If the solve the problem
Question: If $\Delta=\left|\begin{array}{ccc}x-2 2 x-3 3 x-4 \\ 2 x-3 3 x-4 4 x-5 \\ 3 x-5 5 x-8 10 x-17\end{array}\right|=$ $\mathrm{Ax}^{3}+\mathrm{Bx}^{2}+\mathrm{Cx}+\mathrm{D}$, then $\mathrm{B}+\mathrm{C}$ is equal to :$-1$1$-3$9Correct Option: , 3 Solution: $\Delta=\left|\begin{array}{ccc}x-2 2 x-3 3 x-4 \\ 2 x-3 3 x-4 4 x-5 \\ 3 x-5 5 x-8 10 x-17\end{array}\right|$ $=\mathrm{Ax}^{3}+\mathrm{Bx}^{2}+\mathrm{Cx}+\mathrm{D}$ $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1} \quad \m...
Read More →The domain of the function
Question: The domain of the function $f(x)=\sin ^{-1}\left(\frac{|x|+5}{x^{2}+1}\right)$ is $(-\infty,-a] \cup[a, \infty)$. Then a is equal to :$\frac{1+\sqrt{17}}{2}$$\frac{\sqrt{17}-1}{2}$$\frac{\sqrt{17}}{2}+1$$\frac{\sqrt{17}}{2}$Correct Option: 1 Solution: $f(x)=\sin \left(\frac{|x|+5}{x^{2}+1}\right)$ For domain : $-1 \leq \frac{|x|+5}{x^{2}+1} \leq 1$ So for domain : $\frac{|x|+5}{x^{2}+1} \leq 1$ $\Rightarrow|x|+5 \leq x^{2}+1$ $\Rightarrow 0 \leq x^{2}-|x|-4$ $\Rightarrow 0 \leq\left(|x...
Read More →The integral
Question: The integral $\int\left(\frac{x}{x \sin x+\cos x}\right)^{2} d x$ is equal to : (where $\mathrm{C}$ is a constant of integration)$\sec x+\frac{x \tan x}{x \sin x+\cos x}+C$$\sec x-\frac{x \tan x}{x \sin x+\cos x}+C$$\tan x+\frac{x \sec x}{x \sin x+\cos x}+C$$\tan x-\frac{x \sec x}{x \sin x+\cos x}+C$Correct Option: , 4 Solution: $\int\left(\frac{x}{x \sin x+\cos x}\right)^{2} d x=\int\left(\frac{x}{\cos x}\right) \cdot \frac{x \cos x d x}{(x \sin x+\cos x)^{2}}$ $=\frac{x}{\cos x}\left...
Read More →Solve the Following Questions
Question: The value of $\left(2 .{ }^{1} P_{0}-3 .{ }^{2} P_{1}+4 .{ }^{3} P_{2}-\ldots .\right.$ up to $51^{\text {th }}$ term $)+\left(1 !-2 !+3 !-\ldots . .\right.$ up to $51^{\text {th }}$ term $)$ is equal to :$1+(51) !$$1-51(51) !$$1+(52) !$1Correct Option: , 3 Solution: $\mathrm{S}=\left(2 \cdot{ }^{1} \mathrm{p}_{0}-3 \cdot{ }^{2} \mathrm{p}_{1}+4 \cdot{ }^{3} \mathrm{p}_{2} \ldots \ldots \ldots . .\right.$ upto 51 terms $)$ $+(1 !+2 !+3 ! \ldots \ldots \ldots .$ upto 51 terms) $\left[\b...
Read More →A line parallel to the straight line 2 x-y=0 is tangent to the hyperbola
Question: A line parallel to the straight line $2 x-y=0$ is tangent to the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$ at the point $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$. Then $\mathrm{x}_{1}^{2}+5 \mathrm{y}_{1}^{2}$ is equal to :56810Correct Option: , 2 Solution: Slope of tangent is 2 , Tangent of hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$ at the point $\left(x_{1}, y_{1}\right)$ is $\frac{x x_{1}}{4}-\frac{y y_{1}}{2}=1 \quad(T=0)$ Slope : $\frac{1}{2} \frac{x_{1}}{y_{1}}=2 \Righ...
Read More →Solve this
Question: The mean and variance of 8 observations are 10 and $13.5$, respectively. If 6 of these observations are $5,7,10,12,14,15$, then the absolute difference of the remaining two observations is :7359Correct Option: 1 Solution: $\overline{\mathrm{X}}=10$ $\Rightarrow \bar{x}=\frac{63+a+b}{8}=10 \Rightarrow a+b=17$ .........(1) Since, variance is independent of origin. So, we subtract 10 from each observation. So, $\sigma^{2}=13.5=\frac{79+(\mathrm{a}-10)^{2}+(\mathrm{b}-10)^{2}}{8}-(10-10)^{...
Read More →The sum of the first three terms of a G.P.
Question: The sum of the first three terms of a G.P. is $S$ and their product is 27 . Then all such S lie in :$[-3, \infty)$$(-\infty, 9]$$(-\infty,-9] \cup[3, \infty)$$(-\infty,-3] \cup[9, \infty)$Correct Option: , 4 Solution: Let three terms of G.P. are $\frac{\mathrm{a}}{\mathrm{r}}, \mathrm{a}$, ar product $=27$ $\Rightarrow a^{3}=27 \Rightarrow a=3$ $\mathrm{S}=\frac{3}{\mathrm{r}}+3 \mathrm{r}+3$ For $r0$ $\frac{\frac{3}{r}+3 r}{2} \geq \sqrt{3^{2}} \quad($ By $\mathrm{AM} \geq \mathrm{GM}...
Read More →Solve the Following Questions
Question: The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R$, is increasing for all x lying in :$(-\infty, 0) \cup\left(\frac{3}{7}, \infty\right)$$(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$$\left(-\infty, \frac{14}{15}\right)$$\left(-\infty,-\frac{14}{15}\right) \cup(0, \infty)$Correct Option: , 2 Solution: $f(x)=(3 x-7) x^{2 / 3}$ $\Rightarrow \mathrm{f}(\mathrm{x})=3 \mathrm{x}^{5 / 3}-7 \mathrm{x}^{2 / 3}$ $\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=5 \mathrm{x}^{2 / 3}-\frac{14}{3 \...
Read More →If R={(x,y):x,y∈Z,x2+3y2≤8} is a relation on the set of integers Z,
Question: If $R=\left\{(x, y): x, y \in Z, x^{2}+3 y^{2} \leq 8\right\}$ is a relation on the set of integers $Z$, then the domain of $\mathrm{R}^{-1}$ is :$\{-2,-1,1,2\}$$\{-1,0,1\}$$\{-2,-1,0,1,2\}$$\{0,1\}$Correct Option: , 2 Solution: $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{z}, \mathrm{x}^{2}+3 \mathrm{y}^{2} \leq 8\right\}$ For domain of $\mathrm{R}^{-1}$ Collection of all integral of y's For $x=0,3 y^{2} \leq 8$ $\Rightarrow \mathrm{y} \in\{-1,0,1\}$...
Read More →Two vertical poles
Question: Two vertical poles $\mathrm{AB}=15 \mathrm{~m}$ and $\mathrm{CD}=10 \mathrm{~m}$ are standing apart on a horizontal ground with points $\mathrm{A}$ and $\mathrm{C}$ on the ground. If $\mathrm{P}$ is the point of intersection of $\mathrm{BC}$ and $\mathrm{AD}$, then the height of $P$ (in $m$ ) above the line $A C$ is :$20 / 3$5$10 / 3$6Correct Option: , 4 Solution: $\tan \theta=\frac{10}{\mathrm{x}}=\frac{\mathrm{h}}{\mathrm{x}_{2}} \Rightarrow \mathrm{x}_{2}=\frac{\mathrm{hx}}{10}$ $\t...
Read More →Solve the Following Questions
Question: If $y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then :$\left|y^{\prime \prime}(0)\right|=2$$\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=3$$\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=1$$y^{\prime \prime}(0)=0$Correct Option: 1 Solution: $y^{2}+\ln \left(\cos ^{2} x\right)=y \quad x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for $x=0$ $y=0$ or 1 Differentiating wrt $\mathrm{x}$ $\Rightarrow 2 y y^{\pr...
Read More →Let X0 be the point of local maxima of
Question: Let $x_{0}$ be the point of local maxima of $\mathrm{f}(\mathrm{x})=\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}), \quad$ where $\quad \overrightarrow{\mathrm{a}}=\mathrm{x} \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}}=-2 \hat{\mathrm{i}}+x \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$. Then the value of $\vec{a} \c...
Read More →Let X0 be the point of local maxima of
Question: Let $x_{0}$ be the point of local maxima of $\mathrm{f}(\mathrm{x})=\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}), \quad$ where $\quad \overrightarrow{\mathrm{a}}=\mathrm{x} \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}}=-2 \hat{\mathrm{i}}+x \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$. Then the value of $\vec{a} \c...
Read More →Consider the two sets :
Question: Consider the two sets : $A=\{m \in R:$ both the roots of $x^{2}-(m+1) x+m+4=0$ are real $\}$ and $\mathrm{B}=[-3,5)$ Which of the following is not true?$\mathrm{A}-\mathrm{B}=(-\infty,-3) \cup(5, \infty)$$\mathrm{A} \cap \mathrm{B}=\{-3\}$$\mathrm{B}-\mathrm{A}=(-3,5)$$A \cup B=R$Correct Option: 1 Solution: $\mathrm{A}: \mathrm{D} \geq 0$ $\Rightarrow(\mathrm{m}+1)^{2}-4(\mathrm{~m}+4) \geq 0$ $\Rightarrow \mathrm{m}^{2}+2 \mathrm{~m}+1-4 \mathrm{~m}-16 \geq 0$ $\Rightarrow \mathrm{m}^...
Read More →Let alpha and beta be the roots of the equation
Question: Let $\alpha$ and $\beta$ be the roots of the equation $5 x^{2}+6 x-2=0 .$ If $S_{n}=\alpha^{n}+\beta^{n}, n=1,2,3 \ldots .$, then :$5 \mathrm{~S}_{6}+6 \mathrm{~S}_{5}=2 \mathrm{~S}_{4}$$5 \mathrm{~S}_{6}+6 \mathrm{~S}_{5}+2 \mathrm{~S}_{4}=0$$6 \mathrm{~S}_{6}+5 \mathrm{~S}_{5}+2 \mathrm{~S}_{4}=0$$6 \mathrm{~S}_{6}+5 \mathrm{~S}_{5}=2 \mathrm{~S}_{4}$Correct Option: 1 Solution: $\alpha$ and $\beta$ are roots of $5 x^{2}+6 x-2=0$ $\Rightarrow 5 \alpha^{2}+6 \alpha-2=0$ $\Rightarrow 5 ...
Read More →Solve this
Question: Let $u=\frac{2 z+i}{z-k i}, z=x+i y$ and $k0$. If the curve represented by $\operatorname{Re}(u)+\operatorname{Im}(u)=1$ intersects the $y$-axis at the points $P$ and $Q$ where $P Q=5$, then the value of $\mathrm{k}$ is : $3 / 2$42$1 / 2$Correct Option: , 3 Solution: $\mathrm{u}=\frac{2 \mathrm{z}+\mathrm{i}}{\mathrm{z}-\mathrm{ki}}$ $=\frac{2 x^{2}+(2 y+1)(y-k)}{x^{2}+(y-k)^{2}}+i \frac{(x(2 y+1)-2 x(y-k))}{x^{2}+(y-k)^{2}}$ Since $\operatorname{Re}(u)+\operatorname{Im}(u)=1$ $\Righta...
Read More →Solve The Following Questions
Question: $\int_{-\pi}^{\pi}|\pi-| x|| d x$ is equal to :$\pi^{2}$$2 \pi^{2}$$\sqrt{2} \pi^{2}$$\frac{\pi^{2}}{2}$Correct Option: 1 Solution: $\int_{-\pi}^{\pi}|\pi-| x|| d x=2 \int_{0}^{\pi}|\pi-x| d x$ $=2 \int_{0}^{\pi}(\pi-x) d x$ $=2\left[\pi x-\frac{x^{2}}{2}\right]_{0}^{\pi}=\pi^{2}$...
Read More →The plane passing through the points (1,2,1), (2,1,2) and
Question: The plane passing through the points $(1,2,1)$, $(2,1,2)$ and parallel to the line, $2 \mathrm{x}=3 \mathrm{y}, \mathrm{z}=1$ also passes through the point :$(0,6,-2)$$(-2,0,1)$$(0,-6,2)$$(2,0,-1)$Correct Option: , 2 Solution: Two points on the line ( $\mathrm{L}$ say) $\frac{\mathrm{x}}{3}=\frac{\mathrm{y}}{2}, \mathrm{z}=1$ are $(0,0,1) \(3,2,1)$ So dr's of the line is $\langle 3,2,0\rangle$ Line passing through $(1,2,1)$, parallel to $\mathrm{L}$ and coplanar with given plane is $\o...
Read More →For the frequency distribution:
Question: For the frequency distribution: Variate $(\mathrm{x}): \quad \mathrm{x}_{1} \quad \mathrm{x}_{2} \quad \mathrm{x}_{3} \ldots \ldots \mathrm{x}_{15}$ Frequency (f) : $\quad \mathrm{f}_{1} \quad \mathrm{f}_{2} \quad \mathrm{f}_{3} \ldots . \mathrm{f}_{15}$ where $0\mathrm{x}_{1}\mathrm{x}_{2}\mathrm{x}_{3}\ldots .\mathrm{x}_{15}=10$ and $\sum_{i=1}^{15} f_{i}0$, the standard deviation cannot be :2146Correct Option: , 4 Solution: $\because \sigma^{2} \leq \frac{1}{4}(\mathrm{M}-\mathrm{m}...
Read More →Find the value
Question: A survey shows that $63 \%$ of the people in a city read newspaper $\mathrm{A}$ whereas $76 \%$ read newspaper B. If $x \%$ of the people read both the newspapers, then a possible value of $x$ can be: 65372955Correct Option: , 4 Solution: $n(B) \leq n(A \cup B) \leq n(U)$ $\Rightarrow 76 \leq 76+63-x \leq 100$ $\Rightarrow-63 \leq-x \leq-39$ $\Rightarrow 63 \geq x \geq 39$...
Read More →Let P(h,k) be a point on the curve
Question: Let $\mathrm{P}(\mathrm{h}, \mathrm{k})$ be a point on the curve $\mathrm{y}=\mathrm{x}^{2}+7 \mathrm{x}+2$, nearest to the line, $\mathrm{y}=3 \mathrm{x}-3$. Then the equation of the normal to the curve at $P$ is :$x+3 y-62=0$$x-3 y-11=0$$x-3 y+22=0$$x+3 y+26=0$Correct Option: , 4 Solution: Let $\mathrm{L}$ be the common normal to parabola $y=x^{2}+7 x+2$ and line $y=3 x-3$ $\Rightarrow$ slope of tangent of $y=x^{2}+7 x+2$ at $P=3$ $\left.\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}\ri...
Read More →Solve this
Question: Let $f(x)=\int \frac{\sqrt{x}}{(1+x)^{2}} d x(x \geq 0) .$ Then $f(3)-f(1)$ is equal to :$-\frac{\pi}{6}+\frac{1}{2}+\frac{\sqrt{3}}{4}$$\frac{\pi}{6}+\frac{1}{2}-\frac{\sqrt{3}}{4}$ $-\frac{\pi}{12}+\frac{1}{2}+\frac{\sqrt{3}}{4}$$\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}$Correct Option: , 4 Solution: $f(x)=\int_{i}^{3} \frac{\sqrt{x} d x}{(1+x)^{2}}=\int_{1}^{\sqrt{3}} \frac{t .2 t d t}{\left(1+t^{2}\right)^{2}} \quad($ put $\sqrt{x}=t)$ $=\left(-\frac{t}{1+t^{2}}\right)_{1}^{\sq...
Read More →Let P be a point on the parabola,
Question: Let $\mathrm{P}$ be a point on the parabola, $\mathrm{y}^{2}=12 \mathrm{x}$ and $\mathrm{N}$ be the foot of the perpendicular drawn from $\mathrm{P}$ on the axis of the parabola. A line is now drawn through the mid-point $\mathrm{M}$ of PN, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line NQ is $\frac{4}{3}$, then :$\mathrm{MQ}=\frac{1}{3}$$\mathrm{PN}=3$$\mathrm{MQ}=\frac{1}{4}$$\mathrm{PN}=4$Correct Option: , 3 Solution: Let $P=\left(3 t^{2}, 6 t...
Read More →If the tangent to the curve y = x + sin y
Question: If the tangent to the curve $y=x+\sin y$ at a point (a, b) is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right)$, then :$\mathrm{b}=\mathrm{a}$$\mathrm{b}=\frac{\pi}{2}+\mathrm{a}$$\mathrm{lb}-\mathrm{al}=1$$|a+b|=1$Correct Option: 3, Solution: Slope of tangent to the curve $y=x+\sin y$ at $(a, b)$ is $\frac{2-\frac{3}{2}}{\frac{1}{2}-0}=1$ $\left.\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\mathrm{x}=\mathrm{a}}=1$ $\frac{d y}{d x}=1+\...
Read More →Solve this following
Question: Let $\mathrm{P}(3,3)$ be a point on the hyperbola, $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$. If the normal to it at $\mathrm{P}$ intesects the $\mathrm{x}$-axis at $(9,0)$ and $\mathrm{e}$ is its eccentricity, then the ordered pair $\left(a^{2}, e^{2}\right)$ is equal to :$\left(\frac{9}{2}, 3\right)$$\left(\frac{9}{2}, 2\right)$$\left(\frac{3}{2}, 2\right)$$(9,3)$Correct Option: 1 Solution: Since, $(3,3)$ lies on $\frac{x^{2}}{a^{2}}-\frac{y^{2}...
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