The equation
Question: The equation $|\mathrm{z}-\mathrm{i}|=|\mathrm{z}-1|, \mathrm{i}=\sqrt{-1}$, represents:the line through the origin with slope $-1$.a circle of radius 1 .a circle of radius $\frac{1}{2}$.the line through the origin with slope $1 .$Correct Option: , 4 Solution: $|z-i|=|z-1|$ $y=x$...
Read More →If m is the minimum value of
Question: If $\mathrm{m}$ is the minimum value of $\mathrm{k}$ for which the function $\mathrm{f}(\mathrm{x})=\mathrm{x} \sqrt{\mathrm{kx}-\mathrm{x}^{2}}$ is increasing in the interval $[0,3]$ and $\mathrm{M}$ is the maximum value of $\mathrm{f}$ in $[0,3]$ when $\mathrm{k}=\mathrm{m}$, then the ordered pair $(\mathrm{m}, \mathrm{M})$ is equal to :$(4,3 \sqrt{2})$$(4,3 \sqrt{3})$$(3,3 \sqrt{3})$$(5,3 \sqrt{6})$Correct Option: , 2 Solution: $f(x)=x \sqrt{k x-x^{2}}$ $f^{\prime}(x)=\frac{3 k x-4 ...
Read More →The height of a right circular cylinder of maximum volume
Question: The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is$2 \sqrt{3}$$\sqrt{3}$$\sqrt{6}$$\frac{2}{3} \sqrt{3}$Correct Option: 1 Solution: $\mathrm{h}=2 \mathrm{rsin} \theta$ $\mathrm{a}=2 \mathrm{r} \cos \theta$ $\mathrm{v}=\pi(\mathrm{r} \cos \theta)^{2}(2 \mathrm{r} \sin \theta)$ $\mathrm{v}=2 \pi \mathrm{r}^{3} \cos ^{2} \theta \sin \theta$ $\frac{d v}{d \theta}=\pi r^{3}\left(-2 \cos \theta \sin ^{2} \theta+\cos ^{3} \theta\right)=0$ or $\tan \...
Read More →The number of four-digit numbers strictly greater
Question: The number of four-digit numbers strictly greater than 4321 that can be formed using the digits $0,1,2,3,4,5$ (repetition of digits is allowed) is :288306360310Correct Option: , 4 Solution: (1) The number of four-digit numbers Starting with 5 is equal to $6^{3}=216$ (2) Starting with 44 and 55 is equal to $36 \times 2=72$ (3) Starting with 433,434 and 435 is equal to $6 \times 3=18$ (3) Remaining numbers are $4322,4323,4324,4325$ is equal to 4 so total numbers are $216+72+18+4=310$...
Read More →Two vertical poles of heights,
Question: Two vertical poles of heights, $20 \mathrm{~m}$ and $80 \mathrm{~m}$ stand a part on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is :12151618Correct Option: , 3 Solution: by similar triangle $\frac{\mathrm{h}}{\mathrm{x}_{1}}=\frac{80}{\mathrm{x}_{1}+\mathrm{x}_{2}}$ ............(1) by $\frac{h}{x_{2}}=\frac{20}{x_{1}+x_{2}}$ .............(2) by (1) and (2) $\frac{...
Read More →Let f : [ -1 , 3 ] → R be defined as
Question: Let $f:[-1,3] \rightarrow \mathrm{R}$ be defined as $f(x)=\left\{\begin{array}{cc}|x|+[x] , \quad-1 \leq x1 \\ x+|x| , \quad 1 \leq x2 \\ x+[x] , \quad 2 \leq x \leq 3\end{array}\right.$ where [t] denotes the greatest integer less than or equal to $\mathrm{t}$. Then, $f$ is discontinuous at:four or more pointsonly one pointonly two pointsonly three pointsCorrect Option: , 4 Solution: $f(x)=\left\{\begin{array}{ccc}-(x+1) , -1 \leq x0 \\ x , 0 \leq x1 \\ 2 x , 1 \leq x2 \\ x+2 , 2 \leq ...
Read More →Suppose that the points (h, k),(1,2) and
Question: Suppose that the points $(h, k),(1,2)$ and $(-3,4)$ lie on the line $\mathrm{L}_{1}$. If a line $\mathrm{L}_{2}$ passing through the points $(h, k)$ and $(4,3)$ is perpendicular to $\mathrm{L}_{1}$, then $\frac{\mathrm{k}}{\mathrm{h}}$ equals :3$-\frac{1}{7}$$\frac{1}{3}$0Correct Option: , 3 Solution: equation of $L_{1}$ is $y=-\frac{1}{2} x+\frac{5}{2}$ .........(1) equation of $\mathrm{L}_{2}$ is $y=2 x-5$ ..........(2) by $(1)$ and $(2)$ $x=3$ $\mathrm{y}=1 \Rightarrow \mathrm{h}=3,...
Read More →Solve the following equations:
Question: If $\int \frac{\mathrm{dx}}{\mathrm{x}^{3}\left(1+\mathrm{x}^{6}\right)^{2 / 3}}=\mathrm{x} f(\mathrm{x})\left(1+\mathrm{x}^{6}\right)^{\frac{1}{3}}+\mathrm{C}$ where $\mathrm{C}$ is a constant of integration, then the function $f(\mathrm{x})$ is equal to-$-\frac{1}{6 x^{3}}$$\frac{3}{\mathrm{x}^{2}}$$-\frac{1}{2 x^{2}}$$-\frac{1}{2 x^{3}}$Correct Option: , 4 Solution: $\int \frac{\mathrm{dx}}{\mathrm{x}^{3}\left(1+\mathrm{x}^{6}\right)^{2 / 3}}=\mathrm{x} f(\mathrm{x})\left(1+\mathrm{...
Read More →If f(1) = 1, f' (1) = 3, then the derivative
Question: If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^{2}$ at $x=1$ is :1233915Correct Option: , 2 Solution: $\mathrm{y}=f(f(f(\mathrm{x})))+(f(\mathrm{x}))^{2}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=f^{\prime}(f(f(\mathrm{x}))) f^{\prime}(f(\mathrm{x})) f^{\prime}(\mathrm{x})+2 f(\mathrm{x}) f^{\prime}(\mathrm{x})$ $=f^{\prime}(1) f^{\prime}(1) f^{\prime}(1)+2 f(1) f^{\prime}(1)$ $=3 \times 5 \times 3+2 \times 1 \times 3$ $=27+6$ $=33$...
Read More →In an ellipse, with centre at the origin,
Question: In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0,5 \sqrt{3})$, then the length of its latus rectum is:10856Correct Option: , 3 Solution: Let equation of ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $2 \mathrm{a}-2 \mathrm{~b}=10$ ............(1) $\mathrm{ae}=5 \sqrt{3}$ ...............(2) $\frac{2 b^{2}}{a}=?$ $b^{2}=a^{2}\left(1-e^{2}\right)$ $b^{2}=a^{2}-a^{2} e^{2}$ $b^{2}=a^{2}-25 \tim...
Read More →The tangent and the normal lines
Question: The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^{2}+y^{2}=4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is :$\frac{1}{3}$$\frac{4}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$Correct Option: , 4 Solution: Given $x^{2}+y^{2}=4$ equation of tangent $\Rightarrow \sqrt{3} x+y=4$ ...........(1) Equation of normal $x-\sqrt{3} y=0$ ...........(2) Coordinate of $\mathrm{T}\left(\frac{4}{\sqrt{3}}, 0\right)$ $\therefore$ Ar...
Read More →Prove the following identities.
Question: Let $f(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{g}(\mathrm{t}) \mathrm{dt}$, where $\mathrm{g}$ is a non-zero even function. If $f(\mathrm{x}+5)=\mathrm{g}(\mathrm{x})$, then $\int_{0}^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}$ equals-$\int_{\mathrm{x}+5}^{5} \mathrm{~g}(\mathrm{t}) \mathrm{dt}$$5 \int_{x+5}^{5} g(t) d t$$\int_{5}^{x+5} g(t) d t$$2 \int_{5}^{x+5} g(t) d t$Correct Option: 1 Solution: $f(x)=\int_{0}^{x} g(t) d t$ $f(-x)=\int_{0}^{-x} g(t) d t$ put $\mathrm{t}=-\mathrm{u}$ ...
Read More →Prove the following
Question: If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$ then $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ is equal to-110$(-1+2 \mathrm{i})^{9}$Correct Option: 1 Solution: $z=\frac{\sqrt{3}}{2}+\frac{i}{2}=\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}$ $\Rightarrow \mathrm{z}^{5}=\cos \frac{5 \pi}{6}+\mathrm{i} \sin \frac{5 \pi}{6}=\frac{-\sqrt{3}+\mathrm{i}}{2}$ and $z^{8}=\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-\left(\frac{1+i \sqrt{3}}{2}\right)$ $\Rightarrow\left(1+i z+z^{5}+\mathrm{iz}^{8}\ri...
Read More →If a point R ( 4 , y , z) lies on the line segment
Question: If a point $\mathrm{R}(4, \mathrm{y}, \mathrm{z})$ lies on the line segment joining the points $\mathrm{P}(2,-3,4)$ and $\mathrm{Q}(8,0,10)$, then the distance of $\mathrm{R}$ from the origin is :$2 \sqrt{14}$6$\sqrt{53}$$2 \sqrt{21}$Correct Option: 1 Solution: $\frac{4}{2}=\frac{-y}{y+3}=\frac{10-z}{z-4}$ $\Rightarrow \mathrm{z}=6 \ \mathrm{y}=-2$ $\Rightarrow \mathrm{R}(4,-2,6)$ dist. from origin $=\sqrt{16+4+36}=2 \sqrt{14}$...
Read More →The number of integral values of
Question: The number of integral values of $m$ for which the equation $\left(1+\mathrm{m}^{2}\right) \mathrm{x}^{2}-2(1+3 \mathrm{~m}) \mathrm{x}+(1+8 \mathrm{~m})=0$ has no real root is :infinitely many231Correct Option: 1 Solution: $D0$ $4(1+3 m)^{2}-4\left(1+m^{2}\right)(1+8 m)0$ $\Rightarrow \mathrm{m}(2 \mathrm{~m}-1)^{2}0 \Rightarrow \mathrm{m}0$...
Read More →If three distinct numbers a,b,c are in G.P.
Question: If three distinct numbers a,b,c are in G.P. and the equations $a x^{2}+2 b x+c=0$ and $\mathrm{dx}^{2}+2 \mathrm{ex}+f=0$ have a common root, then which one of the following statements is correct?$\mathrm{d}, \mathrm{e}, f$ are in A.P.$\frac{\mathrm{d}}{\mathrm{a}}, \frac{\mathrm{e}}{\mathrm{b}}, \frac{f}{\mathrm{c}}$ are in G.P.$\frac{\mathrm{d}}{\mathrm{a}}, \frac{\mathrm{e}}{\mathrm{b}}, \frac{f}{\mathrm{c}}$ are in A.P.d,e, $f$ are in G.P.Correct Option: , 3 Solution: $\mathrm{a}, ...
Read More →Let the number $2, b, c$ be in an A.P. and
Question: Let the number $2, b, c$ be in an A.P. and $A=\left[\begin{array}{ccc}1 1 1 \\ 2 b c \\ 4 b^{2} c^{2}\end{array}\right] .$ If $\operatorname{det}(A) \in[2,16]$, then $c$ lies in the interval :$[2,3)$$\left(2+2^{3 / 4}, 4\right)$$\left[3,2+2^{3 / 4}\right]$$[4,6]$Correct Option: , 4 Solution: put $\mathrm{b}=\frac{2+\mathrm{c}}{2}$ in determinant of $\mathrm{A}$ $|A|=\frac{c^{3}-6 c^{2}+12 c-8}{4} \in[2,16]$ $\Rightarrow(\mathrm{c}-2)^{3} \in[8,64]$ $\Rightarrow \mathrm{c} \in[4,6]$...
Read More →The tangent to the parabola
Question: The tangent to the parabola $\mathrm{y}^{2}=4 \mathrm{x}$ at the point where it intersects the circle $x^{2}+y^{2}=5$ in the first quadrant, passes through the point :$\left(-\frac{1}{3}, \frac{4}{3}\right)$$\left(-\frac{1}{4}, \frac{1}{2}\right)$$\left(\frac{3}{4}, \frac{7}{4}\right)$$\left(\frac{1}{4}, \frac{3}{4}\right)$Correct Option: , 3 Solution: Given $\mathrm{y}^{2}=4 \mathrm{x}$ .............(1) and $x^{2}+y^{2}=5$ ..............(2) by $(1)$ and (2) $\Rightarrow x=1$ and $y=2$...
Read More →Let f : R → R be a differentiable function
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function satisfying $f^{\prime}(3)+f^{\prime}(2)=0$. Then $\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}$ is equal to$e^{2}$$\mathrm{e}$$\mathrm{e}^{-1}$1Correct Option: , 4 Solution: $\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}\left(1^{\infty}\right.$ form $)$ $\Rightarrow \mathrm{e}^{\lim _{x \rightarrow 0} \frac{f(3+\mathrm{x})-f(2-\mathrm{...
Read More →Which one of the following statements
Question: Which one of the following statements is not a tautology ?$(\mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{p}$$(p \wedge q) \rightarrow(\sim p) \vee q$$\mathrm{p} \rightarrow(\mathrm{p} \vee \mathrm{q})$$(p \vee q) \rightarrow(p \vee(\sim q))$Correct Option: , 4 Solution: Tautology Tautology...
Read More →The vector equation of the plane through the line of intersection
Question: The vector equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$ which is perpendicular to the plane $x-y+z=0$ is :$\overrightarrow{\mathrm{r}} \times(\hat{\mathrm{i}}+\hat{\mathrm{k}})+2=0$$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{k}})-2=0$$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{k}})+2=0$$\overrightarrow{\mathrm{r}} \times(\hat{\mathrm{i}}-\hat{\mathrm{k}})+2=0$Correct Option: , 3 Solution: Le...
Read More →Given that the slope of the tangent to a curve
Question: Given that the slope of the tangent to a curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ at any point $(\mathrm{x}, \mathrm{y})$ is $\frac{2 \mathrm{y}}{\mathrm{x}^{2}}$. If the curve passes through the centre of the circle $x^{2}+y^{2}-2 x-2 y=0$, then its equation is :$x \log _{\mathrm{e}}|\mathrm{y}|=2(\mathrm{x}-1)$$x \log _{e}|y|=x-1$$x^{2} \log _{e}|y|=-2(x-1)$$x \log _{e}|y|=-2(x-1)$Correct Option: 1 Solution: given $\frac{d y}{d x}=\frac{2 y}{x^{2}}$ $\Rightarrow \int \frac{d y}{2 y}...
Read More →Let S(α) = { ( x , y ) :
Question: Let $\mathrm{S}(\alpha)=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{y}^{2} \leq \mathrm{x}, 0 \leq \mathrm{x} \leq \alpha\right\}$ and $\mathrm{A}(\alpha)$ is area of the region $\mathrm{S}(\alpha)$. If for a $\lambda, 0\lambda4$, $\mathrm{A}(\lambda): \mathrm{A}(4)=2: 5$, then $\lambda$ equals$2\left(\frac{4}{25}\right)^{\frac{1}{3}}$$4\left(\frac{4}{25}\right)^{\frac{1}{3}}$$2\left(\frac{2}{5}\right)^{\frac{1}{3}}$$4\left(\frac{2}{5}\right)^{\frac{1}{3}}$Correct Option: , 2 Solution: $S...
Read More →If the fourth term in the binomial expansion of
Question: If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to 200 , and $x1$, then the value of $\mathrm{x}$ is :$10^{3}$100$10^{4}$10Correct Option: , 4 Solution: $200={ }^{6} C_{3}\left(x^{\frac{1}{x+\log _{10} x}}\right)^{\frac{3}{2}} \times x^{\frac{1}{4}}$ $\Rightarrow 10=x^{\frac{3}{2\left(1+\log _{10} x\right)} \cdot \frac{1}{4}}$ $\Rightarrow 1=\left(\frac{3}{2(1+t)}+\frac{1}{4}\right) t$ where $t=\log _{10} ...
Read More →Prove the following
Question: Let $f(\mathrm{x})=\mathrm{a}^{\mathrm{x}}(\mathrm{a}0)$ be written as $f(\mathrm{x})=f_{1}(\mathrm{x})+f_{2}(\mathrm{x})$, where $f_{1}(\mathrm{x})$ is an even function of $f_{2}(x)$ is an odd function. Then $f_{1}(\mathrm{x}+\mathrm{y})+f_{1}(\mathrm{x}-\mathrm{y})$ equals$2 f_{1}(\mathrm{x}) f_{1}(\mathrm{y})$$2 f_{1}(\mathrm{x}) f_{2}(\mathrm{y})$$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{2}(\mathrm{x}-\mathrm{y})$$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{1}(\mathrm{x}-\mathrm{y})$Correct Option:...
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