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Q. Let ƒ be a non-negative function defined on the interval $[0,1] .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} d t=\int_{0}^{x} f(t) d t$ $0 \leq \mathrm{x} \leq 1,$ and $f(0)=0,$ then $-$(A) $f\left(\frac{1}{2}\right)<\frac{1}{2}$ and $f\left(\frac{1}{3}\right)>\frac{1}{3}$(B) $f\left(\frac{1}{2}\right)>\frac{1}{2}$ and $f\left(\frac{1}{3}\right)>\frac{1}{3}$(C) $f\left(\frac{1}{2}\right)<\frac{1}{2}$ and $f\left(\frac{1}{3}\right)<\frac{1}{3}$(D) $f\left(\frac{1}{2}\right)>\frac{1}{2}$ and $f\left(\frac{1}{3}\right)<\frac{1}{3}$ [JEE 2009, 3]
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Sol. (C)$\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \mathrm{d} t=\int_{0}^{x} f(t) d t, 0 \leq x \leq 1$differentiating both the sides & squreing$\Rightarrow 1-\left(f^{\prime}(\mathrm{x})\right)^{2}=f^{2}(\mathrm{x})$$\Rightarrow \frac{f^{\prime}(x)}{\sqrt{1-f^{2}(x)}}=1$$\Rightarrow \sin ^{-1} f(\mathrm{x})=\mathrm{x}+\mathrm{c}$$f(0)=0$$\Rightarrow f(\mathrm{x})=\sin \mathrm{x}$$\Rightarrow \because \sin \mathrm{x} \leq \mathrm{x}$ for $\mathrm{x} \in[0,1]$$\Rightarrow f\left(\frac{1}{2}\right)<\frac{1}{2}$ and $f\left(\frac{1}{3}\right)<\frac{1}{3}$
Q. If $\mathrm{I}_{\mathrm{n}}=\int_{-\pi}^{\pi} \frac{\sin \mathrm{nx}}{\left(1+\pi^{\mathrm{x}}\right) \sin \mathrm{x}} \mathrm{d} \mathrm{x}, \mathrm{n}=0,1,2, \ldots, \mathrm{then}-$(A) $\mathrm{I}_{\mathrm{n}}=\mathrm{I}_{\mathrm{n}+2}$(B) $\sum_{\mathrm{m}=1}^{10} \mathrm{I}_{2 \mathrm{m}+1}=10 \pi$(C) $\sum_{\mathrm{m}=1}^{10} \mathrm{I}_{2 \mathrm{m}}=0$(D) $\mathrm{I}_{\mathrm{n}}=\mathrm{I}_{\mathrm{n}+1}$ [JEE 2009, 4]
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Sol. (A,B,D)$\mathrm{I}_{\mathrm{n}}=\int_{-\pi}^{\pi} \frac{\sin \mathrm{nx}}{\left(1+\pi^{\mathrm{x}}\right) \sin \mathrm{x}} \mathrm{dx}$$\mathrm{I}_{\mathrm{n}}=\int_{-\pi}^{\pi} \frac{\pi^{\mathrm{x}} \sin \mathrm{nx}}{\left(1+\pi^{\mathrm{x}}\right) \sin \mathrm{x}} \mathrm{dx}$$2 \mathrm{I}_{\mathrm{n}}=\int_{-\pi}^{\pi} \frac{\sin \mathrm{nx}}{\sin \mathrm{x}} \mathrm{dx}$ ..(i)$2 \mathrm{I}_{\mathrm{n}+2}=\int_{-\pi}^{\pi} \frac{\sin (\mathrm{n}+2) \mathrm{x}}{\sin \mathrm{x}} \mathrm{dx} \quad \ldots(\mathrm{i})$(ii) – (i)$\Rightarrow 2\left(\ln _{+2}-\mathrm{I}_{\mathrm{n}}\right)=\int_{-\pi}^{\pi} \cos (\mathrm{n}+1) \mathrm{x}=0$$\Rightarrow \quad \mathrm{I}_{\mathrm{n}+2}=\mathrm{I}_{\mathrm{n}}$$\sum_{m=1}^{10} \mathrm{I}_{2 \mathrm{m}}=10 \sum_{\mathrm{m}=1}^{10} \mathrm{I}_{2}=\frac{10}{2} \int_{-\pi}^{\pi} \frac{\sin 2 \mathrm{x}}{\sin \mathrm{x}} \mathrm{d} \mathrm{x}=0$Put n = 1 in equation (i)$2 \mathrm{I}_{1}=\int_{-\pi}^{\pi} \frac{\sin \mathrm{x} \mathrm{d} \mathrm{x}}{\sin \mathrm{x}}=2 \pi$$\mathrm{I}_{1}=\pi$$\sum_{m=1}^{10} I_{2 m+1}=10 \pi$
Q. Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a continuous function which satisfies $\mathrm{f}(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{d} \mathrm{t}$ Then the value of f(ln 5) is…….. [JEE 2009, 4]
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Sol. 0$\mathrm{f}(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}$$\mathrm{f}^{\mathrm{l}}(\mathrm{x})=\mathrm{f}(\mathrm{x})$$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}$$\Rightarrow \int \frac{d y}{y}=\int d x$$\Rightarrow \ln y=x+c$$\Rightarrow y=e^{x+c}$$\Rightarrow y=0$$\left(\begin{array}{c}{\text { at } x=0, y=0} \\ {c \rightarrow-\infty}\end{array}\right)$$f(x)=0$$f(\ell n 5)=0$
Q. The value of $\lim _{x \rightarrow 0} \frac{1}{x^{3}} \int_{0}^{x} \frac{t \ell n(1+t)}{t^{4}+4} d t$ is(A) 0(B) $\frac{1}{12}$(C) $\frac{1}{24}$(D) $\frac{1}{64}$ [JEE 2010, 3 (–1)]
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Sol. (B)Applying L-Hospital rule,
Q. The value(s) of $\int_{0}^{1} \frac{\mathrm{x}^{4}(1-\mathrm{x})^{4}}{1+\mathrm{x}^{2}} \mathrm{dx}$ is (are)(A) $\frac{22}{7}-\pi$(B) $\frac{2}{105}$(C) 0(D) $\frac{71}{15}-\frac{3 \pi}{2}$ [JEE 2010, 3]
Q. Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that$e^{-x} f(x)=2+\int_{0}^{x} \sqrt{t^{4}+1} d t,$ for all $x \in(-1,1),$ and let $f^{-1}$ be the inverse function of $f$ Then $\left(f^{-1}\right)^{\prime}(2)$ is equal to-(A) 1(B) $\frac{1}{3}$(C) $\frac{1}{2}$(D) $\frac{1}{\mathrm{e}}$ [JEE2010, 5 (–2)]
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Sol. (B)
from $(2), f^{-1}(2)=\frac{1}{3}$
Q. For any real number x, let [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 $\mathrm{f}(\mathrm{x})=\left\{\begin{aligned} \mathrm{x}-[\mathrm{x}] & \text { if }[\mathrm{x}] \text { is odd } \\ 1+[\mathrm{x}]-\mathrm{x} & \text { if }[\mathrm{x}] \text { is even } \end{aligned}\right.$ Then the value of $\frac{\pi^{2}}{10} \int_{-10}^{10} \mathrm{f}(\mathrm{x}) \cos \pi \mathrm{x} \mathrm{d} \mathrm{x}$ is [JEE 2010, 3]
Q. The value of $\int_{\sqrt{\mathrm{in} 2}}^{\sqrt{\mathrm{n} 3}} \frac{\mathrm{x} \sin \mathrm{x}^{2}}{\sin \mathrm{x}^{2}+\sin \left(\mathrm{ln} 6-\mathrm{x}^{2}\right)} \mathrm{dx}$ is(A) $\frac{1}{4} \ln \frac{3}{2}$(B) $\frac{1}{2} \ln \frac{3}{2}$(C) $\ln \frac{3}{2}$(D) $\frac{1}{6} \ln \frac{3}{2}$ [JEE 2011, 3 (–1)]
Q. Let $S$ be the area of the region enclosed by $y=e^{-x^{2}}, y=0, x=0,$ and $x=1 .$ Then(A) $\mathrm{S} \geq \frac{1}{\mathrm{e}}$(B) $\mathrm{S} \geq 1-\frac{1}{\mathrm{e}}$(C) $S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{\mathrm{e}}}\right)$(D) $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{\mathrm{e}}}\left(1-\frac{1}{\sqrt{2}}\right)$ [JEE 2012, 4M]
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Sol. (A,B,D)
Q. The value of the integral $\int_{-\pi / 2}^{\pi / 2}\left(\mathrm{x}^{2}+\ln \frac{\pi+\mathrm{x}}{\pi-\mathrm{x}}\right) \cos \mathrm{xd} \mathrm{x}$ is(A) 0(B) $\frac{\pi^{2}}{2}-4$(C) $\frac{\pi^{2}}{2}+4$(D) $\frac{\pi^{2}}{2}$c [JEE 2012, 3M, –1M]
Q. For a $\in \mathrm{R}$ (the set of all real numbers), a\neq-1.$\lim _{n \rightarrow \infty} \frac{\left(1^{a}+2^{a}+\ldots \ldots+n^{a}\right)}{(n+1)^{a-1}[(n a+1)+(n a+2)+\ldots \ldots+(n a+n)]}=\frac{1}{60}$ Then $a=$(A) 5 (B) 7 (C) $\frac{-15}{2}$ (D) $\frac{-17}{2}$ [JEE(Advanced) 2013, 3, (–1)]
Q. Let $f:[\mathrm{a}, \mathrm{b}] \rightarrow[1, \infty)$ be a continuous function and let $\mathrm{g}: \square \rightarrow \square$ be defined as
Then(A) g(x) is continuous but not differentiable at a(B) g(x) is differentiable on (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, 3]
Then(A) g(x) is continuous but not differentiable at a(B) g(x) is differentiable on (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, 3]Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...
Sol. (A,C)
Q. The value of $\int_{0}^{1} 4 x^{3}\left\{\frac{d^{2}}{d x^{2}}\left(1-x^{2}\right)^{5}\right\} d x$ is [JEE(Advanced)-2014, 3]
Q. The following integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}(2 \csc x)^{17} d x$ is equal to(A) $\int_{0}^{\log (1+\sqrt{2})} 2\left(e^{\mathfrak{u}}+e^{-\mathfrak{u}}\right)^{16} \mathrm{d} \mathfrak{u}$(B) $\int_{0}^{\log (1+\sqrt{2})}\left(\mathrm{e}^{\mathrm{u}}+\mathrm{e}^{-\mathrm{u}}\right)^{17} \mathrm{du}$(C) $\int_{0}^{\log (1+\sqrt{2})}\left(e^{\mathfrak{u}}-e^{-\mathfrak{u}}\right)^{17} \mathrm{d} \mathfrak{u}$(D) $\int_{0}^{\log (1+\sqrt{2})} 2\left(e^{\mathfrak{u}}-e^{-\mathfrak{u}}\right)^{16} d \mathfrak{u}$ [JEE(Advanced)-2014, 3(–1)]
Q. Let $f:[0,2] \rightarrow \square$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1 .$ Let $F(x)=\int_{0}^{x^{2}} f(\sqrt{t}) d t$ for $x \in[0,2] .$ If $F^{\prime}(x)=f^{\prime}(x)$ for all $x \in(0,2)$ then $F(2)$ equals $-$(A) $\mathrm{e}^{2}-1$(B) $\mathrm{e}^{4}-1$(C) e – 1(D) e $^{4}$ [JEE(Advanced)-2014, 3(–1)]
Given that for each a $\in(0,1)$, $\lim _{\mathrm{h} \rightarrow 0^{+}} \int_{\mathrm{h}}^{1-\mathrm{h}} \mathrm{t}^{-\mathrm{a}}(1-\mathrm{t})^{\mathrm{a}-1}$ $\mathrm{dt}$ exists. Let this limit be g(a). In addition,it is given that the function g(a) is differentiable on (0,1).
Q. The value of $\mathrm{g}\left(\frac{1}{2}\right)$ is –(A) $\pi$(B) $2 \pi$(C) $\frac{\pi}{2}$(D) $\frac{\pi}{4}$ [JEE(Advanced)-2014, 3(–1)]
Q. The value of $\mathrm{g}^{\prime}\left(\frac{1}{2}\right)$ is-(A) $\frac{\pi}{2}$(B) $\pi$(C) $-\frac{\pi}{2}$(D) 0 [JEE(Advanced)-2014, 3(–1)]
Q. 
[JEE(Advanced)-2014, 3(–1)]
[JEE(Advanced)-2014, 3(–1)]
Q. Let $f: \square \rightarrow \square$ be a function defined by $f(x)$ $=\left\{\begin{array}{ccc}{[\mathrm{x}]} & {,} & {\mathrm{x} \leq 2} \\ {0} & {,} & {\mathrm{x}>2}\end{array}\right.$ where [x] is the greatest integer less than or equal to x. If $\mathrm{I}=\int_{-1}^{2} \frac{\mathrm{x} f\left(\mathrm{x}^{2}\right)}{2+f(\mathrm{x}+1)} \mathrm{dx}$ , then the value of (4I – 1) is [JEE 2015, 4M, –0M]
Q. If $\alpha=\int_{0}^{1}\left(\mathrm{e}^{9 \mathrm{x}+3 \tan ^{-1} \mathrm{x}}\right)\left(\frac{12+9 \mathrm{x}^{2}}{1+\mathrm{x}^{2}}\right) \mathrm{d} \mathrm{x}$ where $\tan ^{-1} \mathrm{x}$ takes only principal values, then the value of $\left(\log _{\mathrm{e}}|1+\alpha|-\frac{3 \pi}{4}\right)$ is [JEE 2015, 4M, –0M]
Q. Let $f: \mathbb{U} \rightarrow \square$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\frac{1}{2}$ Suppose that $\mathrm{F}(\mathrm{x})=\int_{-1}^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}$ for all $\mathrm{x} \in[-1,2]$ and $\mathrm{G}(\mathrm{x})$ $=\int_{-1}^{x} \mathfrak{t}|f(f(\mathfrak{t}))| d \mathfrak{t}$ 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 2015, 4M, –0M]
Q. The option(s) with the values of a and L that satisfy the following equation is(are)
(A) $a=2, L=\frac{e^{4 \pi}-1}{e^{\pi}-1}$(B) $a=2, L=\frac{e^{4 \pi}+1}{e^{\pi}+1}$c(C) $a=4, L=\frac{e^{4 \pi}-1}{e^{\pi}-1}$(D) $a=4, L=\frac{e^{4 \pi}+1}{e^{\pi}+1}$ [JEE 2015, 4M, –0M]
(A) $a=2, L=\frac{e^{4 \pi}-1}{e^{\pi}-1}$(B) $a=2, L=\frac{e^{4 \pi}+1}{e^{\pi}+1}$c(C) $a=4, L=\frac{e^{4 \pi}-1}{e^{\pi}-1}$(D) $a=4, L=\frac{e^{4 \pi}+1}{e^{\pi}+1}$ [JEE 2015, 4M, –0M]
Q. Let $f(\mathrm{x})=7 \tan ^{8} \mathrm{x}+7 \tan ^{6} \mathrm{x}-3 \tan ^{4} \mathrm{x}-3 \tan ^{2} \mathrm{x}$ for all $\mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) .$ Then the correct expression(s)is(are)(A) $\int_{0}^{\pi / 4} \mathrm{x} f(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{1}{12}$(B) $\int_{0}^{\pi / 4} f(\mathrm{x}) \mathrm{d} \mathrm{x}=0$(C) $\int_{0}^{\pi / 4} \mathrm{x} f(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{1}{6}$(D) $\int_{0}^{\pi / 4} f(\mathrm{x}) \mathrm{d} \mathrm{x}=1$ [JEE 2015, 4M, –0M]
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Sol. (A,C)
Q. Let $f^{\prime}(x)=\frac{192 x^{3}}{2+\sin ^{4} \pi x}$ for all $\mathrm{x} \in \square$ with $f$ $\left(\frac{1}{2}\right)$ $=0 .$ If $\mathrm{m} \leq \int_{1 / 2}^{1} f(\mathrm{x}) \mathrm{d} \mathrm{x} \leq \mathrm{M}$ then the possible values of m and M are(A) m = 13, M = 24(B) $\quad \mathrm{m}=\frac{1}{4}, \mathrm{M}=\frac{1}{2}$(C) m = –11, M = 0(D) m = 1, M = 12 [JEE 2015, 4M, –0M]
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Sol. (A,B)
Let $\mathrm{F}: \mathbb{U} \rightarrow \square$ be a thrice differentiable function. Suppose that $\mathrm{F}(1)=0, \mathrm{F}(3)=-4 \mathrm{F}^{\prime}(\mathrm{x})<$ 0 for all $\mathrm{x} \in(1 / 2,3) .$ Let $f(\mathrm{x})=\mathrm{xF}(\mathrm{x})$ for all $\mathrm{x} \in \mathbb{D}$.
Q. The correct statement(s) is(are)(A) $f^{\prime}(1)<0$(B) $f(2)<0$(C) $f^{\prime}(\mathrm{x}) \neq 0$ for any $\mathrm{x} \in(1,3)$(D) $f^{\prime}(x)=0$ for some $x \in(1,3)$ [JEE 2015, 4M, –0M]
Q. If $\int_{1}^{3} \mathrm{x}^{2} \mathrm{F}^{\prime}(\mathrm{x}) \mathrm{d} \mathrm{x}=-12$ and $\int_{1}^{3} \mathrm{x}^{3} \mathrm{F}^{\prime \prime}(\mathrm{x}) \mathrm{d} \mathrm{x}=40,$ then the correct expression(s) is (are)(A) 9ƒ'(3) + ƒ'(1) – 32 = 0(B) $\int_{1}^{3} f(\mathrm{x}) \mathrm{d} \mathrm{x}=12$(C) 9ƒ'(3) – ƒ'(1) + 32 = 0(D) $\left.\int_{1}^{3} f(x) d x=-12\right]$ [JEE 2015, 4M, –0M]
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Sol. (A,B,C)
Q. 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) $\pi^{2}-\mathrm{e}^{\frac{\pi}{2}}$(D) $\pi^{2}+\mathrm{e}^{\frac{\pi}{2}}$ [JEE(Advanced)2016]
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Sol. (C,D)
Q. Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function such that $\mathrm{f}(0)=0, \mathrm{f}\left(\frac{\pi}{2}\right)=3$ and $\mathrm{f}^{\prime}(0)=1$ If $\mathrm{g}(\mathrm{x})=\int_{\mathrm{x}}^{\frac{\pi}{2}}\left[\mathrm{f}^{\prime}(\mathrm{t}) \csc \mathrm{t}-\cot t \csc \mathrm{t} \mathrm{f}(\mathrm{t})\right] \mathrm{d} \mathrm{t}$ for $\mathrm{x} \in\left(0, \frac{\pi}{2}\right],$ then $\lim _{\mathrm{x} \rightarrow 0} \mathrm{g}(\mathrm{x})=$ [JEE(Advanced)-2017]
Q. If $\mathrm{I}=\sum_{\mathrm{k}=1}^{98} \int_{\mathrm{k}}^{\mathrm{k}+1} \frac{\mathrm{k}+1}{\mathrm{x}(\mathrm{x}+1)} \mathrm{d} \mathrm{x},$ then(A) $\mathrm{I}<\frac{49}{50}$(B) $\mathrm{I}<\log _{\mathrm{e}} 99$(C) $\mathrm{I}>\frac{49}{50}$(D) $\mathrm{I}>\log _{\mathrm{e}} 99$ [JEE(Advanced)-2017]
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Sol. (B,C)
Q. If $\mathrm{g}(\mathrm{x})=\int_{\sin \mathrm{x}}^{\sin (2 \mathrm{x})} \sin ^{-1}(\mathrm{t}) \mathrm{dt},$ then(A) $\mathrm{g}^{\prime}\left(\frac{\pi}{2}\right)=-2 \pi$(B) $\mathrm{g}^{\prime}\left(-\frac{\pi}{2}\right)=2 \pi$(C) $\mathrm{g}^{\prime}\left(\frac{\pi}{2}\right)=2 \pi$(D) $\mathrm{g}^{\prime}\left(-\frac{\pi}{2}\right)=-2 \pi$ [JEE(Advanced)-2017]
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Sol. (Bonus)
Q. The value of the integral $\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} d x$ is [JEE(Advanced)-2018]
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some questions in 2014 are missing
Jee advance 2014 Matching wale ques mein Pth me 2 functions hi possible honge.
How u have found 4?
I think 19th answer is -0. 5
Key to question numbers 18,19,20,21 is messed up!please correct it🙃
Key to question numbers 18,19,20,21 is messed up!please correct it🙃
nice👍
please upload 2019 and 2020 questions also