LFM Pure and Mechanics

isi-entrance 2016 Q71 4 marks Definite Integral as a Limit of Riemann Sums View

For each positive integer $n$, define a function $f_n$ on $[0,1]$ as follows: $$f_n(x) = \left\{ \begin{array}{ccc} 0 & \text{if} & x = 0 \\ \sin\frac{\pi}{2n} & \text{if} & 0 < x \leq \frac{1}{n} \\ \sin\frac{2\pi}{2n} & \text{if} & \frac{1}{n} < x \leq \frac{2}{n} \\ \sin\frac{3\pi}{2n} & \text{if} & \frac{2}{n} < x \leq \frac{3}{n} \\ \vdots & \vdots & \vdots \\ \sin\frac{n\pi}{2n} & \text{if} & \frac{n-1}{n} < x \leq 1 \end{array} \right.$$ Then, the value of $\lim_{n \rightarrow \infty} \int_0^1 f_n(x) dx$ is
(A) $\pi$
(B) 1
(C) $\frac{1}{\pi}$
(D) $\frac{2}{\pi}$

isi-entrance 2016 Q71 4 marks Definite Integral as a Limit of Riemann Sums View

For each positive integer $n$, define a function $f _ { n }$ on $[ 0, 1 ]$ as follows: $$f _ { n } ( x ) = \left\{ \begin{array} { c c c } 0 & \text { if } & x = 0 \\ \sin \frac { \pi } { 2 n } & \text { if } & 0 < x \leq \frac { 1 } { n } \\ \sin \frac { 2 \pi } { 2 n } & \text { if } & \frac { 1 } { n } < x \leq \frac { 2 } { n } \\ \sin \frac { 3 \pi } { 2 n } & \text { if } & \frac { 2 } { n } < x \leq \frac { 3 } { n } \\ \vdots & \vdots & \vdots \\ \sin \frac { n \pi } { 2 n } & \text { if } & \frac { n - 1 } { n } < x \leq 1 \end{array} \right.$$ Then, the value of $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { 1 } f _ { n } ( x ) d x$ is
(A) $\pi$
(B) 1
(C) $\frac { 1 } { \pi }$
(D) $\frac { 2 } { \pi }$

isi-entrance 2017 Q20 Average Value of a Function View

Let $f : [0,2] \rightarrow \mathbb{R}$ be a continuous function such that $$\frac{1}{2}\int_0^2 f(x)\,dx < f(2)$$ Then which of the following statements must be true?
(A) $f$ must be strictly increasing.
(B) $f$ must attain a maximum value at $x = 2$.
(C) $f$ cannot have a minimum at $x = 2$.
(D) None of the above.

isi-entrance 2020 Q5 Definite Integral as a Limit of Riemann Sums View

What is the limit of $\sum _ { k = 1 } ^ { n } \frac { e ^ { - k / n } } { n }$ as $n$ tends to $\infty$ ?
(A) The limit does not exist.
(B) $\infty$
(C) $1 - e ^ { - 1 }$
(D) $e ^ { - 0.5 }$

isi-entrance 2022 Q17 Convergence and Evaluation of Improper Integrals View

For $n \in \mathbb { N }$, let $a _ { n }$ be defined as $$a _ { n } = \int _ { 0 } ^ { n } \frac { 1 } { 1 + n x ^ { 2 } } d x$$ Then $\lim _ { n \rightarrow \infty } a _ { n }$
(A) equals 0
(B) equals $\frac { \pi } { 4 }$
(C) equals $\frac { \pi } { 2 }$
(D) does not exist

isi-entrance 2023 Q18 Integral Equation with Symmetry or Substitution View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a twice differentiable one-to-one function. If $f ( 2 ) = 2 , f ( 3 ) = - 8$ and $$\int _ { 2 } ^ { 3 } f ( x ) d x = - 3$$ then $$\int _ { - 8 } ^ { 2 } f ^ { - 1 } ( x ) d x$$ equals
(A) $- 25$.
(B) $25$.
(C) $- 31$.
(D) $31$.

isi-entrance 2026 Q1 Integral Equation with Symmetry or Substitution View

The value of the integral $\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$ equals
(a) 1 .
(B) $\pi$.
(C) $e$.
(D) none of these.

isi-entrance 2026 Q5 Definite Integral as a Limit of Riemann Sums View

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(a) 0 .
(B) 1 .
(C) $e$.
(D) $\sqrt { e } / 2$.

isi-entrance 2026 Q19 Maximizing or Optimizing a Definite Integral View

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(a) $3 / 4$.
(B) $4 / 3$.
(C) $3 / 2$.
(D) $2 / 3$.

jee-advanced 2007 Q66 Integral Inequalities and Limit of Integral Sequences View

Let $I = \int_0^1 \frac{\sin x}{\sqrt{x}}\,dx$ and $J = \int_0^1 \frac{\cos x}{\sqrt{x}}\,dx$. Then which one of the following is true?
(A) $I > \frac{2}{3}$ and $J > 2$
(B) $I < \frac{2}{3}$ and $J < 2$
(C) $I < \frac{2}{3}$ and $J > 2$
(D) $I > \frac{2}{3}$ and $J < 2$

jee-advanced 2007 Q68 Integral Equation with Symmetry or Substitution View

Match the statements in Column I with the values in Column II.
Column I
(A) $\int_{-\pi}^{\pi} \cos^2 x\,\frac{1}{1+a^x}\,dx$, $a > 0$
(B) $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$
(C) $\int_{-2}^{2} \frac{x^2}{1+5^x}\,dx$
(D) $\int_1^2 \frac{\sqrt{\ln(3-x)}}{\sqrt{\ln(3-x)}+\sqrt{\ln(x+1)}}\,dx$
Column II
(p) $\frac{1}{2}$
(q) $0$
(r) $\frac{\pi}{4}$
(s) $\frac{\pi}{2}$

jee-advanced 2008 Q10 Integral Equation with Symmetry or Substitution View

Let $f ( x )$ be a non-constant twice differentiable function defined on $( - \infty , \infty )$ such that $f ( x ) = f ( 1 - x )$ and $f ^ { \prime } \left( \frac { 1 } { 4 } \right) = 0$. Then,
(A) $f ^ { \prime \prime } ( x )$ vanishes at least twice on $[ 0,1 ]$
(B) $f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 0$
(C) $\quad \int _ { - 1 / 2 } ^ { 1 / 2 } f \left( x + \frac { 1 } { 2 } \right) \sin x d x = 0$
(D) $\int _ { 0 } ^ { 1 / 2 } f ( t ) e ^ { \sin \pi t } d t = \int _ { 1 / 2 } ^ { 1 } f ( 1 - t ) e ^ { \sin \pi t } d t$

jee-advanced 2008 Q16 Accumulation Function Analysis View

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Let
$$g ( x ) = \int _ { 0 } ^ { e ^ { x } } \frac { f ^ { \prime } ( t ) } { 1 + t ^ { 2 } } d t$$
Which of the following is true?
(A) $g ^ { \prime } ( x )$ is positive on $( - \infty , 0 )$ and negative on $( 0 , \infty )$
(B) $g ^ { \prime } ( x )$ is negative on $( - \infty , 0 )$ and positive on $( 0 , \infty )$
(C) $g ^ { \prime } ( x )$ changes sign on both $( - \infty , 0 )$ and $( 0 , \infty )$
(D) $g ^ { \prime } ( x )$ does not change sign on $( - \infty , \infty )$

jee-advanced 2008 Q20 Recovering Function Values from Derivative Information View

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
$\int _ { - 1 } ^ { 1 } g ^ { \prime } ( x ) d x =$
(A) $2 g ( - 1 )$
(B) 0
(C) $- 2 g ( 1 )$
(D) $2 g ( 1 )$

jee-advanced 2009 Q24 Integral Equation with Symmetry or Substitution View

If $$I_{n}=\int_{-\pi}^{\pi}\frac{\sin nx}{\left(1+\pi^{x}\right)\sin x}dx,\quad n=0,1,2,\ldots,$$ then
(A) $I_{n}=I_{n+2}$
(B) $\sum_{m=1}^{10}I_{2m+1}=10\pi$
(C) $\sum_{m=1}^{10}I_{2m}=0$
(D) $I_{n}=I_{n+1}$

jee-advanced 2010 Q34 Accumulation Function Analysis View

The value of $\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 3 } } \int _ { 0 } ^ { x } \frac { t \ln ( 1 + t ) } { t ^ { 4 } + 4 } d t$ is
A) 0
B) $\frac { 1 } { 12 }$
C) $\frac { 1 } { 24 }$
D) $\frac { 1 } { 64 }$

jee-advanced 2010 Q41 Definite Integral Evaluation (Computational) View

The value(s) of $\int _ { 0 } ^ { 1 } \frac { x ^ { 4 } ( 1 - x ) ^ { 4 } } { 1 + x ^ { 2 } } d x$ is (are)
A) $\frac { 22 } { 7 } - \pi$
B) $\frac { 2 } { 105 }$
C) 0
D) $\frac { 71 } { 15 } - \frac { 3 \pi } { 2 }$

jee-advanced 2010 Q52 Piecewise/Periodic Function Integration View

For any real number x, let $[ \mathrm { x } ]$ denote the largest integer less than or equal to x. Let $f$ be a real valued function defined on the interval $[ - 10,10 ]$ by $$f ( x ) = \left\{ \begin{array} { c c } x - [ x ] & \text { if } [ x ] \text { is odd } \\ 1 + [ x ] - x & \text { if } [ x ] \text { is even } \end{array} \right.$$ Then the value of $\frac { \pi ^ { 2 } } { 10 } \int _ { - 10 } ^ { 10 } f ( x ) \cos \pi x \, d x$ is

jee-advanced 2013 Q43 Integral Inequalities and Limit of Integral Sequences View

Let $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathbb { R }$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f ^ { \prime } ( x ) < 2 f ( x )$ and $f \left( \frac { 1 } { 2 } \right) = 1$. Then the value of $\int _ { 1/2 } ^ { 1 } f ( x ) d x$ lies in the interval
(A) $( 2 e - 1,2 e )$
(B) $( e - 1,2 e - 1 )$
(C) $\left( \frac { e - 1 } { 2 } , e - 1 \right)$
(D) $\left( 0 , \frac { e - 1 } { 2 } \right)$

jee-advanced 2014 Q44 Finding a Function from an Integral Equation View

Let $f : [0,2] \rightarrow \mathbb{R}$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0) = 1$. Let
$$F(x) = \int_{0}^{x^2} f(\sqrt{t})\, dt$$
for $x \in [0,2]$. If $F'(x) = f'(x)$ for all $x \in (0,2)$, then $F(2)$ equals
(A) $e^2 - 1$
(B) $e^4 - 1$
(C) $e - 1$
(D) $e^4$

jee-advanced 2014 Q45 Properties of Integral-Defined Functions (Continuity, Differentiability) View

Let $f : [a,b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_{a}^{x} f(t)\, dt & \text{if } a \leq x \leq b \\ \int_{a}^{b} f(t)\, dt & \text{if } x > b \end{cases}$$ Then
(A) $g(x)$ is continuous but not differentiable at $a$
(B) $g(x)$ is differentiable on $\mathbb{R}$
(C) $g(x)$ is continuous but not differentiable at $b$
(D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both

jee-advanced 2014 Q58 Integral Equation with Symmetry or Substitution View

List I	List II
P. The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\leq 2$, satisfying $f(0) = 0$ and $\int_{0}^{1} f(x)\,dx = 1$, is	1. 8
Q. The number of points in the interval $[-\sqrt{13}, \sqrt{13}]$ at which $f(x) = \sin(x^2) + \cos(x^2)$ attains its maximum value, is	2. 2
R. $\int_{-2}^{2} \frac{3x^2}{1+e^x}\,dx$ equals	3. 4
S. $\dfrac{\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} \cos 2x \log\left(\frac{1+x}{1-x}\right)dx}{\displaystyle\int_{0}^{\frac{1}{2}} \cos 2x \log\left(\frac{1+x}{1-x}\right)dx}$ equals	4. 0

P Q R S
(A) 3241
(B) 2341
(C) 3214
(D) 2314

jee-advanced 2015 Q46 Piecewise/Periodic Function Integration View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \left\{ \begin{array} { l l } { [ x ] , } & x \leq 2 \\ 0 , & x > 2 \end{array} \right.$, where $[ x ]$ is the greatest integer less than or equal to $x$. If $I = \int _ { - 1 } ^ { 2 } \frac { x f \left( x ^ { 2 } \right) } { 2 + f ( x + 1 ) } d x$, then the value of $( 4 I - 1 )$ is

jee-advanced 2015 Q47 Finding a Function from an Integral Equation View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous odd function, which vanishes exactly at one point and $f ( 1 ) = \frac { 1 } { 2 }$. Suppose that $F ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$ for all $x \in [ - 1,2 ]$ and $G ( x ) = \int _ { - 1 } ^ { x } t | f ( f ( t ) ) | d t$ for all $x \in [ - 1,2 ]$. If $\lim _ { x \rightarrow 1 } \frac { F ( x ) } { G ( x ) } = \frac { 1 } { 14 }$, then the value of $f \left( \frac { 1 } { 2 } \right)$ is

jee-advanced 2015 Q48 Finding a Function from an Integral Equation View

Let $F ( x ) = \int _ { x } ^ { x ^ { 2 } + \frac { \pi } { 6 } } 2 \cos ^ { 2 } t \, d t$ for all $x \in \mathbb { R }$ and $f : \left[ 0 , \frac { 1 } { 2 } \right] \rightarrow [ 0 , \infty )$ be a continuous function. For $a \in \left[ 0 , \frac { 1 } { 2 } \right]$, if $F ^ { \prime } ( a ) + 2$ is the area of the region bounded by $x = 0 , y = 0 , y = f ( x )$ and $x = a$, then $f ( 0 )$ is

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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