Solve the following equations:
Question: If $\quad f(x)=\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x,(x \geq 0)$ and $f(0)=0$, then the value of $f(1)$ is :$-\frac{1}{2}$$\frac{1}{2}$$-\frac{1}{4}$$\frac{1}{4}$Correct Option: , 4 Solution: $\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x$ $=\int \frac{5 x^{-6}+7 x^{-8}}{\left(\frac{1}{x^{7}}+\frac{1}{x^{5}}+2\right)^{2}} d x=\frac{1}{2+\frac{1}{x^{5}}+\frac{1}{x^{7}}}+C$ As $f(0)=0, f(x)=\frac{x^{7}}{2 x^{7}+x^{2}+1}$ $f(1)=\frac{1}{4}$...
Read More →Solve this following
Question: Let $\mathrm{z}=\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)^{5}+\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)^{5}$. If $\mathrm{R}(\mathrm{z})$ and $\mathrm{I}[\mathrm{z}]$ respectively denote the real and imaginary parts of $z$, then : $\mathrm{R}(\mathrm{z})0$ and $\mathrm{I}(\mathrm{z})0$$\mathrm{R}(\mathrm{z})0$ and $\mathrm{I}(\mathrm{z})0$$R(z)=-3$$I(z)=0$Correct Option: , 4 Solution: $z=\left(\frac{\sqrt{3}+i}{2}\right)^{5}+\left(\frac{\sqrt{3}-i}{2}\right)^{5}$ ...
Read More →The area (in sq. units) in the first quadrant bounded by the parabola,
Question: The area (in sq. units) in the first quadrant bounded by the parabola, $\mathrm{y}=\mathrm{x}^{2}+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is :-$\frac{14}{3}$$\frac{187}{24}$$\frac{37}{24}$$\frac{8}{3}$Correct Option: , 3 Solution: Area $=\int_{0}^{2}\left(x^{2}+1\right) d x-\frac{1}{2}\left(\frac{5}{4}\right)(5)=\frac{37}{24}$...
Read More →The value of
Question: Let $A=\{x \varepsilon R: x$ is not a positive integer $\}$ Define a function $f: \mathrm{A} \rightarrow \mathrm{R}$ as $f(\mathrm{x})=\frac{2 \mathrm{x}}{\mathrm{x}-1}$ then $f$ isinjective but not surjectivenot injectivesurjective but not injectiveneither iniective nor suriectiveCorrect Option: 1 Solution: $f(\mathrm{x})=2\left(1+\frac{1}{\mathrm{x}-1}\right)$ $f^{\prime}(\mathrm{x})=-\frac{2}{(\mathrm{x}-1)^{2}}$ $\Rightarrow f$ is one-one but not onto...
Read More →If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13 ,
Question: If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13 , then the eccentricity of the hyperbola is :-2$\frac{13}{6}$$\frac{13}{8}$$\frac{13}{12}$Correct Option: , 4 Solution: $2 \mathrm{~b}=5$ and $2 \mathrm{ae}=13$ $b^{2}=a^{2}\left(e^{2}-1\right) \Rightarrow \frac{25}{4}=\frac{169}{4}-a^{2}$ $\Rightarrow \mathrm{a}=6 \Rightarrow \mathrm{e}=\frac{13}{12}$...
Read More →A hyperbola has its centre at the origin,
Question: A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is:$\frac{2}{\sqrt{3}}$$\frac{3}{2}$$\sqrt{3}$2Correct Option: 1 Solution: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $2 a=4 \quad a=2$ $\frac{x^{2}}{4}-\frac{y^{2}}{b^{2}}=1$ Passes through $(4,2)$ $4-\frac{4}{b^{2}}=1 \Rightarrow b^{2}=\frac{4}{3} \Rightarrow e=\frac{2}{\sqrt{3}}$...
Read More →If in a parallelogram ABDC, the coordinates of A, B and C are respectively
Question: If in a parallelogram $\mathrm{ABDC}$, the coordinates of $\mathrm{A}$, $B$ and $C$ are respectively $(1,2),(3,4)$ and $(2,5)$, then the equation of the diagonal $\mathrm{AD}$ is:-$5 x+3 y-11=0$$3 x-5 y+7=0$$3 x+5 y-13=0$$5 x-3 y+1=0$Correct Option: , 4 Solution: co-ordinates of point $\mathrm{D}$ are $(4,7)$ $\Rightarrow$ line $\mathrm{AD}$ is $5 \mathrm{x}-3 \mathrm{y}+1=0$...
Read More →The number of all possible positive integral values
Question: The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $6 x^{2}-11 x+\alpha=0$ are rational numbers is :2534Correct Option: , 3 Solution: $6 x^{2}-11 x+\alpha=0$ given roots are rational $\Rightarrow$ D must be perfect square $\Rightarrow 121-24 \alpha=\lambda^{2}$ $\Rightarrow$ maximum value of $\alpha$ is 5 $\alpha=1 \Rightarrow \lambda \notin \mathrm{I}$ $\alpha=2 \Rightarrow \lambda \notin \mathrm{I}$ $\alpha=3 \Rightarrow \la...
Read More →A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement.
Question: A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If $X$ be the number of white balls drawn, the $\left(\frac{\text { mean of } X}{\text { standard deviation of } X}\right)$ is equal to:-4$\frac{4 \sqrt{3}}{3}$$4 \sqrt{3}$$3 \sqrt{2}$Correct Option: , 3 Solution: $p$ (probability of getting white ball) $=\frac{30}{40}$ $\mathrm{q}=\frac{1}{4}$ and $\mathrm{n}=16$ mean $=n p=16 \cdot \frac{3}{4}=12$ and standard diviati...
Read More →If the lines x=a y+b, z=c y+d
Question: If the lines $x=a y+b, z=c y+d$ and $x=a^{\prime} z+b^{\prime}$, $\mathrm{y}=\mathrm{c}^{\prime} \mathrm{z}+\mathrm{d}^{\prime}$ are perpendicular, then:$c^{\prime}+a+a^{\prime}=0$$\mathrm{aa}^{\prime}+\mathrm{c}+\mathrm{c}^{\prime}=0$$a b^{\prime}+b c^{\prime}+1=0$$\mathrm{bb}^{\prime}+\mathrm{cc}^{\prime}+1=0$Correct Option: , 2 Solution: Line $x=a y+b, z=c y+d \Rightarrow \frac{x-b}{a}=\frac{y}{1}=\frac{z-d}{c}$ Line $x=a^{\prime} z+b^{\prime}, y=c^{\prime} z+d^{\prime}$ $\Rightarro...
Read More →If the solve the problem
Question: If $\int \frac{\mathrm{x}+1}{\sqrt{2 \mathrm{x}-1}} \mathrm{dx}=\mathrm{f}(\mathrm{x}) \sqrt{2 \mathrm{x}-1}+\mathrm{C}$, where $C$ is a constant of integration, then $\mathrm{f}(\mathrm{x})$ is equal to :-$\frac{1}{3}(x+4)$$\frac{1}{3}(x+1)$$\frac{2}{3}(x+2)$$\frac{2}{3}(x-4)$Correct Option: 1 Solution: $\sqrt{2 \mathrm{x}-1}=\mathrm{t} \Rightarrow 2 \mathrm{x}-1=\mathrm{t}^{2} \Rightarrow 2 \mathrm{dx}=2 \mathrm{t} \cdot \mathrm{dt}$ $\int \frac{x+1}{\sqrt{2 x-1}} d x=\int \frac{\fra...
Read More →Prove the following
Question: If $x=\sin ^{-1}(\sin 10)$ and $y=\cos ^{-1}(\cos 10)$, then $y-x$ is equal to:$\pi$$7 \pi$010Correct Option: 1, Solution: $y=\cos ^{-1}(\cos 10)=4 \pi-10$ $y-x=\pi$...
Read More →If x = 3 tan t and y = 3 sec t, then the value of
Question: If $x=3 \tan t$ and $y=3 \sec t$, then the value of $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ at $\mathrm{t}=\frac{\pi}{4}$, is:$\frac{3}{2 \sqrt{2}}$$\frac{1}{3 \sqrt{2}}$$\frac{1}{6}$$\frac{1}{6 \sqrt{2}}$Correct Option: , 4 Solution: $\frac{\mathrm{dx}}{\mathrm{dt}}=3 \sec ^{2} \mathrm{t}$ $\frac{\mathrm{dy}}{\mathrm{dt}}=3 \sec t \tan t$ $\frac{d y}{d x}=\frac{\tan t}{\sec t}=\sin t$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\cos \mathrm{t} \frac{\mathrm{dt}}{\ma...
Read More →Let A and B be two invertible matrices of order 3 X 3.
Question: Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\operatorname{det}\left(A B A^{T}\right)=8$ and $\operatorname{det}\left(A B^{-1}\right)=8$, then $\operatorname{det}\left(\mathrm{BA}^{-1} \mathrm{~B}^{\mathrm{T}}\right)$ is equal to :-16$\frac{1}{16}$$\frac{1}{4}$1Correct Option: 2, Solution: $|\mathrm{A}|^{2} \cdot|\mathrm{B}|=8$ and $\frac{|\mathrm{A}|}{|\mathrm{B}|}=8 \Rightarrow|\mathrm{A}|=4$ and $|\mathrm{B}|=\frac{1}{2}$ $\therefore \operatorname{det}\left(...
Read More →Let the equations of two sides of a triangle be
Question: Let the equations of two sides of a triangle be $3 x$ $-2 y+6=0$ and $4 x+5 y-20=0$. If the orthocentre of this triangle is at $(1,1)$, then the equation of its third side is :$122 y-26 x-1675=0$$26 x+61 y+1675=0$$122 y+26 x+1675=0$$26 x-122 y-1675=0$Correct Option: , 4 Solution: Equation of $\mathrm{AB}$ is $3 x-2 y+6=0$ equation of $\mathrm{AC}$ is $4 x+5 y-20=0$ Equation of BE is $2 x+3 y-5=0$ Equation of CF is $5 x-4 y-1=0$ $\Rightarrow$ Equation of $\mathrm{BC}$ is $26 \mathrm{x}-...
Read More →The equation of the plane containing the straight line
Question: The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is:$x+2 y-2 z=0$$x-2 y+z=0$$5 \mathrm{x}+2 \mathrm{y}-4 \mathrm{z}=0$$3 x+2 y-3 z=0$Correct Option: , 2 Solution: Vector along the normal to the plane containing the lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is $...
Read More →The solution of the differential equation,
Question: The solution of the differential equation, $\frac{\mathrm{dy}}{\mathrm{dx}}=(\mathrm{x}-\mathrm{y})^{2}$, when $\mathrm{y}(1)=1$, is :-$\log _{\mathrm{e}}\left|\frac{2-\mathrm{y}}{2-\mathrm{x}}\right|=2(\mathrm{y}-1)$$\log _{e}\left|\frac{2-x}{2-y}\right|=x-y$$-\log _{\mathrm{e}}\left|\frac{1+\mathrm{x}-\mathrm{y}}{1-\mathrm{x}+\mathrm{y}}\right|=\mathrm{x}+\mathrm{y}-2$$-\log _{e}\left|\frac{1-x+y}{1+x-y}\right|=2(x-1)$Correct Option: , 4 Solution: $x-y=t \Rightarrow \frac{d y}{d x}=1...
Read More →Solve the following quadratic equations
Question: If $0 \leq x\frac{\pi}{2}$, then the number of values of $x$ for which $\sin x-\sin 2 x+\sin 3 x=0$, is2134Correct Option: 1 Solution: $\sin x-\sin 2 x+\sin 3 x=0$ $\Rightarrow(\sin x+\sin 3 x)-\sin 2 x=0$ $\Rightarrow 2 \sin x \cdot \cos x-\sin 2 x=0$ $\Rightarrow \sin 2 x(2 \cos x-1)=0$ $\Rightarrow \sin 2 x=0$ or $\cos x=\frac{1}{2}$ $\Rightarrow x=0, \frac{\pi}{3}$...
Read More →Contrapositive of the statement
Question: Contrapositive of the statement "If two numbers are not equal, then their squares are not equal." is :-If the squares of two numbers are equal, then the numbers are equal.If the squares of two numbers are equal, then the numbers are not equal.If the squares of two numbers are not equal, then the numbers are equal.If the squares of two numbers are not equal, then the numbers are not equal.Correct Option: 1 Solution: Contrapositive of $\mathrm{p} \rightarrow \mathrm{q}$ is $\sim \mathrm{...
Read More →An urn contains 5 red and 2 green balls.
Question: An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is :$\frac{26}{49}$$\frac{32}{49}$$\frac{27}{49}$$\frac{21}{49}$Correct Option: , 2 Solution: $\mathrm{E}_{1}$ : Event of drawing a Red ba...
Read More →The number of functions
Question: The number of functions $\mathrm{f}$ from $\{1,2,3, \ldots, 20\}$ onto $\{1,2,3, \ldots \ldots, 20\}$ such that $f(k)$ is a multiple of 3 , whenever $\mathrm{k}$ is a multiple of 4 , is :-$(15) ! \times 6 !$$5^{6} \times 15$$5 ! \times 6 !$$6^{5} \times(15) !$Correct Option: 1 Solution: $f(\mathrm{k})=3 \mathrm{~m}(3,6,9,12,15,18)$ for $k=4,8,12,16,20$ $6.5 .4 .3 .2$ ways For rest numbers 15 ! ways Total ways $=6 !(15 !)$...
Read More →The logical statement
Question: The logical statement $[\sim(\sim \mathrm{p} \vee \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{r}) \wedge(\sim \mathrm{q} \wedge \mathrm{r})]$ is equivalent to:$(\mathrm{p} \wedge \mathrm{r}) \wedge \sim \mathrm{q}$$(\sim p \wedge \sim q) \wedge r$$\sim \mathrm{p} \vee \mathrm{r}$$(p \wedge \sim q) \vee r$Correct Option: 1 Solution: $\mathrm{s}[\sim(\sim \mathrm{p} \vee \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})] \cap(\sim \mathrm{q} \wedge \mathrm{r})$ $\equiv[(\mathrm{p} \wedge \...
Read More →Let A(4,-4) and B(9,6) be points on the parabola,
Question: Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, $\mathrm{y}^{2}+4 \mathrm{x}$. Let $\mathrm{C}$ be chosen on the arc $\mathrm{AOB}$ of the parabola, where $O$ is the origin, such that the area of $\triangle A C B$ is maximum. Then, the area (in sq. units) of $\triangle \mathrm{ACB}$, is:$31 \frac{3}{4}$32$30 \frac{1}{2}$$31 \frac{1}{4}$Correct Option: , 4 Solution: Area $=5\left|\mathrm{t}^{2}-\mathrm{t}-6\right|=5\left|\left(\mathrm{t}-\frac{1}{2}\right)^{2}-\frac{25}{4}\right|$...
Read More →If the solve the problem
Question: Two lines $\quad \frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1} \quad$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $x y$-plane has coordinates :-$(2,4,7)$$(-2,4,7)$$(2,-4,-7)$$(2,-4,7)$Correct Option: , 3 Solution: Point on $\mathrm{L}_{1}(\lambda+3,3 \lambda-1,-\lambda+6)$ Point on $\mathrm{L}_{2}(7 \mu-5,-6 \mu+2,4 \mu+3)$ $\Rightarrow \lambda+3=7 \mu-5$ ......(i) $3 \lambda-1=-6 \mu+2$ ......(ii) $\Rightarrow \lambda=-1, \mu=...
Read More →Solve this following
Question: Let $\mathrm{n} \geq 2$ be a natural number and $0\theta\pi / 2$. Then $\int \frac{\left(\sin ^{\mathrm{n}} \theta-\sin \theta\right)^{\frac{1}{n}} \cos \theta}{\sin ^{\mathrm{n}+1} \theta} \mathrm{d} \theta$ is equal to : (Where $\mathrm{C}$ is a constant of integration)$\frac{\mathrm{n}}{\mathrm{n}^{2}-1}\left(1-\frac{1}{\sin ^{\mathrm{n}+1} \theta}\right)^{\frac{\mathrm{n}+1}{\mathrm{n}}}+C$$\frac{n}{n^{2}+1}\left(1-\frac{1}{\sin ^{n-1} \theta}\right)^{\frac{n+1}{n}}+C$$\frac{n}{n^{...
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