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Let a, b, c are the direction ratios of the normal and (x, y, z) be the co-ordinate of the point through which the plane passes. Then the equation of plane is:

${\int }_0^{\pi /2}\sqrt{1-sin2x}dx\left(a\right)2\sqrt{2}\left(b\right)2(\sqrt{2}+1)\left(c\right)2\left(d\right)2(\sqrt{2}-1)$

The equation of the circumcircle of the triangle formed by the lines $xy-3x-2y+6=0$ and $x+y=0$ is

The area bounded by the curves $y=|x|-1$ and $y=-|x|+1$ is

If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3, is

A man has 7 relatives,4 of them are ladies and 3 of them are gentleman, his wife has also 7 relatives 4 of them are gentleman and 3 of them are ladies.In how many ways can they invite a dinner party for 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife' relatives.

Sin^{​​​​2}​​​24° - Sin^{​​​​2}​​​6°

Find the principal values of the following questions:

$cose{c}^{-1}$(2)

Find the equation of the tangent line to the curve $y=x^2-2x+7$ which is parallel to the line 5y-15x=13.

An electri

An integrating factor of the differential equation $x\frac{dy}{dx}+y\log x=x{e}^x{x}^{-\frac{1}{2}\log x},(x>0)$ is

Choose the word/group of words which is MOST SIMILAR in meaning of the words as used in the passage:

Broaden

In $△ABC,1-tan\frac{A}{2}tan\frac{B}{2}=$
[Roorkee 1973]

The argument of complex number Z = $\frac{3+i}{2i+(1+i{)}^2}$

The intercept on the line y = x by the circle $x^2+y^2-2x=0$ is AB. The equation of the circle on AB as a diameter is

If equation $a{x}^2+2cx+b=0$ and $a{x}^2+2bx+c=0$ have one root in common, then a + 4b + 4c equals

If the roots of the equation $x^2-2mx+m^2$ -1=0 lie in the interval (-2,4) then

The foci of the conjugate hyperbola of the hyberbola $\frac{x^2}{4}-\frac{y^2}{12}=1$

Answer the following by appropriately matching the lists based on the information given in Column I and Column II
​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{a.}f:R\to [\frac{3\pi }{4},\pi )\text{and}f\left(x\right)={\cot }^{-1}(2x-x^2-2),& \\ \text{then}f\text{is}& \text{p. one-one}\\ \text{b.}f:R\to R\text{and}f\left(x\right)=e^{px}\sin qx& \\ \text{where}p,q\in R^{+},\text{then}f\text{is}& \text{q. into}\\ \text{c.}f:R^{+}\to [4,\infty )\text{and}f\left(x\right)=4+3x^2,& \\ \text{then}f\text{is}& \text{r. many-one}\\ \text{d.}f:R\to R\text{and}f\left(f\right(x\left)\right)=x,\forall x\in R& \\ \text{then}f\text{is}& \text{s. onto}\end{array}$

If the end points of one axis of an ellipse are $(-12,4)$ and $(14,4)$ and eccentricity $\frac{12}{13}$, then the equation(s) of the ellipse is/are

A plane passes through a fixed point (a,b,c). Then the locus of the foot of the perpendicular to it from the origin is

The locus of midpoint of the chord of contact of $x^2+y^2=2$ from the points on $3x+4y=10$ is a circle whose centre is

if a, b, c are in

In the expansion of $(1+x{)}^n$ the sum of coefficients

of odd powers of x is

Find the equation of the parabola that satisfies the given conditons:

Vertex (0,0) and Focus (-2,0)

A biased ordinary die is loaded in such a way that probability of getting an even outcome is five times the probability of getting an odd outcome. This die is rolled two times. The probability that the sum of outcome will be a prime number is equal to:

The parametric form the curve
$\frac{(x+1{)}^2}{16}-\frac{(y-2{)}^2}{4}=1$ is

A circle is centered at origin. 2 points P and Q lies on the positive x axis outside the circle such that OQ = 2OP. The length of tangent drawn from Q to the circle is thrice the length of tangent from P to the circle. What the radius of circle if OP = a.

If $n$ is an odd integer, $i=\sqrt{-1}$, then $(1+i{)}^{6n}+(1-i{)}^{6n}$ is equal to

General solution of the equation $4\cos x-3\sec x=\tan x,(\cos x\ne 0)$ can be

If |z - 4| < |z - 2|, then:

The differential equation of the family of curves represented by the equation $x^{2}y=a$, is

There are ‘p’ points in space of which ‘q’ points are coplanar. Then the number of planes formed is

If the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ at any point on it cuts the axes OX and OY at P and Q respectively, then OP +OQ is

Differentiate the following functions with respect to x :

$\frac{a{x}^2+bx+c}{p{x}^2+qx+r}$

The angle between the lines represented by the equation $(x^2+y^2)sin\theta +2xy=0$ is

1/2+1/4+1/8+..........+1/2ⁿ=(1-1/2ⁿ)

A coin is tossed 6 times, in how many throws can 4 heads and 2 tails be obtained?

Find the equation of family of circles through the intersection of $x^2+y^2-6x+2y+4=0$ and $x^2+y^2+2x-4y-6=0$ whose center lies on $y=x.$

If we consider only the principle values of the inverse trigonometric functions, then the value of $tan(co{s}^{-1}\frac{1}{5\sqrt{2}}-si{n}^{-1}\frac{4}{\sqrt{17}})$