If \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{b}}=+\lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and the orthogonal projection of \(\overrightarrow{\mathbf{b}}\) on \(\overrightarrow{\mathbf{a}}\) is \(\frac{4}{3}(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}})\), then \(\lambda\) is equal to

The domain of the function \(f(x)=\sqrt{\log _{0.5} x !}\) is

If the tangents drawn from a point \(P\) to the ellipse \(4 x^2+9 y^2-16 x+54 y+61=0\) are perpendicular, then the locus of \(P\) is

Consider the vectors \(u=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}\), \(v=a^2 \hat{\mathbf{i}}+b^2 \hat{\mathbf{j}}+c^2 \hat{\mathbf{k}}\) and \(w=a^3 \hat{\mathbf{i}}+b^3 \hat{\mathbf{j}}+c^3 \hat{\mathbf{k}}\).
These vectors are coplanar if and only if

\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{4 \sqrt{2}-(\cos x+\sin x)^5}{1-\sin 2 x}=\)

The area of the region under the curve \(y=|\sin x-\cos x|\), \(0 \leq x \leq \frac{\pi}{2}\) and above \(x\)-axis, is (in square units)

Suppose a parabola passes through $\left(0,4\right),\left(1,9\right)$ and $\left(4,5\right)$ and has its axis parallel to the $y$-axis. Then the equation of the parabola is

Match the following lists.
\begin{array}{|c|c|c|c|}
\hline & List I & & List II \\
\hline A & \begin{array}{l}\text { Zeroth law of } \\
\text { thermodynamics }\end{array} & 1 & \begin{array}{l}\text { Direction of flow of } \\
\text { heat }\end{array} \\
\hline B & \begin{array}{l}\text { First law of } \\
\text { thermodynamics }\end{array} & \| & Work done is zero \\
\hline\underline{\mathrm{C}} & Free expansion of a gas & III & Thermal equilibrium \\
\hline D & \begin{array}{l}\text { Second law of } \\
\text { Thermodynamics }\end{array} & IV & \begin{array}{l}\text { Law of conservation } \\
\text { of energy }\end{array} \\
\hline
\end{array}
The correct answer is
\(\begin{array}{lccccccccc} & \text { A } & \text { B } & \text { C } & \text { D } \end{array}\)

Find the total number of rectangles on a normal chessboard.

\(\operatorname{In} \frac{x^4-6 x^3+9 x^2+5 x-20}{x^2-x-2}=f(x)+\frac{a}{x+p}+\frac{b}{x+q}\) then \(2(a+b)=\)

At a place where the magnitude of the earth's magnetic field is \(4 \times 10^{-5} \mathrm{~T}\), a short bar magnet is placed with its axis perpendicular to the earth's magnetic field direction. If the resultant magnetic field at a point at a distance of 40 cm from the centre of the magnet on the normal bisector of the magnet is inclined at \(45^{\circ}\) with the earth's field, then the magnetic moment of the magnet is

A segment of wire vibrates with a fundamental frequency of \(450 \mathrm{~Hz}\) under a tension of \(9 \mathrm{~kg} \mathrm{wt}\). Then tension at which the fundamental frequency of the same wire becomes \(900 \mathrm{~Hz}\) is

The perpendicular distance from the point \((-1,1,0)\) to the line joining the points \((0,2,4)\) and \((3,0,1)\) is