These sample questions are taken from JEE 2021 Advanced Mathematics Paper-2
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Let
\(
\begin{aligned}
&S_{1}=\{(i, j, k): i, j, k \in\{1,2, \ldots ., 10\}\}, \\
&S_{2}=\{(i, j): 1 \leq i<j+2 \leq 10, i, j \in\{1,2, \ldots . ., 10\}\}, \\
&S_{3}=\{(i, j, k, l): 1 \leq i<j<k<l, i, j, k, l \in\{1,2, \ldots . .10\}\} \text { and } \\
&S_{4}=\{(i, j, k, l): i, j, k \text { and I are distinct elements in }\{1,2, \ldots . ., 10\}\}
\end{aligned}
\)
If the total number of elements in the set \(S_{r}\) is \(n_{r}, r=1,2,3,4\), then which of the following statements is(are) TRUE?
\(\begin{aligned}
&\mathrm{n}_{1}=10 \times 10 \times 10=10^{3} \\
&\mathrm{n}_{2}={ }^{8} \mathrm{C}_{2}+2{ }^{8} \mathrm{C}_{1}=44 \\
&\mathrm{n}_{3}={ }^{10} \mathrm{C}_{4}=210 \\
&\mathrm{n}_{4}={ }^{10} \mathrm{P}_{4}=5040
\end{aligned}\)
Consider a triangle \(P Q R\) having sides of lengths \(p, q\) and \(r\) opposite to the angles \(P, Q\) and \(R\), respectively. Then which of the following statements is(are) TRUE?
Let \(f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R\) be a continuous function such that \(f(0)=1\) and \(\int_{0}^{\frac{\pi}{3}} f(t) d t=0\). Then which of the following statements is(are) TRUE?
\(
\begin{aligned}
&\int_{0}^{\pi / 3} g(x) d x=\int_{0}^{\pi / 3}(f(x)-3 \cos 3 x) d x=\int_{0}^{\pi / 3} f(x) d x-\int_{0}^{\pi / 3} 3 \cos 3 x d x=0 \\
&\Rightarrow g(x)=0 \text { has at least one solution in }[0, \pi / 3]
\end{aligned}
\)
(B) Let \(\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{x})-3 \sin 3 \mathrm{x}+6 / \pi\)
\(\int_{0}^{\pi / 3} h(x)=\int_{0}^{\pi / 3}\left(f(x)-3 \sin 3 x+\frac{6}{\pi}\right) d x=0\) \(\Rightarrow h(x)=0\) has atleast one solution in \([0, \pi / 3]\)
(C) \(\lim _{x \rightarrow 0} \frac{\int_{0}^{x} f(t) d t}{x}=\lim _{x \rightarrow 0} \frac{f(x)}{1}=f(0)=1\)
\(
\Rightarrow \lim _{x \rightarrow 0} \frac{x \int_{0}^{x} f(t) d t}{x^{2}\left(\frac{1-e^{x^{2}}}{x^{2}}\right)}=\lim _{x \rightarrow 0} \frac{-\int_{0}^{x} f(t)}{x} \cdot \frac{x^{2}}{\left(e^{x^{2}}-1\right)}=-1
\)
(D) \(\lim _{x \rightarrow 0} \frac{\sin x}{x} \cdot \frac{\int_{0}^{x} f(t)}{x}=1\)
For any real numbers \(\alpha\) and \(\beta\), let \(y_{\alpha, \beta}(x), x \in R\), be the solution of the differential equation
\(
\frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1 .
\)
Let \(S=\left\{y_{\alpha, \beta}(x): \alpha, \beta \in R\right\}\). Then which of the following functions belong(s) the set \(S\) ?
\(\begin{aligned}
&\frac{d y}{d x}+\alpha y=x e^{\beta x}\\
&d\left(e^{\alpha x} \cdot y\right) x e^{\alpha x} \cdot e^{\beta x}\\
&d\left(e^{\alpha x} \cdot y\right)=x e^{(\alpha+\beta) x}\\
&\text { Case-I : } \alpha+\beta \neq 0\\
&d\left(e^{\alpha x} \cdot y\right)=x e^{(\alpha+\beta) x}\\
&e^{\alpha x} \cdot y=\frac{x e^{(\alpha+\beta) x}}{(\alpha+\beta)}-\frac{e^{(\alpha+\beta) x}}{(\alpha+\beta)^{2}}+C\\
&y=\frac{x e^{\beta x}}{(\alpha+\beta)}-\frac{e^{\beta x}}{(\alpha+\beta)^{2}}+C e^{-\alpha x}\\
&\alpha=1, \beta=1 \Rightarrow y=\frac{x e^{x}}{2}-\frac{e^{x}}{4}+C e^{-x}\\
&\text { as } y(1)=1 \Rightarrow C=e\left(1-\frac{e}{4}\right)\\
&y(x)=\frac{x e^{x}}{2}-\frac{e^{x}}{4}+\left(e-\frac{e^{2}}{4}\right) e^{-x}\\
&\text { Case-II }: \alpha+\beta=0\\
&\Rightarrow \quad \frac{d y}{d x}-\beta y=x e^{\beta x}\\
&d\left(e^{-\beta x} y\right)=x\\
&e^{-\beta x} \cdot y=\frac{x^{2}}{2}+C\\
&y=\frac{e^{\beta x} x^{2}}{2}+C e^{\beta x}\\
&\mathrm{y}(1)=1\\
&\Rightarrow C=\left(1-\frac{e}{2}\right) \frac{1}{e} \Rightarrow y=e^{\beta x} \cdot \frac{x^{2}}{2}+\left(1-\frac{e}{2}\right) \frac{1}{e} \cdot e^{\beta x}\\
&\text { Take } \beta=-1 \Rightarrow y=\frac{x^{2}}{2} e^{-x}+\left(1-\frac{e}{2}\right) e^{-x}
\end{aligned}\)
Let \(\mathrm{O}\) be the origin and \(\overrightarrow{\mathrm{OA}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}[latex] and [latex]\overrightarrow{\mathrm{OC}}=\frac{1}{2}(\overrightarrow{\mathrm{OB}}-\lambda \overrightarrow{\mathrm{OA}})\) for some \(\lambda>0\). If \(|\overrightarrow{O B} \times \overrightarrow{O C}|=\frac{9}{2}\), then which of the following statements is(are) TRUE?
\(\overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}=0 \Rightarrow \overrightarrow{\mathrm{OA}} \perp \overrightarrow{\mathrm{OB}}\) \(\overrightarrow{\mathrm{OC}}=\frac{1}{2}((1-2 \lambda) \hat{\mathrm{i}}+(-2-2 \lambda) \hat{\mathrm{j}}+(2-\lambda) \hat{\mathrm{k}})\) \(|\overrightarrow{\mathrm{OB}} \times \overrightarrow{\mathrm{OC}}|=\frac{9|\lambda|}{2}=\frac{9}{2}\) \(\Rightarrow \quad \lambda=\pm 1\), as \(\lambda>0, \lambda=1\) \(\overrightarrow{\mathrm{OC}}=\frac{\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}}{2}\)
(A) Projection \(\overrightarrow{\mathrm{OC}}\) on \(\overrightarrow{\mathrm{OA}}\); \(\overrightarrow{\mathrm{OC}} \cdot \overrightarrow{\mathrm{OA}}=-\frac{3}{2}\)
(B) Area of \(\triangle \mathrm{OAB}=9 / 2\)
(C) Area of \(\Delta \mathrm{ABC}=9 / 2\)
(D) \(\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}}=\frac{3 \hat{\mathrm{i}}+3 \overrightarrow{\mathrm{OB}}=9 / 2}{2}, \overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OC}}=\frac{5 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{2}\) \((\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}})(\overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OC}})=|\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}}||\overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OC}}| \cos \theta\) \(\cos \theta=\frac{1}{\sqrt{5}}\)
Let \(\mathrm{E}\) denote the parabola \(\mathrm{y}^{2}=8 \mathrm{x}\). Let \(\mathrm{P}=(-2,4)\) and let \(\mathrm{Q}\) and \(\mathrm{Q}^{\prime}\) be two distinct points on \(\mathrm{E}\) such that the lines \(P Q\) and \(P Q^{\prime}\) are tangents to \(E\). Let \(F\) be the focus of \(E\). Then which of the following statements is(are) TRUE?
Consider the region \(R=\left\{(x, y) \in R \times R: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(F\) be the family of all circles that are contained in \(\mathrm{R}\) and have centers on the \(\mathrm{x}\)-axis. Let \(\mathrm{C}\) be the circle that has largest radius among the circles in \(\mathrm{F}\). Let \((\alpha, \beta)\) be a point where the circle C meets the curve \(\mathrm{y}^{2}=4-\mathrm{x}\).
The radius of the circle \(C\) is ______.
Consider the region \(R=\left\{(x, y) \in R \times R: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(F\) be the family of all circles that are contained in \(\mathrm{R}\) and have centers on the \(\mathrm{x}\)-axis. Let \(\mathrm{C}\) be the circle that has largest radius among the circles in \(\mathrm{F}\). Let \((\alpha, \beta)\) be a point where the circle C meets the curve \(\mathrm{y}^{2}=4-\mathrm{x}\).
The radius of the circle \((\alpha)\) is ______.
Let \(\mathrm{g}_{\mathrm{i}}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathrm{R}, \mathrm{i}=1,2\) and \(\mathrm{f}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathrm{R}\) be functions such that \(g_{1}(x)=1, g_{2}(x)=|4 x-\pi|\) and \(f(x)=\sin ^{2} x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] .Define S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2\)
The value of \(\frac{16 S_{1}}{\pi}\) is ______.
Let \(\mathrm{g}_{\mathrm{i}}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathrm{R}, \mathrm{i}=1,2\) and \(\mathrm{f}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathrm{R}\) be functions such that \(g_{1}(x)=1, g_{2}(x)=|4 x-\pi|\) and \(f(x)=\sin ^{2} x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] .Define S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2\)
The value of \(\frac{48 S_{2}}{\pi^{2}}\) is ______.
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