If \(A(2,-3)\) and \(B(-2,1)\) are two vertices of a \(\triangle A B C\) and if the centroid of \(\triangle A B C\) lies on the line \(2 x+3 y=1\), then the locus of vertex \(C\) of \(\triangle A B C\) is equal to

The derivative of \(f(x)=\cos ^{-1}\left[\sin \sqrt{\frac{1+x}{2}}\right]+x^x\) with respect to \(x\) at \(x=1\) is equal to

If the point p represents the comple \(x\) number \(z=x+i y\) in the argand plane and if \(\frac{z+i}{z-1}\) is a purely imaginary number then the locus of \(p\) is

The equation of a chord \(A B\) of an ellipse \(2 x^2+y^2=1\) is \(x-y+1=0\). If 0 is the origin, then \(\angle \mathrm{AOB}=\)

\(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-10 x-24=\) 0 . If \(\alpha(\beta+\gamma), \beta(\gamma+\alpha)\) and \(\gamma(\alpha+\beta)\) are the roots of the equation \(x^3+p x^2+q x+r=0\), then \(q=\)

If the mean of the data $p, 6,6,7,8,11,15,16$, is $3$ times $p$, then the mean deviation of the data from its mean is

Eight different letters of an alphabet are given. Words of four letters from these are formed. The number of such words with at least one letter repeated is :

In an examination, the maximum marks for each of three subjects is \(n\) and that for the fourth subject is \(2 n\). The number of ways in which candidates can get \(3 n\) marks is

If \(\alpha \neq 0\) and the mean deviation of the observations \(\{k \alpha\}\), for \(k=1,2, \ldots \ldots, 50\) about its median is 50 , then \(|\alpha|=\)

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

Identify the reaction in which oxygen is not one of the products

Consider the following statements:
I. In diamond, each carbon atom is \(s p^3\)-hybridised.
II. Graphite has planar hexagonal layers of carbon atoms.
III. Silicones being surrounded by non-polar alkyl groups are water repelling in nature
IV. The order of catenation in group 14 elements is \(\mathrm{Si}>\mathrm{C}>\mathrm{Sn}>\mathrm{Ge}>\mathrm{Pb}\).

The correct statements are

A pipe having an internal diameter \(D\) is connected to another pipe of same size. Water flows into the second pipe through ' \(n\) ' holes, each of diameter \(d\). If the water in the first pipe has speed \(v\), the specd of water leaving the second pipe is