If the line segment joining the points \((1,0)\) and \((0,1)\) subtends an angle of \(45^{\circ}\) at a variable point P , then the equation of the locus of P is

If \(-\frac{\pi}{4} < x < \frac{\pi}{4}\), then the general solution of the differential equation \(\cos ^2 x \cdot \frac{d y}{d x}-(\tan 2 x) y=\cos ^4 x\) is

A question paper consists of two sections having 3 and 4 questions respectively. The following note is given on the paper. "It is not necessary to attempt all the questions. One question from each section is compulsory". In how many ways can a candidate select the questions?

The number of numbers between 2000 and 5000 that can be formed with the digits 0,1 , 2, 3, 4 (repetition of digits not allowed) and are multiple of 3 is

Let \(L\) be the line joining the origin to the point of intersection of the lines represented by \(2 x^2-3 x y\) \(-2 y^2+10 x+5 y=0\). If \(L\) is perpendicular to the line \(k x+y+3=0\), then \(k\) is equal to

If \(\frac{1+3 P}{3}, \frac{1-2 P}{2}\) are probabilities of two mutually exclusive events, then \(P\) lies in the interval.

\(\tanh (\log x)=\)

A physical quantity \(P\) is related to four observables \(a, b, c\) and \(d\) as \(P=\frac{\sqrt{a b} \cdot d^\alpha}{\sqrt{c}}(\alpha\) is constant) The percentage errors in \(a, b, c\) and \(d\) are \(0.5 \%\) in each. If the percentage error in \(P\) is \(2 \%\), then \(\alpha\) is

Identify the correct set of molecules with different geometries and central atoms with different hybridisations.

If \(M\) is a point on the line \(y=x\) and points \(P(0,1), Q(2,0)\) are such that \(P M+Q M\) is minimum, the coordinates of \(M\) are

As shown in the figure, two infinitely long straight parallel wires \(P\) and \(Q\) carrying equal currents in opposite directions are arranged parallel to \(\mathrm{Y}\)-axis. If the magnetic field due to wire \(\mathrm{P}\) at the origin ' \(\mathrm{O}\) ' of the co-ordinate system is B, then match the resultant magnetic fields at various points given in column \(A\) with the points given in column \(B\).
\(\begin{array}{ll}
\text{Column A} & \text{Column B} \\
\text{A)} \frac{B}{4} & \text{i)} (0,0) \\
\text{B)} \frac{B}{2} & \text{ii)} (a, 0) \\
\text{C)} \frac{2 \mathrm{~B}}{3} & \text{iii)} (2a, 0) \\
\text{D)} 2 \mathrm{~B} & \text{iv)} (3 \mathrm{a}, 0) \\
\end{array}\)


The correct answer is

A monoatomic ideal gas goes through a cyclic process as shown in the figure. The efficiency of this process is

The equation of the tangent to the parabola \(y^2=12 x\) at \((3,-6)\) is