Which one of the following Boolean expressions is a tautology ?
Question: Which one of the following Boolean expressions is a tautology ?$(\mathrm{P} \vee \mathrm{q}) \wedge(\sim \mathrm{p} \vee \sim \mathrm{q})$$(P \wedge q) \vee(p \wedge \sim q)$$(P \vee q) \wedge(p \vee \sim q)$$(P \vee q) \vee(p \vee \sim q)$Correct Option: , 4 Solution: (1) $(p \vee q) \wedge(\sim p \vee \sim q) \equiv(p \vee q) \wedge \sim(p \wedge q) \rightarrow$ Not tautology (Take both $p$ and $q$ as $T$ ) (2) $(p \wedge q) \vee(p \wedge \sim q) \equiv p \wedge(q \vee \sim q) \equiv...
Read More →Let the sum of the first n terms of a non
Question: Let the sum of the first n terms of a non constant A.P., $\quad a_{1}, \quad a_{2}, \quad a_{3}, \ldots \ldots$ be $50 \mathrm{n}+\frac{\mathrm{n}(\mathrm{n}-7)}{2} \mathrm{~A}$, wherre $\mathrm{A}$ is a constant. If $\mathrm{d}$ is the common difference of this A.P., then the ordered pair $\left(d, a_{50}\right)$ is equal to $(\mathrm{A}, 50+46 \mathrm{~A})$$(\mathrm{A}, 50+45 \mathrm{~A})$$(50,50+46 \mathrm{~A})$$(50,50+45 \mathrm{~A})$Correct Option: 1 Solution: $\mathrm{S}_{\mathrm...
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Question: Let $f(\mathrm{x})=\mathrm{e}^{\mathrm{x}}-\mathrm{x}$ and $\mathrm{g}(\mathrm{x}) \mathrm{x}^{2}-\mathrm{x}, \forall \mathrm{x} \in \mathrm{R}$. Then the set of all $x \in R$, where the function $h(x)=(f \circ g)(x)$ is increasing, is :$\left[-1, \frac{-1}{2}\right] \cup\left[\frac{1}{2}, \infty\right)$$\left[0, \frac{1}{2}\right] \cup[1, \infty)$$\left[\frac{-1}{2}, 0\right] \cup[1, \infty)$$[0, \infty)$Correct Option: , 2 Solution: $h(x)=f(g(x))$ $\Rightarrow h^{\prime}(x)=f^{\prime...
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Question: The integral $\int \sec ^{2 / 3} x \operatorname{cosec}{ }^{4 / 3} x d x$ is equal to (Hence $\mathrm{C}$ is a constant of integration)$3 \tan ^{-1 / 3} x+C$$-\frac{3}{4} \tan ^{-4 / 3} x+C$$-3 \cot ^{-1 / 3} x+C$$-3 \tan ^{-1 / 3} x+C$Correct Option: , 4 Solution: $I=\int \frac{d x}{(\sin x)^{4 / 3} \cdot(\cos x)^{2 / 3}}$ $I=\int \frac{d x}{\left(\frac{\sin x}{\cos x}\right)^{4 / 3} \cdot \cos ^{2} x}$ $\Rightarrow I=\int \frac{\sec ^{2} x}{(\tan x)^{4 / 3}} d x$ put $\tan x=t \Right...
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Question: Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ respectively and $\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\frac{1}{2} \overrightarrow{\mathrm{b}}$, the...
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Question: If $\mathrm{n}_{4}, \mathrm{n}_{5}$ and $\mathrm{nC}_{6}$ are in A.P., then $\mathrm{n}$ can be:1411912Correct Option: 1 Solution: $2 \cdot \mathrm{n}_{5}=\mathrm{n}_{4}+\mathrm{n}_{6}$ 2. $\frac{\lfloor n}{|5| n-5}=\frac{\mid n}{|4| n-4}+\frac{\lfloor n}{|6| n-6}$ $\frac{2}{5} \cdot \frac{1}{n-5}=\frac{1}{(n-4)(n-5)}+\frac{1}{30}$ $\mathrm{n}=14$ satisfying equation....
Read More →The set of all values of
Question: The set of all values of $\lambda$ for which the system of linear equations. $x-2 y-2 z=\lambda x$ $x+2 y+z=\lambda y$ $-x-y=\lambda z$ has a non-trivial solution.contains more than two elementsis a singletonis an empty setcontains exactly two elementsCorrect Option: , 2 Solution: $\left|\begin{array}{ccc}\lambda-1 2 2 \\ 1 2-\lambda 1 \\ 1 1 1\end{array}\right|=0 \Rightarrow(\lambda-1)^{3}=0 \Rightarrow \lambda=1$...
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Question: $\lim _{n \rightarrow \infty}\left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\frac{n}{n^{2}+3^{2}}+\ldots .+\frac{1}{5 n}\right)$$\frac{\pi}{4}$$\tan ^{-1}(2)$$\tan ^{-1}(3)$$\frac{\pi}{2}$Correct Option: , 2 Solution: $\lim _{x \rightarrow \infty} \sum_{r=1}^{2 n} \frac{n}{n^{2}+r^{2}}$\ $\lim _{x \rightarrow \infty} \sum_{r=1}^{2 n} \frac{1}{n\left(1+\frac{r^{2}}{n^{2}}\right)}=\int_{0}^{2} \frac{d x}{1+x^{2}}=\tan ^{-1} 2$...
Read More →The integral
Question: The integral $\int_{1}^{\mathrm{e}}\left\{\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{2 \mathrm{x}}-\left(\frac{\mathrm{e}}{\mathrm{x}}\right)^{\mathrm{x}}\right\} \log _{\mathrm{e}} \mathrm{xdx}$ is equal to:$\frac{1}{2}-\mathrm{e}-\frac{1}{\mathrm{e}^{2}}$$\frac{3}{2}-\frac{1}{\mathrm{e}}-\frac{1}{2 \mathrm{e}^{2}}$$-\frac{1}{2}+\frac{1}{\mathrm{e}}-\frac{1}{2 \mathrm{e}^{2}}$$\frac{3}{2}-\mathrm{e}-\frac{1}{2 \mathrm{e}^{2}}$Correct Option: , 4 Solution: $\int_{1}^{\mathrm{e}}\left(...
Read More →Solve this following
Question: A plane passing through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi}{4}$ with the plane $y-z+5=0$, also passes through the point$(-\sqrt{2}, 1,-4)$$(\sqrt{2}, 1,4)$$(\sqrt{2},-1,4)$$(-\sqrt{2},-1,-4)$Correct Option: , 2 Solution: Let $a x+b y+c z=1$ be the equation of the plane $\Rightarrow 0-b+0=1$ $\Rightarrow b=-1$ $0+0+c=1$ $\frac{1}{\sqrt{2}}=\frac{|0-1-1|}{\sqrt{\left(a^{2}+1+1\right)} \sqrt{0+1+1}}$ $\Rightarrow \mathrm{a}^{2}+2=4$ $\Rightarrow \mathrm{a}=...
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Question: Le $f(\mathrm{x})=\mathrm{x}^{2}, \mathrm{x} \in \mathrm{R}$. For any $\mathrm{A} \subseteq \mathrm{R}$, define $\mathrm{g}(\mathrm{A})=\{\mathrm{x} \in \mathrm{R}, f(\mathrm{x}) \in \mathrm{A}\}$. If $\mathrm{S}=[0,4]$, then which one of the following statements is not true ?$f(\mathrm{~g}(\mathrm{~S})) \neq f(\mathrm{~S})$$f(g(S))=S$$g(f(S))=g(S)$$\mathrm{g}(f(\mathrm{~S})) \neq \mathrm{S}$Correct Option: , 3 Solution: $\mathrm{g}(\mathrm{S})=[-2,2]$ So, $\mathrm{f}(\mathrm{g}(\mathr...
Read More →The solution of the differential equation
Question: The solution of the differential equation $x \frac{d y}{d x}+2 y=x^{2}(x \neq 0)$ with $y(1)=1$, is $y=\frac{x^{3}}{5}+\frac{1}{5 x^{2}}$$y=\frac{4}{5} x^{3}+\frac{1}{5 x^{2}}$$y=\frac{3}{4} x^{2}+\frac{1}{4 x^{2}}$$y=\frac{x^{2}}{4}+\frac{3}{4 x^{2}}$Correct Option: , 4 Solution: $x \frac{d y}{d x}+2 y=x^{2}: y(1)=1$ $\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{2}{\mathrm{x}}\right) \mathrm{y}=\mathrm{x}$ (LDE in $\left.\mathrm{y}\right)$ $I F=e^{\int \frac{2}{x} d x}=e^{2(n x}=x^{2}$...
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Question: If the circles $x^{2}+y^{2}+5 K x+2 y+K=0$ and $2\left(x^{2}+y^{2}\right)+2 K x+3 y-1=0,(K \in R)$, intersect at the points $\mathrm{P}$ and $\mathrm{Q}$, then the line $4 x+5 y-K=0$ passes through $P$ and $Q$ for :exactly two values of Kexactly one value of Kno value of K.infinitely many values of KCorrect Option: , 3 Solution: Equation of common chord $4 \mathrm{kx}+\frac{1}{2} \mathrm{y}+\mathrm{k}+\frac{1}{2}=0$ .....(1) and given line is $4 x+5 y-k=0$ .......(2) On comparing (1) (...
Read More →If the fourth term in the binomial expansion of
Question: If the fourth term in the binomial expansion of $\left(\frac{2}{x}+x^{\log _{8} x}\right)^{6}(x0)$ is $20 \times 8^{7}$, then a value of $x$ is :8$8^{2}$$8^{-2}$$8^{3}$Correct Option: , 2 Solution: $\mathrm{T}_{4}=\mathrm{T}_{3+1}=\left(\begin{array}{l}6 \\ 3\end{array}\right)\left(\frac{2}{\mathrm{x}}\right)^{3} \cdot\left(\mathrm{x}^{\log _{\mathrm{s}} \mathrm{x}}\right)^{3}$ $20 \times 8^{7}=\frac{160}{x^{3}} \cdot x^{3 \log _{8} x}$ $8^{6}=x^{\log _{2} x}-3$ $2^{18}=x^{\log _{2} x-...
Read More →If the system of linear equations
Question: If the system of linear equations $x+y+z=5$ $x+2 y+2 z=6$ $x+3 y+\lambda z=\mu,(\lambda, \mu \in R)$, has infinitely many solutions, then the value of $\lambda+\mu$ is :121097Correct Option: , 2 Solution: $x+3 y+\lambda z-\mu=p(x+y+z-5)+$ $q(x+2 y+2 z-6)$ on comparing the coefficient; $p+q=1$ and $p+2 q=3$ $\Rightarrow(\mathrm{p}, \mathrm{q})=(-1,2)$ Hence $x+3 y+\lambda z-\mu=x+3 y+3 z-7$ $\Rightarrow \lambda=3, \mu=7$...
Read More →A committee of 11 members is to be formed from 8 males and 5 females.
Question: A committee of 11 members is to be formed from 8 males and 5 females. If $\mathrm{m}$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then :$\mathrm{m}=\mathrm{n}=78$$\mathrm{n}=\mathrm{m}-8$$m+n=68$$\mathrm{m}=\mathrm{n}=68$Correct Option: 1 Solution: Since there are 8 males and 5 females. Out of these 13 , if we select 11 persons, then there will be at least 6 males and atleast 3 female...
Read More →If the angle of elevation of a cloud from a point
Question: If the angle of elevation of a cloud from a point $\mathrm{P}$ which is $25 \mathrm{~m}$ above a lake be $30^{\circ}$ and the angle of depression of reflection of the cloud in the lake from $P$ be $60^{\circ}$, then the height of the cloud (in meters) from the surface of the lake is :42455060Correct Option: , 2 Solution: $\tan 30^{\circ}=\frac{x}{y} \Rightarrow y=\sqrt{3} x$ .............(i) $\tan 60^{\circ}=\frac{25+x+25}{y}$ $\Rightarrow \sqrt{3} y=50+x$ $\Rightarrow 3 x=50+x$ $\Righ...
Read More →Slope of a line passing through
Question: Slope of a line passing through $\mathrm{P}(2,3)$ and intersecting the line, $x+y=7$ at a distance of 4 units from $\mathrm{P}$, is$\frac{\sqrt{5}-1}{\sqrt{5}+1}$$\frac{1-\sqrt{5}}{1+\sqrt{5}}$$\frac{1-\sqrt{7}}{1+\sqrt{7}}$$\frac{\sqrt{7}-1}{\sqrt{7}+1}$Correct Option: , 3 Solution: $x=2+r \cos \theta$ $y=3+r \sin \theta$ $\Rightarrow 2+r \cos \theta+3+r \sin \theta=7$ $\Rightarrow r(\cos \theta+\sin \theta)=2$ $\Rightarrow \sin \theta+\cos \theta=\frac{2}{r}=\frac{2}{\pm 4}=\pm \frac...
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Question: If $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}$, then $k$ is :$\frac{3}{8}$$\frac{3}{2}$$\frac{4}{3}$$\frac{8}{3}$Correct Option: , 4 Solution: $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}$ $\Rightarrow \lim _{x \rightarrow 1}(x+1)\left(x^{2}+1\right)=\frac{k^{2}+k^{2}+k^{2}}{2 k}$ $\Rightarrow \mathrm{k}=8 / 3$...
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Question: If $\Delta_{1}=\left|\begin{array}{ccc}x \sin \theta \cos \theta \\ -\sin \theta -x 1 \\ \cos \theta 1 x\end{array}\right|$ and $\Delta_{2}=\left|\begin{array}{ccc}\mathrm{x} \sin 2 \theta \cos 2 \theta \\ -\sin 2 \theta -\mathrm{x} 1 \\ \cos 2 \theta 1 \mathrm{x}\end{array}\right|, \mathrm{x} \neq 0 ;$ then for all $\theta \in\left(0, \frac{\pi}{2}\right):$$\Delta_{1}-\Delta_{2}=\mathrm{x}(\cos 2 \theta-\cos 4 \theta)$$\Delta_{1}+\Delta_{2}=-2 \mathrm{x}^{3}$$\Delta_{1}-\Delta_{2}=-2 ...
Read More →Solve this following
Question: Let $p, q \in R$. If $2-\sqrt{3}$ is a root of the quadratic equation, $x^{2}+p x+q=0$, then :$\mathrm{q}^{2}+4 \mathrm{p}+14=0$$\mathrm{p}^{2}-4 \mathrm{q}-12=0$$q^{2}-4 p-16=0$$p^{2}-4 q+12=0$Correct Option: , 2 Solution: In given question $\mathrm{p}, \mathrm{q} \in \mathrm{R}$. If we take other root as any real number $\alpha$, then quadratic equation will be $x^{2}-(\alpha+2-\sqrt{3}) x+\alpha \cdot(2-\sqrt{3})=0$ Now, we can have none or any of the options can be correct dependin...
Read More →The equation of a tangent to the parabola,
Question: The equation of a tangent to the parabola, $\mathrm{x}^{2}=8 \mathrm{y}$, which makes an angle $\theta$ with the positive direction of $\mathrm{x}$-axis, is :$x=y \cot \theta+2 \tan \theta$$x=y \cot \theta-2 \tan \theta$$y=x \tan \theta-2 \cot \theta$$y=x \tan \theta+2 \cot \theta$Correct Option: 1 Solution: $x^{2}=8 y$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{4}=\tan \theta$ $\therefore \quad \mathrm{x}_{1}=4 \tan \theta$ $\mathrm{y}_{1}=2 \tan ^{2} \theta$ Equat...
Read More →Solve this following
Question: Let $f(\mathrm{x})=15-|\mathrm{x}-10| ; \mathrm{x} \in \mathrm{R}$. Then the set of all values of $\mathrm{x}$, at which the function, $\mathrm{g}(\mathrm{x})=f(f(\mathrm{x}))$ is not differentiable, is :$\{5,10,15,20\}$$\{10,15\}$$\{5,10,15\}$$\{10\}$Correct Option: , 3 Solution: $f(\mathrm{x})=15-|\mathrm{x}-10|, \mathrm{x} \in \mathrm{R}$ $f(f(\mathrm{x}))=15-|f(\mathrm{x})-10|$ $=15-|15-| \mathrm{x}-10|-10|$ $=15-|5-| \mathrm{x}-10||$ $x=5,10,15$ are points of non differentiability...
Read More →1. If for some x E R, the frequency distribution of the marks obtained by 20 students in a test is :
Question: If for some $\mathrm{x} \in \mathrm{R}$, the frequency distribution of the marks obtained by 20 students in a test is : then the mean of the marks is :2.83.23.02.5Correct Option: 1 Solution: $\sum \mathrm{f}_{\mathrm{i}}=20=2 \mathrm{x}^{2}+2 \mathrm{x}-4$ $\Rightarrow \mathrm{x}^{2}+2 \mathrm{x}-24=0$ $\mathrm{x}=3,-4$ (rejected) $\overline{\mathrm{x}}=\frac{\sum \mathrm{x}_{\mathrm{i}} f_{\mathrm{i}}}{\sum f_{\mathrm{i}}}=2.8$...
Read More →If a circle of radius R passes through the origin O
Question: If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $\mathrm{AB}$ is :$\left(x^{2}+y^{2}\right)^{2}=4 R x^{2} y^{2}$$\left(x^{2}+y^{2}\right)(x+y)=R^{2} x y$$\left(x^{2}+y^{2}\right)^{3}=4 R^{2} x^{2} y^{2}$$\left(x^{2}+y^{2}\right)^{2}=4 R^{2} x^{2} y^{2}$Correct Option: , 3 Solution: Slope of $\mathrm{AB}=\frac{-\mathrm{h}}{\mathrm{k}}$ Equation of $\mathrm{AB}$ is $\mathrm{...
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