A bag contains 3 red, 5 black and 7 blue balls. If three balls are drawn at random simultaneously from the bag then the probability of getting at least two blue balls is

For \(a \neq b \neq c\), if the lines \(x+2 a y+a=0\), \(x+3 b y+b=0\) and \(x+4 c y+c=0\) are concurrent, then \(a, b, c\) are in

In a \(\triangle A B C,(a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}\) is equal to

The eccentricity of the conic \(36 x^2+144 y^2-36 x-96 y-119=0\) is

If \(x\) is real, then the interval in which no value of the expression \(\frac{2\left(x^2+2 x-11\right)}{2 x-5}\) lies, is

If \(\log y=y^{\log \mathrm{x}}\) then \(\frac{d y}{d x}=\)

The angle between the curves \(y=\sin x\) and \(y=\cos x\) is

If \(\mathrm{x}=\log \mathrm{p}\) and \(\mathrm{y}=\frac{1}{p}\) then \(\frac{d y}{d x}=\)

Two identical bar magnets of magnetic moment \(M\) each, are placed along \(X\) and \(Y\)-axes, respectively at a distance \(d\) from the origin (as shown in the figure). The origin lies on perpendicular bisector of magnet placed on \(X\)-axis and on the magnetic axis of magnet placed on \(Y\)-axis. If the magnitude of total magnetic field at the origin is \(B=\alpha\left[\frac{\mu_0}{4 \pi} \frac{M}{d^3}\right]\), then the value of constant \(\alpha\) will be \(\langle d>>l\), where \(l\) is the length of the bar magnets and direction of \(\mathrm{N}\) to \(\mathrm{S}\) in magnets is opposite with respect to each other)

Two point charges \(-10 \mu \mathrm{C}\) and \(+5 \mu \mathrm{C}\) are situated on \(X\)-axis at \(x=0\) and \(x=\sqrt{2} \mathrm{~m}\). The point along the \(X\)-axis where the electric field becomes zero is

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

If the equations \(x^2+p x+2=0\) and \(x^2+x+2 p=0\) have a common root then the sum of the roots of the equation \(x^2+2 p x+8=0\) is

In the pulley system shown in the figure, the mass of \(A\) is half of that of \(\operatorname{rod} B\). The rod length is \(500 \mathrm{~cm}\). The mass of pulleys and the threads may be neglected. The mass \(A\) is set at the same level as the lower end of the rod and then released. After releasing the mass \(A\), it would reach the top end of the rod \(B\) in time (Assume, \(g=10 \mathrm{~m} / \mathrm{s}^2\) )