) $( 3,3 )$
(C) $( 2,2 )$
(D) $( 0,0 )$

isi-entrance 2018 Q14 Find absolute extrema on a closed interval or domain View

Let $S = \left\{ x - y \mid x , y \text{ are real numbers with } x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Then the maximum number in the set $S$ is
(A) 1
(B) $\sqrt { 2 }$
(C) $2 \sqrt { 2 }$
(D) $1 + \sqrt { 2 }$.

isi-entrance 2018 Q16 Geometric or applied optimisation problem View

Let $A B C D$ be a rectangle with its shorter side $a > 0$ units and perimeter $2 s$ units. Let $P Q R S$ be any rectangle such that vertices $A , B , C$ and $D$ respectively lie on the lines $P Q , Q R , R S$ and $S P$. Then the maximum area of such a rectangle $P Q R S$ in square units is given by
(A) $s ^ { 2 }$
(B) $2 a ( s - a )$
(C) $\frac { s ^ { 2 } } { 2 }$
(D) $\frac { 5 } { 2 } a ( s - a )$.

isi-entrance 2019 Q8 Geometric or applied optimisation problem View

Consider the following subsets of the plane: $$C_{1} = \left\{(x, y) : x > 0,\ y = \frac{1}{x}\right\}$$ and $$C_{2} = \left\{(x, y) : x < 0,\ y = -1 + \frac{1}{x}\right\}$$ Given any two points $P = (x, y)$ and $Q = (u, v)$ of the plane, their distance $d(P, Q)$ is defined by $$d(P, Q) = \sqrt{(x - u)^{2} + (y - v)^{2}}$$ Show that there exists a unique choice of points $P_{0} \in C_{1}$ and $Q_{0} \in C_{2}$ such that $$d(P_{0}, Q_{0}) \leq d(P, Q) \quad \text{for all } P \in C_{1} \text{ and } Q \in C_{2}.$$

isi-entrance 2019 Q15 Find critical points and classify extrema of a given function View

Let $f$ be a real-valued differentiable function defined on the real line $\mathbb { R }$ such that its derivative $f ^ { \prime }$ is zero at exactly two distinct real numbers $\alpha$ and $\beta$. Then,
(A) $\alpha$ and $\beta$ are points of local maxima of the function $f$.
(B) $\alpha$ and $\beta$ are points of local minima of the function $f$.
(C) one must be a point of local maximum and the other must be a point of local minimum of $f$.
(D) given data is insufficient to conclude about either of them being local extrema points.

isi-entrance 2020 Q5 Geometric or applied optimisation problem View

Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius 1 is regular (i.e., has equal sides).

isi-entrance 2020 Q26 Geometric or applied optimisation problem View

Let $S$ be the set consisting of all those real numbers that can be written as $p - 2 a$ where $p$ and $a$ are the perimeter and area of a right-angled triangle having base length 1 . Then $S$ is
(A) $( 2 , \infty )$
(B) $( 1 , \infty )$
(C) $( 0 , \infty )$
(D) the real line $\mathbb { R }$.

isi-entrance 2021 Q6 Determine intervals of increase/decrease or monotonicity conditions View

Let $f ( x ) = \sin x + \alpha x , x \in \mathbb { R }$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha > 1$ or $\alpha < - 1$.
(B) $\alpha \geq 1$ or $\alpha \leq - 1$.
(C) $\alpha \geq 1$ or $\alpha < - 1$.
(D) $\alpha > 1$ or $\alpha \leq - 1$.

isi-entrance 2021 Q15 Count or characterize roots using extremum values View

The polynomial $x ^ { 4 } + 4 x + c = 0$ has at least one real root if and only if
(A) $c < 2$.
(B) $c \leq 2$.
(C) $c < 3$.
(D) $c \leq 3$.

isi-entrance 2022 Q2 Find critical points and classify extrema of a given function View

Consider the function $$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$ where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.

isi-entrance 2022 Q12 Find absolute extrema on a closed interval or domain View

If $x , y$ are positive real numbers such that $3 x + 4 y < 72$, then the maximum possible value of $12 x y ( 72 - 3 x - 4 y )$ is:
(A) 12240
(B) 13824
(C) 10656
(D) 8640

isi-entrance 2022 Q24 Find critical points and classify extrema of a given function View

The function $x ^ { 2 } \log _ { e } x$ in the interval $( 0,2 )$ has:
(A) exactly one point of local maximum and no points of local minimum.
(B) exactly one point of local minimum and no points of local maximum.
(C) points of local maximum as well as local minimum.
(D) neither a point of local maximum nor a point of local minimum.

isi-entrance 2023 Q16 Find critical points and classify extrema of a given function View

Suppose $F : \mathbb { R } \rightarrow \mathbb { R }$ is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) $F$ cannot have a local minimum.
(B) $F$ must have exactly one local minimum.
(C) $F$ must have at least two local minima.
(D) $F$ must have either a global maximum or a local minimum.

isi-entrance 2023 Q27 Find critical points and classify extrema of a given function View

Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$?
(A), (B), (C), (D) [graphs as provided in the figure]

jee-advanced 2007 Q48 Determine intervals of increase/decrease or monotonicity conditions View

Let $f(x) = x^x$ for $x > 0$. Then $f$ is
(A) increasing on $(0, \infty)$
(B) decreasing on $(0, \infty)$
(C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$
(D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$

jee-advanced 2007 Q60 Determine intervals of increase/decrease or monotonicity conditions View

Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then
(A) $f$ is an increasing function of $x$
(B) $f$ is a decreasing function of $x$
(C) $f$ is neither increasing nor decreasing function of $x$
(D) $f'$ is not a monotonic function of $x$

jee-advanced 2007 Q65 Find critical points and classify extrema of a given function View

Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then
(A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$
(B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$
(C) $f$ has local minima at $x = -1$ and at $x = 2$
(D) $f$ has local maxima at $x = -1$ and at $x = 2$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40