If \(\sin ^{-1}\left(\frac{3}{x}\right)+\sin ^{-1}\left(\frac{4}{x}\right)=\frac{\pi}{2}\), then \(x\) is equal to

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)

Let \(f:[2,5] \rightarrow \mathbf{R}\) be a differentiatiable function and \(\frac{f(5)}{f(2)}=1\). If there is a \(c \in(2,5)\) such that \(c f^{\prime}(c)=2 f(c)-2 c^3\), then \(f(x)=\)

If \(y=x+\tan x\), then \(\cos ^2 x \frac{d^2 y}{d x^2}+2 x\) is equal to

The mean deviation from the mean of the discrete data $1,3,4,7,11,18,29,47,78$ is

If the origin of a coordinate system is shifted to \((-\sqrt{2}, \sqrt{2})\) and the coordinate system is rotated anti-clockwise through an angle \(45^{\circ}\), then the point \(P(1,-1)\) in the original system has new coordinates

In a \(\triangle A B C\), if \(3 a=b+c\), then \(\cot \frac{B}{2} \cot \frac{C}{2}\) is equal to :

If \(P_1, P_2, P_3, \ldots, P_n\) are \(n\) points on the line \(y=x\) all lying in the first quadrant, such that \(\left(O P_n\right)=n\left(O P_{n-1}\right)(O\) is origin \(), O P_1=1\) and \(P_n=(2520 \sqrt{2}, 2520 \sqrt{2})\), then \(n=\)

If the kinetic energy of a body moving with a velocity of \((2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}) \mathrm{ms}^{-1}\) is 87 J, then the mass of the body is

The Young's modulus of a material is \(2 \times 10^{11} \mathrm{~N} / \mathrm{m}^2\) and its elastic limit is \(1 \times 10^8 \mathrm{~N} / \mathrm{m}^2\). For a wire of \(1 \mathrm{~m}\) length of this material, the maximum elongation achievable is

If \(\frac{1-10 i \cos \theta}{1-10 \sqrt{3} i \sin \theta}\) is purely real, then one of the values of \(\theta\) is

If the points \((1,1, \lambda)\) and \((-3,0,1)\) are equidistant from the plane \(3 x+4 y-12 z+13=0\), then the values of \(\lambda\) are

Let the angles \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) of a triangle ABC be in arithmetic progression. If the exradii \(r_1, r_2, r_3\) of triangle ABC satisfy the condition \(r_3^2=r_1 r_2+r_2 r_3+r_3 r_1\), then \(b=\)