Two bad eggs are mixed accidentally with 10 good ones. If three eggs are drawn at random from this lot in succession without replacement, then the variance of the probability distribution of the number of bad eggs drawn is

If \(A\) and \(B\) are events of a sample space such that \(P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}\) and \(P(\bar{A})=\frac{2}{3}\), then \(P(\bar{A} \cap B)\) is

If the line segment joining the vertex of the parabola \(y^2=4 a x\) and a point on the parabola, makes an angle \(\theta\) with the positive \(X\)-axis, then the length of that line segment is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

Let \(\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\). If \(\mathbf{p}\) is a unit vector such that \([\mathbf{a b p}]\) is maximum. then \(\mathbf{p}=\)

If \(\mathrm{I}_1=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, \mathrm{I}_2=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x\), then \(\mathrm{I}_2-\mathrm{I}_1=\)

The diameters of a circle are along \(2 x+y-7=0\) and \(x+3 y-11=0\). Then, the equation of this circle, which also passes through \((5,7)\), is

The sensitivity of a galvanometer is 60 division/A. When a shunt is used, its sensitivity becomes 10 division/A. If the galvanometer is of resistance \(20 \Omega\), the value of shunt used is

If \(A\) and \(B\) are two independent events such that \(P(B)=\frac{2}{7}\) and \(P\left(A \cup B^C\right)=0.8\), then \(P(A \cup B)\) \(=\)

If the peak value of the magnetic field of an electromagnetic wave is \(30 \times 10^{-9} \mathrm{~T}\), then the peak value of the electric field is

An iron rod of length \(1.5 \mathrm{~m}\) lying on a horizontal table is pulled up form one end along a vertical line so as to move it with a constant velocity \(3 \mathrm{~m} / \mathrm{s}\), while the other end of the rod slides along the floor. After how much time the speed of the end sliding on the floor equals to the speed of the end being pulled up.

The magnetic field of an electromagneticwave obeys the relation in a certain region is \(B=10^{12} \sin \left(5 \times 10^6 t\right) \mathrm{T}\), where \(t\) is the time. Then, the induced emf, in a 300 tums in coil of area \(20 \mathrm{~cm}^2\) oriented perpendicular to the field is

Let \(A=(2,0)\) and \(B=(0,-2)\). Let \(P\) be any point such that the sum of the distances of \(P\) from \(A\) and \(B\) is 4 . Then the equation of the locus of the point \(P\) is