\(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) will be purely imaginary, if \(\theta\) is equal to

If three dice are thrown, then the mean of the sum of the numbers appearing on them is

Let \(\mathrm{A}, \mathrm{B}\) be two \(3 \times 3\) matrices and \(\mathrm{C}\) be a \(3 \times 3\) unit matrix such that \(A B-C\) is a non-singular matrix. Let \(\mathrm{D}=(\mathrm{AB}-\mathrm{C})^{-1}\). Then, consider the following statements.
Statement I : \(\operatorname{det}(\mathrm{BA})=\operatorname{det}(\mathrm{BA}-\mathrm{C}) \operatorname{det}(\mathrm{BDA})\)
Statement II : ABD = DAB
Which of the above statements is (are) true?

\(\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^8+\left(\frac{1+\cos \theta-i \sin \theta}{1+\cos \theta+i \sin \theta}\right)^{16}=\)

When \(3^{2023}\) is divided by 16 , the remainder obtained is

Let \(P\) be the point of intersection of the lines \(L_1 \equiv x-y-7=0\) and \(L_2 \equiv x+y-5=0\). \(A\left(x_1, y_1\right)\) and \(B\left(x_2, y_2\right)\) are points on the lines \(L_1=0\) and \(L_2=0\) respectively such that \(P A=3 \sqrt{2}, P B=\sqrt{2}, x_1, y_1 \geq 0, x_2, y_2 \geq 0\), then the angle made by the line segment \(A B\) at the origin is

The set of solutions of the equation \((\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2\) is

A body is projected from the earth at angle \(30^{\circ}\) with the horizontal with some initial velocity. If its range is \(20 \mathrm{~m}\), the maximum height reached by it is : (in metres)

The charge of "SnO \({ }_2\)-sol" in alkaline and acidic medium are respectively (Hint \(\mathrm{SnO}_2\) is amphoteric oxide)

\(f:[2,10] \rightarrow R\) is defined as \[ f(x)=\left\{\begin{array}{cc} \frac{1}{2}(x-6)^2-3, & x \leq 4 \ x-5, & x>4 \end{array}\right. \] Which of the following is true?

If the family of straight lines $ax+by+c=0$, where $2a+3b=4c$ is concurrent at the point $P\left(l, m\right)$, then the foot of the perpendicular drawn from $P$ to the line $x+y+1=0$ is

\((p, q)\) is the point of intersection of a latus rectum and an asymptote of the hyperbola \(9 x^2-16 y^2=144\). If \(p\gt0\) and \(q\gt0\) then \(q=\)

Two identical bar magnets of magnetic moment \(M\) each, are placed along \(X\) and \(Y\)-axes, respectively at a distance \(d\) from the origin (as shown in the figure). The origin lies on perpendicular bisector of magnet placed on \(X\)-axis and on the magnetic axis of magnet placed on \(Y\)-axis. If the magnitude of total magnetic field at the origin is \(B=\alpha\left[\frac{\mu_0}{4 \pi} \frac{M}{d^3}\right]\), then the value of constant \(\alpha\) will be \(\langle d>>l\), where \(l\) is the length of the bar magnets and direction of \(\mathrm{N}\) to \(\mathrm{S}\) in magnets is opposite with respect to each other)