If the normal to the curve \(y=f(x)\) at the point \((3,4)\) makes an angle \(\frac{3 \pi}{4}\) with the positive \(x\)-axis, then \(f^{\prime}(3)=\) ________ .

In a game of 100 points, \(A\) can give 20 points to \(B\) and 28 points to \(C\). How many points can \(B\) give \(C\) ?

For the problem \(\max Z=a x+b y, x \geq 0, y \geq 0\), which of the following is NOT a valid constraint to make it a linear programming problem?

Match List - I with List - II.

LIST - I	LIST - II
(A) \(x^x\) has a stationary point at \(x\) equal to	(I) \(e\)
(B) For \(x>0\), minimum value of \(x^x\)	(II) \(1 / e\)
(C) The greatest value of \(\left(\frac{1}{x}\right)^x\)	(III) \(e^{1 / e}\)
(D) The stationary point of \(\frac{\log x}{x}\) for \(x\), where \(x>0\), is	(IV) \(e^{-1 / e}\)

Choose the correct answer from the otions given below:

If \(\theta \in[0, \pi]\) is the angle between any two non-zero vectors \(\vec{a}\) and \(\vec{b}\), such that \(|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|\), then \(\theta=\)

The direction cosines of a line which makes equal angles with co-ordinate axes are:

\(\cos ^{-1}\left(\frac{3}{5} \cos x+\frac{4}{5}\ sinx \right)=\)

Arrange the following substances in the increasing order of magnetic susceptibility.
(A) Paramagnetic substances
(B) Superconductors
(C) Diamagnetic substances
(D) Ferromagnetic substances
Choose the correct answer from the options given below :

Read the passage carefully and answer the questions
The resistance ( \(R\) ) of an object is given by the expression \(R=\rho \frac{l}{A} .\)
The inverse of \(R\) is called conductance and the inverse of \(\rho\) is called conductivity or specific conductance represented by \(\kappa\).
The conductivity depends upon the nature of the material (insulator, semiconductor or conductor) and the temperature of the substance.
Similarly, the conductance due to the ions present in the solution (ionic conductance) depends upon the nature of the electrolyte, size of ions, the nature of the solvent and the temperature.
Molar conductivity ( \(\Lambda_m\) ) of a solution at a given concentration is the conductance of volume \(V\) containing one mole of electrolyte kept between two electrodes of area \(A\) and unit distance apart.
It is given by the expression \(\Lambda_m=\kappa V .\)
Molar conductivity increases with decrease in concentration and when concentration approaches zero, it is called limiting molar conductivity.
The decrease is linear for a strong electrolyte. However, for a weak electrolyte the change is gradual at higher concentrations and sharp near zero concentration.
Using limiting molar conductivity values of strong electrolytes we can calculate the limiting molar conductivity of weak electrolytes using Kohlrausch law of independent migration of ions.
From molar conductivity and limiting molar conductivity we can determine the degree of dissociation \(\left(\frac{\Lambda_m}{\Lambda_m^0}\right)\) and the ionization constant of a weak acid.
Select the correct statement.

Area of region bounded by the curves \(x=y^3, x=0\) between \(y=-1\) and \(y=2\) is :

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:

Match List - I with List - II.

List - I	List - II
(A) Human skin colour	(I) Multiple allelism
(B) Pair of alleles expressing contrasting traits	(II) Polygenic inheritance
(C) ABO blood group	(III) Incomplete dominance
(D) Antirrhinum flower colour	(IV) Heterozygous

Choose the correct answer from the options given below :

Which of the following is caused due to substitution of glutamic acid to valine on 6th position of B-globin chain of Haemoglobin molecule ?