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Let $L$ be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$. If point $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular from $(3,2,1)$ on $L$, then the value of $21(\alpha +\beta +\gamma )$ equals:

Describe the sample space for the indicated experiment.
A Coin is tossed four times.

If $\int \frac{d\theta }{\left({\cos }^2\theta \right(\tan 2\theta +\sec 2\theta )}=\lambda \tan \theta +2{\log }_e|f\left(\theta \right)|+C$ where $C$ is constant of integration, then the ordered pair $(\lambda ,f(\theta \left)\right)$ is equal to:

Let $A=\{1,2,3\},B=\{1,3,5\}.$ A relation $R$ is defined from $A$ to $B$ as $R=\left\{\right(1,3),(1,5),(2,1\left)\right\}.$ Then $R^{-1}=$

Complete values of $a$ for which the equation $|x-1|+|x-2|+x-a>0$ has two solutions, is

The integral ${\int }_1^3\frac{1}{x}dx$, when evaluated by using Simpson's $1/3$ rule on two equal subintervales each of length 1, equal

The set of real values of x for which $2^{{log}_{\sqrt{2}}(x-1)}>x+5$ is

If $f^{\prime }\left(x\right)=2-\frac{5}{x^4}$ and $f\left(1\right)=\frac{14}{3},$ then $f(-1)=$

An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps D, E and F, whose requirements are 4500 L, 3000 L and 3500 L respectively. The distance (in~km)between the depots and the petrol pumps in given in the following table:

$\begin{array}{ccc}From/To& A& B\\ D& 7& 3\\ E& 6& 4\\ F& 3& 2\end{array}$

Assuming that the transportation cost of 10 L oil is Rs 1 per km. How should that delivery be scheduled in order the transportation cost is minimum? What is the minimum cost?

If $z_1,z_2,\cdots ,z_n$ lie on the circle $|z|=2,$ then the value of $|z_1+z_2+\cdots +z_n|-4|\frac{1}{z_1}+\frac{1}{z_2}+\cdots +\frac{1}{z_n}|$ is

If the line
$(3x+14y+7)+k(5x+7y+6)=0$
is parallel to the y-axis, then the value of $k$ is

Let 4 $x^2$ - 4(a - 2) x + a - 2 = 0, a R be a quadratic equation with real roots. Atleast one root of this equation lies in (0, 0.5) if

${lim}_{x\to 0}\frac{(a+x{)}^2-a^2}{x}$

The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y-$axis and lie in the first quadrant, is:

The set of real values of $t\in [-\frac{\pi }{2},\frac{\pi }{2}]$ satisfying $2\sin t=\frac{1-2x+5x^2}{3x^2-2x-1}$, $\forall x\in R-\{-\frac{1}{3},1\}$ lies in the interval

The general solution of the inequality $-7\le \frac{3-2x}{5}\le 3$ is

If $\underset{x\to 0}{lim}\frac{a\sin x-bx+c{x}^2+x^3}{2x^2\log (1+x)-2x^3+x^4}$ exists and is finite, then

How to send pictures of a math problem? Please give a tutorial

The value of the expression $\frac{-5+10i}{3+4i}$

Let $ABCD$ be a square. $E$ and $F$ be points on $AC$ such that $AE=EF=FC=\frac{AC}{3}$. Then $\tan \left(DEBF\right)$ equals:

The value of $\underset{0}{\overset{5}{\int }}{e}^{|x-3|}dx$ is

Show that the function f: R
→ R given by f(x) = x³ is
injective.

Find the
general solution of the differential equation

A unit vector perpendicular to the plane determined by the points P(1,-1,2), Q(2,0,-1) and R(0,2,1) is

If $f\left(x\right)=x^3+b{x}^2+ax$ satisfies the condition of Rolle's theorem on $[1,3]$ with $c=2+\frac{1}{\sqrt{3}}.$ Then $(a+b)=$

The sum of all the $4$ digited numbers that can be formed using the digits $1,2,5,6,7$ and are divisible by $2$ is

cot (pie/24) = 2^1/2 + 3^1/2 + 4^1/2 +6^1/2

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed ?

Evaluate $\int ({\text{cosec}}^{2}x\cdot \ln |\sec x|)dx$

The particular integral of the differential equation $\frac{d^{4}y}{d{x}^4}+4y=x^4$ will be at x = 2

1. 2.5

Find the sum of the following series up to n terms: