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Equation of line of shortest distance between the skew lines. $\overline{r}=(3\hat{i}+5\hat{j}+7\hat{k})+\lambda (\hat{i}-2\hat{j}+\hat{k})$
$\overline{r}=(-\hat{i}-\hat{j}-\hat{k})+\mu (7\hat{i}-6\hat{j}+\hat{k})$

The tangent of the graph of the function y = f(x) at the point with abscissa x = 1 form an angle of $\frac{\pi }{6}$ and at the point x = 2 an angle of $\frac{\pi }{3}$ and at the point x = 3 angle of
​$\frac{\pi }{4}$. The value of ${\int }_1^3{f}^{\prime }\left(x\right)f"\left(x\right)dx+{\int }_1^{3}f"\left(x\right)dx(f"(x\left)\right)$ is suppose to be continuous) is

$\text{In triangle ABC, internal angle bisector of}\angle A\text{makes an angle}\theta \text{with side BC}.\text{Value of sin}\theta \text{is equal to}$

In $\Delta ABC$, foot of the altitudes from vertices to the opposite sides are (0, 0) (3, 0) and (0, 4) then the orthocenter of $\Delta ABC$ is

If $tan(\frac{\pi }{4}+\theta )+tan(\frac{\pi }{4}-\theta )=\lambda sec2\theta$, then

If the lengths of the sides of a triangle be $7,4\sqrt{3}and\sqrt{13}cm$, then the smallest angle is

The equation of the directrix of the parabola with vertex at the origin and having the axis along $x$ − axis and a common tangent of slope $2$ with the circle $x^2+y^2=5$ is⁄are

Consider $f\left(x\right)=1+\underset{0}{\overset{2x}{\int }}\frac{e^{\frac{t}{2}}\cdot f\left(\frac{t}{2}\right)\left(2t\right)}{\sqrt{16-t^4}}dt-\underset{x}{\overset{0}{\int }}f\left(t\right)\cdot e^t{\sin }^{-1}\left(t^2\right)dt$ and $h\left(x\right)=\sin \left(e^{-x}\ln \right(f\left(x\right)\left)\right)$.
Then the range of $y=h\left(x\right)+4x+5$ is

The equationof one side of an equilateral triangle is $x-y=0$ and one vertex is $(2+\sqrt{3},5)$. Prove that the second side is $y+(2-\sqrt{3})x=6$ and find the equation of the third side.

The line $\frac{x}{a}+\frac{y}{b}=1$ moves in such a way that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$ where $c$ is a constant. The locus of foot of perpendicular from the origin on the given line will be

If $p,q$ be two prime numbers such that $p+q=31$, then the possible value(s) of $3p+q$ is/are

Find the equation of a line passing through $(3,-2)$ and perpendicular to the line $x-3y+5=0$

If the point $(\alpha ,\alpha )$ lies between the lines $|2x+y|=5$ then select one of the most appropriate option:

Let O be the vertex and Q be any point on the parabola $x^2=8y$. If the point P divides the line segement OQ internally in the ratio 1 : 3, then the locus of P is

A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn at random. From the box, what is the probability that :

(i) all are blue?

(ii) at least one is green?

A variable circle passes through the point $P(1,2)$ and touches the $x-$axis. The locus of the other end of the diameter through $P$ is

If A and B are square matrices of same order and A is non-singular, then for a positive integer n, $(A^{-1}BA{)}^n$ is equal to

The equation of a straight line(s) passing through $(1,2)$ and having intercept of length $3$ units between the straight lines $3x+4y=24$ and $3x+4y=12$, is

Find area of the triangle with vertices at the points in each of the following

(2,7), (1,1), (10,8)

The minimum value of $|z-1|+|z-3|$ is

The range of the function $f\left(x\right)=2|x-1|+|x+2|$, $-1\le x\le 2$ is

$f\left(x\right)=\{\begin{array}{c}{x}^{2}x\le x_0\\ ax+b,x>x_0\end{array}$. If f(x) is differentiable at $x_0$. Then

Lt _x→π/2 cot(x) –cos(x)/(π/2 –x)³

How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?

, the solution set of equation 4sin theta into cos theta - 2 cos theta minus 2 root 3 sin theta + root 3 is equal to zero in the interval 0 to 2 pie is

Value of cos1410^o

Given f(1) = 2 and f(n + 1) $=\frac{f\left(n\right)-1}{f\left(n\right)+1}\forall nϵN$ then which of the following is/ are correct?

If $\frac{T_2}{T_3}$ in the expansion of $(a+b{)}^n$ and $\frac{T_3}{T_4}$ in the expansion of $(a+b{)}^{n+3}$ are equal, then n =

If the centre of the circle $2x^2+pxy+q{y}^2+2gx+2fy+3$ = 0 is (1,-3), then the radius of the circle is

The shortest distance from the line 3x + 4y = 25 to the circle $x^2+y^2-6x+8y=0$ is equal to (in units)

Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability, can leave the cabin at any floor beginning with the first. Find out the probability of all five persons leaving at different floors.

[cos⁴ A - sin⁴A] is equal to:

The larger of ${99}^{50}+{100}^{50}and{101}^{50}$ is . . .

Find the value of ${}^n{C}_{n-r}$.

The equation of the circle which touches both the axes and whose radius is a, is

The angle between the two curves $x^3–3x{y}^2=4\text{and}3x^{2}y–y^3=4$ is

If the line, $y=\sqrt{3}x+k$ touches the circle $x^2+y^2=16$, then k =

The sum of the series
$2\cdot {}^{20}{C}_0+5\cdot {}^{20}{C}_1+8\cdot {}^{20}{C}_2+11\cdot {}^{20}{C}_3+\cdots +62\cdot {}^{20}{C}_{20}$
is equal to:

${2.7}^n+{3.5}^n-5$ is divisible by 24 for all $nϵN$.

$n^3-7n+3$ is divisible by 3 for all $nϵN$.