If the coefficients \(a\) and \(b\) of a quadratic expression \(x^2+a x+b\) are chosen from the sets \(A=\{3,4,5\}\) and \(B=\{1,2,3,4\}\) respectively, then the probability that the equation \(x^2+a x+b=0\) has real roots is

Bag A contains 4 white and 2 black balls, bag B contains 3 white and 3 black balls and bag C contains 2 white and 4 black balls. If a bag is chosen at random and a ball is chosen at random from it, then the probability that the ball drawn is black is

If \(y=\left(\sin ^{-1} 2 x\right)^2+\left(\cos ^{-1} 2 x\right)^2\), then \(\left(1-4 x^2\right) y_2-4 x y_1=\)

Let $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}+2\hat{j}-3\hat{k}, \overrightarrow{c}=\hat{i}-\hat{j}$ and $\overrightarrow{d}=\hat{i}+\hat{j}+x\hat{k}$. If $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}$ is perpendicular to $\overrightarrow{d}$, then $x=$

A random variable \(\mathrm{X}\) has the following probability distribution \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=\mathrm{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \ \hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.15 & 0.23 & \mathrm{k} & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \ \hline \end{array} For the events \(E=\{x / x\) is a prime number \(\}\) and \(F=\{x / x < 4\}\) then \(P(E \cup F)=\)

\(\int_0^\pi \frac{\theta \sin \theta}{1+\cos ^2 \theta} d \theta\) is equal to

\(\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\)

A black coloured element with \(\mathrm{ns}^2 \mathrm{np}^1\) outer electronic configuration cannot react with air in its crystalline form. However, in amorphous form, it gives an oxide in air which is acidic in nature. Identify the element.

The dibasic oxoacid of phosphorus on disproportionation gives two products A and B. \(A\) and \(B\) are respectively

If the number of turns per unit length of a solenoid is doubled, what happens to the magnetic field in the solenoid?

Assertion (A): \(\mathrm{Be}\) and \(\mathrm{Mg}\) do not impart any colour to the flame. Reason (R): The electrons in them are strongly bound to get excited the by flame. The correct option among the following is:

Identify the correct statements from the following: (i) A closed vessel containing \(90 \% \mathrm{CO}_2\) and \(10 \% \mathrm{O}_2\) is an aerosol. (ii) Milk is an emulsion. (iii) Smoke is an aerosol. (iv) Peptísation is a method of purification of colloidal solution. (v) Ultrafilteration is a method of purification of colloidal solution.

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 \(\begin{array}{llllllll}\text { A } & \text { B } & \text { C } & \text { D } & \text { A } & \text { B } & \text { C } & \text { D }\end{array}\)