For all twice differentiable functions
Question: For all twice differentiable functions $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=f^{\prime}(0)=0$$f^{\prime \prime}(x)=0$, for some $x \in(0,1)$$f^{\prime \prime}(0)=0$$f^{\prime \prime}(\mathrm{x}) \neq 0$ at every point $\mathrm{x} \in(0,1)$$f^{\prime \prime}(x)=0$ at every point $x \in(0,1)$Correct Option: 1 Solution: $f(0)=f(1)=f^{\prime}(0)=0$ Apply Rolles theorem on $\mathrm{y}=f(\mathrm{x})$ in $\mathrm{x} \in[0,1]$ $f(0)=f(1)=0$ $\Rightarrow f^{\prime}(\alpha)=0$ ...
Read More →There are 3 sections in a question paper and each section contains 5 questions.
Question: There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:1500225530002250Correct Option: , 4 Solution: $=\left({ }^{5} \mathrm{C}_{1}{ }^{5} \mathrm{C}_{2}{ }^{5} \mathrm{C}_{2}\right) \cdot 3+\left({ }^{5} \mathrm{C}_{1}{ }^{5} \mathrm{C}_{1}{ }^{5} \mathrm{C}_{3}\right) \cdot 3$ $=5 \cdo...
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Question: The set of all real values of $\lambda$ for which the function $f(x)=\left(1-\cos ^{2} x\right) \cdot(\lambda+\sin x)$, $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, has exactly one maxima and exactly one minima, is : $\left(-\frac{1}{2}, \frac{1}{2}\right)-\{0\}$$\left(-\frac{1}{2}, \frac{1}{2}\right)$$\left(-\frac{3}{2}, \frac{3}{2}\right)$$\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$Correct Option: , 4 Solution: $f(x)=\left(1-\cos ^{2} x\right)(\lambda+\sin x)$ $x \in\left(\fr...
Read More →If the line y=m x+c is a common tangent to
Question: If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36$, then which one of the following is true?$5 \mathrm{~m}=4$$4 \mathrm{c}^{2}=369$$c^{2}=369$$8 m+5=0$Correct Option: , 2 Solution: $\mathrm{y}=\mathrm{m} \mathrm{x}+\mathrm{c}$ is tangent to $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and $x^{2}+y^{2}=36$ $c^{2}=100 m^{2}-64 \mid c^{2}=36\left(1+m^{2}\right)$ $\Rightarrow 100 \mathrm{~m}^{2}-64=36+36 \mathrm{~m}^{...
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Question: If $\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^{2} \theta} d \theta=A \log _{e}|B(\theta)|+C$ where $C$ is a constant of integration, then $\frac{B(\theta)}{A}$ can be :$\frac{2 \sin \theta+1}{5(\sin \theta+3)}$$\frac{2 \sin \theta+1}{\sin \theta+3}$$\frac{5(\sin \theta+3)}{2 \sin \theta+1}$$\frac{5(2 \sin \theta+1)}{\sin \theta+3}$Correct Option: , 4 Solution: $\int \frac{\cos \theta d \theta}{5+7 \sin \theta-2 \cos ^{2} \theta}$ $\int \frac{\cos \theta d \theta}{3+7 \sin \theta+...
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Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as of $\lambda$ for which $f^{\prime \prime}(0)$ exists, is Solution: $f(\mathrm{x})=\mathrm{x}^{5} \cdot \sin \frac{1}{\mathrm{x}}+5 \mathrm{x}^{2} \quad$ if $\mathrm{x}0$ $f(x)=0$ if $x=0$ $f(x)=x^{5} \cdot \cos \frac{1}{x}+\lambda x^{2} \quad$ if $x0$ LHD of $f^{\prime}(\mathrm{x})$ at $\mathrm{x}=0$ is 10 RHD of $f^{\prime}(\mathrm{x})$ at $\mathrm{x}=0$ is $2 \lambda$ if $f^{\prime \prime}(0)$ exists then $2 \lambda=10$ $\Right...
Read More →If a+x=b+y=c+z+1, where a, b, c, x, y , z
Question: If $a+x=b+y=c+z+1$, where $a, b, c, x$, $\mathrm{y}, \mathrm{z}$ are non-zero distinct real numbers, then $\left|\begin{array}{lll}x a+y x+a \\ y b+y y+b \\ z c+y z+c\end{array}\right|$ is equal to :0$y(a-b)$$\mathrm{y}(\mathrm{b}-\mathrm{a})$$\mathrm{y}(\mathrm{a}-\mathrm{c})$Correct Option: , 2 Solution: $a+x=b+y=c+z+1$ $\left|\begin{array}{ccc}x a+y x+a \\ y b+y y+b \\ z c+y z+c\end{array}\right| \quad \quad C_{3} \rightarrow C_{3}-C_{1}$ $\left|\begin{array}{lll}x a+y a \\ y b+y b ...
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Question: If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is Solution: $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ $=\sqrt{3}(\sqrt{2+2 \cos \theta})+\sqrt{2-2 \cos \theta}$ $=\sqrt{6}(\sqrt{1+\cos \theta})+\sqrt{2}(\sqrt{1-\cos \theta})$ $=2 \sqrt{3}\left|\cos \frac{\theta}{2}\right|+2\left|\sin \frac{\theta}{2}\right|$ $\leq \sqrt{(2 \sqrt{3})^{2}+(2)^{2}}=4$...
Read More →If the mean and the standard deviation of the data
Question: If the mean and the standard deviation of the data $3,5,7, \mathrm{a}, \mathrm{b}$ are 5 and 2 respectively, then $\mathrm{a}$ and $\mathrm{b}$ are the roots of the equation :$2 x^{2}-20 x+19=0$$x^{2}-10 x+19=0$$x^{2}-10 x+18=0$$x^{2}-20 x+18=0$Correct Option: , 2 Solution: Mean $=5$ $\frac{3+5+7+a+b}{5}=5$ $a+b=10$.....(i) S.d. $=2 \Rightarrow \sqrt{\frac{\sum_{i=1}^{5}\left(x_{i}-\bar{x}\right)^{2}}{5}}=2$ $(3-5)^{2}+(5-5)^{2}+(7-5)^{2}+(a-5)^{2}+(b-5)^{2}=20$ $\Rightarrow 4+0+4+(a-5...
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Question: Set A has m elements and Set B has n elements. If the total number of subsets of $\mathrm{A}$ is 112 more than the total number of subsets of B, then the value of $m . n$ is.. Solution: $2^{m}-2^{n}=112$ $\mathrm{m}=7, \mathrm{n}=4$ $\left(2^{7}-2^{4}=112\right)$ $\mathrm{m} \times \mathrm{n}=7 \times 4=28$...
Read More →The angle of elevation of the top of a hill from
Question: The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $45^{\circ}$. After walking a distance of 80 meters towards the top, up a slope inclined at an angle of $30^{\circ}$ to the horizontal plane, the angle of elevation of the top of the hill becomes $75^{\circ}$. Then the height of the hill (in meters) is.. Solution: $\mathrm{h}+40-40 \sqrt{3}$ $\tan 75^{\circ}=\frac{h}{h+40-40 \sqrt{3}}$ $\frac{2+\sqrt{3}}{...
Read More →If x=1 is a critical point of the function
Question: If $x=1$ is a critical point of the function $f(x)=\left(3 x^{2}+a x-2-a\right) e^{x}$, then :$x=1$ is a local minima and $x=-\frac{2}{3}$ is a local maxima of $f$.$x=1$ is a local maxima and $x=-\frac{2}{3}$ is a local minima of $f$.$x=1$ and $x=-\frac{2}{3}$ are local minima of $f$.$x=1$ and $x=-\frac{2}{3}$ are local maxima of $f$.Correct Option: 1 Solution: $f(x)=\left(3 x^{2}+a x-2-a\right) e^{x}$ $f^{\prime}(x)=\left(3 x^{2}+a x-2-a\right) e^{x}+e^{x}(6 x+a)$ $=e^{x}\left(3 x^{2}...
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Question: Let $\mathrm{AD}$ and $\mathrm{BC}$ be two vertical poles at $\mathrm{A}$ and B respectively on a horizontal ground. If $\mathrm{AD}=8 \mathrm{~m}, \mathrm{BC}=11 \mathrm{~m}$ and $\mathrm{AB}=10 \mathrm{~m}$; then the distance (in meters) of a point $\mathrm{M}$ on $\mathrm{AB}$ from the point $A$ such that $M D^{2}+M C^{2}$ is minimum is. Solution: $\left(\mathrm{MD}^{2}+(\mathrm{MC})^{2}=\mathrm{h}^{2}+64+(\mathrm{h}-10)^{2}+121\right.$ $=2 \mathrm{~h}^{2}-20 \mathrm{~h}+64+100+121$...
Read More →If the length of the chord of the circle,
Question: If the length of the chord of the circle, $x^{2}+y^{2}=r^{2}(r0)$ along the line, $y-2 x=3$ is $\mathrm{r}$, then $\mathrm{r}^{2}$ is equal to :$\frac{9}{5}$$\frac{12}{5}$12$\frac{24}{5}$Correct Option: , 2 Solution: Let chord $\mathrm{AB}=\mathrm{r}$ $\because \Delta \mathrm{AOM}$ is right angled triangle $\therefore \mathrm{OM}=\frac{\mathrm{r} \sqrt{3}}{2}=$ perpendicular distance of line AB from $(0,0)$ $\frac{r \sqrt{3}}{2}=\left|\frac{3}{\sqrt{5}}\right|$ $r^{2}=\frac{12}{5}$...
Read More →Let m and M be respectively the minimum and
Question: Let $\mathrm{m}$ and $\mathrm{M}$ be respectively the minimum and maximum values of $\left|\begin{array}{ccc}\cos ^{2} x 1+\sin ^{2} x \sin 2 x \\ 1+\cos ^{2} x \sin ^{2} x \sin 2 x \\ \cos ^{2} x \sin ^{2} x 1+\sin 2 x\end{array}\right| .$ Then the ordered pair $(\mathrm{m}, \mathrm{M})$ is equal to$(-3,-1)$$(-4,-1)$$(1,3)$$(-3,3)$Correct Option: Solution: $\left|\begin{array}{ccc}\cos ^{2} x 1+\sin ^{2} x \sin 2 x \\ 1+\cos ^{2} x \sin ^{2} x \sin 2 x \\ \cos ^{2} x \sin ^{2} x 1+\si...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $A=\{(x, y):(x-1)[x] \leq y \leq 2 \sqrt{x}, 0 \leq x \leq 2\}$ where $[t]$ denotes the greatest integer function, is :$\frac{8}{3} \sqrt{2}-\frac{1}{2}$$\frac{8}{3} \sqrt{2}-1$$\frac{4}{3} \sqrt{2}-\frac{1}{2}$$\frac{4}{3} \sqrt{2}+1$Correct Option: 1 Solution: $(x-1)[x] \leq y \leq 2 \sqrt{x}, \quad 0 \leq x \leq 2$ Draw $y=2 \sqrt{x} \Rightarrow y^{2}=4 x \quad x \geq 0$ $y=(x-1)[x]=\left\{\begin{array}{cc}0 , 0 \leq x1 \\ x-1, 1 \leq x2 \\ 2, x...
Read More →The shortest distance between the lines
Question: The shortest distance between the lines $\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$ and $x+y+z+1=0$ $2 x-y+z+3=0$ is : $\frac{1}{2}$1$\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{3}}$Correct Option: , 4 Solution: Line of intersection of planes $x+y+z+1=0$ ........(1) $2 x-y+z+3=0$ ............(2) eliminate $y$ $3 x+2 z+4=0$ $x=\frac{-2 z-4}{3}$ .........(3) put in equaiton (1) $z=-3 y+1$ ..............(4) from $(3)$ and $(4)$ $\frac{3 x+4}{-2}=-3 y+1=z$ $\frac{x-\left(-\frac{4}{3}\right)}{-\frac{...
Read More →The derivative of
Question: The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)$ at $x=\frac{1}{2}$ is :$\frac{\sqrt{3}}{12}$$\frac{\sqrt{3}}{10}$$\frac{2 \sqrt{3}}{5}$$\frac{2 \sqrt{3}}{3}$Correct Option: , 2 Solution: Let $f=\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ Put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$ $f=\tan ^{-1}\left(\frac{\sec \theta-1}{\tan \theta}\right)$ $f=\tan ^{-1}\left(\frac{1-\cos ...
Read More →The value of
Question: The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is :$2^{15} \mathrm{i}$$-2^{15}$$-2^{15} \mathrm{i}$$6^{5}$Correct Option: , 3 Solution: $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}=\left(\frac{2 \omega}{1-i}\right)^{30}$ $=\frac{2^{30} \cdot \omega^{30}}{\left((1-i)^{2}\right)^{30}}$ $=\frac{2^{30} \cdot 1}{\left(1+i^{2}-2 i\right)^{15}}$ $=\frac{2^{30}}{-2^{15} \cdot \mathrm{i}^{15}}$ $=-2^{15} \mathrm{i}$...
Read More →If the sum of the second, third and fourth terms of a positive term G.P.
Question: If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243 , then the sum of the first 50 terms of this G.P. is :$\frac{2}{13}\left(3^{50}-1\right)$$\frac{1}{26}\left(3^{50}-1\right)$$\frac{1}{13}\left(3^{50}-1\right)$$\frac{1}{26}\left(3^{49}-1\right)$Correct Option: , 2 Solution: Let first term $=a0$ Common ratio $=r0$ $a r+a r^{2}+a r^{3}=3$$\ldots(i)$ $a r^{5}+a r^{6}+a r^{7}=243$ $\ldots$ (ii) $r^{4}\left...
Read More →Solve this following
Question: If $\alpha$ and $\beta$ be two roots of the equation $x^{2}-64 x+256=0$ Then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is1342Correct Option: , 4 Solution: $x^{2}-64 x+256=0$ $\alpha+\beta=64, \alpha \beta=256$ $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{1 / 8}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{1 / 8}$ $=\frac{\alpha^{3 / 8}}{\beta^{5 / 8}}+\frac{\beta^{3 / 8}}{\alpha^{5 / 8}}$ $=\frac{\alpha+\...
Read More →solve the following
Question: $\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^{2}+x^{4}}-1\right) / x}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$does not exist.is equal to $\sqrt{\mathrm{e}}$.is equal to 0 .is equal to 1 .Correct Option: , 4 Solution: $\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^{2}+x^{4}}-1\right) / x}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$ $\because \lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}+x^{4}}-1}{x}\left(\frac{0}{0}\right.$ from) $\lim _{x \rightarrow 0} \frac{\left(1+x^{2}+x^{4}\r...
Read More →The region represented by
Question: The region represented by $\{z=x+$ iy $\in C:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality : $y^{2} \geq x+1$$\mathrm{y}^{2} \geq 2(\mathrm{x}+1)$$y^{2} \leq x+\frac{1}{2}$$y^{2} \leq 2\left(x+\frac{1}{2}\right)$Correct Option: , 4 Solution: $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ $|\mathrm{z}|-\mathrm{ke}(\mathrm{z}) \leq 1$ $\Rightarrow \sqrt{x^{2}+y^{2}}-x \leq 1$ $\Rightarrow \sqrt{x^{2}+y^{2}} \leq 1+x$ $\Rightarrow x^{2}+y^{2} \leq 1+2 x+x^{2}$ $\Rightarrow y^{2}...
Read More →If the sum of the first 20 terms of the series
Question: If the sum of the first 20 terms of the series $\log _{\left(7^{1 / 2}\right)} x+\log _{\left(7^{1 / 3}\right)} x+\log _{\left(7^{1 / 4}\right)} x+\ldots$ is 460 , then $x$ is equal to : $746 / 21$$7^{1 / 2}$$\mathrm{e}^{2}$$7^{2}$Correct Option: , 4 Solution: $460=\log _{7} x \cdot(2+3+4+\ldots \ldots+20+21)$ $\Rightarrow 460=\log _{7} x \cdot\left(\frac{21 \times 22}{2}-1\right)$ $\Rightarrow 460=230 \cdot \log _{7} x$ $\Rightarrow \log _{7} x=2 \Rightarrow x=49$...
Read More →The position of a moving car at time t is given
Question: The position of a moving car at time $t$ is given by $f(\mathrm{t})=\mathrm{at}^{2}+\mathrm{bt}+\mathrm{c}, \mathrm{t}0$, where $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are real numbers greater than 1 . Then the average speed of the car over the time interval $\left[\mathrm{t}_{1}, \mathrm{t}_{2}\right]$ is attained at the point :$\mathrm{a}\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right)+\mathrm{b}$$\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right) / 2$$2 \mathrm{a}\left(\mathrm{t}_{1}+\mathrm{t}...
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