Solve the Following Questions
Question: Let $J_{n, m}=\int_{0}^{\frac{1}{2}} \frac{x^{n}}{x^{m}-1} d x, \quad \forall nm$ and $n, m \in N$ Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ where $a_{i j}=\left\{\begin{array}{cl}J_{6+i, 3}-J_{i+3,3}, i \leq j \\ 0, ij\end{array} .\right.$ Then $\left|\operatorname{adj} \mathrm{A}^{-1}\right|$ is :$(15)^{2} \times 2^{42}$$(15)^{2} \times 2^{34}$$(105)^{2} \times 2^{38}$$(105)^{2} \times 2^{36}$Correct Option: , 3 Solution: $\left[\begin{array}{ccc}a v v \\ a_{11} a_{12} ...
Read More →Let f : R→R be defined as
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)= \begin{cases}\frac{x^{3}}{(1-\cos 2 x)^{2}} \log _{e}\left(\frac{1+2 x e^{-2 x}}{\left(1-x e^{-x}\right)^{2}}\right) , \quad x \neq 0 \\ \alpha , \quad x=0\end{cases}$ If $f$ is continuous at $\mathrm{x}=0$, then $\alpha$ is equal to :1302Correct Option: 1 Solution: For continuity $\lim _{x \rightarrow 0} \frac{x^{3}}{4 \sin ^{4} x}\left(\ell n\left(1+2 x e^{-2 x}\right)-2 \ell n\left(1-x e^{-x}\right)\right)$ $=\alpha$ $\...
Read More →The function
Question: The function $f(x)=x^{3}-6 x^{2}+a x+b$ is such that $f(2)=f(4)=0 .$ Consider two statements. $(\mathrm{S} 1)$ there exists $\mathrm{x}_{1}, \mathrm{x}_{2} \in(2,4), \mathrm{x}_{1}\mathrm{x}_{2}$, such that $f^{\prime}\left(x_{1}\right)=-1$ and $f^{\prime}\left(x_{2}\right)=0 .$ $(\mathrm{S} 2)$ there exists $\mathrm{x}_{3}, \mathrm{x}_{4} \in(2,4), \mathrm{x}_{3}\mathrm{x}_{4}$, such that $f$ is decreasing in $\left(2, x_{4}\right)$, increasing in $\left(x_{4}, 4\right)$ and $2 f^{\pr...
Read More →If the area of the bounded region
Question: If the area of the bounded region $R=\left\{(x, y): \max \left\{0, \log _{e} x\right\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}$ is, $\alpha\left(\log _{\mathrm{e}} 2\right)^{-1}+\beta\left(\log _{\mathrm{e}} 2\right)+\gamma$, then the value of $(\alpha+\beta-2 \gamma)^{2}$ is equal to :8241Correct Option: , 2 Solution: $R=\left\{(x, y): \max \left(0, \log _{e} x\right) \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}$ $\int_{\frac{1}{2}}^{2} 2^{x} d x-\int_{1}^{2} \ell...
Read More →There are 5 students in class 10,6 students in class 11 and 8
Question: There are 5 students in class 10,6 students in class 11 and 8 students in class 12 . If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is $100 \mathrm{k}$, then $\mathrm{k}$ is equal to_______. Solution: ' $\Rightarrow$ Total number of ways $=23800$ According to question $100 \mathrm{~K}=23800$ $\Rightarrow \mathrm{K}=238$...
Read More →If the domain of the function
Question: If the domain of the function $f(x)=\frac{\cos ^{-1} \sqrt{x^{2}-x+1}}{\sqrt{\sin ^{-1}\left(\frac{2 x-1}{2}\right)}}$ is the interval $(\alpha, \beta]$, then $\alpha+\beta$ is equal to :$\frac{3}{2}$2$\frac{1}{2}$1Correct Option: 1 Solution: $O \leq x^{2}-x+1 \leq 1$ $\Rightarrow x^{2}-x \leq 0$ $\Rightarrow x \in[0,1]$ Also, $0\sin ^{-1}\left(\frac{2 x-1}{2}\right) \leq \frac{\pi}{2}$ $\Rightarrow 0\frac{2 x-1}{2} \leq 1$ $\Rightarrow 02 \mathrm{x}-1 \leq 2$ $12 x \leq 3$ $\frac{1}{2...
Read More →solve that
Question: Let $M=\left\{A=\left(\begin{array}{ll}a b \\ c d\end{array}\right): a, b, c, d \in\{\pm 3, \pm 2, \pm 1,0\}\right\}$ Define $f: M \rightarrow Z$, as $f(A)=\operatorname{det}(A)$, for all $A \in M$, where $\mathbf{Z}$ is set of all integers. Then the number of $A \in M$ such that $f(A)=15$ is equal to_________. Solution: $|A|=a d-b c=15$ where $a, b, c, d \in\{\pm 3, \pm 2, \pm 1,0\}$ Case I ad $=9 \ \mathrm{bc}=-6$ For ad possible pairs are $(3,3),(-3,-3)$ For bc possible pairs are $(...
Read More →Let C be the set of all complex numbers.
Question: Let C be the set of all complex numbers. Let $\mathrm{S}_{1}=\left\{\mathrm{z} \in \mathrm{C}|| \mathrm{z}-3-\left.2 \mathrm{i}\right|^{2}=8\right\}$ $\mathrm{S}_{2}=\{\mathrm{z} \in \mathrm{C} \mid \operatorname{Re}(\mathrm{z}) \geq 5\}$ and $\mathrm{S}_{3}=\{\mathrm{z} \in \mathrm{C}|| \mathrm{z}-\overline{\mathrm{z}} \mid \geq 8\}$ Then the number of elements in $S_{1} \cap S_{2} \cap S_{3}$ is equal to102InfiniteCorrect Option: 1 Solution: $\mathrm{S}_{1}:|\mathrm{z}-3-2 \mathrm{i}...
Read More →If n is the number of solutions
Question: If $\mathrm{n}$ is the number of solutions of the equation $2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]$ and $S$ is the sum of all these solutions, then the ordered pair $(\mathrm{n}, \mathrm{S})$ is :$(3,13 \pi / 9)$$(2,2 \pi / 3)$$(2,8 \pi / 9)$$(3,5 \pi / 3)$Correct Option: 1 Solution: $2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1$ $2 \cos x\left(4\left(\sin ^{2} \frac...
Read More →The value of the definite integral
Question: The value of the definite integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{\left(1+e^{x \cos x}\right)\left(\sin ^{4} x+\cos ^{4} x\right)}$ is equal to :$-\frac{\pi}{2}$$\frac{\pi}{2 \sqrt{2}}$$-\frac{\pi}{4}$$\frac{\pi}{\sqrt{2}}$Correct Option: , 2 Solution: $\mathrm{I}=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\mathrm{dx}}{\left(1+\mathrm{e}^{\mathrm{x} \cos \mathrm{x}}\right)\left(\sin ^{4} \mathrm{x}+\cos ^{4} \mathrm{x}\right)}$ .....(1) Using $\int_{a}^{b} f(x) d x=...
Read More →The number of solutions of
Question: The number of solutions of $\sin ^{7} x+\cos ^{7} x=1$, $x \in[0,4 \pi]$ is equal to11759Correct Option: , 3 Solution: $\sin ^{7} x \leq \sin ^{2} x \leq 1$ ..........(1) and $\cos ^{7} x \leq \cos ^{2} x \leq 1$ ...........(2) also $\sin ^{2} x+\cos ^{2} x=1$ $\Rightarrow$ equality must hold for (1) \ (2) $\Rightarrow \sin ^{7} x=\sin ^{2} x \ \cos ^{7}=\cos ^{2} x$ $\Rightarrow \sin x=0 \ \cos x=1$ or $\cos x=0 \ \sin x=1$ $\Rightarrow x=0,2 \pi, 4 \pi, \frac{\pi}{2}, \frac{5 \pi}{2}...
Read More →The ratio of the coefficient of the middle term in the expansion
Question: The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is________. Solution: Coeff. of middle term in $(1+x)^{20}={ }^{20} C_{10}$ \ Sum of Coeff. of two middle terms in $(1+x)^{19}={ }^{19} C_{9}+{ }^{19} C_{10}$ So required ratio $=\frac{{ }^{20} \mathrm{C}_{10}}{{ }^{19} \mathrm{C}_{9}+{ }^{19} \mathrm{C}_{10}}=\frac{{ }^{20} \mathrm{C}_{10}}{{ }^{20} \mathrm{C}_{10}}=1$...
Read More →If y=y(x) is the solution curve
Question: If $y=y(x)$ is the solution curve of the differential equation $x^{2} d y+\left(y-\frac{1}{x}\right) d x=0 \quad ; x0$ and $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to :$\frac{3}{2}-\frac{1}{\sqrt{\mathrm{e}}}$$3+\frac{1}{\sqrt{\mathrm{e}}}$$3+\mathrm{e}$$3-e$Correct Option: , 4 Solution: $x^{2} d y+\left(y-\frac{1}{x}\right) d x=0: x0, y(1)=1$ $x^{2} d y+\frac{(x y-1)}{x} d x=0$ $x^{2} d y=\frac{(x y-1)}{x} d x$ $\frac{d y}{d x}=\frac{1-x y}{x^{3}}$ $\frac{d y}{d x}=\frac{1}...
Read More →Prove the following
Question: Let $\overrightarrow{\mathrm{p}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{q}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ be two vectors. If a vector $\overrightarrow{\mathrm{r}}=(\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}})$ is perpendicular to each of the vectors $(\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}})$ and $(\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}})$, and $|\overrightar...
Read More →Solve this
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. Then the vector product $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \times((\overrightarrow{\mathrm{a}} \times((\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{b}})) \times \overrightarrow{\mathrm{b}})$ is equal to:$5(34 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+...
Read More →Two squares are chosen
Question: Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is. $\frac{2}{7}$$\frac{1}{18}$$\frac{1}{7}$$\frac{1}{9}$Correct Option: , 2 Solution: Total ways of choosing square $={ }^{64} \mathrm{C}_{2}$ $=\frac{64 \times 63}{2 \times 1}=32 \times 63$ ways of choosing two squares having common side $=2(7 \times 8)=112$ Required probability $=\frac{112}{32 \times 63}=\frac{16}{32 \times 9}=\frac{1}{18}$...
Read More →Which of the following
Question: Which of the following is equivalent to the Boolean expression $p \wedge \sim q$ ?$\sim(\mathrm{q} \rightarrow \mathrm{p})$$\sim \mathrm{p} \rightarrow \sim \mathrm{q}$$\sim(\mathrm{p} \rightarrow \sim \mathrm{q})$$\sim(\mathrm{p} \rightarrow \mathrm{q})$Correct Option: , 4 Solution: $\mathrm{p} \wedge \sim \mathrm{q} \equiv \sim(\mathrm{p} \rightarrow \mathrm{q})$ Option (4)...
Read More →Consider the following frequency distribution :
Question: Consider the following frequency distribution : If the sum of all frequencies is 584 and median is 45 , then $|\alpha-\beta|$ is equal to__________. Solution: $\because$ Sum of frequencies $=584$ $\Rightarrow \alpha+\beta=390$ Now, Median is at $\frac{584}{2}=292^{\text {th }}$ $\because$ Median $=45($ lies in class $40-50)$ $\Rightarrow \alpha+110+54+15=292$ $\Rightarrow \alpha=113, \beta=277$ $\Rightarrow|\alpha-\beta|=164$...
Read More →Let n denote the number of solutions of the equation
Question: Let $n$ denote the number of solutions of the equation $\mathrm{z}^{2}+3 \overline{\mathrm{Z}}=0$, where $\mathrm{z}$ is a complex number. Then the value of $\sum_{\mathrm{k}=0}^{\infty} \frac{1}{\mathrm{n}^{\mathrm{k}}}$ is equal to1$\frac{4}{3}$$\frac{3}{2}$2Correct Option: , 2 Solution: $\mathrm{z}^{2}+3 \overline{\mathrm{z}}=0$ Put $z=x+i y$ $\Rightarrow x^{2}-y^{2}+2 i x y+3(x-i y)=0$ $\Rightarrow\left(x^{2}-y^{2}+3 x\right)+i(2 x y-3 y)=0+i 0$ $\therefore \quad x^{2}-y^{2}+3 x=0$...
Read More →Let the acute angle bisector
Question: Let the acute angle bisector of the two planes $x-2 y-2 z+1=0$ and $2 x-3 y-6 z+1=0$ be the plane P. Then which of the following points lies on P?$\left(3,1,-\frac{1}{2}\right)$$\left(-2,0,-\frac{1}{2}\right)$$(0,2,-4)$$(4,0,-2)$Correct Option: , 2 Solution: $P_{1}: x-2 y-2 z+1=0$ $P_{2}: 2 x-3 y-6 z+1=0$ $\left|\frac{x-2 y-2 z+1}{\sqrt{1+4+4}}\right|=\left|\frac{2 x-3 y-6 z+1}{\sqrt{2^{2}+3^{2}+6^{2}}}\right|$ $\frac{x-2 y-2 z+1}{3}=\pm \frac{2 x-3 y-6 z+1}{7}$ Since $a_{1} a_{2}+b_{1...
Read More →If the value of
Question: If the value of $\left(1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots \ldots \text { upto } \infty\right)^{\log _{(0.25)}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \ldots \text { upto } \infty\right)}$ is $l$, then $l^{2}$ is equal to_________. Solution: $\ell=(\underbrace{1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}}_{\mathrm{s}}+\ldots)^{\log _{025}\left(\frac{1}{3}+\frac{1}{3^{2}}+\ldots\right)}$ $\mathrm{S}=1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldot...
Read More →Solve this
Question: The value of $\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}} \sum_{\mathrm{j}=1}^{\mathrm{n}} \frac{(2 \mathrm{j}-1)+8 \mathrm{n}}{(2 \mathrm{j}-1)+4 \mathrm{n}}$ is equal to :$5+\log _{e}\left(\frac{3}{2}\right)$$2-\log _{e}\left(\frac{2}{2}\right)$$3+2 \log _{e}\left(\frac{2}{3}\right)$$1+2 \log _{e}\left(\frac{3}{2}\right)$Correct Option: , 4 Solution: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{n} \frac{\left(\frac{2 j}{n}-\frac{1}{n}+8\right)}{\left(\frac{2 j...
Read More →Consider the system of linear equations
Question: Consider the system of linear equations $-x+y+2 z=0$ $3 x-a y+5 z=1$ $2 x-2 y-a z=7$ Let $S_{1}$ be the set of all a $\in \mathbf{R}$ for which the system is inconsistent and $\mathrm{S}_{2}$ be the set of all $\mathrm{a} \in \mathbf{R}$ for which the system has infinitely many solutions. If $n\left(S_{1}\right)$ and $n\left(S_{2}\right)$ denote the number of elements in $S_{1}$ and $\mathrm{S}_{2}$ respectively, then$n\left(S_{1}\right)=2, n\left(S_{2}\right)=2$$\mathrm{n}\left(\mathr...
Read More →Let the circle
Question: Let the circle $S: 36 x^{2}+36 y^{2}-108 x+120 y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2 y=4$ and $2 x-y=5$ lies inside the circle $\mathrm{S}$, then :$\frac{25}{9}C\frac{13}{3}$$100\mathrm{C}165$$81C156$$100\mathrm{C}156$Correct Option: , 4 Solution: $S: 36 x^{2}+36 y^{2}-108 x+120 y+C=0$ $\Rightarrow x^{2}+y^{2}-3 x+\frac{10}{3} y+\frac{C}{36}=0$ Centre $\equiv(-g,-f) \equiv\left(\frac{3}{2}, \frac{-10}...
Read More →Let y = y(x) be solution of the following
Question: Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be solution of the following differential equation $e^{y} \frac{d y}{d x}-2 e^{y} \sin x+\sin x \cos ^{2} x=0, y\left(\frac{\pi}{2}\right)=0$ If $y(0)=\log _{e}\left(\alpha+\beta e^{-2}\right)$, then $4(\alpha+\beta)$ is equal to_______. Solution: $\operatorname{Let} e^{y}=t$ $\Rightarrow \frac{\mathrm{dt}}{\mathrm{dx}}-(2 \sin \mathrm{x}) \mathrm{t}=-\sin x \cos ^{2} \mathrm{x}$ I.F. $=e^{2 \cos x}$ $\Rightarrow \mathrm{t} \cdot \mathrm{e}^{2 \c...
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