If \(\sec \theta+\tan \theta=\frac{1}{3}\), then the quadrant in which \(2 \theta\) lies is

\(\log _4 2-\log _8 2+\log _{16} 2-\ldots\) is equal to

For \(a, b, h>0\), if the slope of one of the lines represented by \(a^2 x^2+2 h x y+b^2 y^2=0\) is twice that of the other, then the value of \(\frac{h}{a b}\) is

If \(f(x)\) is a polynomial of the second degree in \(x\) such that \(f(0)=f(1)=3, f(2)=-3\). Then, \(\int \frac{f(x)}{x^3-1} d x=\)

Let \(f(x)\) be a real valued function. If \(f^{\prime}(x)\) is a constant for all \(x \in \mathbb{R}, f(0)=2\) and \(f^{\prime}(0)=1\), then

Choose the correct option about the matrices given below
\(\begin{aligned} & A=\left[\begin{array}{ccc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1\end{array}\right] \\ & B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{array}\right] \\ & C=\left[\begin{array}{ccc}\cos \frac{\pi}{6} & 0 & \sin \frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\end{array}\right]\end{aligned}\)
\(D=\left[\begin{array}{ccc}
\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\
-\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\)

If the harmonic mean of the roots of the equation \(\sqrt{2} \mathrm{x}^2-\mathrm{bx}+(8-2 \sqrt{5})=0\) is 4, then the value of \(b\) is

A ball is dropped from a tower of height \(80 \mathrm{~m}\). The time it takes to cover the last \(50 \%\) of its fall is
(acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )

The set of values that \(\beta\) can assume so that the point \((0, \beta)\) should lie on or inside the triangle having sides \(3 x+y+2=0\), \(2 x-3 y+5=0\) and \(x+4 y-14=0\), is

\(F_{\mathrm{pp}}, F_{\mathrm{nn}}\) and \(F_{\mathrm{np}}\) are the nuclear forces between proton-proton, neutron-neutron and neutron-proton respectively. Then relation between them is

If the tangent at the point $P$ on the circle $x^2+y^2+6x+6y=2$ meets $$ the $$ straight $$ line $5x-2y+6=0$ at a point $Q$ on the $Y$ -axis, then the length of $PQ$ is

Suppose that a bag \(A\) contains \(n\) red and 2 black balls and another bag \(B\) contains 2 red and \(n\) black balls. One of the two bags is selected at random and two balls are drawn from it at a time. When it is known that the two balls drawn are red, if the probability that those two balls drawn are from bag \(A\) is \(\frac{6}{7}\), then \(n=\)

In a triangle $ABC,$ if the mid points of sides $AB,BC,CA$ are $(3,0,0),(0,4,0),(0,0,5)$, respectively, then $A{B}^2+B{C}^2+C{A}^2=$