The integral
Question: The integral $\int_{1}^{2} e^{x} \cdot x^{x}\left(2+\log _{e} x\right) d x$ equal : $e(4 e+1)$$e(2 e-1)$$4 e^{2}-1$$e(4 e-1)$Correct Option: , 4 Solution: $\int_{1}^{2} e^{x} \cdot x^{x}\left(2+\log _{e} x\right) d x$ $\int_{1}^{2} e^{x}\left(2 x^{x}+x^{x} \log _{e} x\right) d x$ $\int_{1}^{2} \mathrm{e}^{\mathrm{x}}(\underbrace{\mathrm{x}^{\mathrm{x}}}_{f(\mathrm{x})}+\underbrace{\mathrm{x}^{\mathrm{x}}\left(1+\log _{\mathrm{e}} \mathrm{x}\right)}_{f^{\prime}(\mathrm{x})}) d \mathrm{x...
Read More →if the constant term in the binomial expansion
Question: if the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is 405, then $|k|$ equals : 2139Correct Option: , 3 Solution: $\left(\sqrt{\mathrm{x}}-\frac{\mathrm{k}}{\mathrm{x}^{2}}\right)^{10}$ $\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}}(\sqrt{\mathrm{x}})^{10-\mathrm{r}}\left(\frac{-\mathrm{k}}{\mathrm{x}^{2}}\right)^{\mathrm{r}}$ $\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}} \cdot \mathrm{x}^{\frac{10-\mathrm{r}}{2}} \c...
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Question: The probabilities of three events A, B and C are given by $P(A)=0.6, P(B)=0.4$ and $P(C)=0.5$. If $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.8, \mathrm{P}(\mathrm{A} \cap \mathrm{C})=0.3, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap$ C) $=0.2, \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\beta$ and $\mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\alpha$, where $0.85 \leq \alpha \leq 0.95$, then $\beta$ lies in the interval:$[0.36,0.40]$$[0.35,0.36]$$[0.25,0.35]$$[0.20,0.25]$Correct Opti...
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Question: The area (in sq. units) of the region enclosed by the curves $y=x^{2}-1$ and $y=1-x^{2}$ is equal to : $\frac{4}{3}$$\frac{8}{3}$$\frac{16}{3}$$\frac{7}{2}$Correct Option: Solution: $y=x^{2}-1$ and $y=1-x^{2}$ $A=\int_{-1}^{1}\left(\left(1-x^{2}\right)-\left(x^{2}-1\right)\right) d x$ $A=\int_{-1}^{1}\left(2-2 x^{2}\right) d x=4 \int_{0}^{1}\left(1-x^{2}\right) d x$ $\mathrm{A}=4\left(\mathrm{x}-\frac{\mathrm{x}^{3}}{3}\right)_{0}^{1}=4\left(\frac{2}{3}\right)=\frac{8}{3}$...
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Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=\max \left\{x, x^{2}\right\}$. Let $S$ denote the set of all points in R, where $f$ is not differentiable. Then :$\{0,1\}$$\{0\}$ $\phi($ an empty set $)$$\{1\}$Correct Option: 1 Solution: $f(x)=\max \left(x, x^{2}\right)$ Non-differentiable at $x=0,1$ $\mathrm{S}=\{0,1\}$...
Read More →For a suitably chosen real constant a, let a
Question: For a suitably chosen real constant a, let a function, $f: \mathrm{R}-\{-\mathrm{a}\} \rightarrow \mathrm{R}$ be defined by $f(x)=\frac{a-x}{a+x} .$ Further suppose that for any real number $x \neq-a$ and $f(x) \neq-a,(f \circ f)(x)=x$. Then $f\left(-\frac{1}{2}\right)$ is equal to :$\frac{1}{3}$3$-3$$-\frac{1}{3}$Correct Option: , 2 Solution: $f(x)=\frac{a-x}{a+x}$ $\mathrm{x} \in \mathrm{R}-\{-\mathrm{a}\} \rightarrow \mathrm{R}$ $f(f(\mathrm{x}))=\frac{\mathrm{a}-f(\mathrm{x})}{\mat...
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Question: Let $\theta=\frac{\pi}{5}$ and $\mathrm{A}=\left[\begin{array}{cc}\cos \theta \sin \theta \\ -\sin \theta \cos \theta\end{array}\right] .$ If $\mathrm{B}=\mathrm{A}$ $+\mathrm{A}^{4}$, then $\operatorname{det}(\mathrm{B}):$ is onelies in $(1,2)$is zerolies in $(2,3)$Correct Option: , 2 Solution: $A=\left[\begin{array}{cc}\cos \theta \sin \theta \\ -\sin \theta \cos \theta\end{array}\right]$ $A^{2}=\left[\begin{array}{cc}\cos \theta \sin \theta \\ -\sin \theta \cos \theta\end{array}\rig...
Read More →Let the vectors vector a, vector b, vector c be such that |vector a|=2,|vector b|=4 and |vector c|=4.
Question: Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the projection of $\vec{c}$ on $\vec{a}$ and $\vec{b}$ is perpendicular to $\vec{c}$, then the value of $|\vec{a}+\vec{b}-\vec{c}|$ is Solution: Projection of $\overrightarrow{\mathrm{b}}$ on $\overrightarrow{\mathrm{a}}=$ projection of $\overrightarrow{\mathrm{c}}$ on $\overrightarrow{\mathrm{a}}$ $\Rightarrow \frac{\vec{b} \cdot...
Read More →The angle of elevation of the summit of a
Question: The angle of elevation of the summit of a mountain from a point on the ground is $45^{\circ}$. After climding up one $\mathrm{km}$ towards the summit at an inclination of $30^{\circ}$ from the ground, the angle of elevation of the summit is found to be $60^{\circ}$. Then the height (in $\mathrm{km}$ ) of the summit from the ground is :$\frac{1}{\sqrt{3}-1}$$\frac{1}{\sqrt{3}+1}$$\frac{\sqrt{3}-1}{\sqrt{3}+1}$$\frac{\sqrt{3}+1}{\sqrt{3}-1}$Correct Option: 1 Solution: $\sin 30^{\circ}=x ...
Read More →If the lines x+y=a and x-y=b touch the curve
Question: If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^{2}-3 x+2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to________. Solution: $y=x^{2}-3 x+2$ At $x$-axis $y=0=x^{2}-3 x+2$ $x=1,2$ $\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}-3$ $\mathrm{A}(1,0) \mathrm{B}(2,0)$ $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{x=1}=-1$ and $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{x=2}=1$ $\# x+y=a \Rightarrow \frac{d y}{d x}=-1$ So $A(1,0)$ lies on it $\R...
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Question: The common difference of the A.P. $b_{1}, b_{2}, \ldots$, $\mathrm{b}_{\mathrm{m}}$ is 2 more than the common difference of A.P. $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{n}}$. If $\mathrm{a}_{40}=-159, \mathrm{a}_{100}=-399$ and $\mathrm{b}_{100}=\mathrm{a}_{70}$, then $\mathrm{b}_{1}$ is equal to : $-127$$-81$81127Correct Option: , 2 Solution: $a_{1}, a_{2}, \ldots, a_{n} \rightarrow(C D=d)$ $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{\mathrm{m}} \rightar...
Read More →The coefficient of x^4 in the expansion of
Question: The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x$, is_________. Solution: $\left(1+x+x^{2}+x^{3}\right)^{6}=\left((1+x)\left(1+x^{2}\right)\right)^{6}$ $=(1+x)^{6}\left(1+x^{2}\right)^{6}$ $=\sum_{r=0}^{6}{ }^{6} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}} \sum_{\mathrm{r}=0}^{6}{ }^{6} \mathrm{C}_{\mathrm{t}} \mathrm{x}^{2 \mathrm{t}}$ $=\sum_{r=0}^{6} \sum_{t=0}^{6}{ }^{6} C_{r}{ }^{6} C_{t} x^{r+2 t}$ $=\sum_{r=0}^{6} \sum_{t=0}^{...
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Question: Let $z=x+$ iy be a non-zero complex number such that $z^{2}=i|z|^{2}$, where $i=\sqrt{-1}$, then $z$ lies on the :imaginary axisreal axisline, $y=x$line, $y=-x$Correct Option: , 3 Solution: $z=x+i y$ $z^{2}=i|z|^{2}$ $(x+i y)^{2}=i\left(x^{2}+y^{2}\right)$ $\left(x^{2}-y^{2}\right)-i\left(x^{2}+y^{2}-2 x y\right)=0$ $(x-y)(x+y)-i(x-y)^{2}=0$ $(x-y)((x+y)-i(x-y))=0$ $\Rightarrow x=y$ $z$ lies on $y=x$...
Read More →Let A = { a, b, c } and B = { 1, 2, 3, 4 }.
Question: Let $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$ and $\mathrm{B}=\{1,2,3,4\}$. Then the number of elements in the set $\mathrm{C}=\{\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B} \mid 2 \in \mathrm{f}(\mathrm{A})$ and $\mathrm{f}$ is not one-one $\}$ is Solution: $\mathrm{C}=\{\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B} \mid 2 \in \mathrm{f}(\mathrm{A})$ and $\mathrm{f}$ is not one-one $\}$ Case-I : If $\mathrm{f}(\mathrm{x})=2 \forall \mathrm{x} \in \mathrm{A}$ then number of ...
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Question: If $\alpha$ and $\beta$ are the roots of the equation $2 x(2 x+1)=1$, then $\beta$ is equal to :$2 \alpha^{2}$$2 \alpha(\alpha+1)$$-2 \alpha(\alpha+1)$$2 \alpha(\alpha-1)$Correct Option: , 3 Solution: $\alpha$ and $\beta$ are the roots of the equation $4 x^{2}+2 x-1=0$. $4 \alpha^{2}+2 \alpha=1 \Rightarrow \frac{1}{2}=2 \alpha^{2}+\alpha$ .........(1) $\beta=\frac{-1}{2}-\alpha$ using equation (1) $\beta=-\left(2 \alpha^{2}+\alpha\right)-\alpha$ $\beta=-2 \alpha^{2}-2 \alpha$ $\beta=-2...
Read More →In a bombing attack, there is 50% chance that a bomb will hit the target.
Question: In a bombing attack, there is $50 \%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least $99 \%$ chance of completely destroying the target, is Solution: $\mathrm{P}(\mathrm{H})=\frac{1}{2}$ $\mathrm{P}(\overline{\mathrm{H}})=\frac{1}{2}$ Let total 'n' bomb are required to destroy the target $1-{ }^{n} C_{n}\left(\frac{1}{2}\right)^{n...
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Question: A plane $\mathrm{P}$ meets the coordinate axes at $\mathrm{A}, \mathrm{B}$ and C respectively. The centroid of $\triangle \mathrm{ABC}$ is given to be $(1,1,2)$. Then the equation of the line through this centroid and perpendicular to the plane $P$ is :$\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$$\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-2}{1}$$\frac{x-1}{2}=\frac{y-1}{1}=\frac{z-2}{1}$$\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-2}{2}$Correct Option: , 2 Solution: $\frac{x}{a}+\frac{y}{b}+\frac{z}{c...
Read More →If the normal at an end of a latus rectum of an ellipse passes through an extremity of the
Question: If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies :$\mathrm{e}^{2}+2 \mathrm{e}-1=0$$e^{2}+e-1=0$$\mathrm{e}^{4}+2 \mathrm{e}^{2}-1=0$$e^{4}+e^{2}-1=0$Correct Option: Solution: $\frac{a^{2} x}{x_{1}}-\frac{b^{2} y}{y_{1}}=a^{2} e^{2}$ $\frac{a^{2} x}{a e}-\frac{b^{2} y}{b^{2}} \cdot a=a^{2} e^{2}$ $\frac{a x}{e}-a y=a^{2} e^{2} \Rightarrow \frac{x}{e}-y=a e^{2}$ passes through $(0, \ma...
Read More →If L = sin squer ( π/16 ) - sin squer ( π/8) and
Question: If $L=\sin ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$ and $\mathrm{M}=\cos ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$, then :$\mathrm{M}=\frac{1}{2 \sqrt{2}}+\frac{1}{2} \cos \frac{\pi}{8}$$L=\frac{1}{4 \sqrt{2}}-\frac{1}{4} \cos \frac{\pi}{8}$$M=\frac{1}{4 \sqrt{2}}+\frac{1}{4} \cos \frac{\pi}{8}$$L=-\frac{1}{2 \sqrt{2}}+\frac{1}{2} \cos \frac{\pi}{8}$Correct Option: 1 Solution: $L=\sin ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(...
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Question: Consider the statement : "For an integer $\mathrm{n}$, if $\mathrm{n}^{3}-1$ is even, then $\mathrm{n}$ is odd." The contrapositive statement of this statement is :For an integer $\mathrm{n}$, if $\mathrm{n}^{3}-1$ is not even, then $\mathrm{n}$ is not odd.For an integer $\mathrm{n}$, if $\mathrm{n}$ is even, then $\mathrm{n}^{3}-1$ is odd.For an integer $\mathrm{n}$, if $\mathrm{n}$ is odd, then $\mathrm{n}^{3}-1$ is even.For an integer $n$, if $n$ is even, then $n^{3}-1$ is even.Corr...
Read More →The statement ( p → ( q → p ) ) → ( p → ( p v q ) ) is :
Question: The statement $(\mathrm{p} \rightarrow(\mathrm{q} \rightarrow \mathrm{p})) \rightarrow(\mathrm{p} \rightarrow(\mathrm{p} \vee \mathrm{q}))$ is:a contradictionequivalent to $(p \wedge q) \vee(\sim q)$a tautologyequivalent to $(p \vee q) \wedge(\sim p)$Correct Option: , 3 Solution:...
Read More →Which of the following points lies on the tangent to
Question: Which of the following points lies on the tangent to the curve $x^{4} e^{y}+2 \sqrt{y+1}=3$ at the point $(1,0)$ ?$(2,2)$$(-2,6)$$(-2,4)$$(2,6)$Correct Option: , 2 Solution: $x^{4} e^{y}+2 \sqrt{y+1}=3$ d.W.r. to $\mathrm{X}$ $x^{4} e^{y} y^{\prime}+e^{y} 4 x^{3}+\frac{2 y^{\prime}}{2 \sqrt{y+1}}=0$ at $\mathrm{P}(1,0)$ $y_{P}^{\prime}+4+y_{P}^{\prime}=0$ $\Rightarrow y_{P}^{\prime}=-2$ Tangent at $\mathrm{P}(1,0)$ is $y-0=-2(x-1)$ $2 x+y-2$ $(-2,6)$ lies on it...
Read More →Let y=y(x) be the solution of the differential
Question: Let y=y(x) be the solution of the differential equation $\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x$, $x \in\left(0, \frac{\pi}{2}\right) .$ If $y(\pi / 3)=0$, then $y(\pi / 4)$ is equal to :$\sqrt{2}-2$$\frac{1}{\sqrt{2}}-1$$2-\sqrt{2}$$2+\sqrt{2}$Correct Option: 1 Solution: $\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x$ $\frac{d y}{d x}+\frac{2 \sin x}{\cos x} y=2 \sin x$ I.F. $=e^{\int 2 \frac{\sin x}{\cos x} d x}$ $=e^{2} \ln \sec x=\sec ^{2} x$ $y \cdot \sec ^{2} x=\int 2 \sin x \cd...
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Question: If the tangent to the curve, $y=f(x)=x \log _{e} x$, $(x0)$ at a point $(c, f(c))$ is parallel to the line - segement joining the points $(1,0)$ and $(e, e)$, then $\mathrm{c}$ is equal to :$\frac{1}{e-1}$$e^{\left(\frac{1}{1-e}\right)}$$e^{\left(\frac{1}{e-1}\right)}$$\frac{\mathrm{e}-1}{\mathrm{e}}$Correct Option: , 3 Solution: $f(x)=x \log _{e} x$ $\left.f^{\prime}(\mathrm{X})\right|_{(\mathrm{c}, f(\mathrm{c}))}=\frac{\mathrm{e}-0}{\mathrm{e}-1}$ $f^{\prime}(\mathrm{x})=1+\log _{\m...
Read More →If for some α ∈ R, the lines
Question: If for some $\alpha \in \mathrm{R}$, the lines $\mathrm{L}_{1}: \frac{\mathrm{x}+1}{2}=\frac{\mathrm{y}-2}{-1}=\frac{\mathrm{z}-1}{1}$ and $\mathrm{L}_{2}: \frac{\mathrm{x}+2}{\alpha}=\frac{\mathrm{y}+1}{5-\alpha}=\frac{\mathrm{z}+1}{1}$ are coplanar, then the line $L_{2}$ passes through the point :$(-2,10,2)$$(10,2,2)$$(10,-2,-2)$$(2,-10,-2)$Correct Option: , 4 Solution: $\mathrm{L}_{1} \equiv \frac{\mathrm{x}+1}{2}=\frac{\mathrm{y}-2}{-1}=\frac{\mathrm{z}-1}{1}$ $\mathrm{L}_{2} \equi...
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