\[
(1+i)^{2024}+(1-i)^{2024}=
\]

If the harmonic mean of the roots \(\sqrt{2} x^2-b x+(8-2 \sqrt{d})=0\) is 4 , then the value of \(b\) is

The equation of the normal drawn to the parabola \(y^2=6 x\) at the point \((24,12)\) is

Let the equation of the tangent at a point \(\mathrm{P}\) on the parabola \(x^2-4 x-4 y+16=0\) be \(2 x-y-5=0\). If the equation of the normal drawn at \(\mathrm{P}\) to this parabola is \(\mathrm{ax}+\mathrm{y}+\mathrm{c}=0\), then \(\mathrm{ac}=\).0

The value of \(\sum_{n=0}^{\infty}\left(\frac{2 i}{3}\right)^n\) is

Let \(f\) be a polynomial function defined on \([2,7]\). If \(f(2)=3\) and \(f^{\prime}(x) \leq 5\) for all \(x\) in \((2,7)\), then the maximum possible value attained by \(f\) at \(x=7\) is

If set A has 5 elements, set B has 7 elements then the number of many one functions that can be defined from \(A\) to \(B\) is

The work done by a gas is maximum when it expands from a volume \(V_1\) to \(V_2\), then the expansion is

The relation between the pressure \((\mathrm{P})\), volume \((\mathrm{V})\) and average kinetic energy (E) of a gas is

If \(\int \frac{\cos 4 x+1}{\cot x-\tan x} d x=k \cos 4 x+c\), then \(k\) is

Three charged particles of each mass \(0.1 \mathrm{~g}\) and charge ' \(\mathrm{q}\) ' are suspended from a common rigid point by insulated massless threads of each \(1 \mathrm{~m}\) long. If the three particles are in equilibrium and are located at the corners of an equilateral triangle of side \(3 \mathrm{~cm}\), the charge ' \(q\) ' on each particle is ____ \(\mathrm{nC}\). (The angle made by the line joining the centroid of the triangle and the point of suspension with the vertical is very small).
(Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) and \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\) )

The efficiency of a Carnot's engine is \(25 \%\), when the temperature of sink is 300 K . The increase in the temperature of source required for the efficiency to become \(50 \%\) is

The molecular formula of a crystalline solid is \(\mathrm{X}_3 \mathrm{Y}_2\). Atoms of Y form ccp lattice, and atoms of X occupy 50\% octahedral voids and \(x \%\) of tetrahedral voids. What is the percentage of unoccupied tetrahedral voids?