If \(\Delta=a^2-(b-c)^2\), is the area of the \(\triangle A B C\), then \(\tan A\) is equal to

If \(0 < y < 2^{1 / 3}\) and \(x\left(y^3-1\right)=1\), then \(\frac{2}{x}+\frac{2}{3 x^3}+\frac{2}{5 x^5}+\ldots\) is equal to:

The numbers \(a_n=6^n-5 n\) for \(n=1,2,3, \ldots\) when divided by 25 leave the remainder

If the mean and variance of a binomial variate \(X\) are 8 and 4 respectively, then \(P(X < 3)\) equals to

If $\theta$ is the angle between the lines joining the origin to the points of intersection of the curve $2x^2+3y^2=6$ and the line $x+y=1,$ then $\sin \theta =$

The number of ways in which 6 men and 4 women can be seated around a table so that a particular man and a particular woman never sit adjacent to each other is

The numbers \(2,3,5,7,11,13\) are written on six distinct paper chits. If 3 of them are chosen at random, then the probability that the sum of the numbers on the obtained chits is divisible by 3 , is

Which one of the following statements is correct?

If the range of a random variable \(X\) is \(\{0,1,2,3,4, \ldots \ldots\}\) with \(P(X=k)=\frac{(k+1) a}{3^k}\) for \(k \geq 0\), then \(a\) is equal to

If \(\alpha, \beta\) are the roots of the equation \(x^2+3 x+\mathrm{k}=0\) and \(\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}\) are the roots of the equation \(4 x^2+p x+18=0\) then k satisfies the equation

\((1,1)\) is the vertex and \(x+y+1=0\) is the directrix of a parabola. If \((a, b)\) is its focus and \((c, d)\) is the point of intersection of the directrix and the axis of the parabola, then \(a+b+c+d=\)

Match the following
The correct answer is

Match the following methods for treating hardness of water with their corresponding reagents used. \begin{array}{|c|c|c|} \hline List-I & & List II \ \hline A. Treatment with washing soda & \mathrm{I}. & \mathrm{Ca}(\mathrm{OH})_2 \ \hline B. Calgon's method & II. & \mathrm{Na}_2 \mathrm{CO}_3 \ \hline C. Clark's method & III & \mathrm{NaAlSiO}_4 \ \hline D. Zeolite/permutit process & IV & \mathrm{Na}_6 \mathrm{P}_6 \mathrm{O}_{18} \ \hline \end{array} The correct match is \(\begin{array}{llllllll}\text { A } & \text { B } & \text { C } & \text { D } & \text { A } & \text { B } & \text { C } & \text { D }\end{array}\)