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The volume of the tetrahedron whose co-terminous edges are $(-12\hat{i}+\lambda \hat{k}),(3\hat{j}-\hat{k})$ and $(2\hat{i}+\hat{j}-15\hat{k})$ is $546$ cubic units.Then the value of $\lambda$ is

If $lo{g}_{|sinx|}(x^2-8x+23)>\frac{3}{lo{g}_2|sinx|}$

Let $\alpha$ and $\beta$ are two real roots of the equation $(k+1){\tan }^{2}x-\sqrt{2}\lambda \tan x=1-k,$ where $k\ne -1$ and $\lambda$ are real numbers. If ${\tan }^2(\alpha +\beta )=50,$ then the value of $\lambda$ is

If $si{n}^2$z = 1 + $co{s}^2$ y, find the value of $co{s}^2$ z + $si{n}^2$ y

There are $12$ points $(A_1,A_2,...,A_{12})$ in a plane, where $(A_1,A_2,A_3,A_4)$ are collinear to each other and $(A_5,A_6,A_7,A_8)$ are collinear to each other. If no points other than these two set of points are collinear, then the total number of straight lines that can be formed using these $12$ points is

The equation of the image of circle $x^2+y^2-2x-4y=0$ in the line $3x-4y+30=0$ is

If $\int \frac{dx}{\sin x{\cos }^{3}x}=\ln |f\left(x\right)|+g\left(x\right)+C$, then the number of solution(s) of the equation $g\left(x\right)-f\left(x\right)=0$ in $[0,(2n+1\left)\frac{\pi }{2}\right],(n\in W)$ is

1. 1415000

If a plane passes through the point (1,1,1) and is perpendicular to the line $\frac{x-1}{3}=\frac{y-1}{0}=\frac{z-1}{4}$ then its perpendicular distance from the origin is

Which of the following function(s) has/have point of inflection?

In $(\sqrt[3]{32}+\frac{1}{\sqrt[3]{33}}{)}^n$ if the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}$, then n =

If $x^{2}ta{n}^2{60}^{\circ }-4xco{s}^2{45}^{\circ }=4sin{30}^{\circ }cos{60}^{\circ }tan{45}^{\circ }$ and x is not an integer then x is equal to

Three waves of equal frequency having amplitude 10 mm, 4 mm and 7 mm arrive at a given point with successive phase difference 90°. The amplitude of the resulting wave in mm is given by

Let $x^2-(m-3)x+m=0,m\in R$ be a quadratic equation. Then the set of value(s) of $m$ for which roots are real and distinct is/are

Let, $f\left(x\right)=\{\begin{array}{c}\left[x\right],-2\le x\le -1\\ |x|+1,-1<x\le 2\end{array}$ and $g\left(x\right)=\{\begin{array}{c}\left[x\right],-\pi \le x\le 0\\ \sin x,0<x\le \pi \end{array},$ then the exhaustive domain of $g\left(f\right(x\left)\right)$ is

Show that the function given by f(x)
= e^2x is strictly increasing on
R.

In a triangle ABC, $\angle c=\frac{\pi }{2}$ and $si{n}^{-1}x=si{n}^{-1}\left(\frac{ax}{c}\right)+si{n}^{-1}\left(\frac{bx}{c}\right)$ where a, b are the sides of the triangle,then total number of different values of x is

$co{s}^2{48}^{\circ }-si{n}^2{12}^{\circ }=$

Which of the following
can not be valid assignment of probabilities for outcomes of sample
space S =

What is the logarithmic form of E=1/2mv^2

Sort the given values in descending order.

If $f\left(\frac{x-4}{x+2}\right)=2x+1,(x\in R-\{1,-2\left\}\right),$ then $\int f\left(x\right)dx$ is equal to :

(where $C$ is a constant of integration)

Prove that

$\left|\begin{array}{ccc}-a^2& ab& ac\\ ba& -b^2& bc\\ ca& cb& -c^2\end{array}\right|=4a^2{b}^2{c}^2$

Find the integrals of the functions.
$\int sin3xcos4x$ dx.

If $sin\theta =\frac{a^2-b^2}{a^2+b^2}$, find the values of tan $\theta ,sec\theta$ and $cosec\theta .$

The acute angle between two lines whose direction cosines are given by the relation between $l+m+n=0$ and $l^2+m^2-n^2=0$ is

Which of the following points are extrema for f(x) = sin(x) ?

The vectors $\overrightarrow{a}=\hat{i}+2\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}+\hat{j},\overrightarrow{c}=-\hat{i}$ are such that $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b},$ then the value of $\lambda +\mu$ is

A rod of length l slides between the two perpendicular lines.Find the locus of the point on the rod which divedes it in the ratio 1 : 2.

If $k\in R$ lies between the roots of $a{x}^2+bx+c=0,a<0$. Consider $f\left(x\right)=a{x}^2+bx+c=0$, then $f\left(k\right)$ $\&$