If \(\left|\begin{array}{ccc}a+b+2 c & a & b \\ c & 2 a+b+c & b \\ c & a & a+2 b+c\end{array}\right|=2\), then \(a^3+b^3+c^3-3 a b c=\)

If \(x=a\) is a root of multiplicity two of a polynomial equation \(f(x)=0\), then

\(y=A e^x+B e^{2 x}+C e^{3 x}\) satisfies the differential equation

Solve the equation, \(3^{x^2-x}=25-4^{x^2-x}\)

The angle between any two diagonals of a cube is

If \((\alpha, \beta)\) is the orthocentre of the triangle with the vertices \((2,2),(5,1),(4,4)\), then \(\alpha+\beta=\)

A stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is

Let \(S\) be the set of all quadratic equations of the form \(x^2+b x+c=0\), where \(b, c \in\{1,2,3\), \(4,5,6\}\). If an equation is selected at random from \(S\), then the probability that the equation has real roots is

A conductor moving in the magnetic field \(B\) and the induced current is \(I\). If the magnetic field is doubled the induced current will

One end each of a resistance \(r\) capacitor \(C\) and resistance \(2 r\) are connected together. The other ends are respectively connected to the positive terminals of batteries, \(P, Q, R\) having respectively emf's \(E, E\) and \(2 E\). The negative terminals of the batteries are then connected together. In this circuit, with steady current the potential drop across the capacitor is :

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

If \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) defined by
\(f(x)=\left\{\begin{array}{ll}\frac{\sin x-\sin \frac{x}{2}}{x}, & x < 0 \\ \frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{array}\right.\) is continuous on \(\mathbb{R}\), then \(f(0)=\)

A particle is executing simple harmonic motion with an amplitude of \(2 \mathrm{~m}\). The difference in the magnitudes of its maximum acceleration and maximum velocity is 4 . The time-period of its oscillation and its velocity when it is 1 \(\mathrm{m}\) away from the mean position are respectively