The number of solutions, of the equation $$e^{\sin x}-2e^{-\sin x}=2$$, is :

For $$\alpha ,\beta \in (0,\pi /2)$$, let $$3\sin (\alpha +\beta )=2\sin (\alpha -\beta )$$ and a real number $$k$$ be such that $$\tan \alpha =k\tan \beta$$. Then, the value of $$k$$ is equal to

If the value of $$\frac{3\cos 36^{\circ }+5\sin 18^{\circ }}{5\cos 36^{\circ }-3\sin 18^{\circ }}$$ is $$\frac{a\sqrt{5}-b}{c}$$, where $$a,b,c$$ are natural numbers and $$\gcd (a,c)=1$$, then $$a+b+c$$ is equal to :

If $$\sin x=-\frac{3}{5}$$, where $$$\pi$

Suppose $$\theta \in \left[0,\frac{\pi }{4}\right]$$ is a solution of $$4\cos \theta -3\sin \theta =1$$. Then $$\cos \theta$$ is equal to :

Let the system of equations $x+2y+3z=5,2x+3y+z=9,4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda +2\mu$ is equal to :

If $\mathrm{A}=\left[\begin{array}{cc}\sqrt{2}& 1\\ -1& \sqrt{2}\end{array}\right],\mathrm{B}=\left[\begin{array}{ll}1& 0\\ 1& 1\end{array}\right],\mathrm{C}={\mathrm{ABA}}^{\mathrm{T}}$ and $\mathrm{X}={\mathrm{A}}^{\mathrm{T}}{\mathrm{C}}^2\mathrm{A}$, then $\det \mathrm{X}$ is equal to :

If the system of equations $$\begin{array}{rl}& 2x+3y-z=5\\ & \\ & x+\alpha y+3z=-4\\ & \\ & 3x-y+\beta z=7\end{array}$$ has infinitely many solutions, then $13\alpha \beta$ is equal to :

Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$. Given below are two statements : Statement I : $ f(-x)$ is the inverse of the matrix $f(x)$. Statement II : $f(x) f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below :

The values of $$\alpha$$, for which $$\left|\begin{array}{ccc}1& \frac{3}{2}& \alpha +\frac{3}{2}\\ 1& \frac{1}{3}& \alpha +\frac{1}{3}\\ 2\alpha +3& 3\alpha +1& 0\end{array}\right|=0$$, lie in the interval

Let $$A$$ be a $$3\times 3$$ real matrix such that $$A\left(\begin{array}{l}1\\ 0\\ 1\end{array}\right)=2\left(\begin{array}{l}1\\ 0\\ 1\end{array}\right),A\left(\begin{array}{l}-1\\ 0\\ 1\end{array}\right)=4\left(\begin{array}{l}-1\\ 0\\ 1\end{array}\right),A\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)=2\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)\text{. }$$ Then, the system $$(A-3I)\left(\begin{array}{l}x\\ y\\ z\end{array}\right)=\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right)$$ has :

If the system of linear equations $$\begin{array}{rl}& x-2y+z=-4\\ & 2x+\alpha y+3z=5\\ & 3x-y+\beta z=3\end{array}$$ has infinitely many solutions, then $$12\alpha +13\beta$$ is equal to

Let $$\mathrm{A}$$ be a square matrix such that $${\mathrm{AA}}^{\mathrm{T}}=\mathrm{I}$$. Then $$\frac{1}{2}A\left[{\left(A+A^T\right)}^2+{\left(A-A^T\right)}^2\right]$$ is equal to

Let $$A=\left[\begin{array}{ccc}2& 1& 2\\ 6& 2& 11\\ 3& 3& 2\end{array}\right]$$ and $$P=\left[\begin{array}{lll}1& 2& 0\\ 5& 0& 2\\ 7& 1& 5\end{array}\right]$$. The sum of the prime factors of $$\left|P^{-1}AP-2I\right|$$ is equal to

Let $$R=\left(\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right)$$ be a non-zero $$3\times 3$$ matrix, where $$x\sin \theta =y\sin \left(\theta +\frac{2\pi }{3}\right)=z\sin \left(\theta +\frac{4\pi }{3}\right)\ne 0,\theta \in (0,2\pi )$$. For a square matrix $$M$$, let trace $$(M)$$ denote the sum of all the diagonal entries of $$M$$. Then, among the statements: (I) Trace $$(R)=0$$ (II) If trace $$(\mathrm{adj}(\mathrm{adj}(R))=0$$, then $$R$$ has exactly one non-zero entry.

Consider the system of linear equations $$x+y+z=5,x+2y+{\lambda }^{2}z=9,x+3y+\lambda z=\mu$$, where $$\lambda ,\mu \in \mathbb{R}$$. Then, which of the following statement is NOT correct?

Consider the system of linear equations $$x+y+z=4\mu ,x+2y+2\lambda z=10\mu ,x+3y+4{\lambda }^{2}z={\mu }^2+15$$ where $$\lambda ,\mu \in \mathbf{R}$$. Which one of the following statements is NOT correct ?

Let $$B=\left[\begin{array}{ll}1& 3\\ 1& 5\end{array}\right]$$ and $$A$$ be a $$2\times 2$$ matrix such that $$A{B}^{-1}=A^{-1}$$. If $$BC{B}^{-1}=A$$ and $$C^4+\alpha C^2+\beta I=O$$, then $$2\beta -\alpha$$ is equal to

Let $$\lambda ,\mu \in \mathbf{R}$$. If the system of equations $$\begin{array}{rl}& 3x+5y+\lambda z=3\\ & 7x+11y-9z=2\\ & 97x+155y-189z=\mu \end{array}$$ has infinitely many solutions, then $$\mu +2\lambda$$ is equal to :