Let \(\pi\) be the plane passing through the point \((3,-3,1)\) and perpendicular to the line joining the points \((3,4,-1)\) and \((2,-1,5)\). If the equation of the plane containing the points \((3,4,-1),(-1,2,5)\) and perpendicular to the plane \(\pi\) is \(a x+y+c z-d=0\), then \(3(a+c)=\)

The probability distribution of a random variable X is as follows. Then the mean of X is

\(\mathrm{X} = \mathrm{x}_{\mathrm{i}}\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(\mathrm{P}\left(\mathrm{X} = \mathrm{x}_{\mathrm{i}}\right)\)	\(\mathrm{k}^2 / 3\)	\(\mathrm{k}^2\)	\(2 \mathrm{k}^2 / 3\)	\(k / 2\)	\(k / 2\)

If the straight line \(L \equiv 3 x+4 y-k=0\) cuts the line segment joining the points \(P(2,-1)\) and \(Q(1,1)\) in the ratio \(4: 1\), then the equation of the line parallel to the line \(y=x\) and concurrent with the lines \(P Q\) and \(L=0\) is

The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If two more normals to this parabola also pass through$(5,0),$  then the centroid of the triangle formed by the feet of these three normals is

\(\frac{1}{2 \cdot 3}+\frac{1}{4 \cdot 5}+\frac{1}{6 \cdot 7}+\frac{1}{8 \cdot 9}+\ldots\) is equal to

If the coefficients of \(r\) th and \((r+1)\) th terms in the expansion of \((1+x)^{24}\) are in the ratio \(12: 13\), then \(r\) is the root of the quadratic equation

If the line \(2 b x+3 c y+4 d=0\) passes through the points of intersection of \(y^2=4 a x\) and \(x^2=4 a y\), then

(i) \(\mathrm{H}_3 \mathrm{PO}_4(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{H}_2 \mathrm{PO}_4^{-}(a q)\)
(ii) \(\mathrm{H}_2 \mathrm{PO}_4^{-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{HPO}_4^{2-}(a q)\)
(iii) \(\mathrm{HPO}_4^{2-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{PO}_4^{3-}(a q)\)
The equilibrium constants for the above reactions at a certain temperature are \(K_1, K_2\) and \(K_3\) respectively. The equilibrium constant for the reaction \(\mathrm{H}_3 \mathrm{PO}_4(a q) \rightleftharpoons 3 \mathrm{H}^{+}(a q)+\mathrm{PO}_4^{3-}(a q)\) in terms of \(K_1, K_2\) and \(K_3\) is

If $z=\frac{\sqrt{3}+i}{2}$ then ${\left(z^{101}+i^{103}\right)}^{105}=$

The relation between efficiency \(\eta\) of a heat engine and the coefficient of performance \(\alpha\) of a refrigerator is

If force \(=\frac{\alpha}{\text { density }+\beta^3}\), then the dimensional formulae of \(\alpha\) and \(\beta\) are respectively

A point moves in the \(x y\)-plane such that the sum of its distance from two mutually perpendicular lines is always equal to 5 units. The area (in sq units) enclosed by the locus of the point, is

Consider the following statements \(A\) and \(B\) and identify the correct answer

A. In an elastic collision, if a body suffers a head on collision with another of same mass at rest, then the first body comes to rest while the other starts moving with the velocity of the first one.

B. Two bodies of equal mass suffering a head on elastic collision merely exchanges their velocities