The locus of the midpoint of the portion of the line \(x \cos \alpha+y \sin \alpha=p\) intercepted by the coordinate axes, where \(p\) is a constant, is

\(\frac{\cos x}{\cos (x-2 y)}=\lambda \Rightarrow \tan (x-y) \tan y\) is equal to

A circle is such that \((x-2) \cos \theta+(y-2) \sin \theta=1\) touches it for all values of \(\theta\). Then, the circle is

If \(\operatorname{det}(\mathrm{AB})=(\operatorname{det} \mathrm{A})(\operatorname{det} \mathrm{B})\) and \(\mathrm{A}\) is a non-singular matrix of \(\operatorname{order} 3 \times 3\), then \(\operatorname{det}(\operatorname{adj} A)=\)

If \(x^3+y^3=3 a x y\), then at \(\left(\frac{3 a}{2}, \frac{3 a}{2}\right)\) the value of \(3 a y^{\prime \prime}+40\) is

The sum of the squares of the intercepts made by the line $5x-2y=10$ on the co-ordinate axes equals

If [.] denotes the greatest integer function, then \(f(x)=[x]^2-\left[x^2\right]\) is discontinuous at

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

Let \(\mathrm{x} \in \mathbf{R}\) and \(|\mathrm{x}| < 1\). Then \(\tanh ^{-1} x=\)

A car of mass 2000 kg is accelerating from rest. If its engine is supplying constant power of 10 kW, then the velocity of the car at a time of 10 s is

If \(f: R \rightarrow R\) is defined by \(f(x)=x-[x]-\frac{1}{2}\) for \(x \in R\), where \([x]\) is the greatest integer not exceeding \(x\), then \(\left\{x \in R: f(x)=\frac{1}{2}\right\}\) is equal to :

An ideal gas is found to obey \(\mathrm{Pv}^{\frac{3}{2}}=\) constant during an adiabatic process. If such a gas initially at a temperature T is aḍiabatically compressed to \(\frac{1}{4}\) th of its volume, then its final temperature is

The pairs of straight lines \(x^2-3 x y+2 y^2=0\) and \(x^2-3 x y+2 y^2+x-2=0\) form a