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If $I=\int (\sec x-\sin x\tan x)dx,$ then the value of $I$ is

(where $C$ is constant of integration)

If $[\overrightarrow{a}\times \overrightarrow{b}\overrightarrow{b}\times \overrightarrow{c}\overrightarrow{c}\times \overrightarrow{a}]=\lambda {\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}^2$ then $\lambda$ is equal to

The range of $f\left(x\right)=2x^2-3x+2$ if $x\in [0,2]$ is

Find the equation of the plane which is perpendicular to the plane 5x+3y+6z+8=0 and which contains the line of intersection of the planes x+2y+3z-4=0 and 2x+y-z+5=0.

The co-ordinates of the point which divides the join of the points (2, –1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by

If $y=y\left(x\right)$ is the solution of the differential equation, $e^y(\frac{dy}{dx}-1)=e^x$ such that $y\left(0\right)=0$, then $y\left(1\right)$ is equal to

$\int \frac{x{e}^x}{(1+x{)}^2}dx=$

In a $\Delta ABC,\cot \frac{A}{2}=30,\cot \frac{B}{2}=50,\cot \frac{C}{2}=70$ then the sides $a,b,c$ in ascending order is:

Let the lines $(2–i)z=(2+i)\overline{z}$ and $(2+i)z+(i–2)\overline{z}–4i=0,$ $($here $i^2=–1$$)$ be normal to a circle $C$. If the line $iz+\overline{z}+1+i=0$ is tangent to this circle $C$, then its radius is :

The value of $\int \frac{dx}{x^2+x+1}$ is equal to

(where $c$ is integration constant)

If A(1, 2, -1) and B(-1, 0, 1) are given, then the co-ordinates of P which divides AB externally in the ratio 1:2, are [MP PET 1989]

Find inverse, by elementary row operations (if possible), of the following matrix.

$\left[\begin{array}{cc}1& -3\\ -2& 6\end{array}\right]$

If $A,B,C$ are angles of a triangle, then the value of $\left|\begin{array}{ccc}{e}^{-2iA}& e^{iC}& e^{iB}\\ e^{iC}& e^{-2iB}& e^{iA}\\ e^{iB}& e^{iA}& e^{-2iC}\end{array}\right|$ is

Show that $1+i^{10}+i^{20}+i^{30}$ is a real number.

Negation of the statement $p\to (q\vee r)$ is

Three lines through origin having direction ratios $(1,a,a^2),$ $(1,b,b^2)$ and $(1,c,c^2)$ are non- coplanar. But the lines with direction ratios $(a,a^2,1+a^3),$ $(b,b^2,1+b^3)$ and $(c,c^2,1+c^3)$ are coplanar. Then the value of $abc$ is

The number of ways in which we can get a sum of $11$ by throwing three dice is :

The common tangent for $y=\sin x$ and $(y-1)=(x-2{)}^2$ is

Which term of the following sequences:

(a) (b) (c)

The eccentricity of the hyperbola $-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

If $A=\left[\begin{array}{ccc}{a}^2& ab& ac\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right]\text{and}{a}^2+b^2+c^2=1,$ then $A^2=$

If $x^4+3\cos (a{x}^2+bx+c)=2(x^2-2)$ has two solutions with $a,b,c\in (2,5),$ then which of the following can be TRUE ?

If $i=\sqrt{-1},$ then $4+5{(\frac{-1}{2}+i\frac{\sqrt{3}}{2})}^{334}-3{(\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{365}$ is equal to

Let R be a relation over the set $N\times n$ and it is defined by (a, b) R (c, d) $\Rightarrow$ a+ d = b + c. Then, R is