The length of the common chord of the circles \(x^2+y^2+3 x+5 y+4=0\) and \(x^2+y^2+5 x+3 y+4=0\) is

\(\cos ^3 \theta+\cos ^3\left(120^{\circ}+\theta\right)+\cos ^3\left(\theta-120^{\circ}\right)=\)

\({ }^{29} C_5+\sum_{r=0}^4{ }^{(33-r)} C_4=\)

The harmonic conjugate of the point \((2,3,4)\) with respect to the point \((3,-2,2)\) and \((6,-17,-4)\) is

If $2x-y+c\log \left(\left(x-2y-4\right)\right)=k$ is the general solution of $\frac{dy}{dx}=\frac{2x-4y-5}{x-2y+2}$ then $c=$

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

If \(\mathrm{A}(4,7,8), \mathrm{B}(2,3,4)\) and \(\mathrm{C}(2,5,7)\) are the position vectors of the vertices of a triangle \(\mathrm{ABC}\) and if the internal bisector of \(\angle A\) meets \(\mathrm{BC}\) at \(\mathrm{D}\) then \(\mathrm{AD}=\)

A ball of mass \(2 \mathrm{~kg}\) collides with another ball of mass \(\mathrm{M}\) at rest. If the collision is elastic and after the collision, the first ball moves with \(\frac{1}{3}\) of its initial velocity in the same direction. then the mass of second ball is

The distance of a point \((2,3,-5)\) from the plane \(\vec{r} \cdot(4 \hat{i}-3 \hat{j}+2 \hat{k})=4\) is

An object of mass \(20 \mathrm{~kg}\). is displaced by \(x=5 t^2 \mathrm{~m}\) (here \(t\) is time) by the application of a force. Then the ratio of the work done in times \(3 \mathrm{~s}\) and \(5 \mathrm{~s}\) is

Identify \(B\) in the following reaction
\(\mathrm{H}_4 \mathrm{SiO}_4 \underset{-\mathrm{H}_2 \mathrm{O}}{\stackrel{1000{ }^{\circ} \mathrm{C}}{\longrightarrow}} A \underset{\Delta}{\stackrel{\text { Carbon }}{\longrightarrow}} B+\mathrm{CO}\)

For $n\ge 2,$ if $I_n=\int (\sin x+\cos x{)}^{n}dx$, then $n{\mathrm{I}}_n-2(n-1)I_{n-2}=$

The area (in sq unit) of the region bounded by the curves \(2 x=y^2-1\) and \(x=0\) is