Solve the Following Questions
Question: Let $S=\{1,2,3,4,5,6,9\} .$ Then the number of elements in the set $\mathrm{T}=\{\mathrm{A} \subseteq \mathrm{S}: \mathrm{A} \neq \phi$ and the sum of all the elements of $\mathrm{A}$ is not a multiple of 3$\}$ is Solution: $3 \mathrm{n}$ type $\rightarrow 3,6,9=\mathrm{P}$ $3 \mathrm{n}-1$ type $\rightarrow 2,5=\mathrm{Q}$ $3 \mathrm{n}-2$ type $\rightarrow 1,4=\mathrm{R}$ number of subset of $\mathrm{S}$ containing one element which are not divisible by $3={ }^{2} \mathrm{C}_{1}+{ }^...
Read More →Let S={1,2,3,4,5,6}. Then the probability that
Question: Let $S=\{1,2,3,4,5,6\} .$ Then the probability that a randomly chosen onto function g from $S$ to $S$ satisfies $g(3)=2 g(1)$ is :$\frac{1}{10}$$\frac{1}{15}$$\frac{1}{5}$$\frac{1}{30}$Correct Option: 1 Solution: $\mathrm{g}(3)=2 \mathrm{~g}(1)$ can be defined in 3 ways number of onto functions in this condition $=3 \times 4 !$ Total number of onto functions $=6 !$ Required probability $=\frac{3 \times 4 !}{6 !}=\frac{1}{10}$...
Read More →If a line along a chord of the circle
Question: If a line along a chord of the circle $4 x^{2}+4 y^{2}+120 x+675=0$, passes through the point $(-30,0)$ and is tangent to the parabola $y^{2}=30 x$, then the length of this chord is :57$5 \sqrt{3}$$3 \sqrt{5}$Correct Option: , 4 Solution: Equation of tangent to $y^{2}=30 x$ $y=m x+\frac{30}{4 m}$ Pass thru $(-30,0): \mathrm{a}=-30 \mathrm{~m}+\frac{30}{4 \mathrm{~m}} \Rightarrow \mathrm{m}^{2}=1 / 4$ $\Rightarrow \mathrm{m}=\frac{1}{2}$ or $\mathrm{m}=-\frac{1}{2}$ At $m=\frac{1}{2}: y...
Read More →The domain of the function
Question: The domain of the function $f(x)=\sin ^{-1}\left(\frac{3 x^{2}+x-1}{(x-1)^{2}}\right)+\cos ^{-1}\left(\frac{x-1}{x+1}\right)$ is :$\left[0, \frac{1}{4}\right]$$[-2,0] \cup\left[\frac{1}{4}, \frac{1}{2}\right]$$\left[\frac{1}{4}, \frac{1}{2}\right] \cup\{0\}$$\left[0, \frac{1}{2}\right]$Correct Option: , 3 Solution: $f(x)=\sin ^{-1}\left(\frac{3 x^{2}+x-1}{(x-1)^{2}}\right)+\cos ^{-1}\left(\frac{x-1}{x+1}\right)$ $-1 \leq \frac{x-1}{x+1} \leq 1 \Rightarrow 0 \leq x\infty$........(1) $-1...
Read More →Prove the following
Question: Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\overrightarrow{\mathrm{r}}$ satisfies. $\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\ov...
Read More →Solve the Following Questions
Question: Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\arg \left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)=\frac{\pi}{4}$ and $\mathrm{z}_{1}, \mathrm{z}_{2}$ satisfy the equation $|z-3|=\operatorname{Re}(z)$. Then the imaginary part of $z_{1}+z_{2}$ is equal to Solution: $|z-3|=\operatorname{Re}(z)$ let $Z=x=$ iy $\Rightarrow(x-3)^{2}+y^{2}=x^{2}$ $\Rightarrow x^{2}+9-6 x+y^{2}=x^{2}$ $\Rightarrow y^{2}=6 x-9$ $\Rightarrow y^{2}=6\left(x-\frac{3}{2}\right)$ $\Rightarrow z_{1}$ and $z_{...
Read More →If α + γ = 2π, then the system of equations
Question: If $\alpha+\beta+\gamma=2 \pi$, then the system of equations $x+(\cos \gamma) y+(\cos \beta) z=0$ $(\cos \gamma) x+y+(\cos \alpha) z=0$ $(\cos \beta) x+(\cos \alpha) y+z=0$ has :no solutioninfinitely many solutionexactly two solutionsa unique solutionCorrect Option: , 2 Solution: $\alpha+\beta+\gamma=2 \pi$ $\left|\begin{array}{ccc}1 \cos \gamma \cos \beta \\ \cos \gamma 1 \cos \alpha \\ \cos \beta \cos \alpha 1\end{array}\right|$ $=1+2 \cos \alpha \cdot \cos \beta \cdot \cos \gamma-\c...
Read More →Solve this
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}}$ is a vectorsuch that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=3$, then $\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$ is equal to :...
Read More →The probability distribution of random
Question: The probability distribution of random variable $\mathrm{X}$ is given by: Let $\mathrm{p}=\mathrm{P}(1\mathrm{X}4 \mid \mathrm{X}3) .$ If $5 \mathrm{p}=\lambda \mathrm{K}$, then $\lambda$ equal to Solution: $\sum \mathrm{P}(\mathrm{X})=1 \Rightarrow \mathrm{k}+2 \mathrm{k}+2 \mathrm{k}+3 \mathrm{k}+\mathrm{k}=1$ $\Rightarrow \mathrm{k}=\frac{1}{9}$ Now, $\mathrm{p}=\mathrm{P}\left(\frac{\mathrm{k} \mathrm{X}4}{\mathrm{X}3}\right)=\frac{\mathrm{P}(\mathrm{X}=2)}{\mathrm{P}(\mathrm{X}3)}...
Read More →Let S be the mirror image of the point
Question: Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2 \mathrm{x}-\mathrm{y}+\mathrm{z}+3=0$ and let $\mathrm{R}(3,5, \gamma)$ be a point of this plane. Then the square of the length of the line segment SR is Solution: Since $\mathrm{R}(3,5, \gamma)$ lies on the plane $2 \mathrm{x}-\mathrm{y}+\mathrm{z}+3=0$ Therefore, $6-5+\gamma+3=0$ $\Rightarrow \gamma=-4$ Now, dr's of line QS are $2,-1,1$ equation of line $\mathrm{QS}$ is $\frac{x-1}{2}=\frac{y-3}{-1}=\fra...
Read More →Let S be the sum of all solutions
Question: Let $S$ be the sum of all solutions (in radians) of the equation $\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0$ in $[0,4 \pi]$. Then $\frac{8 \mathrm{~S}}{\pi}$ is equal to Solution: Given equation $\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0$ $\Rightarrow 1-\sin ^{2} \theta \cos ^{2} \theta-\sin \theta \cos \theta=0$ $\Rightarrow 2-(\sin 2 \theta)^{2}-\sin 2 \theta=0$ $\Rightarrow(\sin 2 \theta)^{2}+(\sin 2 \theta)-2=0$ $\Rightarrow(\sin 2 \theta+2)(\sin...
Read More →Solve this
Question: The equation arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with:(1) centre at $(0,-1)$ and radius $\sqrt{2}$(2) centre at $(0,1)$ and radius $\sqrt{2}$(3) centre at $(0,0)$ and radius $\sqrt{2}$(4) centre at $(0,1)$ and radius 2Correct Option: , 2 Solution: In $\triangle \mathrm{OAC}$ $\sin \left(\frac{\pi}{4}\right)=\frac{1}{\mathrm{AC}}$ $\Rightarrow \mathrm{AC}=\sqrt{2}$ Also, $\tan \frac{\pi}{4}=\frac{\mathrm{OA}}{\mathrm{OC}}=\frac{1}{\mathrm{OC}}$ $\Rightar...
Read More →Solve the Following Questions
Question: If $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-x+1}-a x\right)=b$, then the ordered pair $(\mathrm{a}, \mathrm{b})$ is:$\left(1, \frac{1}{2}\right)$$\left(1,-\frac{1}{2}\right)$$\left(-1, \frac{1}{2}\right)$$\left(-1,-\frac{1}{2}\right)$Correct Option: , 2 Solution: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-x+1}\right)-a x=b \quad(\infty-\infty)$ $\Rightarrow a0$ Now, $\lim _{x \rightarrow \infty} \frac{\left(x^{2}-x+1-a^{2} x^{2}\right)}{\sqrt{x^{2}-x+1}+a x}=b$ $\Rightarrow \...
Read More →An electric instrument consists of two units.
Question: An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is $0.9$ and that of the second unit is $0.8$. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $\mathrm{p}$, then $98 \mathrm{p}$ is equal to Solution: $\mathrm{I}_{1}=$ first unit is functioning $\mathrm{I}_{2}=$ second unit is function...
Read More →The value of the integral
Question: The value of the integral $\int_{0}^{1} \frac{\sqrt{x} d x}{(1+x)(1+3 x)(3+x)}$ is:$\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{2}\right)$$\frac{\pi}{4}\left(1-\frac{\sqrt{3}}{6}\right)$$\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{6}\right)$$\frac{\pi}{4}\left(1-\frac{\sqrt{3}}{2}\right)$Correct Option: 1 Solution: $\mathrm{I}=\int_{0}^{1} \frac{\sqrt{\mathrm{x}}}{(1+\mathrm{x})(1+3 \mathrm{x})(3+\mathrm{x})} \mathrm{d} \mathrm{x}$ Let $x=t^{2} \Rightarrow d x=2 t \cdot d t$ $I=\int_{0}^{1} \frac{t(...
Read More →Solve the following equations:
Question: If $\left(\frac{3^{6}}{4^{4}}\right) \mathrm{k}$ is the term, independent of $\mathrm{x}$, in the binomial expansion of $\left(\frac{\mathrm{x}}{4}-\frac{12}{\mathrm{x}^{2}}\right)^{12}$, then $\mathrm{k}$ is equal to______. Solution: $\left(\frac{x}{4}-\frac{12}{x^{2}}\right)^{12}$ $\mathrm{T}_{\mathrm{r}+1}=(-1)^{\mathrm{r}} \cdot{ }^{12} \mathrm{C}_{\mathrm{r}}\left(\frac{\mathrm{x}}{4}\right)^{12-\mathrm{r}}\left(\frac{12}{\mathrm{x}^{2}}\right)^{\mathrm{r}}$ $\mathrm{T}_{\mathrm{r...
Read More →Out of all the patients in a hospital 89 % are found to be suffering from heart ailment and 98 % are suffering from lungs infection. If K % of them are suffering from both ailments, then K can not belong to the set :
Question: Out of all the patients in a hospital $89 \%$ are found to be suffering from heart ailment and $98 \%$ are suffering from lungs infection. If $\mathrm{K} \%$ of them are suffering from both ailments, then $K$ can not belong to the set :$\{80,83,86,89\}$$\{84,86,88,90\}$$\{79,81,83,85\}$$\{84,87,90,93\}$Correct Option: , 3 Solution: $n(A \cup B) \geq n(A)+n(B)-n(A \cap B)$ $100 \geq 89+98-n(A \cup B)$ $n(A \cup B) \geq 87$ $87 \leq \mathrm{n}(\mathrm{A} \cup \mathrm{B}) \leq 89$ Option ...
Read More →Solve the Following Questions
Question: If $0x1$ and $y=\frac{1}{2} x^{2}+\frac{2}{3} x^{3}+\frac{3}{4} x^{4}+\ldots$, then the value of $e^{1+y}$ at $x=\frac{1}{2}$ is:$\frac{1}{2} \mathrm{e}^{2}$$2 \mathrm{e}$$\frac{1}{2} \sqrt{\mathrm{e}}$$2 \mathrm{e}^{2}$Correct Option: 1 Solution: $y=\left(1-\frac{1}{2}\right) x^{2}+\left(1-\frac{1}{3}\right) x^{3}+\ldots .$ $=\left(x^{2}+x^{3}+x^{4}+\ldots \ldots\right)-\left(\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}+\ldots .\right)$ $=\frac{x^{2}}{1-x}+x-\left(x+\frac{x^{2}}{2}...
Read More →Prove the following
Question: If $x \phi(x)=\int_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x-2$, and $\phi(0)=4$ then $\phi(2)$ is Solution: $x \phi(x)=\int_{5}^{x} 3 t^{2}-2 \phi^{\prime}(t) d t$ $x \phi(x)=x^{3}-125-2[\phi(x)-\phi(5)]$ $x \phi(x)=x^{3}-125-2 \phi(x)-2 \phi(5)$ $\phi(0)=4 \Rightarrow \phi(5)=-\frac{133}{2}$ $\phi(x)=\frac{x^{3}+8}{x+2}$ $\phi(2)=4$...
Read More →The number of six letter words (with or without meaning),
Question: The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is_________. Solution: All Consonants should not be together $=$ Total $-$ All consonants together $=6 !-3 ! 4 !=576$...
Read More →Two poles, AB of length
Question: Two poles, $\mathrm{AB}$ of length a metres and $\mathrm{CD}$ of length $a+b(b \neq a)$ metres are erected at the same horizontal level with bases at $\mathrm{B}$ and $\mathrm{D}$. If $\mathrm{BD}=\mathrm{x}$ and $\tan \left\lfloor\mathrm{ACB}=\frac{1}{2}\right.$, then:$x^{2}+2(a+2 b) x-b(a+b)=0$$x^{2}+2(a+2 b) x+a(a+b)=0$$x^{2}-2 a x+b(a+b)=0$$x^{2}-2 a x+a(a+b)=0$Correct Option: , 3 Solution: $\tan \theta=\frac{1}{2}$ $\tan (\theta+\alpha)=\frac{\mathrm{X}}{\mathrm{b}}, \tan \alpha=\...
Read More →If the variable line 3 x+4 y= α
Question: If the variable line $3 x+4 y=\alpha$ lies between the two $\operatorname{circles}(x-1)^{2}+(y-1)^{2}=1$ and $(x-9)^{2}+(y-1)^{2}=4$ without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is_______. Solution: Both centres should lie on either side of the line as well as line can be tangent to circle. $(3+4-\alpha) \cdot(27+4-\alpha)0$ $(7-\alpha) \cdot(31-\alpha)0 \Rightarrow \alpha \in(7,31) \ldots(1)$ $\mathrm{d}_{1}=$ distance of $(1,1)$ f...
Read More →if solve this problem
Question: If ${ }^{20} \mathrm{C}$ is the co-efficient of $\mathrm{x}^{\varepsilon}$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20} r^{2}{ }^{20} C_{r}$ is equal to :$420 \times 2^{19}$$380 \times 2^{19}$$380 \times 2^{18}$$420 \times 2^{18}$Correct Option: , 4 Solution: $\sum_{r=0}^{20} r^{2}{ }^{20} C_{r}$ $\sum(4(\mathrm{r}-1)+\mathrm{r}){ }^{20} \mathrm{C}_{\mathrm{r}}$ $\sum r(r-1) \cdot \frac{20 \times 19}{r(r-1)} \cdot{ }^{18} C_{r}+r \cdot \frac{20}{r} \cdot \sum{ }...
Read More →Solve the Following Questions
Question: If $y(x)=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right), x \in\left(\frac{\pi}{2}, \pi\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{5 \pi}{6}$ is:$-\frac{1}{2}$$-1$$\frac{1}{2}$0Correct Option: 1 Solution: $y(x)=\cot ^{-1}\left[\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\sin \frac{x}{2}+\cos \frac{x}{2}}\right]$ $y(x)=\cot ^{-1}\left(\tan \frac{x}{2}...
Read More →The area of the region bounded
Question: The area of the region bounded by the parabola $(y-2)^{2}=(x-1)$, the tangent to it at the point whose ordinate is 3 and the $x$-axis is :91046Correct Option: 1 Solution: $y=3 \Rightarrow x=2$ Point is $(2,3)$ Diff. w.r.t $\quad x$ $2(y-2) y^{\prime}=1$ $\Rightarrow y^{\prime}=\frac{1}{2(y-2)}$ $\Rightarrow \mathrm{y}_{(2,3)}^{\prime}=\frac{1}{2}$ $\Rightarrow \frac{y-3}{x-2}=\frac{1}{2} \Rightarrow x-2 y+4=0$ Area $=\int_{0}^{3}\left((y-2)^{2}+1-(2 y-4)\right) d y$ $=9 \mathrm{sq} .$ ...
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