If \(x, y\) and \(z\) are real numbers such that \(x+y+z=0\), then the value of \(\left|\begin{array}{ccc} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z \end{array}\right| \) is

The point which provides the optimal solution of the linear programming problem maximize \(z=21 x+35 y\)
\(\begin{array}{l}3 x+2 y \leq 30 \\4 x+5 y \leq 60 \\x \geq 0, y \geq 0\end{array}\)
has the coordinates

Which of the following functions \(f(x)\) are differentiable at \(x=0\) ?
(A) \(|x|\)
(B) \(|x-1|\)
(C) \(|\sin x|\)
(D) \(|\cos x|\)
(E) \(x^2\)
Choose the correct answer from the options given below :

Let \(\vec{a}=\hat{i}+4 \hat{j}, \vec{b}=4 \hat{j}+\hat{k}\) and \(\vec{c}=\hat{i}-2 \hat{k}\). If \(\vec{d}\) is a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) such that \(\vec{c} \cdot \vec{d}=16\), then \(|\vec{d}|\) is equal to:

Which of the following statements are correct about the "Central Limit Theorem"?
(A) The sampling distribution of the sample mean approaches the normal distribution as the sample size gets larger.
(B) A sample size of 30 or more is considered to be sufficient to hold the "Central Limit Theorem".
(C) As the sample size becomes larger, the prediction of characteristics of the population becomes more accurate.
(D) The sampling distribution of the sample mean approaches bell shaped curveas the sample size gets larger.
Choose the correct answer from the options given below :

Value of \(\int \frac{2}{(x-3) \sqrt{x+1}} d x\) is : (Here \(C\) is an arbitrary constant)

The value of \(\tan \frac{1}{2}\left\{\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)+\cos ^{-1}\left(\frac{1-y^2}{1+y^2}\right)\right\}\) if \(|x|<1, y>0\) and \(x y<1\) is:

Which one of the following is a sequence of mRNA?

A gaussian surface encloses three point charges \( q_1 = -14 \text{ nC}, q_2 = 78.85 \text{ nC} \) and \( q_3 = -56 \text{ nC} \). The electric flux for the gaussian surface due to these charges is

Read the Passage carefully and answer the questions
When a ligand is bound to a metal ion through a single donor atom, it is said to be a unidentate ligand but when it bind through two donor atoms, then it is said to be didentate ligand and when several donor atoms are present in a single ligand, it is said to be a polydentate ligand. When a dior polydentate ligand uses its two or more donor atoms simultaneously to bind a single metal ion, it is said to be a chelate ligand. The number of such ligating groups is called the denticity of the ligand. ligand which has two different donor atoms and either of the two ligetes in the complex is called ambidentate ligands. The number of ligand donor atoms to which the metal is directly bonded, is called coordination number of the metal ion in the complex. Complexes in which a metal is bound to only one kind of donor groups are known as homoleptic and in which a metal is bound to more than one kind of donor groups are known as heteroleptic complexes.
The coordination number of Co and Al in \(\left[ CoCl ( en )_2\right] Cl\) and \(K _2\left[ Al \left( C _2 O _4\right)_3\right]\), respectively, are

The shortest distance between the lines \(l_1\) and \(l_2\) given by
\(l_1: \vec{r}=\hat{i}+2 \hat{j}-4 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})\) and \(l_2: \vec{r}=3 \hat{i}+3 \hat{j}-5 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+6 \hat{k})\) is :

The electric field in a region is \( \vec{E} = (6\hat{i} + 3\hat{j} + 5\hat{k}) \text{ N/C} \).
The electric flux due to this electric field through an area of \( 100 \text{ cm}^2 \) lying in the X-Y plane is :

In Freundlich adsorption isotherm, the value of \(1/\text{n}\) is: