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If the red line divides the parallelogram such that a is ${\frac{1}{3}}^{rd}$ of b and ${\frac{2}{3}}^{rd}$ of c, what is the measure of d?

If $y=(1+x)(1+x^2)(1+x^4).....(1+x^{2n}),\text{then}{\left(\frac{dy}{dx}\right)}_{x=0}=$

The area of the curve $x^2+y^2=2ax$ is

The value of $\int \frac{(x-2)dx}{{\left\{\right(x-2{)}^2(x+3{)}^7\}}^{1/3}}$ is

In a series of 20 observations , 10 observations are each equal to k and each of the remaining half is equal to -k. If the standard deviation of the observations is 2, then write value of k.

The middle term(s) in the expansion of ${(3x-\frac{x^3}{6})}^7$ is/are:

The value of the integral $\int \cos \sqrt{x}dx=$

Find the
values of k for which the lineis

(a) Parallel
to the x-axis,

(b) Parallel
to the y-axis,

(c) Passing
through the origin.

If $f\left(x\right)=x^3$, then the graph of $g\left(x\right)=f(x+2)$ is

Let $P$ be a non-singular matrix and $I+P+P^2+\cdots +P^n=O$. Then $P^{-1}$ is equal to (where $n\in Z^{+}$)

In following figure, the feasible region (shaded) for a LPP is shown. Determine the maximum and minimum value of Z =x + 2y.

If velocity of a particle is given by $v=2t-1$, then find the acceleration of particle at $t=2s$.

The equation of the straight line through the intersection of line $2x+y=1$ and $3x+2y=5$ and passes through the origin is

If ${}^8{C}_r{-}^7{C}_3{=}^7{C}_2,$ find r.

A radioactive nucleus can decay by two different processes. If the half lives of the first and second decay processes are $5\times {10}^3\text{years}$ and ${10}^5\text{years}$ respectively, then the effective half life of the nucleus is-(nearly)

Find the maximum and minimum values of f (x) = 2x³ – 24x + 107 in the interval [1, 3].

The value of ${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}$ is:

The equation of plane whose perpendicular distance from origin is $2\text{units}$ and vector $\hat{i}-2\hat{j}-2\hat{k}$ is normal to the plane, is

A line $L_1$ passes through the point $(1,1)$ and $(2,0)$ and another line $L_2$ passes through $(\frac{1}{2},0)$ and perpendicular to $L_1.$ Then the area (in sq. units) of the triangle formed by the lines $L_1,$ $L_2$ and $y-$axis is

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type
$\int \left[f\right(x\left)h\right(x)+g(x\left)\right]dx$
Let $\int f\left(x\right)h\left(x\right)dx=I_1$ and $\int g\left(x\right)dx=I_2$
Integrating $I_1$ by parts, we get
$I_1=f\left(x\right)\int h\left(x\right)dx-\int \left\{f^{\prime }\right(x)\int h(x\left)dx\right\}dx$

$\int \frac{x{e}^x}{(1+x{)}^2}dx$ is equal to

The equation of normal to the parabola $x^2=4y$ drawn from the point $(1,2)$ is

Planes are drawn through the points d(5, 0, 2) and (3, -2, 5)

parallel to the coordinate planes. Find the lengths of the edges of

the rectangular parallelopiped so formed.

The coefficient of $x^{10}$ in the expansion of $[1+x^2(1-x){]}^8$ is

$\begin{array}{cccccc}& ColumnI& & Column2& & Column3\\ \left(I\right)& cos\left(si{n}^{-1}\right(sin\frac{5\pi }{6}\left)\right)=& \left(i\right)& 1& \left(P\right)& f\left(x\right)=3x^3-13x^2+14x-2has\\ & & & & & threeroots\alpha ,\beta ,\gamma ,thenvalueof\\ & & & & & |sin(ta{n}^{-1}\alpha +ta{n}^{-1}\beta +ta{n}^{-1}\gamma )|=\\ \left(II\right)& cos\left(co{s}^{-1}\right(-sin\frac{7\pi }{6}\left)\right)=& \left(ii\right)& \frac{1}{\sqrt{2}}& \left(Q\right)& \frac{(\sqrt{3}cosec{20}^{\circ }-sec{20}^{\circ })}{8}=\\ \left(III\right)& \left|cos\left\{ta{n}^{-1}\right(tan\frac{15\pi }{4}\left)\right\}\right|=& \left(iii\right)& \frac{1}{2}& \left(R\right)& 4\sqrt{2}\left(sin{12}^{\circ }\right)\left(sin{48}^{\circ }\right)\left(sin{54}^{\circ }\right)=\\ \left(IV\right)& sin({(co{s}^{-1}\left\{\frac{1}{\sqrt{2}}(cos\frac{9\pi }{10}-sin\frac{9\pi }{10})\right\}+\frac{7\pi }{20})}^{-1}\times {\pi }^2)& \left(iv\right)& \frac{\sqrt{3}}{2}& \left(S\right)& IftheanglesA,BandCofa\\ & & & & & triangleareinanarithmeticprogression\\ & & & & & andifa,bandcdenotethelengthsof\\ & & & & & sidesoppositetoA,BandCrespectively,\\ & & & & & thenthevalueoftheexpression\\ & & & & & \frac{1}{2}(\frac{a}{c}sin2C+\frac{c}{a}sin2A)is\end{array}$

Which of the following options is only INCORRECT combination?

$x^2+y^2-4x+6y-12=0$ and $x^2+y^2+6x+18y+26=0$ __________.

The value of $\underset{1}{\overset{2}{\int }}(4x^3-5x^2+6x+9)dx$ is:

The equations $\lambda x-y=2,2x-3y=-\lambda ,3x-2y=-1$ are consistent for

Number of ways in which $10$ different diamonds can be arranged to make a necklace is