The sum of the series
Question: The sum of the series $\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots . .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is :$1+\frac{2^{101}}{4^{101}-1}$$1+\frac{2^{100}}{4^{101}-1}$$1-\frac{2^{100}}{4^{100}-1}$$1-\frac{2^{101}}{4^{101}-1}$Correct Option: , 4 Solution: $S=\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots \frac{2^{100}}{x^{2^{100}}+1}$ $S+\frac{1}{1-x}=\frac{1}{1-x}+\frac{1}{x+1}+\ldots .=\frac{2}{1-x^{2}}+\frac{2}{1+x^{2}}+\ldots$ $S+\frac{1}{1-x}=...
Read More →The mean of 10 numbers
Question: The mean of 10 numbers $7 \times 8,10 \times 10,13 \times 12,16 \times 14, \ldots .$ is________. Solution: $7 \times 8,10 \times 10,13 \times 12,16 \times 14 \ldots \ldots$ $\mathrm{T}_{\mathrm{n}}=(3 \mathrm{n}+4)(2 \mathrm{n}+6)=2(3 \mathrm{n}+4)(\mathrm{n}+3)$ $=2\left(3 n^{2}+13 n+12\right)=6 n^{2}+26 n+24$ $S_{10}=\sum_{n=1}^{10} T_{n}=6 \sum_{n=1}^{10} n^{2}+26 \sum_{n=1}^{10} n+24 \sum_{n=1}^{10} 1$ $=\frac{6(10 \times 11 \times 21)}{6}+26 \times \frac{10 \times 11}{2}+24 \times...
Read More →If ' R ' is the least value of 'a' such
Question: If ' $R$ ' is the least value of 'a' such that the function $f(x)=x^{2}+a x+1$ is increasing on $[1,2]$ and 'S' is the greatest value of 'a' such that the function $f(x)=x^{2}+a x+1$ is decreasing on $[1,2]$, then the value of $|R-S|$ is______. Solution: $f(x)=x^{2}+a x+1$ $f^{\prime}(x)=2 x+a$ when $f(\mathrm{x})$ is increasing on $[1,2]$ $2 \mathrm{x}+\mathrm{a} \geq 0 \quad \forall \mathrm{x} \in[1,2]$ $a \geq-2 x \forall x \in[1,2]$ $R=-4$ when $f(\mathrm{x})$ is decreasing on $[1,...
Read More →Let Z be the set of all integers,
Question: Let $\mathbb{Z}$ be the set of all integers, $\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2} \leq 4\right\}$ $\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\}$ and $\mathrm{C}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+(\mathrm{y}-2)^{2} \leq 4\right\}$ If the total number of relation from $\mathrm{A} \cap \math...
Read More →Solve this
Question: $\operatorname{Let} f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$ $0x1$. Then :(1) $(1-x)^{2} f^{\prime}(x)-2(f(x))^{2}=0$(2) $(1+x)^{2} f^{\prime}(x)+2(f(x))^{2}=0$(3) $(1-x)^{2} f^{\prime}(x)+2(f(x))^{2}=0$(4) $(1+x)^{2} f^{\prime}(x)-2(f(x))^{2}=0$Correct Option: , 3 Solution: $f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$ $\cot ^{-1} \sqrt{\frac{1-x}{x}}=\sin ^{-1} \sqrt{x}$ or $\mathrm{f}(\mathrm{x})...
Read More →The square of the distance of the point of intersection
Question: The square of the distance of the point of intersection of the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and the plane $2 \mathrm{x}-\mathrm{y}+\mathrm{z}=6$ from the point $(-1,-1,2)$ is______. Solution: $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}=\lambda$ $x=2 \lambda+1, y=3 \lambda+2, z=6 \lambda-1$ for point of intersection of line \ plane $2(2 \lambda+1)-(3 \lambda+2)+(6 \lambda-1)=6$ $7 \lambda=7 \Rightarrow \lambda=1$ point: $(3,5,5)$ $(\text { distance })^{2}=(3+1)^{2}+(5...
Read More →A point z moves in the complex plane such
Question: A point $\mathrm{z}$ moves in the complex plane such that $\arg \left(\frac{\mathrm{z}-2}{\mathrm{z}+2}\right)=\frac{\pi}{4}$, then the minimum value of $|\mathrm{z}-9 \sqrt{2}-2 i|^{2}$ is equal to______. Solution: Let $z=x+i y$ $\arg \left(\frac{x-2+i y}{x+2+i y}\right)=\frac{\pi}{4}$ $\arg (x-2+i y)-\arg (x+2+i y)=\frac{\pi}{4}$ $\tan ^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}-2}\right)-\tan ^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}+2}\right)=\frac{\pi}{4}$ $\frac{\frac{\mathrm{y}}{\math...
Read More →The Boolean expression
Question: The Boolean expression $(p \wedge q) \Rightarrow((r \wedge q) \wedge p)$ is equivalent to :$(\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \wedge \mathrm{q})$$(\mathrm{q} \wedge \mathrm{r}) \Rightarrow(\mathrm{p} \wedge \mathrm{q})$$(\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \vee \mathrm{q})$$(\mathrm{p} \wedge \mathrm{r}) \Rightarrow(\mathrm{p} \wedge \mathrm{q})$Correct Option: 1 Solution: $(p \wedge q) \Rightarrow((r \wedge q) \wedge p)$ $\sim(p \wedge q) \vee((r \...
Read More →Solve this
Question: Let $\theta \in\left(0, \frac{\pi}{2}\right) .$ If the system of linear equations $\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$ $\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$ $\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$ has a non-trivial solution, then the value of $\theta$ is :$\frac{4 \pi}{9}$$\frac{7 \pi}{18}$$\frac{\pi}{18}$$\frac{5 \pi}{18}$Correct Option: , 2 Solution: Case-I $\left|\begin{array}{ccc}1+\c...
Read More →Let [t] denote the greatest integer
Question: Let $[\mathrm{t}]$ denote the greatest integer $\leq \mathrm{t}$. Then the value of $8 \cdot \int_{-\frac{1}{2}}^{1}([2 x]+|x|) d x$ is__________. Solution: $I=\int_{-1 / 2}^{1}([2 x]+|x|) d x$ $=\int_{-1 / 2}^{1}[2 x] d x+\int_{-1 / 2}^{1}|x| d x$ $=0+\int_{-1 / 2}^{0}(-x) d x+\int_{0}^{1} x d x$ $=\left(-\frac{x^{2}}{2}\right)_{-1 / 2}^{0}+\left(\frac{x^{2}}{2}\right)_{0}^{1}$ $=\left(0+\frac{1}{8}\right)+\frac{1}{2}$ $=\frac{5}{8}$ $8 I=5$...
Read More →A box open from top is made from a rectangular sheet
Question: A box open from top is made from a rectangular sheet of dimension $\mathrm{a} \times \mathrm{b}$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $\mathrm{x}$ is equal to :$\frac{a+b-\sqrt{a^{2}+b^{2}-a b}}{12}$$\frac{a+b-\sqrt{a^{2}+b^{2}+a b}}{6}$$\frac{a+b-\sqrt{a^{2}+b^{2}-a b}}{6}$$\frac{a+b+\sqrt{a^{2}+b^{2}-a b}}{6}$Correct Option: , 3 Solution: $\mathrm{V}=\ell . \mathrm{b} . \mathrm{h}=(\mathr...
Read More →The line 12x cosθ + 5y sinθ = 60 is tangent
Question: The line $12 x \cos \theta+5 y \sin \theta=60$ is tangent to which of the following curves?$x^{2}+y^{2}=169$$144 x^{2}+25 y^{2}=3600$$25 x^{2}+12 y^{2}=3600$$x^{2}+y^{2}=60$Correct Option: , 2 Solution: $12 x \cos \theta+5 y \sin \theta=60$ $\frac{x \cos \theta}{5}+\frac{y \sin \theta}{12}=1$ is tangent to $\frac{x^{2}}{25}+\frac{y^{2}}{144}=1$ $144 x^{2}+25 y^{2}=3600$...
Read More →The set of all values of
Question: The set of all values of $\mathrm{k}-1$, for which the equation $\left(3 x^{2}+4 x+3\right)^{2}-(k+1)\left(3 x^{2}+4 x+3\right)$ $\left(3 x^{2}+4 x+2\right)+k\left(3 x^{2}+4 x+2\right)^{2}=0$ has real roots, is :$\left(1, \frac{5}{2}\right]$$[2,3)$$\left[-\frac{1}{2}, 1\right)$$\left(\frac{1}{2}, \frac{3}{2}\right]-\{1\}$Correct Option: 1 Solution: $\left(3 x^{2}+4 x+3\right)^{2}-(k+1)\left(3 x^{2}+4 x+3\right)\left(3 x^{2}+4 x+2\right)$ $+k\left(3 x^{2}+4 x+2\right)^{2}=0$ Let $3 x^{2...
Read More →Let A and B be independent events such that
Question: Let $\mathrm{A}$ and $\mathrm{B}$ be independent events such that $P(A)=p, P(B)=2 p .$ The largest value of $p$, for which $\mathrm{P}$ (exactly one of $\mathrm{A}, \mathrm{B}$ occurs) $=\frac{5}{9}$, is :$\frac{1}{3}$$\frac{2}{9}$$\frac{4}{9}$$\frac{5}{12}$ Correct Option: , 4 Solution: $\mathrm{P}($ Exactly one of $\mathrm{A}$ or $\mathrm{B})$ $=\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})=\frac{5}{9}$ $=\mathrm{P}(\mathrm{A}) \m...
Read More →If a, = cos2rπ/9 + isin2rπ/9, r = 1, 2, 3, ......, i
Question: If $\mathrm{a}_{\mathrm{r}}=\cos \frac{2 \mathrm{r} \pi}{9}+i \sin \frac{2 \mathrm{r} \pi}{9}, \mathrm{r}=1,2,3, \ldots, i=\sqrt{-1}$ then the determinant $\left|\begin{array}{lll}a_{1} a_{2} a_{3} \\ a_{4} a_{5} a_{6} \\ a_{7} a_{8} a_{9}\end{array}\right|$ is equal to :$a_{2} a_{6}-a_{4} a_{8}$$\mathrm{a}_{9}$$a_{1} a_{9}-a_{3} a_{7}$$\mathrm{a}_{5}$Correct Option: , 3 Solution: $a_{r}=e^{\frac{i 2 \pi r}{9}}, r=1,2,3, \ldots a_{1}, a_{2}, a_{3}, \ldots$ are in G.P. $\left|\begin{arr...
Read More →Solve the Following Questions
Question: Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3 x+2 y+5 z=3$, $9 x+4 y+(28+[\lambda]) z=[\lambda]$ has a solution is:$\mathbf{R}$$(-\infty,-9) \cup(-9, \infty)$$[-9,-8)$$(-\infty,-9) \cup[-8, \infty)$Correct Option: 1 Solution: $\mathrm{D}=\left|\begin{array}{llc}1 1 1 \\ 3 2 5 \\ 9 4 28+[\lambda]\end{array}\right|=-24-[\lambda]+15=-[\lambda]-9$ if $[\lambda]+9 \neq 0$ then u...
Read More →A vertical pole fixed to the horizontal ground is divided
Question: A vertical pole fixed to the horizontal ground is divided in the ratio $3: 7$ by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground $18 \mathrm{~m}$ away from the base of the pole, then the height of the pole (in meters) is:$12 \sqrt{15}$$12 \sqrt{10}$$8 \sqrt{10}$$6 \sqrt{10}$Correct Option: , 2 Solution: Let height of pole $=10 \ell$ $\tan \alpha=\frac{3 \ell}{18}=\frac{\ell}{6}$ $\tan 2 \alpha=\frac{10 \ell}{18}$ ...
Read More →Solve the Following Questions
Question: Let $A(a, 0), B(b, 2 b+1)$ and $C(0, b), b \neq 0,|b| \neq 1$, be points such that the area of triangle $A B C$ is 1 sq. unit, then the sum of all possible values of a is :$\frac{-2 b}{b+1}$$\frac{2 b}{b+1}$$\frac{2 b^{2}}{b+1}$$\frac{-2 b^{2}}{b+1}$Correct Option: , 4 Solution: $\Rightarrow\left|\begin{array}{ccc}\mathrm{a} 0 1 \\ \mathrm{~b} 2 \mathrm{~b}+1 1 \\ 0 \mathrm{~b} 1\end{array}\right|=\pm 2$ $\Rightarrow a(2 b+1-b)-0+1\left(b^{2}-0\right)=\pm 2$ $\Rightarrow a=\frac{\pm 2-...
Read More →Solve this
Question: Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be a solution curve of the differential equation $(y+1) \tan ^{2} x d x+\tan x d y+y d x=0$,\ $x \in\left(0, \frac{\pi}{2}\right) .$ If $\lim _{x \rightarrow 0+} x y(x)=1$, then the value of $y\left(\frac{\pi}{4}\right)$ is:$-\frac{\pi}{4}$$\frac{\pi}{4}-1$$\frac{\pi}{4}+1$$\frac{\pi}{4}$Correct Option: 4, Solution: $(y+1) \tan ^{2} x d x+\tan x d y+y d x=0$ or $\frac{d y}{d x}+\frac{\sec ^{2} x}{\tan x} \cdot y=-\tan x$ $\mathrm{IF}=\mathrm{e}^{...
Read More →Prove the following
Question: $\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$ is equal to :$\pi^{2}$$2 \pi^{2}$$4 \pi^{2}$$4 \pi$Correct Option: , 3 Solution: $\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$ $\lim _{x \rightarrow 0} \frac{1-\cos \left(2 \pi \cos ^{4} x\right)}{2 x^{4}}$ $\lim _{x \rightarrow 0} \frac{1-\cos \left(2 \pi-2 \pi \cos ^{4} x\right)}{\left[2 \pi\left(1-\cos ^{4} x\right)\right]^{2}} 4 \pi^{2} \cdot \frac{\sin ^{4} x}{2 x^{4}}\lef...
Read More →If the solution curve of the differential equation
Question: If the solution curve of the differential equation $\left(2 x-10 y^{3}\right) d y+y d x=0$, passes through the points $(0,1)$ and $(2, \beta)$, then $\beta$ is a root of the equation:$y^{5}-2 y-2=0$$2 y^{5}-2 y-1=0$$2 y^{5}-y^{2}-2=0$$y^{5}-y^{2}-1=0$Correct Option: , 4 Solution: $\left(2 x-10 y^{3}\right) d y+y d x=0$ $\Rightarrow \frac{\mathrm{dx}}{\mathrm{dy}}+\left(\frac{2}{\mathrm{y}}\right) \mathrm{x}=10 \mathrm{y}^{2}$ I. F. $=e^{\int_{y}^{2} d y}=e^{2 \ln (y)}=y^{2}$ Solution o...
Read More →if dy/dx = 2x+y - 2x/2y, y(0) = 1,
Question: If $\frac{d y}{d x}=\frac{2^{x+y}-2^{x}}{2^{y}}, y(0)=1$, then $y(1)$ is equal to :$\log _{2}(2+\mathrm{e})$$\log _{2}(1+\mathrm{e})$$\log _{2}(2 \mathrm{e})$$\log _{2}\left(1+\mathrm{e}^{2}\right)$Correct Option: , 2 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{\mathrm{x}} 2^{\mathrm{y}}-2^{\mathrm{x}}}{2^{\mathrm{y}}}$ $2^{y} \frac{d y}{d x}=2^{x}\left(2^{y}-1\right)$ $\int \frac{2^{y}}{2^{y}-1} d y=\int 2^{x} d x$ $\frac{\ln \left(2^{y}-1\right)}{\ln 2}=\frac{2^{x}}{\ln 2}+C$...
Read More →The equation of the plane passing
Question: The equation of the plane passing through the line of intersection of the planes $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=1$ and $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})+4=0$ and parallel to the $x$-axis is:$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{j}}-3 \hat{\mathrm{k}})+6=0$$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+3 \hat{\mathrm{k}})+6=0$$\overrightarrow{\mathrm{r}} \cdot(...
Read More →If the function f (x) = { 1/x log e (1+x/a / 1-x/a) , x< 0
Question: is continuous at $x=0$, then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :$-5$5$-4$4Correct Option: 1 Solution: If $f(x)$ is continuous at $x=0, R H L=L H L=f(0)$ $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} \frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1} \cdot \frac{\sqrt{x^{2}+1}+1}{\sqrt{x^{2}+1}+1}$ (Rationalisation) $\lim _{x \rightarrow 0^{+}}-\frac{2 \sin ^{2} x}{x^{2}} \cdot\left(\sqrt{x^{2}+1}+1\right)=-4$ $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \r...
Read More →Solve this
Question: On the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$ let P be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x+2 y=0 .$ Let $S$ and $S^{\prime}$ be the foci of the ellipse and e be its eccentricity. If $\mathrm{A}$ is the area of the triangle SPS' then, the value of $\left(5-\mathrm{e}^{2}\right) . \mathrm{A}$ is :6121424Correct Option: 1, Solution: Equation of tangent : $\mathrm{y}=2 \mathrm{x}+6$ at $\mathrm{P}$ $\therefore \mat...
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