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A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2).$ If $O$ is the origin and $P$ is $(2,-1,1),$ then the projection of $\overrightarrow{OP}$ on this plane is of length :

If y = (1 + tan A)(1 - tan B) where A - B = $\frac{\pi }{4}$, then

$(y+1{)}^{y+1}$ is equal to

[J & K 2005]

Let $f\left(x\right)=A\sin \left(\frac{\pi x}{2}\right)+B,f^{\prime }\left(\frac{1}{2}\right)=\sqrt{2}$ and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{2A}{\pi }$, then $A$ and $B$ are

The equations of circles with radius $3$ units and touching the circle $x^2+y^2-2x-4y-20=0$ at $(5,5)$ is/are

The point from which the tangents to the circles $x^2+y^2–8x+40=0,5x^2+5y^2–25x+80=0,x^2+y^2–8x+16y+160=0$ are equal in length is

If non-zero real numbers b and c are such that min.(f(x)) > max.(g(x)), where $f\left(x\right)=x^2+2bx+2c^2$,where $g\left(x\right)=-x^2-2cx+b^2$, $\left(xϵR\right)$, then $\left|\frac{c}{b}\right|$ belongs to

The radius of a circle is 30 cm. Find the length of an arc of this circle, if the length of the chord of the arc is 30 cm.

Given A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find $(A\times B)\cap (B\times C).$

A variable plane at a constant distance p from origin meets the co-ordinate axes in A, B, C. Through these points planes are drawn parallel to co-ordinate planes.Then locus of the point of intersection is

Let $Z_1,Z_2,Z_3$ be three points $A,B$ and $P$ respectively in the argand plane. Let $P$ moves in the plane such that $|Z-Z_1|+|Z-Z_2|=K.$ Let $Z_1=–Z_2=i$

If $K=6$ and maximum area of the triangle $ABP$ is $\sqrt{8}$ sq. units, then possible number of positions of $P$ is

The distance of the point $(1,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$

The coordinates of focus of parabola $9y^2-9x-12y-57=0$ is

A particle is fired vertically upward from earth's surface and it goes up to a maximum height of 6400 km. Find the initial speed of particle.

Find the value of $ta{n}^{-1}\left(tan\frac{5\pi }{6}\right)+co{s}^{-1}\left(cos\frac{13\pi }{6}\right).$

If $ta{n}^{-1}x+ta{n}^{-1}y+ta{n}^{-1}z=\pi ,$ then $x+y+z$ is equal to
[Kerala (Engg.) 2002]

Three dice are thrown simultaneously. What is the probability of getting 15 as the sum ?

The equation of the line through the point $(0,1,2)$ and perpendicular to the line
$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :

If $f\left(x\right)$ is a quadratic expression such that $f\left(1\right)+f\left(2\right)=0$, and $-1$ is a root of $f\left(x\right)=0$ then the other root of $f\left(x\right)=0$ is:

If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is $e$, then :

Let A = {$x:xϵR,x>4$} and B = {$xϵR:x<5$}. Then $A\cap B$ =

If the tangents drawn to the parabola at the extremities of a common chord $AB$ of the circle $x^2+y^2=5$ and the parabola $y=b{x}^2$ intersect at the point $T$ which lies on the directrix of the parabola, then $\frac{1}{b}=$

A committee of 11 members is to be formed out of 8 males and 5 females. If m is the number of ways the committee is formed with atleast 6 males and n is the number of ways with atleast 3 females, then

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99\%$ is :

In question 1 (iii), (iv), (v) find the number of observations lying between $\overline{X}-M.D.$ and $\overline{X}-M.D,$ where M.D. is the mean deviation from the mean.

Write the least positive integeral vlaue of n for which ${\left(\frac{1+i}{1-i}\right)}^n$ is real.

If $\frac{(a+i{)}^2}{2a-i}=p+iq$, where $a\in R$, then the value of $p^2+q^2$ is

The maximum value of the term independent of $t$ in the expansion of ${(t{x}^{1/5}+\frac{(1-x{)}^{1/10}}{t})}^{10}$ where $x\in (0,1)$ is :

${lim}_{x\to 1}(4x^2+2)$

Let the natural numbers be divided into groups as $\left(1\right),(2,3,4),(5,6,7,8,9),...$ and so on. Then the sum of the numbers in the $n^{th}$ group is

Evaluate the following limits:

${lim}_{x\to 1}\frac{x^2+1}{x+1}$

A real valued differentiable function defined on $[1,\infty )$ where $f\left(1\right)=1.$
If $f^{\prime }\left(x\right)=\frac{1}{x^2+f^2\left(x\right)}$ then the maximum value of $\left[f\right(x\left)\right]$ is
(where $[.]$ is greatest integer function)

A circle passes through the points $(2,3)$ and $(4,5)$. If its centre lies on the line, $y-4x+3=0$, then its radius is equal to :

A manufacturer sells the products x,y,z in two markets, annual sales are indicated below.
$\begin{array}{cccc}Market& Products& & \\ I& 10000& 2000& 18000\\ II& 6000& 20000& 8000\end{array}$
(a)If unit sale prices of x,y and z are Rs 2.50, Rs 1.50 and Rs 1.00 respectively. Find the total revenue in each market with the help of matrix algebra.

(b)If the unit costs of the above three commodities are Rs 2.00m Rs 1.00 and 50 paise respectively. Find the gross profit.

Let $z=x+iy$ be a complex number satisfying equation $|z-(2+i)|=|Re\left(z\right)-4|$
Which of the following options describes the above equation ?

Find the point on the curve $y=(x-2{)}^2$ at which the tangent is parallel to the chord joining the points (2,0) and (4,4).

${\int }_0^{\pi /2}\frac{dx}{(a^{2}co{s}^{2}x+b^{2}si{n}^{2}x{)}^2}$

In any $\Delta ABC,\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}$ is equal to

Let A, B ,C be three sets . If A is subset of B abd B belnngs to c, is it true that A belongs to C ? If not given qn example

Value of the following expression is $\underset{n\to \infty }{lim}\frac{1}{n^3}(1^2+2^2+3^2+\cdots +n^2)$

If ${\sin }^{2}x+\sin x-1=0$, then the value of ${\cos }^{12}x+3{\cos }^{10}x+3{\cos }^{8}x+{\cos }^{6}x$ is

Number of ways to choose an order pair (a,b) of numbers from the set {1,2,3..........10} such that |a-b|≤5 is

The equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13, is