If \(\sin \alpha, \cos \alpha\) be the roots of the equation \(x^{2}-b x+c=0 .\) Then, which of the following statements is/are correct?

Let N be the number of quadratic equations with coefficients from {0, 1, 2, ....., 9) such that 0 is a solution of each equation. Then the value of N is

If the equation \(x^{2}-ca+d=0\) has roots equal to the fourth powers of the roots of \(x^{2}+a x+b=0,\) where \(a^{2}>4 b,\) then the roots of \(x^{2}-4 b x+2 b^{2}-c=0\) will be

A rectangle \(A B C D\) has its side parallel to the line \(y=2 x\) and vertices \(A, B, D\) are on lines \(y=1, x=1\) and \(x=-1\) respectively. The coordinate of \(C\) can be

\(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar vectors and \(\lambda\) is a real number then the vectors \(\vec{a}+2 \vec{b}+3 \vec{c}, \lambda \vec{b}+4 \vec{c}\) and \((2 \lambda-1) \vec{c}\) are non-coplanar for

\(\int \frac{\mathrm{dx}}{\sin x+\sqrt{3} \cos x}\) equals
where \(\mathrm{c}\) is an arbitrary constant

Consider the equation \(y-y_1=m\left(x-x_1\right)\). If \(m\) and \(x_1\) are fixed and different lines are drawn for different values of \(y_1\), then

The \(\mathrm{sp}^3 \mathrm{~d}^2\) hybridization of central atom of a molecule would lead to

An electron in Hydrogen atom jumps from the second Bohr orbit to the ground state and the difference between the energies of the two states is radiated in the form of a photon. This photon strikes a material. If the work function of the material is 4.2 eV , then the stopping potential is (Energy of electron in n -th orbit \(=-\frac{13.6}{\mathrm{n}^2} \mathrm{eV}\) )

A glass slab consists of thin uniform layers of progressively decreasing refractive indices RI (see figure) such that the RI of any layer is \(\mu-m \Delta \mu\). Here, \(\mu\) and \(\Delta \mu\) denote the \(\mathrm{RI}\) of 0 th layer and the difference in RI between any two consecutive layers, respectively. The integer \(m=0,1,2,3, \ldots\) denotes the numbers of the successive layers. A ray of light from the oth layer enters the 1st layer at an angle of incidence of \(30^{\circ}\). After undergoing the mth refraction, the ray emerges parallel to the interface. If \(\mu=1.5\) and \(\Delta \mu=0.015,\) the value of \(m\) is

A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to

3 moles of mono-atomic gas \((\gamma=5 / 3)\) is mixed with 1 mole of a diatomic gas \((\gamma=7 / 3) .\) The value of \(\gamma\) for the mixture will be

Calculate \(\mathrm{K}_{\mathrm{C}}\) for the reversible process given below if \(\mathrm{K}_{\mathrm{P}}=167\) and \(\mathrm{T}=800^{\circ} \mathrm{C}\)
\(\mathrm{CaCO}_3(\mathrm{~s}) \rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_2(\mathrm{~g})\)