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Given $\Delta$$=\left|\begin{array}{ccc}P& 2-i& i+1\\ 2+i& q& 3+i\\ 1-i& 3-i& r\end{array}\right|$, $\Delta$ is

If $x\in (\pi ,\frac{3\pi }{2})$, then $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to

If x + 2y = $\left[\begin{array}{ccc}2& 0& 3\\ 1& -1& 1\end{array}\right]$ and 2x – y = $\left[\begin{array}{ccc}1& -4& -5\\ 0& 2& -1\end{array}\right]$, then



यदि x + 2y = $\left[\begin{array}{ccc}2& 0& 3\\ 1& -1& 1\end{array}\right]$ तथा 2x – y = $\left[\begin{array}{ccc}1& -4& -5\\ 0& 2& -1\end{array}\right]$, तब

Let $f:[0,1]\to R$(the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable,$f\left(0\right)=f\left(1\right)=0$ and satisfies $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x$, $x\in [0,1]$

Which of the following is true for $0<x<1$?

Let M and N be two $3\times 3$ non-singular skew-symmetric matrices such that MN = NM. If $P^T$ denotes the transpose of P, then $M^2{N}^2\left(M^{T}N{)}^1\right(M{N}^{-1}{)}^T$ is equal to

Let $f\left(x\right)=\frac{x^4-\lambda x^3-3x^2+3x\lambda }{x-\lambda },x\in R-\left\{\lambda \right\}$. If the range of $f\left(x\right)$ is $R$, then the complete set of values of $\lambda$ is

​​​​​​​(correct answer + 1, wrong answer - 0.25)

The inverse of the number 5 with respect to the binary operation * defined by $a\ast b=4ab$ is .

The domain of the function $f\left(x\right)={\sin }^{-1}\left(5x\right)$ is

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2\sin \theta -x(\sin \theta \cos \theta +1)+\cos \theta =0(0<\theta <{45}^{\circ })$, and $\alpha <\beta$. Then

$\sum _{\infty }^{n=0}({\alpha }^n+\frac{(-1{)}^n}{{\beta }^n})$ is equal to :

If odds in favoui-e(gn event be 2 : 3, find the probability of occurrence of this event.

The domain of the real function $f\left(x\right)=\sqrt{3-2^x-2^{1-x}}+\sqrt{si{n}^{-1}x}$ is

Number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in $x\in [0,2\pi ]$ is

The arbitrary constant on which the value of the determinant

The solution of $(y+x+5)dy=(y-x+1)dx$ is

(where $C$ is integration constant)

Find the values of

The value of $\int (\sqrt{\tan x}+\sqrt{\cot x})dx$ is:

(where $C$ is integration constant)

Two straight lines are perpendicular to each other. One of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b)$. Their point of intersection lies on the line

The equation $|\sqrt{(x-2{)}^2+(y-1{)}^2}-\sqrt{(x+2{)}^2+y^2}|=c$ will represent a hyperbola if

If the vertices of a triangle are $(1,2),(4,-6)$ and $(3,5)$, then the area (in sq. units) of the triangle is

The value of $\cos {12}^{\circ }\cos {24}^{\circ }\cos {36}^{\circ }\cos {48}^{\circ }\cos {60}^{\circ }\cos {72}^{\circ }\cos {84}^{\circ }$ is

If a, b,
c, d are in G.P, prove that

are in G.P.

1. -0.5