Let f be a differentiable function from R to R
Question: Let $\mathrm{f}$ be a differentiable function from $\mathrm{R}$ to $\mathrm{R}$ such that $|f(x)-f(y)| \leq 2|x-y|^{\frac{3}{2}}$, for all $x, y \varepsilon R$. If $f(0)=1$ then $\int_{0}^{1} f^{2}(\mathrm{x}) \mathrm{dx}$ is equal to0$\frac{1}{2}$21Correct Option: , 4 Solution: $|f(\mathrm{x})-f(\mathrm{y})| \leq 2|\mathrm{x}-\mathrm{y}|^{3 / 2}$ divide both sides by $|x-y|$ $\left|\frac{f(x)-f(y)}{x-y}\right| \leq 2 .|x-y|^{1 / 2}$ apply limit $x \rightarrow y$ $\left|f^{\prime}(\mat...
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Question: If $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{3}{\cos ^{2} \mathrm{x}} \mathrm{y}=\frac{1}{\cos ^{2} \mathrm{x}}, \mathrm{x} \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$, and $\mathrm{y}\left(\frac{\pi}{4}\right)=\frac{4}{3}$, then $\mathrm{y}\left(-\frac{\pi}{4}\right)$ equals :$\frac{1}{3}+e^{6}$$\frac{1}{3}$$-\frac{4}{3}$$\frac{1}{3}+e^{3}$Correct Option: 1 Solution: $\frac{d y}{d x}+3 \sec ^{2} x \cdot y=\sec ^{2} x$ I.F. $=e^{3 \int \sec ^{2} x d x}=e^{3 \tan x}$ or $y \cdot e^{3 \t...
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Question: A point $P$ moves on the line $2 x-3 y+4=0$. If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus of the centroid of $\triangle \mathrm{PQR}$ is a line : parallel to $x$-axiswith slope $\frac{2}{3}$with slope $\frac{3}{2}$parallel to $\mathrm{y}$-axisCorrect Option: , 2 Solution: Let the centroid of $\triangle \mathrm{PQR}$ is $(\mathrm{h}, \mathrm{k}) \ \mathrm{P}$ is $(\alpha, \beta)$, then $\frac{\alpha+1+3}{3}=\mathrm{h} \quad$ and $\quad \frac{\beta+4-2}{3}=\mathrm{k}$ $\alp...
Read More →Two integers are selected at random from the set
Question: Two integers are selected at random from the set $\{1,2, \ldots, 11\}$. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :$\frac{2}{5}$$\frac{1}{2}$$\frac{3}{5}$$\frac{7}{10}$Correct Option: 1 Solution: Since sum of two numbers is even so either both are odd or both are even. Hence number of elements in reduced samples space $={ }^{5} C_{2}+{ }^{6} C_{2}$ so required probability $=\frac{{ }^{5} \mathrm{C}_{2}}{{ }^{5} \mathrm...
Read More →Let f : R → R be a function defined as:
Question: Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined as : $f(x)=\left\{\begin{array}{ccc}5, \text { if } x \leq 1 \\ a+b x, \text { if } 1x3 \\ b+5 x, \text { if } 3 \leq x5 \\ 30, \text { if } x \geq 5\end{array}\right.$ Then, $\mathrm{f}$ is :continuous if $a=5$ and $b=5$continuous if $a=-5$ and $b=10$continuous if $a=0$ and $b=5$not continuous for any values of $a$ and $b$Correct Option: , 4 Solution: $f(x)=\left\{\begin{array}{ccc}5 \text { if } x \leq 1 \\ a+b...
Read More →If tangents are drawn to the ellipse
Question: If tangents are drawn to the ellipse $x^{2}+2 y^{2}=2$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted betwen the coordinate axes lie on the curve :$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$$\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1$$\frac{1}{4 x^{2}}+\frac{1}{2 y^{2}}=1$Correct Option: , 3 Solution: Equation of general tangent on ellipse $\frac{\mathrm{x}}{\mathrm{a} \sec \theta}+\frac{\mathrm{y}}{\mathr...
Read More →The mean of five observations is 5 and their variance is 9.20.
Question: The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1,3 and 8, then a ratio of other two observations is :$4: 9$$6: 7$$5: 8$$10: 3$Correct Option: 1 Solution: Let two observations are $\mathrm{x}_{1}$ \ $\mathrm{x}_{2}$ mean $=\frac{\sum \mathrm{x}_{\mathrm{i}}}{5}=5 \Rightarrow 1+3+8+\mathrm{x}_{1}+\mathrm{x}_{2}=25$ $\Rightarrow x_{1}+x_{2}=13$ ..................(1) variance $\left(\sigma^{2}\right)=\frac{\sum x_{i}^{2}}{5}-25=9....
Read More →Prove the following
Question: $\lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}}$exists and equals $\frac{1}{4 \sqrt{2}}$does not existexists and equals $\frac{1}{2 \sqrt{2}}$exists and equals $\frac{1}{2 \sqrt{2}(\sqrt{2}+1)}$Correct Option: 1 Solution: $\lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}}$ $=\lim _{y \rightarrow 0} \frac{1+\sqrt{1+y^{4}}-2}{y^{4}\left(\sqrt{1+\sqrt{1+y^{4}}}+\sqrt{2}\right)}$ $=\lim _{y \rightarrow 0} \frac{\left(\sqrt{1+y^{4}}-1\right)\left...
Read More →The plane containing the line
Question: The plane containing the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$ and also containing its projection on the plane $2 x+3 y-z=5$, contains which one of the following points ?(2, 0, 2)(2, 2, 2)(0, 2, 2)(2, 2, 0)Correct Option: 1 Solution: The normal vector of required plane $=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \times(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ $=-8 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}$ So, direction ratio of n...
Read More →Solve this following
Question: If the line $3 x+4 y-24=0$ intersects the $x$-axis at the point $\mathrm{A}$ and the $\mathrm{y}$-axis at the point $\mathrm{B}$, then the incentre of the triangle $\mathrm{OAB}$, where $\mathrm{O}$ is the origin, is$(3,4)$$(2,2)$$(4,4)$$(4,3)$Correct Option: , 2 Solution: $\left|\frac{3 r+4 r-24}{5}\right|=r$ $7 r-24=\pm 5 r$ $2 r=24$ or $12 r+24$ $r=14, \quad r=2$ then incentre is $(2,2)$...
Read More →If the Boolean expression
Question: If the Boolean expression $(p \oplus q)^{\wedge}(\sim p \odot q)$ is equivalent to $p^{\wedge} q$, where $\oplus, \odot \in\{\wedge, \vee\}$, then the ordered pair $(\oplus, \odot)$ is:$(\wedge, \vee)$$(\vee, v)$$(\wedge, \wedge)$$(\vee, \wedge)$Correct Option: 1 Solution: $(p \oplus q) \wedge(\sim p \square q) \equiv p \wedge q($ given $)$ from truth table $(\oplus, \square)=(\wedge, \vee)$...
Read More →The value of
Question: The value of $\int_{0}^{\pi}|\cos x|^{3} d x$$2 / 3$0$-4 / 3$$4 / 3$Correct Option: , 4 Solution: $\int_{0}^{\pi}|\cos x|^{3} d x=\int_{0}^{\pi / 2} \cos ^{3} x d x-\int_{\pi / 2}^{\pi} \cos ^{3} x d x$ $=\int_{0}^{\pi / 2}\left(\frac{\cos 3 x+3 \cos x}{4}\right) d x-\int_{\pi / 2}^{\pi}\left(\frac{\cos 3 x+3 \cos x}{4}\right) d x$ $=\frac{1}{4}\left[\left(\frac{\sin 3 x}{3}+3 \sin x\right)_{0}^{\pi / 2}-\left(\frac{\sin 3 x}{3}+3 \sin x\right)_{\pi / 2}^{\pi}\right]$ $=\frac{1}{4}\lef...
Read More →Equation of a common tangent to the parabola
Question: Equation of a common tangent to the parabola $y^{2}=4 x$ and the hyperbole $x y=2$ is :x + 2y + 4 = 0x 2y + 4 = 0x + y + 1 = 04x + 2y + 1 = 0Correct Option: 1 Solution: Let the equation of tangent to parabola $y^{2}=4 x$ be $y=m x+\frac{1}{m}$ It is also a tangent to hyperbola $x y=2$ $\Rightarrow x\left(m x+\frac{1}{m}\right)=2$ $\Rightarrow x^{2} m+\frac{x}{m}-2=0$ $\mathrm{D}=0 \Rightarrow \mathrm{m}=-\frac{1}{2}$ So tangent is $2 y+x+4=0$...
Read More →If the third term in the binomial expansion of
Question: If the third term in the binomial expansion of $\left(1+x^{\log _{2} x}\right)^{5}$ equals 2560 , then a possible value of $x$ is:$2 \sqrt{2}$$\frac{1}{8}$$4 \sqrt{2}$$\frac{1}{4}$Correct Option: , 4 Solution: $\left(1+x^{\log _{2} x}\right)^{5}$ $\mathrm{T}_{3}={ }^{5} \mathrm{C}_{2} \cdot\left(\mathrm{x}^{\log _{2} \mathrm{x}}\right)^{2}=2560$ $\Rightarrow 10 \cdot x^{2 \log _{2} x}=2560$ $\Rightarrow x^{2 \log _{2} x}=256$ $\Rightarrow 2\left(\log _{2} x\right)^{2}=\log _{2} 256$ $\...
Read More →Two circles with equal radii are intersecting at the points (0, 1) and (0, –1).
Question: Two circles with equal radii are intersecting at the points (0, 1) and (0, 1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :1$\sqrt{2}$$2 \sqrt{2}$2Correct Option: , 4 Solution: So distance between centres $=\sqrt{2} r=2$...
Read More →Prove the following
Question: If $\cos ^{-1}\left(\frac{2}{3 x}\right)+\cos ^{-1}\left(\frac{3}{4 x}\right)=\frac{\pi}{2}\left(x\frac{3}{4}\right)$ then $x$ is equal to:$\frac{\sqrt{145}}{12}$$\frac{\sqrt{145}}{10}$$\frac{\sqrt{146}}{12}$$\frac{\sqrt{145}}{11}$Correct Option: 1 Solution: $\cos ^{-1}\left(\frac{2}{3 x}\right)+\cos ^{-1}\left(\frac{3}{4 x}\right)=\frac{\pi}{2}\left(x\frac{3}{4}\right)$ $\cos ^{-1}\left(\frac{3}{4 x}\right)=\frac{\pi}{2}-\cos ^{-1}\left(\frac{2}{3 x}\right)$ $\cos ^{-1}\left(\frac{3}{...
Read More →Solve this following
Question: Let $\mathrm{d} \in \mathrm{R}$, and $A=\left[\begin{array}{lll}-2 4+d (\sin \theta)-2 \\ 1 (\sin \theta)+2 d \\ 5 (2 \sin \theta)-d (-\sin \theta)+2+2 d\end{array}\right]$ $\theta \in[0,2 \pi]$. If the minimum value of $\operatorname{det}(\mathrm{A})$ is 8 , then a value of $d$ is :$-7$$2(\sqrt{2}+2)$$-5$$2(\sqrt{2}+1)$Correct Option: , 3 Solution: $\operatorname{det} A=\left|\begin{array}{ccc}-2 4+d \sin \theta-2 \\ 1 \sin \theta+2 d \\ 5 2 \sin \theta-d -\sin \theta+2+2 d\end{array}...
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Question: $\sum_{\mathrm{i}=1}^{20}\left(\frac{{ }^{20} \mathrm{C}_{\mathrm{i}-1}}{{ }^{20} \mathrm{C}_{\mathrm{i}}+{ }^{20} \mathrm{C}_{\mathrm{i}-1}}\right)=\frac{\mathrm{k}}{21}$, then $\mathrm{k}$ equals : 20050100400Correct Option: , 3 Solution: $\sum_{i=1}^{20}\left(\frac{{ }^{20} \mathrm{C}_{\mathrm{i}-1}}{{ }^{20} \mathrm{C}_{\mathrm{i}}+{ }^{20} \mathrm{C}_{\mathrm{i}-1}}\right)^{3}=\frac{\mathrm{k}}{21}$ $\Rightarrow \sum_{\mathrm{i}=1}^{20}\left(\frac{{ }^{20} \mathrm{C}_{\mathrm{i}-1...
Read More →If one real root of the quadratic equation
Question: If one real root of the quadratic equation $81 x^{2}+k x+256=0$ is cube of the other root, then a value of k is81100300144Correct Option: , 3 Solution: $81 x^{2}+k x+256=0 ; x=\alpha, \alpha^{3}$ $\Rightarrow \alpha^{4}=\frac{256}{81} \Rightarrow \alpha=\pm \frac{4}{3}$ Now $-\frac{\mathrm{k}}{81}=\alpha+\alpha^{3}=\pm \frac{100}{27}$ $\Rightarrow k=\pm 300$...
Read More →The maximum value of the function
Question: The maximum value of the function $f(x)=3 x^{3}-18 x^{2}+27 x-40$ on the set $S=\left\{x \in R: x^{2}+30 \leq 11 x\right\}$ is :122222122222Correct Option: 1 Solution: $\mathrm{S}=\left\{\mathrm{x} \in \mathrm{R}, \mathrm{x}^{2}+30-11 \mathrm{x} \leq 0\right\}$ $=\{x \in R, 5 \leq x \leq 6\}$ Now $f(x)=3 x^{3}-18 x^{2}+27 x-40$ $\Rightarrow f^{\prime}(x)=9(x-1)(x-3)$, which is positive in $[5,6]$ $\Rightarrow f(x)$ increasing in $[5,6]$ Hence maximum value $=f(6)=122$...
Read More →Consider the quadratic equation
Question: Consider the quadratic equation $(c-5) x^{2}-2 c x+(c-4)=0, c \neq 5$. Let $S$ be the set of all integral values of $\mathrm{c}$ for which one root of the equation lies in the interval $(0,2)$ and its other root lies in the interval $(2,3)$. Then the number of elements in $S$ is : 11181012Correct Option: 1 Solution: Let $f(x)=(c-5) x^{2}-2 c x+c-4$ $\therefore f(0) f(2)0$ ...................(1) $\ f(2) f(3)0$ ..............(2) from (1) \ (2) $(c-4)(c-24)0$ $\(c-24)(4 c-49)0$ $\Rightarr...
Read More →For any θ ∈ (π/4 , π/2), the expression
Question: For any $\theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$, the expression $3(\sin \theta-\cos \theta)^{4}+6(\sin \theta+\cos \theta)^{2}+4 \sin ^{6} \theta$$13-4 \cos ^{6} \theta$$13-4 \cos ^{4} \theta+2 \sin ^{2} \theta \cos ^{2} \theta$$13-4 \cos ^{2} \theta+6 \cos ^{4} \theta$$13-4 \cos ^{2} \theta+6 \sin ^{2} \theta \cos ^{2} \theta$Correct Option: 1 Solution: We have, $3(\sin \theta-\cos \theta)^{4}+6(\sin \theta+\cos \theta)^{2}+4 \sin ^{6} \theta$ $=3(1-\sin 2 \theta)^{2}+6(...
Read More →If y(x) is the solution of the differential equation
Question: If y(x) is the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{2 \mathrm{x}+1}{\mathrm{x}}\right) \mathrm{y}=\mathrm{e}^{-2 \mathrm{x}}, \mathrm{x}0$ where $y(1)=\frac{1}{2} e^{-2}$, then :y(x) is decreasing in (0, 1)$y(x)$ is decreasing in $\left(\frac{1}{2}, 1\right)$$y\left(\log _{e} 2\right)=\frac{\log _{e} 2}{4}$$y\left(\log _{e} 2\right)=\log _{e} 4$Correct Option: , 2 Solution: $\frac{d y}{d x}+\left(\frac{2 x+1}{x}\right) y=e^{-2 x}$ I.F. $=e^...
Read More →If the solve the problem
Question: If $\mathrm{q}$ is false and $\mathrm{p} \wedge \mathrm{q} \leftrightarrow \mathrm{r}$ is true, then which one of the following statements is a tautology?$(\mathrm{p} \vee \mathrm{r}) \rightarrow(\mathrm{p} \wedge \mathrm{r})$$p \vee r$$p \wedge r$$(p \wedge r) \rightarrow(p \vee r)$Correct Option: , 4 Solution: Given $q$ is $F$ and $(p \wedge q) \leftrightarrow r$ is $T$ $\Rightarrow \mathrm{p} \wedge \mathrm{q}$ is $\mathrm{F}$ which implies that $\mathrm{r}$ is $\mathrm{F}$ $\Righta...
Read More →Solve this following
Question: The shortest distance between the point $\left(\frac{3}{2}, 0\right)$ and the curve $y=\sqrt{x},(x0)$ is :$\frac{\sqrt{5}}{2}$$\frac{5}{4}$$\frac{3}{2}$$\frac{\sqrt{3}}{2}$Correct Option: 1 Solution: Let points $\left(\frac{3}{2}, 0\right),\left(t^{2}, t\right), t0$ Distance $=\sqrt{\mathrm{t}^{2}+\left(\mathrm{t}^{2}-\frac{3}{2}\right)^{2}}$ $=\sqrt{t^{4}-2 t^{2}+\frac{9}{4}}=\sqrt{\left(t^{2}-1\right)^{2}+\frac{5}{4}}$ So minimum distance is $\sqrt{\frac{5}{4}}=\frac{\sqrt{5}}{2}$...
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