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If Lines $OA,OB$ are drawn from $O$(origin) with direction cosines proportional to $(1,-2,-1),(3,-2,3)$. Then the direction cosines of the normal to the plane $AOB$ is

If $\alpha$ and $\beta$ are the roots of $x(x+1)+(x+1)(x+2)+\cdots +(x+10)(x+11)=110,$ then the value of $|\alpha -\beta |$ is

Rakhi and Shikha are partners in a firm, with capitals of Rs 2,00,000 and Rs 3,00,000 respectively. The profit of the firm, for the year ended 2006-07 is Rs 23,200. As per the partnership agreement, they share the profit in their capital ratio, after allowing a salary of Rs 5,000 per month to Shikha and interest on Partner's capital at the rate of 10% pa. During the year Rakhi withdrew Rs 7,000 and Shikha Rs 10,000 for their personal use. You are required to prepare profit and loss appropriation account and partner's capital accounts.

If $|z-2|=min\{|z-1|,|z-5|\}$, where $z$ is a complex number, then possible value(s) of $Re\left(z\right)$ is/are

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& \text{The equation x log x =}3-x\text{has atleast one root in}& \text{(p)}& (0,1)\\ \text{(b)}& \text{If 27 a+ 9b+ 3c+ d= 0, then the equation}4a{x}^3+3b{x}^2+2cx+d=0\text{has atleast one}& \text{(q)}& (1,3)\\ & \text{root in}& & \\ \text{(c)}& \text{If c}=\sqrt{3}\text{and}f\left(x\right)=x+\frac{1}{x},\text{then interval of x in which LMVT is applicable for}& \text{(r)}& (0,3)\\ & f\left(x\right)\text{is}& & \\ \text{(d)}& \text{If c}=\frac{1}{2}\text{and}f\left(x\right)=2x-x^2,\text{then interval of x in which LMVT is applicable for}& \text{(s)}& (-1,1)\\ & f\left(x\right)\text{is}& & \end{array}$

Which of the following is correct?

If the function $f\left(x\right)=\{\begin{array}{cc}x-7,& x<1\\ k,& x=1\\ x^2+2,& x>1\end{array}$

is strictly increasing at $x=1,$ then $k$ can't be

The number of solution(s) of the equation $\tan x\tan 4x=1$ for $0<x<\pi$ is

Equation of curve which passes through point $(1,1)$ and satisfies the differential equation $3x{y}^{2}dy=(x^2+2y^3)dx$ is

Write the maximum and minimum values of cos (cos x).

The coefficient of x in the equation $x^2$ + p$x$ + q = 0 was taken as 17 in place of 13, its roots were found to be -2 and -15, the roots of the original equation are

Show that

PSQ is a focal chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}$ = 1 then $\frac{1}{SP}+\frac{1}{SQ}$ =

The value of $\sqrt{3}\text{cosec}{20}^{\circ }-\sec {20}^{\circ }$ is

The number of elements in a 3 x 2 matrix is not the same as ___

Let $f:(0,\infty )$ and F(x)=${\int }_1^{x}f\left(t\right).IfF\left(x^2\right)=x^2(1+x)$ then f(4) equals

Let $\left[x\right]$ denote the greatest integer $\le x$, where $x\in R$. If the domain of the real valued function $f\left(x\right)=\sqrt{\frac{\left|\right[x\left]\right|-2}{|\left[x\right]|-3}}$ is $(-\infty ,a)\cup [b,c)\cup [4,\infty ),$ $a<b<c,$ then the value of $a+b+c$ is

1. 5.65

Consider the following system of equations :

$ax+by+cz=0az+bx+cy=0ay+bz+cx=0$

$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If}a+b+c\ne 0\text{and}& \text{(P)}& \text{Planes meet only at one point}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0.& & \\ \text{(II)}& \text{If}a+b+c=0\text{and}& \left(Q\right)& \text{Equations represent the line}x=y=z\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(III)}& \text{If}a+b+c\ne 0\text{and}& \text{(R)}& \text{Equations represent identical planes}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(IV)}& \text{If}a+b+c=0\text{and}& \text{(S)}& \text{The solution of the system represents}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0& & \text{whole of the three dimensional space}\end{array}$



Which of the following is the "CORRECT" option?

Let ${(1-x+x^4)}^{10}=a_0+a_{1}x+a_2{x}^2+.....+a_{40}{x}^{40}$, then the correct option(s) is/are

If $y=(2x^2+6x)(2x^3+5x^2),$ then $\frac{dy}{dx}=$

Observe the sales table below and fill in the missing values using the correct relation between $P$ and $Q$.

The solution set of the inequation ${\sin }^{-1}\left(\sin \frac{2x^2+3}{x^2+1}\right)\le (\pi -\frac{5}{2})$ is

Let $L$ be an end of the latus rectum of $y^2=4x$. If the normal at $L$ meets the curve again at $M$ and the normal at $M$ meets the curve again at $N,$ then area of $△LMN$ (in sq. units) is

The locus of the vertices of the family of parabola $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$, $a$ being parameter is :

In a survey of 300 android mobile users, who make video calls, 75 people said they use viber, 45 use skype and 90 people use both. The number of persons who use neither viber nor skype is :-

The equation of circle touching the line $2x+3y+1=0$ at $(1,-1)$ and orthogonally cutting the cirlce whose endpoints of diameter are $(0,3)$ and $(-2,-1)$ is

Prove that : $cos\frac{\pi }{5}cos\frac{2\pi }{5}cos\frac{4\pi }{5}cos\frac{8\pi }{5}=-\frac{1}{16}$

The Cartesian equation
of a line is
.
Write its vector form.

Write the sum of the series $i+i^2+i^3+.......$ upto 1000 terms.

If a, b, and c are nonzero real numbers such that ab = 2 (a + b), bc = 3( b + c) and ca = 4 (c + a) then the value of 5a + 7b + c is :

In the given figure chord AB extended meets tangent DC at C. If AB = 7cm and BC =9 cm and area of $△CBD=18c{m}^2$,

then find the area of $△CAD.$