If the vectors \(\vec{a}=3 \hat{i}-p \hat{j}+5 \hat{k}\) and \(\vec{b}=-6 \hat{i}+14 \hat{j}+q \hat{k}\) are collinear, then the value of \(p\) and \(q\) are :

Choose the correct answer from the options given below:
A. Equation of the line passing through the point (1, 2, 3) and parallel to the vector \(3 \hat{i}+2 \hat{j}-2 \hat{k}\) is \(\frac{x-1}{3}=\frac{y-2}{2}=\frac{y-3}{-2}\)
B. Equation of line passing through (1, 2, 3) and parallel to the line given by \(\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}\) is \(\frac{x-1}{3}=\frac{y-2}{5}=\frac{z+3}{6}\).
C. Equation of line passing through the origin and (5, -2,3) is \(\frac{x}{5}=\frac{y}{-2}=\frac{z}{3}\).
D. Equation of plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratio's 2, 3, -1 is 2(x - 1) + 3(y - 2) - (z - 3) = 0.
E. Equation of plane with intercepts 2, 3 and 4 on x, y and z-axis respectively is 2x + 3y + 4z = 1

If the lines \(\frac{1-x}{3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{x-1}{3 \lambda}=\frac{y-1}{1}=\frac{6-z}{5}\) are perpendicular, then \(\lambda\) is equal to

In reference to the Inferential Statistics, which of the following is NOT correct?

There are two values of a for which the determinant, \(\Delta = \begin{vmatrix} 1 & -2 & 5 \\ 0 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86\), then the sum of these values of a is:

The value of \(\tan ^{-1}(2)+\tan ^{-1}(3)\) is equal to

If the matrix \(A=\left[\begin{array}{ccc}x & 2 & y \\ -2 & 0 & 3 \\ -1 & z & 0\end{array}\right]\) is skew-symmetric, then the value of \(2 x-3 y+5 z\) is equal to

Read the passage carefully and answer the Questions
Werner in 1898, propounded his theory of coordination compounds. In coordination compounds metals show two types of linkages (valences)-primary and secondary. The primary valences are normally ionisable and are satisfied by negative ions. The secondary valences are non-ionisable. These are satisfied by neutral molecules or negative ions. The secondary valence is equal to the coordination number and is fixed for a metal. The ions or group bound by the secondary linkages to the metal have characteristic spatial arrangements corresponding to the different coordination numbers. Both double salts as well as complexes are formed by the combination of two or more stable compounds in stoichiometric ratio. However, they differ in the fact that double salts such as carnallite, Mohr's salt, potash alum etc. dissociate into simple ions completely when dissolved in water. But complex ions do not dissociate completely into simple ions.
Identify the correct formula of Mohr's salt and potash alum, respectively.

A battery of emf 3 V and internal resistance \( r \) is connected in series with a resistance of \( R = 55 \text{ } \Omega \) through an ammeter of resistance \( R_A = 1 \text{ } \Omega \). The ammeter reads a current \( I = 50 \text{ mA} \). Find the internal resistance \( r \) of the battery.

Match List - I with List - II.

List - I	List - II
(A) Plasmid of Salmonella	(I) Join foreign DNA to plasmid
(B) Restriction enzymes	(II) Insertional inactivation
(C) Ligases	(III) Construction of \(1^{s t}\) recombinant DNA
(D) \(\beta\)-galactosidase	(IV) Molecular scissors

Choose the correct answer from the options given below:

The marginal cost (MC) and marginal revenue (MR) functions of a product are \(M C=20+\frac{x}{20}\) and \(M R=30\) respectively. If the fixed cost is 200 , then the maximum value of the profit is :

The decomposition of \(NH_3\) on platinum surface is a zero order reaction. The rates of production of \(N_2\) and \(H_2\) respectively are : (in \(mol^{-1}Ls^{-1}\)) Given that \(2.5 \times 10^{-4}\text{ mol}^{-1}\text{L s}^{-1}\) is the rate of reaction.

During the development of human embryo, which of the following hormone is NOT produced by placenta?