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A line passing through the point $(4,6)$, then the locus of the mid point of the portion of line between the co-ordinate axes is

The point dividing the line joining the points (1, 2, 3) and (3, -5, 6) in the ratio 3 : -5 is

Prove that:

$(\text{cos x}-\text{cos y}{)}^2+(\text{sin x}-\text{sin y}{)}^2=4{\text{sin}}^2\frac{x-y}{2}$

For circles $x^2+y^2+2x-8y+13=0$ and $x^2+y^2-12x-14y+76=0$ equation of all the common tangents are:

At least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8, is

Suppose OABC is a rectangle in the xy-plane where O is the origin and A, B lie on the parabola $y=x^2$. Then C must lie on the curve.

Evaluate the integrals using substitution.
${\int }_{-1}^1\frac{1}{x^2+2x+5}dx.$

If is a prime number, then $n^p$ - n is divisible by p

when n is a

$x=sin\theta +\theta cos\theta ,y=cos\theta -\theta sin\theta ,then{\left(\frac{dy}{dx}\right)}_{\theta =\frac{\pi }{2}}$

In a game, a man wins Rs. $1000$ if he gets an even number $\ge 4$ on a fair die and loses Rs. $200$ for getting any other number on the die. If he decides to throw the die until he wins or maximum of three times, then his expected gain/loss (in Rupees) is

Integrate the following functions.
$\int \frac{e^{2x}-1}{e^{2x}+1}dx.$

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

If $tanx=\frac{b}{a}$, then find the value of $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}$

Let $T_r$ be the $r^{th}$ term of A.P for r = 1, 2, 3…. If for some positive integer m and n we have $T_m=\frac{1}{n}and{T}_n=\frac{1}{m}$ , then $T_{mn}$

The number of real solutions of the equation ${27}^{\frac{1}{x}}+{12}^{\frac{1}{x}}=2\left(8^{\frac{1}{x}}\right)$ is

Vector of magnitude 1 is called.

From the following equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.​

$y^2=a(b^2-x^2)$

If $x+y+z=4,x^2+y^2+z^2=14$ and $x^3+y^3+z^3=34$, then the number of ordered triplet solution (x, y, z) is given by

If the functions $g\left(x\right)=\{\begin{array}{cc}{x}^2,& -1\le x\le 2\\ x+2,& 2<x\le 3\end{array}$

and $f\left(x\right)=\{\begin{array}{cc}x+4,& x\le 1\\ 2x+1,& 1<x\le 2\end{array}$

then, the number of roots fo the equation $f\left(g\right(x\left)\right)=0$ is

The radius of an air bubble is increasing at the rate of $\frac{1}{2}$ cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Values of $x$ satisfying the inequality $x^{\left(\frac{1}{{\log }_{10}x}\right)}.{\log }_{10}x<1$ is -

Area of region bounded by the curve $y=\frac{16-x^2}{4}$ and $y=se{c}^{-1}[-si{n}^{2}x]$ (where [.] denotes greatest integer function) is

If $p_1$ and $p_2$ are the lengths of the perpendiculars from the origin to the straight lines $x\sec \theta +y\text{cosec}\theta =a$ and $x\cos \theta -y\sin \theta =a\cos 2\theta$ respectively, then the value of $4p_1^2+p_2^2$ is

An urn contains m white and n black balls. A ball is drawn at randow and is put back into the urn along with K additional balls of the same colour as that of the ball drawn. A ball is again drawn at randow. Show that the probability of drawing a white ball now does not depend on k.

If $tanx=\frac{2b}{a-c}(a\ne c),y=aco{s}^{2}x+2bsinxcosx+csi{n}^{2}xandz=asi{n}^{2}x-2bsinxcosx+co{s}^{2}x,$ then

If f(x) $=2x^2,$find $\frac{f\left(3.8\right)-f\left(4\right)}{3.8-4}$

If $A(2,8)$ is an interior point of a circle $x^2+y^2-2x+4y-P=0$ which neither touches nor intersects the axes, then set for $P$ is

The coefficient of $x^6$ in $\left\{\right(1+x{)}^6+(1+x{)}^7+.....+(1+x{)}^{15}\}$is

If a,b,c are in G.P, then the equation $a{x}^2$ + 2bx + c = 0 and $d{x}^2$ + 2ex + f = 0 have a common root if $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in

Write the value of $\theta \epsilon \left(\frac{\pi }{2}\right)$ for which area of the triangle formed by points O(0,0), A$(acos\theta ,bsin\theta )$ and $(acos\theta ,bsin\theta )$ is maximam.

Number of ways committee of 12 can be formed from 9 women and 8 men in which at least 7 women have to be included in a committee is ______

In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is

A line makes angles $a,b,c,d$ with the four diagonals of a cube, then ${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c+{\cos }^{2}d=$

If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2x+3y-z+13=0$ at a point $P$ and the plane $3x+y+4z=16$ at a point $Q,$ then $PQ$ is equal to :

All possible numbers are formed using the digits $1,1,2,2,2,2,3,4,4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is:

If z be a non- real complex number satisfying |z|=2, then which of the following is/are true?

m men and n women are to be seated in a row so that no two women sit together. If m> n then show that the number of ways in which they can be seated as $\frac{m!(m+1)!}{(m-n+1)!}$ factorial.

Find the eqns of the tangent to curve 3x²-y²=8 which passes through the point (4/3,0).

After how many turns will the given figure look the same?

Find the equation of the circle with:
(i) Centre (-2,3 and radius 4.
(ii) Centre (a, b) and radius $\sqrt{a^2+b^2}$
(iii) Centre (0,-1) and radius 1.
(iv) Centre $(\alpha cos\alpha ,\alpha sin\alpha )$ and radius $\alpha$.
(v) Centre $(\alpha ,a)$ and radius $\sqrt{2}a$.

If the equation x^{​​​​​​2}+bx+c=0 and bx^{​​​​​2}+cx+1=0 then

A) b+c+1=0 B)b^{​​​​​​2}+c^{​​​​​​2}+1=bc C)(b-c)²+(b-1)²+(c-1)²=0 D)b+c+1=bc

(Multi option correct)

The order of the differential equation of all circles of radius r, having centre on y-axis and passing through the origin is