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The ends of latus rectum of parabola $x^2+8y=0$ are

The solution(s) of the equation $9{\cos }^{12}x+{\cos }^{2}2x+1=6{\cos }^{6}x\cos 2x+6{\cos }^{6}x-2\cos 2x$ is/are $(n\in I)$.

In a city $20\%$ of the population travels by car, $50\%$ travels by bus and $10\%$ travels by both car and bus. Then, persons travelling by car or bus is

If $(1+i\sqrt{3}{)}^9$=a+ib, then b is equal to

If the four complex numbers $z$, $\overline{z},\overline{z}-2Re\left(\overline{z}\right)$ and $z-2Re\left(z\right)$ represent the vertices of a square of side $4$ units in the Argand plane, then $\left|z\right|$ is equal to:

Find the equation of the circle which touches the axes and whose' centre lies on x - 2y = 3.

Find the shortest distance between the lines whose vector equations are $r=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-3\hat{j}+2\hat{k})$
and $r=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (2\hat{i}+3\hat{j}+\hat{k})$

The point $(-2m,m+1)$ is an interior point of the smaller region bounded by the circle $x^2+y^2=4$ and the parabola $y^2=4x$. Then $m$ belongs to the intyerval

The equation of the plane passing through the point $(3,-6,9)$ and perpendicular to the $x$ axis is:

If $se{c}^{-1}\sqrt{1-x^2}+cose{c}^{-1}\frac{\sqrt{1+y^2}}{4}+co{t}^{-1}\frac{1}{z}=\pi$ ,then x+y+z is equal to

Distinguish between convenience and shopping products.

The factor of the determinant

$\left|\begin{array}{ccc}x& 5& 2\\ x^2& 9& 4\\ x^3& 16& 8\end{array}\right|$ is (x - .......)

The exhaustive domain of the function ${\sin }^{-1}\left[{\log }_2\right(x^2/2\left)\right]$ is

√3 cosec 20° - sec 20° =

(A) 2

(B) 2 sin 20° / sin 40°

(C) 4

(D) 4 sin 20°/ sin 40°

The assignment problem in Linear Programming is also an example of a discrete optimization problem. How many feasible solutions are there to this problem defined on n jobs and n persons?

If the locus of the circumcentre of a variable triangle having sides $y-$axis, $y=2$ and $lx+my=1$, where $(l,m)$ lies on the parabola $y^2=4ax$ is a curve $C$, then the length of smallest focal chord of this curve $C$ (in units) is

Find the values of a,b,c and d, if

$3\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& 6\\ -1& 2d\end{array}\right]+\left[\begin{array}{cc}4& a+b\\ c+d& 3\end{array}\right]$

A point moves so that the sum of its distances from the points (4, 0, 0) and (-4, 0, 0) remains 10. The locus of the point is
[MP PET 1988]

Let f and g are two real valued differentiable functions satisfying.
$f\left(x\right)=\underset{Lt}{\alpha \to 0}\frac{1}{{\alpha }^4}{\int }_0^{\alpha }\frac{(e^{x+t}-e^x)\left(l{n}^2\right(t+1\left)\right)}{2t^2+3}dtand{\int }_0^{x}g\left(t\right)dt=3x+{\int }_x^{0}co{s}^{2}tg\left(t\right)dt$
Range of g(x) is

The distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5 = 0 measured parallel to the line $\frac{x+3}{3}=\frac{y-2}{6}=\frac{z}{2}$ is

Which of the following is the correct expansion of the bracket for the multiplication of the given numbers?

$15\times 102$

The value of $\underset{x\to \frac{\pi }{6}}{lim}\frac{2{\sin }^{2}x+\sin x-1}{2{\sin }^{2}x-3\sin x+1}$ is

The locus of the point, whose chord of contact w.r.t the circle $x^2+y^2=a^2$ makes an angle $2\alpha$ at the centre of the circle is

At time t = 0, a material is composed of two radioactive atoms A and B, where $N_A\left(0\right)=2N_B\left(0\right)$. The decay constant of both kind of radioactive atoms is $\lambda$. However, A disintegrates to B and B disintegrates to C. Which of the following figures represents the evolution of $N_B\left(t\right)/N_B\left(0\right)$ with respect to time t ?

$\left[N_A\right(0)=\text{No. of atoms at t = 0}{N}_B\left(0\right)=\text{No. of B atoms at t = 0}]$

A ray of light is sent along the line $x-2y-3=0$. Upon reaching the line $3x-2y-5=0$ the ray is reflected from it. The equation of the reflected ray is

If $y={\cos }^{-1}\frac{1-(\log x{)}^2}{1+(\log x{)}^2},$ then $f^{\prime }\left(e\right)=$

If a, b, c are positive real numbers such that a + b + c = 18, then maximum value of $a^2{b}^3{c}^4$ will be

The total number of ways in which a beggar can be given at least one rupee from two $1$ rupee coins, three $50$ paisa coins and four $25$ paisa coins is

If $f\left(x\right)$ is quadratic in $x$, then ${\int }_0^{1}f\left(x\right)dx$ is

Suppose a,b,c are positive integers such that $2^a+4^b+8^c=328$ Then $\frac{\alpha +2b+3c}{abc}$ is equal to

The number of distinct real roots of $\left|\begin{array}{ccc}sinx& cosx& cosx\\ cosx& sinx& cosx\\ cosx& cosx& sinx\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is

(a) 0
(b) 2
(c) 1
(d) 3