If b is very small as compared to the value of a,
Question: If b is very small as compared to the value of a, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b}+\ldots .+\frac{1}{a-n b}=\alpha n+\beta n^{2}+\gamma n^{3}$ then the value of $\gamma$ is :$\frac{\mathrm{a}^{2}+\mathrm{b}}{3 \mathrm{a}^{3}}$$\frac{\mathrm{a}+\mathrm{b}}{3 \mathrm{a}^{2}}$$\frac{\mathrm{b}^{2}}{3 \mathrm{a}^{3}}$$\frac{a+b^{2}}{3 a^{3}}$Correct Option: , 4 Solution: $\frac{\mathrm{...
Read More →Consider the line L given by the equation
Question: Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1} .$ Let $Q$ be the mirror image of the point $(2,3,-1)$ with respect to $L$. Let a plane $P$ be such that it passes through $Q$, and the line $L$ is perpendicular to P. Then which of the following points is on the plane $\mathrm{P}$ ?$(-1,1,2)$$(1,1,1)$$(1,1,2)$$(1,2,2)$Correct Option: , 4 Solution: Plane $p$ is $\perp^{\mathrm{r}}$ to line $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}$ \ passes through ...
Read More →A circle C touches the line x=2 y at the point (2,1) and intersects
Question: A circle $C$ touches the line $x=2 y$ at the point $(2,1)$ and intersects the circle $C_{1}: x^{2}+y^{2}+2 y-5=0$ at two points $P$ and $Q$ such that $P Q$ is a diameter of $\mathrm{C}_{1}$. Then the diameter of $\mathrm{C}$ is :$7 \sqrt{5}$15$\sqrt{285}$$4 \sqrt{15}$Correct Option: 1 Solution: $(x-2)^{2}+(y-1)^{2}+\lambda(x-2 y)=0$ $\mathrm{C}: \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{x}(\lambda-4)+\mathrm{y}(-2-2 \lambda)+5=0$ $\mathrm{C}_{1}: \mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{y}-...
Read More →Let y = y(x)
Question: Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ satisfies the equation $\frac{\mathrm{dy}}{\mathrm{dx}}-|\mathrm{A}|=0$ for all $x0$, where $A=\left[\begin{array}{ccc}y \sin x 1 \\ 0 -1 1 \\ 2 0 \frac{1}{x}\end{array}\right] .$ If $y(\pi)=\pi+2$, then the value of $y\left(\frac{\pi}{2}\right)$ is :$\frac{\pi}{2}+\frac{4}{\pi}$$\frac{\pi}{2}-\frac{1}{\pi}$$\frac{3 \pi}{2}-\frac{1}{\pi}$$\frac{\pi}{2}-\frac{4}{\pi}$Correct Option: 1 Solution: $|A|=-\frac{y}{x}+2 \sin x+2$ $\frac{d y}{d x}=|A|$ $...
Read More →Four dice are thrown simultaneously and the numbers
Question: Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2 \times 2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is :$\frac{45}{162}$$\frac{23}{81}$$\frac{22}{81}$$\frac{43}{162}$Correct Option: , 4 Solution: $A=\left|\begin{array}{ll}a b \\ c d\end{array}\right| \quad|A|=a d-b c$ Total case $=6^{4}$ For non-singular matrix $|\mathrm{A}| \neq 0 \Rightarrow \mathrm{ad}-\mathrm{bc} \neq 0$ $\Rightarr...
Read More →The value of the definite integral
Question: The value of the definite integral $\int_{\pi / 24}^{5 \pi / 24} \frac{d x}{1+\sqrt[3]{\tan 2 x}} i s$$\frac{\pi}{3}$$\frac{\pi}{6}$$\frac{\pi}{12}$$\frac{\pi}{18}$Correct Option: , 3 Solution: Let $\mathrm{I}=\int_{\pi / 24}^{5 \pi / 24} \frac{(\cos 2 \mathrm{x})^{1 / 3}}{(\cos 2 \mathrm{x})^{1 / 3}+(\sin 2 \mathrm{x})^{1 / 3}} \mathrm{dx}$....(i) $\Rightarrow I=\int_{\pi / 24}^{5 \pi / 24} \frac{\left(\cos \left\{2\left(\frac{\pi}{4}-x\right)\right\}\right)^{\frac{1}{3}}}{\left(\cos ...
Read More →Solve this
Question: The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1+\sin ^{2} x}{1+\pi^{\sin x}}\right) d x$ is$\frac{\pi}{2}$$\frac{5 \pi}{4}$$\frac{3 \pi}{4}$$\frac{3 \pi}{2}$Correct Option: , 3 Solution: $I=\int_{0}^{\pi / 2} \frac{\left(1+\sin ^{2} x\right)}{\left(1+\pi^{\sin x}\right)}+\frac{\pi^{\sin x}\left(1+\sin ^{2} x\right)}{\left(1+\pi^{\sin x}\right)} d x$ $I=\int_{0}^{\pi / 2}\left(1+\sin ^{2} x\right) d x$ $I=\frac{\pi}{2}+\frac{\pi}{2} \cdot \frac{1}{2}=\frac{3 \pi}{4}$...
Read More →Let f : R → R be defined as
Question: Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)= \begin{cases}\frac{\lambda\left|x^{2}-5 x+6\right|}{\mu\left(5 x-x^{2}-6\right)}, x2 \\ e^{\frac{\tan (x-2)}{x-[x]}} , x2 \\ \mu , x=2\end{cases}$ where $[\mathrm{x}]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x=2$, then $\lambda+\mu$ is equal to:$\mathrm{e}(-\mathrm{e}+1)$e(e-2)1$2 \mathrm{e}-1$Correct Option: 1 Solution: $\lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{...
Read More →Let in a right angled triangle,
Question: Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin \theta$ is equal to :$\frac{\sqrt{5}+1}{4}$$\frac{\sqrt{5}-1}{2}$$\frac{\sqrt{2}-1}{2}$$\frac{\sqrt{5}-1}{4}$Correct Option: , 2 Solution: $\mathrm{A}=\theta$ $\mathrm{B}=90-\theta$ $a=$ smallest side $\mathrm{c}^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}$ $\frac{1}{a^{2}}=\frac{1}{b^{2}}+\frac{1}{c^{2}}$ $\frac{\mathrm{b}^{2} \ma...
Read More →A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top
Question: A 10 inches long pencil $\mathrm{AB}$ with mid point $\mathrm{C}$ and a small eraser P are placed on the horizontal top of a table such that $\quad \mathrm{PC}=\sqrt{5} \quad$ inches and $\angle \mathrm{PCB}=\tan ^{-1}(2)$. The acute angle through which the pencil must be rotated about $\mathrm{C}$ so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is: $\tan ^{-1}\left(\frac{3}{4}\right)$$\tan ^{-1}(1)$$\tan ^{-1}\left(\frac{4}{3}\right)$$\tan ^{-1}\lef...
Read More →Let y=y(x) be the solution of the differential equation
Question: Let $y=y(x)$ be the solution of the differential equation $\operatorname{cosec}^{2} x d y+2 d x=(1+y \cos 2 x) \operatorname{cosec}^{2} x d x$, with $y\left(\frac{\pi}{4}\right)=0 .$ Then, the value of $(y(0)+1)^{2}$ is equal to :$e^{1 / 2}$$\mathrm{e}^{-1 / 2}$$\mathrm{e}^{-1}$eCorrect Option: , 3 Solution: $\frac{d y}{d x}+2 \sin ^{2} x=1+y \cos 2 x$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}+(-\cos 2 \mathrm{x}) \mathrm{y}=\cos 2 \mathrm{x}$ I.F. $=e^{\int-\cos 2 x d x}=e^{-\frac{...
Read More →The sum of all the local minimum
Question: The sum of all the local minimum values of the twice differentiable function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-\frac{3 f^{\prime \prime}(2)}{2} \mathrm{x}+f^{\prime \prime}(1)$ is :-225-270Correct Option: , 3 Solution: $f(x)=x^{3}-3 x^{2}-\frac{3}{2} f^{\prime \prime}(2) x+f^{\prime \prime}(1) \ldots(1)$ $f^{\prime}(x)=3 x^{2}-6 x-\frac{3}{2} f^{\prime \prime}(2) \ldots(2)$ $f^{\prime \prime}(x)=6 x-6$..(3) Now is $3^{\text...
Read More →Let the vectors
Question: Let the vectors $(2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},(1+b) \hat{i}+2 b \hat{j}-b \hat{k}$ and $(2+\mathrm{b}) \hat{\mathrm{i}}+2 \mathrm{~b} \hat{\mathrm{j}}+(\mathrm{I}-\mathrm{b}) \hat{\mathrm{k}} \mathrm{a}, \mathrm{b}, \mathrm{c}, \in \mathbf{R}$ be co-planar. Then which of the following is true?$2 \mathrm{~b}=\mathrm{a}+\mathrm{c}$$3 \mathrm{c}=\mathrm{a}+\mathrm{b}$$a=b+2 c$$2 \mathrm{a}=\mathrm{b}+\mathrm{c}$Correct Option: 1 Solution: If the vectors are co-planar, $...
Read More →Let A, B and C be three events
Question: Let $A, B$ and $C$ be three events such that the probability that exactly one of $\mathrm{A}$ and $\mathrm{B}$ occurs is $(1-\mathrm{k})$, the probability that exactly one of $\mathrm{B}$ and $\mathrm{C}$ occurs is $(1-2 \mathrm{k})$, the probability that exactly one of $C$ and $A$ occurs is $(1-k)$ and the probability of all $A, B$ and $C$ occur simultaneously is $k^{2}$, where $0\mathrm{k}1$. Then the probability that at least one of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ occur is...
Read More →The locus of the centroid of the triangle formed
Question: The locus of the centroid of the triangle formed by any point $\mathrm{P}$ on the hyperbola $16 x^{2}-9 y^{2}+32 x+36 y-164=0$, and its foci is :$16 x^{2}-9 y^{2}+32 x+36 y-36=0$$9 x^{2}-16 y^{2}+36 x+32 y-144=0$$16 x^{2}-9 y^{2}+32 x+36 y-144=0$$9 x^{2}-16 y^{2}+36 x+32 y-36=0$Correct Option: 1 Solution: Given hyperbola is $16(x+1)^{2}-9(y-2)^{2}=164+16-36=144$ $\Rightarrow \frac{(x+1)^{2}}{9}-\frac{(y-2)^{2}}{16}=1$ Eccentricity, $\mathrm{e}=\sqrt{1+\frac{16}{9}}=\frac{5}{3}$ $\Right...
Read More →Let P be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes
Question: Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}})=16$ and $\overrightarrow{\mathrm{r}} \cdot(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=6 .$ Then which of the following points does NOT lie on P?$(3,3,2)$$(6,-6,2)$$(4,2,2)$$(-8,8,6)$Correct Option: , 3 Solution: $(x+y+4 z-16)+\lambda(-x+y+z-6)=0$ Passes through $(1,2,3)$ $-1+\lambda(-2) \Ri...
Read More →Let f: R → R be defined as
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\left\{\begin{array}{ll}-\frac{4}{3} x^{3}+2 x^{2}+3 x , \quad x0 \\ 3 x e^{x} \quad, \quad x \leq 0\end{array}\right.$ Then $f \quad$ is increasing function in the interval$\left(-\frac{1}{2}, 2\right)$$(0,2)$$\left(-1, \frac{3}{2}\right)$$(-3,-1)$Correct Option: , 3 Solution: $f^{\prime}(x)\left\{\begin{array}{rr}-4 x^{2}+4 x+3 x0 \\ 3 e^{x}(1+x) x \leq 0\end{array}\right.$ For $x0, f^{\prime}(x)=-4 x^{2}+4 x+3$ $f(x)$ is...
Read More →Solve the Following Questions
Question: If $f: \mathbf{R} \rightarrow \mathbf{R}$ is given by $f(\mathrm{x})=\mathrm{x}+1$, then the value of $\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}}\left[f(0)+f\left(\frac{5}{\mathrm{n}}\right)+f\left(\frac{10}{\mathrm{n}}\right)+\ldots .+f\left(\frac{5(\mathrm{n}-1)}{\mathrm{n}}\right)\right]$ is :$\frac{3}{2}$$\frac{5}{2}$$\frac{1}{2}$$\frac{7}{2}$Correct Option: , 4 Solution: $I=\sum_{r=0}^{n-1} f\left(\frac{5 r}{n}\right) \frac{1}{n}$ $I=\int_{0}^{1} f(5 x) d x$ $I=\in...
Read More →A hall has a square floor of dimension 10m x10m (see the figure) and vertical walls.
Question: A hall has a square floor of dimension $10 \mathrm{~m} \times 10 \mathrm{~m}$ (see the figure) and vertical walls. If the angle GPH between the diagonals $\mathrm{AG}$ and $\mathrm{BH}$ is $\cos ^{-1} \frac{1}{5}$, then the height of the hall (in meters) is : 5$2 \sqrt{10}$$5 \sqrt{3}$$5 \sqrt{2}$Correct Option: , 4 Solution: $\mathrm{A}(\hat{\mathrm{j}}) \cdot \mathrm{B}(10 \hat{\mathrm{i}})$ $\mathbf{H}(\hat{\mathrm{h}}+10 \hat{\mathrm{k}})$ $\mathbf{G}(10 \hat{\mathrm{i}}+\mathrm{h}...
Read More →Solve the Following Questions
Question: Let $f: \mathbf{R}-\left\{\frac{\alpha}{6}\right\} \rightarrow \mathbf{R}$ be defined by $f(x)=\frac{5 x+3}{6 x-\alpha}$ Then the value of $\alpha$ for which $(f o f)(x)=x$, for all $x \in \mathbf{R}-\left\{\frac{\alpha}{6}\right\}$, is :No such $\alpha$ exists586Correct Option: , 2 Solution: $f(x)=\frac{5 x+3}{6 x-\alpha}=y$..(1) $5 x+3=6 x y-\alpha y$ $x(6 y-5)=\alpha y+3$ $x=\frac{\alpha y+3}{6 y-5}$ $\mathrm{f}^{-1}(\mathrm{x})=\frac{\alpha \mathrm{x}+3}{6 \mathrm{x}-5}$...(2) fo $...
Read More →If the real part of the complex number
Question: If the real part of the complex number $(1-\cos \theta+2 \text { isin } \theta)^{-1}$ is $\frac{1}{5}$ for $\theta \in(0, \pi)$, then the value of the integral $\int_{0}^{\theta} \sin x d x$ is equal to :12-10Correct Option: 1 Solution: $\mathrm{z}=\frac{1}{1-\cos \theta+2 \mathrm{i} \sin \theta}$ $=\frac{2 \sin ^{2} \frac{\theta}{2}-2 \mathrm{i} \sin \theta}{(1-\cos \theta)^{2}+4 \sin ^{2} \theta}$ $=\frac{\sin \frac{\theta}{2}-2 \mathrm{i} \cos \frac{\theta}{2}}{4 \sin \frac{\theta}{...
Read More →Solve this
Question: If $(\sqrt{3}+\mathrm{i})^{100}=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})$, then $\mathrm{p}$ and $\mathrm{q}$ are roots of the equation:$x^{2}-(\sqrt{3}-1) x-\sqrt{3}=0$$x^{2}+(\sqrt{3}+1) x+\sqrt{3}=0$$x^{2}+(\sqrt{3}-1) x-\sqrt{3}=0$$x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$Correct Option: 1 Solution: $\left(2 e^{i \pi / 6}\right)^{100}=2^{99}(p+i q)$ $2^{100}\left(\cos \frac{50 \pi}{3}+\mathrm{i} \sin \frac{50 \pi}{3}\right)=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})$ $p+i q=2\left(\cos \frac{...
Read More →If [x] denotes the greatest integer
Question: If $[x]$ denotes the greatest integer less than or equal to $\mathrm{x}$, then the value of the integral $\int_{-\pi / 2}^{\pi / 2}[[x]-\sin x] d x$ is equal to :$-\pi$$\pi$01Correct Option: 1 Solution: $I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}([x]+[-\sin x]) d x$..(1) $I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}([-x]+[\sin x]) d x \ldots(2)$ (King property) $2 \mathrm{I}=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}(\underbrace{[\mathrm{x}]+[-\mathrm{x}]}_{-1})+(\underbrace{[\sin \mathrm{x}]+[-\si...
Read More →Let S_n be the sum of the first n terms of an arithmetic progression.
Question: Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3 \mathrm{n}}=3 S_{2 \mathrm{n}}$, then the value of $\frac{S_{4 n}}{S_{2 n}}$ is :6428Correct Option: 1 Solution: Let a be first term and d be common diff. of this A.P. Given $S_{3 \mathrm{n}}=3 S_{2 \mathrm{n}}$ $\Rightarrow \frac{3 \mathrm{n}}{2}[2 \mathrm{a}+(3 \mathrm{n}-1) \mathrm{d}]=3 \frac{2 \mathrm{n}}{2}[2 \mathrm{a}+(2 \mathrm{n}-1) \mathrm{d}]$ $\Rightarrow 2 \mathrm{a}+(3 \mathrm{n}-1) \mat...
Read More →Let S_n denote the sum of first n-terms of an arithmetic progression.
Question: Let $S_{n}$ denote the sum of first n-terms of an arithmetic progression. If $S_{10}=530, S_{5}=140$, then $\mathrm{S}_{20}-\mathrm{S}_{6}$ is equal to :1862184218521872Correct Option: 1, Solution: $\mathrm{S}_{10}=530 \Rightarrow \frac{10}{2}\{2 \mathrm{a}+9 \mathrm{~d}\}=530$ $\Rightarrow 2 \mathrm{a}+9 \mathrm{~d}=106$ and $S_{5}=140 \Rightarrow \frac{5}{2}\{2 a+4 d\}=140$ $\Rightarrow 2 \mathrm{a}+4 \mathrm{~d}=56$ $\Rightarrow 5 \mathrm{~d}=50 \Rightarrow \mathrm{d}=10 \Rightarrow...
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