Let $A=\left(3,4,0\right), B=\left(4,4,4\right),C=\left(-6,2,3\right)$ and $D=\left(1,1,2\right)$. If $\theta$ is the acute angle between the lines $AB$ and $CD$ then $\cos \theta =$

In a triangle ABC, if \(c^2-a^2=b(\sqrt{3} c-b)\) and \(b^2-a^2=c(c-a)\), then \(\lfloor\mathrm{ACB}=\)

If \(x^2+y^2=t+\frac{2}{t}\) and \(x^4+y^4=t^2+\frac{4}{t^2}\), then \(x^3 y \frac{d y}{d x}\) equals

In a game, two dice are thrown simultaneously by a person \(\mathrm{A}\) and two cards are drawn at random simultaneously from a pack of 52 playing cards by a person B. They win the game if A gets a prime score as the sum of the numbers appear on both the dice and \(B\) gets a face card and a card having a prime number. Then the probability. that both \(\mathrm{A}\) and \(\mathrm{B}\) win is

If \(f(x)=\frac{x}{1+|x|}\), for \(x \in R\), then \(f^{\prime}(0)\) is equal to :

If \(z=\cos \alpha+i \sin \alpha ; 0 < \alpha < \frac{\pi}{4}\), then \(\left|\frac{1+z^4}{1-z^3}\right|=\)

A is a \(3 \times 3\) matrix satisfying \(\mathrm{A}^3-5 \mathrm{~A}^2+7 \mathrm{~A}+\mathrm{I}=0\).
If \(\mathrm{A}^5-6 \mathrm{~A}^4+12 \mathrm{~A}^3-6 \mathrm{~A}^2+2 \mathrm{~A}+2 \mathrm{I}=l \mathrm{~A}+m \mathrm{I}\), then \(l+m=\)

With reference to the redox properties of hydrogen peroxide \(\left(\mathrm{H}_2 \mathrm{O}_2\right)\), which of these reactions are feasible? (i) \(2 \mathrm{Fe}^{2+} 2 \mathrm{H}^{+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{H}_2 \mathrm{O}\) (ii) \(2 \mathrm{MnO}_4^{-}+6 \mathrm{H}^{+}+5 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Mn}^{2+}\) \(+8 \mathrm{H}_2 \mathrm{O}+5 \mathrm{O}_2\) (iii) \(2 \mathrm{Fe}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{OH}^{-}\) (iv)\(2 \mathrm{MnO}_4^{-}+3 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O}\) \(+3 \mathrm{O}_2+2 \mathrm{OH}^{-}\)

If the Wheatstone's bridge with four resistors \(R_1, R_2\) and \(R_3, R_4\) is balanced, then the correct expression is

Two wires of the same material and length but diameters in the ratio \(1: 2\) are stretched by the same force. The elastic potential energy per unit volume for the wires, when stretched by the same force will be in the ratio.

A whistle of frequency \(540 \mathrm{~Hz}\) rotates in a horizontal circle of radius \(2 \mathrm{~m}\) at an angular speed of \(15 \mathrm{rad} / \mathrm{s}\). The highest frequency heard by a listener at rest with respect to the centre of circle (velocity of sound in air \(=330 \mathrm{~ms}^{-1}\) )

A pair of straight lines drawn through the origin forms an isosceles triangle right angled at the origin with the line \(2 x+3 y=6\). The area (in sq. units) of the triangle so formed is

If \(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x=A \sin 2 x+B\), then \(A\) is equal to