LFM Pure and Mechanics

jee-advanced 2015 Q53 Integral Equation with Symmetry or Substitution View

The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)
$$\frac { \int _ { 0 } ^ { 4 \pi } e ^ { t } \left( \sin ^ { 6 } a t + \cos ^ { 4 } a t \right) d t } { \int _ { 0 } ^ { \pi } e ^ { t } \left( \sin ^ { 6 } a t + \cos ^ { 4 } a t \right) d t } = L ?$$
(A) $\quad a = 2 , L = \frac { e ^ { 4 \pi } - 1 } { e ^ { \pi } - 1 }$
(B) $\quad a = 2 , L = \frac { e ^ { 4 \pi } + 1 } { e ^ { \pi } + 1 }$
(C) $\quad a = 4 , L = \frac { e ^ { 4 \pi } - 1 } { e ^ { \pi } - 1 }$
(D) $\quad a = 4 , L = \frac { e ^ { 4 \pi } + 1 } { e ^ { \pi } + 1 }$

jee-advanced 2015 Q55 Definite Integral Evaluation (Computational) View

Let $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the correct expression(s) is(are)
(A) $\quad \int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 12 }$
(B) $\quad \int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 0$
(C) $\int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 6 }$
(D) $\int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 1$

jee-advanced 2015 Q56 Integral Inequalities and Limit of Integral Sequences View

Let $f ^ { \prime } ( x ) = \frac { 192 x ^ { 3 } } { 2 + \sin ^ { 4 } \pi x }$ for all $x \in \mathbb { R }$ with $f \left( \frac { 1 } { 2 } \right) = 0$. If $m \leq \int _ { 1 / 2 } ^ { 1 } f ( x ) d x \leq M$, then the possible values of $m$ and $M$ are
(A) $m = 13 , M = 24$
(B) $\quad m = \frac { 1 } { 4 } , M = \frac { 1 } { 2 }$
(C) $m = - 11 , M = 0$
(D) $m = 1 , M = 12$

jee-advanced 2015 Q60 Definite Integral Evaluation (Computational) View

If $\int _ { 1 } ^ { 3 } x ^ { 2 } F ^ { \prime \prime } ( x ) d x = - 12$ and $\int _ { 1 } ^ { 3 } x ^ { 3 } F ^ { \prime \prime } ( x ) d x = 40$, then the correct expression(s) is(are)
(A) $9 f ^ { \prime } ( 3 ) + f ^ { \prime } ( 1 ) - 32 = 0$
(B) $\int _ { 1 } ^ { 3 } f ( x ) d x = 12$
(C) $9 f ^ { \prime } ( 3 ) - f ^ { \prime } ( 1 ) + 32 = 0$
(D) $\int _ { 1 } ^ { 3 } f ( x ) d x = - 12$

jee-advanced 2016 Q40 Integral Equation with Symmetry or Substitution View

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { x ^ { 2 } \cos x } { 1 + e ^ { x } } d x$ is equal to
(A) $\frac { \pi ^ { 2 } } { 4 } - 2$
(B) $\frac { \pi ^ { 2 } } { 4 } + 2$
(C) $\pi ^ { 2 } - e ^ { \frac { \pi } { 2 } }$
(D) $\pi ^ { 2 } + e ^ { \frac { \pi } { 2 } }$

jee-advanced 2016 Q52 Maximizing or Optimizing a Definite Integral View

The total number of distinct $x \in [0,1]$ for which $\int_0^x \frac{t^2}{1+t^4}\,dt = 2x - 1$ is

jee-advanced 2017 Q40 Integral Inequalities and Limit of Integral Sequences View

Let $f : \mathbb{R} \rightarrow (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$?
[A] $x^9 - f(x)$
[B] $x - \int_0^{\frac{\pi}{2} - x} f(t)\cos t\, dt$
[C] $e^x - \int_0^x f(t)\sin t\, dt$
[D] $f(x) + \int_0^{\frac{\pi}{2}} f(t)\sin t\, dt$

jee-advanced 2017 Q45 Definite Integral Evaluation (Computational) View

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$, $f\left(\frac{\pi}{2}\right) = 3$ and $f'(0) = 1$. If $$g(x) = \int_x^{\frac{\pi}{2}} \left[f'(t)\operatorname{cosec} t - \cot t\operatorname{cosec} t\, f(t)\right] dt$$ for $x \in \left(0, \frac{\pi}{2}\right]$, then $\lim_{x \rightarrow 0} g(x) =$

jee-advanced 2019 Q6 Definite Integral as a Limit of Riemann Sums View

For $a \in \mathbb{R}$, $|a| > 1$, let $$\lim_{n\rightarrow\infty}\left(\frac{1 + \sqrt[3]{2} + \cdots + \sqrt[3]{n}}{n^{7/3}\left(\frac{1}{(an+1)^2} + \frac{1}{(an+2)^2} + \cdots + \frac{1}{(an+n)^2}\right)}\right) = 54$$
Then the possible value(s) of $a$ is/are
(A) $-9$
(B) $-6$
(C) $7$
(D) $8$

jee-advanced 2019 Q13 Integral Equation with Symmetry or Substitution View

The value of the integral $$\int_0^{\pi/2} \frac{3\sqrt{\cos\theta}}{(\sqrt{\cos\theta} + \sqrt{\sin\theta})^5}\,d\theta$$ equals

jee-advanced 2019 Q17 Integral Equation with Symmetry or Substitution View

If $$I = \frac { 2 } { \pi } \int _ { - \pi / 4 } ^ { \pi / 4 } \frac { d x } { \left( 1 + e ^ { \sin x } \right) ( 2 - \cos 2 x ) }$$ then $27 I ^ { 2 }$ equals

jee-advanced 2020 Q12 Integral Inequalities and Limit of Integral Sequences View

Which of the following inequalities is/are TRUE?
(A) $\int _ { 0 } ^ { 1 } x \cos x \, d x \geq \frac { 3 } { 8 }$
(B) $\int _ { 0 } ^ { 1 } x \sin x \, d x \geq \frac { 3 } { 10 }$
(C) $\int _ { 0 } ^ { 1 } x ^ { 2 } \cos x \, d x \geq \frac { 1 } { 2 }$
(D) $\int _ { 0 } ^ { 1 } x ^ { 2 } \sin x \, d x \geq \frac { 2 } { 9 }$

jee-advanced 2020 Q16 Integral Equation with Symmetry or Substitution View

Let the function $f: [0,1] \rightarrow \mathbb{R}$ be defined by $$f(x) = \frac{4^{x}}{4^{x} + 2}$$ Then the value of $$f\left(\frac{1}{40}\right) + f\left(\frac{2}{40}\right) + f\left(\frac{3}{40}\right) + \cdots + f\left(\frac{39}{40}\right) - f\left(\frac{1}{2}\right)$$ is $\_\_\_\_$

jee-advanced 2020 Q17 Definite Integral Evaluation (Computational) View

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$.
If $F: [0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x) = \int_{0}^{x} f(t)\, dt$, and if $$\int_{0}^{\pi} \left(f'(x) + F(x)\right) \cos x\, dx = 2$$ then the value of $f(0)$ is $\_\_\_\_$

jee-advanced 2021 Q3 Integral Inequalities and Limit of Integral Sequences View

Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be a continuous function such that $$f ( 0 ) = 1 \text { and } \int _ { 0 } ^ { \frac { \pi } { 3 } } f ( t ) d t = 0$$ Then which of the following statements is (are) TRUE ?
(A) The equation $f ( x ) - 3 \cos 3 x = 0$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$
(B) The equation $f ( x ) - 3 \sin 3 x = - \frac { 6 } { \pi }$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$
(C) $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { 1 - e ^ { x ^ { 2 } } } = - 1$
(D) $\lim _ { x \rightarrow 0 } \frac { \sin x \int _ { 0 } ^ { x } f ( t ) d t } { x ^ { 2 } } = - 1$

jee-advanced 2021 Q9 2 marks Definite Integral Evaluation (Computational) View

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{16 S_1}{\pi}$ is ____.

jee-advanced 2021 Q10 2 marks Definite Integral Evaluation (Computational) View

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{48 S_2}{\pi^2}$ is ____.

jee-advanced 2021 Q11 4 marks Piecewise/Periodic Function Integration View

For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. If $$I = \int_0^{10} \left\lfloor \frac{10x}{x+1} \right\rfloor dx,$$ then the value of $9I$ is ____.

jee-advanced 2021 Q11 Definite Integral Evaluation (Computational) View

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 16 S _ { 1 } } { \pi }$ is $\_\_\_\_$.

jee-advanced 2021 Q12 Definite Integral Evaluation (Computational) View

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 48 S _ { 2 } } { \pi ^ { 2 } }$ is $\_\_\_\_$.

jee-advanced 2021 Q19 Piecewise/Periodic Function Integration View

For any real number $x$, let $[ x ]$ denote the largest integer less than or equal to $x$. If $$I = \int _ { 0 } ^ { 10 } \left[ \sqrt { \frac { 10 x } { x + 1 } } \right] d x$$ then the value of $9 I$ is $\_\_\_\_$.

jee-advanced 2022 Q3 3 marks Definite Integral Evaluation (Computational) View

The greatest integer less than or equal to
$$\int _ { 1 } ^ { 2 } \log _ { 2 } \left( x ^ { 3 } + 1 \right) d x + \int _ { 1 } ^ { \log _ { 2 } 9 } \left( 2 ^ { x } - 1 \right) ^ { \frac { 1 } { 3 } } d x$$
is $\_\_\_\_$ .

jee-advanced 2024 Q13 4 marks Accumulation Function Analysis View

Let the function $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f ( t ) = \left\{ \begin{array} { c c } ( - 1 ) ^ { n + 1 } 2 , & \text { if } t = 2 n - 1 , n \in \mathbb { N } , \\ \frac { ( 2 n + 1 - t ) } { 2 } f ( 2 n - 1 ) + \frac { ( t - ( 2 n - 1 ) ) } { 2 } f ( 2 n + 1 ) , & \text { if } 2 n - 1 < t < 2 n + 1 , n \in \mathbb { N } . \end{array} \right.$$ Define $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t , x \in ( 1 , \infty )$. Let $\alpha$ denote the number of solutions of the equation $g ( x ) = 0$ in the interval $( 1,8 ]$ and $\beta = \lim _ { x \rightarrow 1 + } \frac { g ( x ) } { x - 1 }$. Then the value of $\alpha + \beta$ is equal to $\_\_\_\_$ .

jee-advanced 2024 Q16 3 marks Definite Integral Evaluation (Computational) View

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $2 \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x - \int _ { 0 } ^ { \frac { \pi } { 2 } } g ( x ) d x$ is $\_\_\_\_$ .

jee-advanced 2024 Q17 3 marks Definite Integral Evaluation (Computational) View

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $\frac { 16 } { \pi ^ { 3 } } \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x$ 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