The value of
Question: The value of $\cot \frac{\pi}{24}$ is:$\sqrt{2}+\sqrt{3}+2-\sqrt{6}$$\sqrt{2}+\sqrt{3}+2+\sqrt{6}$$\sqrt{2}-\sqrt{3}-2+\sqrt{6}$$3 \sqrt{2}-\sqrt{3}-\sqrt{6}$Correct Option: , 2 Solution: $\cot \theta=\frac{1+\cos 2 \theta}{\sin 2 \theta}=\frac{1+\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)}{\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)}$ $\theta=\frac{\pi}{24}$ $\Rightarrow \cot \left(\frac{\pi}{24}\right)=\frac{1+\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)}{\left(\frac{\sqrt{3}-1}{2 \sqrt{...
Read More →A man starts walking from
Question: A man starts walking from the point $\mathrm{P}(-3,4)$, touches the $\mathrm{x}$-axis at $\mathrm{R}$, and then turns to reach at the point $Q(0,2)$. The man is walking at a constant speed. If the man reaches the point $Q$ in the minimum time, then $50\left((\mathrm{PR})^{2}+(\mathrm{RQ})^{2}\right)$ is equal to Solution: $50\left(\mathrm{PR}^{2}+\mathrm{RQ}^{2}\right)$ $50(20+5)$ $50(25)$ $=1250$...
Read More →Two tangents are drawn from the point
Question: Two tangents are drawn from the point $\mathrm{P}(-1,1)$ to the circle $x^{2}+y^{2}-2 x-6 y+6=0$. If these tangents touch the circle at points $A$ and $B$, and if $D$ is a point on the circle such that length of the segments $\mathrm{AB}$ and $\mathrm{AD}$ are equal, then the area of the triangle ABD is equal to:2$(3 \sqrt{2}+2)$4$3(\sqrt{2}-1)$Correct Option: , 3 Solution: $\triangle \mathrm{ABD}=\frac{1}{2} \times 2 \times 4$ $=4$...
Read More →Let [t] denote the greatest integer
Question: Let $[t]$ denote the greatest integer $\leq t$. The number of points where the function $$ f(x)=[x]\left|x^{2}-1\right|+\sin \left(\frac{\pi}{[x]+3}\right)-[x+1], x \in(-2,2) $$ is not continuous is Solution: $f(x)=[x]\left|x^{2}-1\right|+\sin \frac{\pi}{[x+3]}-[x+1]$ $f(x)=\left\{\begin{array}{cc}3-2 x^{2}, -2x-1 \\ x^{2}, -1 \leq x0 \\ \frac{\sqrt{3}}{2}+1 0 \leq x1 \\ x^{2}+1+\frac{1}{\sqrt{2}}, 1 \leq x2\end{array}\right.$ discontinuous at $x=0,1$...
Read More →Solve the Following Questions
Question: Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$. Let a vector $\overrightarrow{\mathrm{v}}$ be in the plane containing $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$. If $\overrightarrow{\mathrm{v}}$ is perpendicular to the vector $3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and its projection on $\vec{a}$ is 19 units, then $|2 \ve...
Read More →If the sum of the coefficients
Question: If the sum of the coefficients in the expansion of $(x+y)^{n}$ is 4096 , then the greatest coefficient in the expansion is Solution: $(x+y)^{n} \Rightarrow 2^{n}=4096 \quad 2^{10}=1024 \times 2$ $\Rightarrow 2^{n}=2^{12} \quad 2^{11}=2048$ $\mathrm{n}=12 \quad 2^{12}=\underline{4096}$ ${ }^{12} C_{6}=\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$ $=11 \times 3 \times 4 \times 7$ $=924$...
Read More →All the arrangements
Question: All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is Solution: FARMER (6) $\mathrm{A}, \mathrm{E}, \mathrm{F}, \mathrm{M}, \mathrm{R}, \mathrm{R}$ $\frac{\lfloor}{\lfloor 2}-4=60-24=36$ $\frac{\underline{3}}{L 2}-\lfloor 2=3-2=1$ $=1$ $=2$ $=1$ _______________...
Read More →Let f(x) be a polynomial of degree
Question: Let $f(x)$ be a polynomial of degree 3 such that $\mathrm{f}(\mathrm{k})=-\frac{2}{\mathrm{k}}$ for $\mathrm{k}=2,3,4,5 .$ Then the value of $52-10 \mathrm{f}(10)$ is equal to : Solution: $\mathrm{k} \mathrm{f}(\mathrm{k})+2=\lambda(\mathrm{x}-2)(\mathrm{x}-3)(\mathrm{x}-4)(\mathrm{x}-5) \ldots(1)$ put $x=0$ we get $\lambda=\frac{1}{60}$ Now put $\lambda$ in equation (1) $\Rightarrow \mathrm{kf}(\mathrm{k})+2=\frac{1}{60}(\mathrm{x}-2)(\mathrm{x}-3)(\mathrm{x}-4)(\mathrm{x}-5)$ Put $x=...
Read More →Let the points of intersections
Question: Let the points of intersections of the lines $x-y+1=0$, $x-2 y+3=0$ and $2 x-5 y+11=0$ are the mid points of the sides of a triangle $\mathrm{ABC}$. Then the area of the triangle $A B C$ is Solution: intersection point of give lines are $(1,2),(7,5)$, $(2,3)$ $\Delta=\frac{1}{2}\left|\begin{array}{lll}1 2 1 \\ 7 5 1 \\ 2 3 1\end{array}\right|$ $=\frac{1}{2}[1(5-3)-2(7-2)+1(21-10)]$ $=\frac{1}{2}[2-10+11]$ $\Delta \mathrm{DEF}=\frac{1}{2}(3)=\frac{3}{2}$ $\Delta \mathrm{ABC}=4 \Delta \m...
Read More →If for the complex numbers
Question: If for the complex numbers $z$ satisfying $|z-2-2 i| \leq 1$, the maximum value of $|3 i z+6|$ is attained at $\mathrm{a}+i \mathrm{~b}$, then $\mathrm{a}+\mathrm{b}$ is equal to Solution: $|z-2-2 i| \leq 1$ $|x+i y-2-2 i| \leq 1$ $|(x-2)+i(y-2)| \leq 1$ $(x-2)^{2}+(y-2)^{2} \leq 1$ $|3 i z+6|_{\max }$ at $a+i b$ |3il $\left|z+\frac{6}{3 i}\right|$ $3|z-2 i|_{\max }$ From Figure maximum distance at $3+2 \mathrm{i}$ $a+i b=3+2 i=a+b=3+2=5$ Ans....
Read More →Solve the Following Questions
Question: Let $f(x)=x^{6}+2 x^{4}+x^{3}+2 x+3, x \in \mathbf{R}$. Then the natural number $n$ for which $\lim _{x \rightarrow 1} \frac{x^{n} f(1)-f(x)}{x-1}=44$ is Solution: $f(n)=x^{6}+2 x^{4}+x^{3}+2 x+3$ $\lim _{x \rightarrow 1} \frac{x^{n} f(1)-f(x)}{x-1}=44$ $\lim _{x \rightarrow 1} \frac{9 x^{n}-\left(x^{6}+2 x^{4}+x^{3}+2 x+3\right)}{x-1}=44$ $\lim _{x \rightarrow 1} \frac{9 n x^{n-1}-\left(6 x^{5}+8 x^{3}+3 x^{2}+2\right)}{1}=44$ $\Rightarrow 9 \mathrm{n}-(19)=44$ $\Rightarrow 9 \mathrm{...
Read More →Let X be a random variable with distribution.
Question: Let $X$ be a random variable with distribution. If the mean of $X$ is $2.3$ and variance of $X$ is $\sigma^{2}$, then $100 \sigma^{2}$ is equal to : Solution: $\bar{X}=2.3$. $-a+6 b=\frac{9}{10}$..(1) $\sum P_{i}=\frac{1}{5}+a+\frac{1}{3}+\frac{1}{5}+b=1$ $a+b=\frac{4}{15}$..(2) From equation (1) and (2) $a=\frac{1}{10}, \quad b=\frac{1}{6}$ $\sigma^{2}=\Sigma p_{i} x_{i}^{2}-(\bar{X})^{2}$ $\frac{1}{5}(4)+a(1)+\frac{1}{3}(9)+\frac{1}{5}(16)+b(36)-(2.3)^{2}$ $=\frac{4}{5}+a+3+\frac{16}...
Read More →The number of elements in the set
Question: The number of elements in the set $\{n \in\{1,2,3, \ldots . .100\} \mid$ $\left.(11)^{\mathrm{n}}(10)^{\mathrm{n}}+(9)^{\mathrm{n}}\right\}$ is__________. Solution: $11^{n}10^{n}+9^{n}$ $\Rightarrow 11^{\mathrm{n}}-9^{\mathrm{n}}10^{\mathrm{n}}$ $\Rightarrow(10+1)^{\mathrm{n}}-(10-1)^{\mathrm{n}}10^{\mathrm{n}}$ $\Rightarrow\left\{{ }^{n} C_{1} \cdot 10^{n-1}+{ }^{n} C_{3} 10^{n-0}+{ }^{n} C_{5} 10^{n-5}+\cdots \cdots\right\}10^{n}$ $\Rightarrow 2 \mathrm{n} \cdot 10^{\mathrm{n}-1}+2\l...
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 $\left((x+2) e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right) d x=(x+2) d y$ $y(1)=1$. If the domain of $y=y(x)$ is an open interval $(\alpha, \beta)$, then $|\alpha+\beta|$ is equal to Solution: $\mathrm{y}+1=\mathrm{Y} \Rightarrow \mathrm{dy}=\mathrm{d} \mathrm{Y}$ $x+2=X \Rightarrow d x=d X$ $\Rightarrow\left(X e^{\frac{Y}{X}}+Y\right) d X=X d Y$ $\Rightarrow \mathrm{Xd} \mathrm{Y}-\mathrm{YdX}=\mathrm{Xe}^{\mathrm{Y} / \mathrm{...
Read More →Let the plane passing through the point $(-1,0,-2)$
Question: Let the plane passing through the point $(-1,0,-2)$ and perpendicular to each of the planes $2 x+y-z=2$ and $x-y-z=3$ be $a x+b y+c z+8=0$. Then the value of $a+b+c$ is equal to:3854Correct Option: , 4 Solution: Normal of req. plane $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}) \times(\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})$ $=-2 \hat{i}+\hat{j}-3 \hat{k}$ Equation of plane $-2(x+1)+1(y-0)-3(z+2)=0$ $-2 x+y-3 z-8=0$ $2 x-y+3 z+8=0$ $a+b+c=4$...
Read More →If the constant term, in binomial expansion
Question: If the constant term, in binomial expansion of $\left(2 x^{r}+\frac{1}{x^{2}}\right)^{10}$ is 180 , then $r$ is equal to Solution: $\left(2 x^{r}+\frac{1}{x^{2}}\right)^{10}$ General term $={ }^{10} \mathrm{C}_{\mathrm{R}}\left(2 \mathrm{x}^{2}\right)^{10-\mathrm{R}} \mathrm{x}^{-2 \mathrm{R}}$ $\Rightarrow 2^{10-\mathrm{R}^{10} \mathrm{C}_{\mathrm{R}}}=180$..........(1) $\(10-R) r-2 R=0$ $\mathrm{r}=\frac{2 \mathrm{R}}{10-\mathrm{R}}$ $r=\frac{2(R-10)}{10-R}+\frac{20}{10-R}$ $\Rightar...
Read More →Solve the Following Questions
Question: Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to :$\frac{5}{2 \sqrt{3}}$$\frac{2}{\sqrt{3}}$$\frac{4}{\sqrt{3}}$2Correct Option: , 2 Solution: The point of intersection of the curves $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and $x^{2}+y^{2}=3$ in the first quadrant is $\left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right)$ Now slop...
Read More →Let f : R→R be a function defined as
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function defined as $f(x)=\left\{\begin{array}{cl}3\left(1-\frac{|x|}{2}\right) \text { if } \quad|x| \leq 2 \\ 0 \text { if } \quad|x|2\end{array}\right.$ Let $g: \mathbf{R} \rightarrow \mathbf{R}$ be given by $g(x)=f(x+2)-f(x-2)$ If $\mathrm{n}$ and $\mathrm{m}$ denote the number of points in $\mathbf{R}$ where $g$ is not continuous and not differentiable, respectively, then $\mathrm{n}+\mathrm{m}$ is equal to Solution: $f(x-2) \begin{c...
Read More →Let y=y(x) be solution of the differential equation
Question: Let $y=y(x)$ be solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$, with $y(0)=0$ If $\mathrm{y}\left(-\frac{2}{3} \log _{\mathrm{e}} 2\right)=\alpha \log _{\mathrm{e}} 2$, then the value of $\alpha$ is equal to:$-\frac{1}{4}$$\frac{1}{4}$2$-\frac{1}{2}$Correct Option: 1 Solution: $\frac{d y}{d x}=e^{3 x} \cdot e^{4 y} \Rightarrow \int e^{-4 y} d y=\int e^{3 x} d x$ $\frac{\mathrm{e}^{-4 \mathrm{y}}}{-4}=\frac{\mathrm{e}^{3 \mathrm{x}}}{3}+\mathrm{C} ...
Read More →The function f(x),
Question: The function $f(x)$, that satisfies the condition $f(x)=x+\int_{0}^{\pi / 2} \sin x \cdot \cos y f(y) d y$, is :$x+\frac{2}{3}(\pi-2) \sin x$$x+(\pi+2) \sin x$$x+\frac{\pi}{2} \sin x$$x+(\pi-2) \sin x$Correct Option: , 4 Solution: $f(x)=x+\int_{0}^{\pi / 2} \sin x \cos y f(y) d y$ $f(x)=x+\sin x \underbrace{\int_{0}^{\pi / 2} \cos y f(y) d y}_{K}$ $\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{x}+\mathrm{K} \sin \mathrm{x}$ $\Rightarrow \mathrm{f}(\mathrm{y})=\mathrm{y}+\mathrm{K} \sin \m...
Read More →The area (in sq. units) of the region bounded by the curves
Question: The area (in sq. units) of the region bounded by the curves $x^{2}+2 y-1=0, y^{2}+4 x-4=0$ and $y^{2}-4 x-4=0$, in the upper half plane is__________. Solution: Required Area (shaded) $=2\left[\int_{0}^{2}\left(\frac{4-y^{2}}{4}\right) d y-\int_{0}^{1}\left(\frac{1-x^{2}}{2}\right) d x\right]$ $=2\left[\frac{4}{3}-\frac{1}{3}\right]=(2)$...
Read More →Consider the statement "The match will be played only if the weather
Question: Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following:The match will not be played and weather is not good and ground is wet.If the match will not be played, then either weather is not good or ground is wet.The match will be played and weather is not good or ground is wet.The match will not be played or weather is good and ground is not wet.Correct Option: , 3 Solution: $\mathrm{p}:$ weather i...
Read More →Solve this
Question: Let $\mathrm{f}:\left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow \mathbf{R}$ be defined as $f(x)=\left\{\begin{array}{ccc}(1+|\sin x|)^{\frac{3 a}{\sin x \mid}} , -\frac{\pi}{4}x0 \\ b , x=0 \\ e^{\cot 4 x / \cot 2 x} , 0x\frac{\pi}{4}\end{array}\right.$ If $f$ is continuous at $x=0$, then the value of $6 a+b^{2}$ is equal to:$1-\mathrm{e}$e $-1$$1+\mathrm{e}$eCorrect Option: , 3 Solution: $\lim _{x \rightarrow 0} f(x)=b$ $\lim _{x \rightarrow 0^{+}} x e^{\frac{\cot 4 x}{\cot 2 x...
Read More →The sum of all the elements in the set
Question: The sum of all the elements in the set $\{n \in\{1,2, \ldots \ldots, 100\}$ । H.C.F. of $\mathrm{n}$ and 2040 is 1$\}$ is equal to Solution: $2040=2^{3} \times 3 \times 5 \times 17$ $\mathrm{n}$ should not be multiple of $2,3,5$ and 17 . Sum of all $n=(1+3+5 \ldots \ldots+99)-(3+9+15+$ $21+\ldots \ldots+99)-(5+25+35+55+65+85+95)$ $-(17)$ $=2500-\frac{17}{2}(3+99)-365-17$ $=1251$...
Read More →Consider the following frequency distribution :
Question: Consider the following frequency distribution : $\begin{array}{lccccc}\text { Class: } 0-6 6-12 12-18 18-24 24-30 \\ \text { Frequency: } \text { a } \text { b } 12 9 5\end{array}$ If mean $=\frac{309}{22}$ and median $=14$, then the value $(a-b)^{2}$ is equal to Solution: Mean $=\frac{3 a+9 b+180+189+135}{a+b+26}=\frac{309}{22}$ $\Rightarrow 66 a+198 b+11088=309 a+309 b+8034$ $\Rightarrow 243 a+111 b=3054$ Now, Median $=12+\frac{\frac{\mathrm{a}+\mathrm{b}+26}{2}-(\mathrm{a}+\mathrm{b...
Read More →