Two biased dice are thrown together. For the first die \(P(6)=1 / 2\), other scores are equally likely. While for the second die, \(P (1)=2 / 5\) and other scores are equally likely then the mean for the probability distribution of 'the number of ones seen', will be

The solution of the differential equation \(\frac{d y}{d x}=\sqrt{\frac{y}{x}}\) is

Derivative of \(\cos \left(\sin \left(x^2\right)\right)\) with respect to x is:

If \(A^T=\left[\begin{array}{cc}-2 & 3 \\ 1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{cc}-1 & 0 \\ 1 & 2\end{array}\right]\), then the matrix \((A+2 B)^T\) is

A box contains 3 white and 2 red balls. If we draw one ball and without replacing the first ball, the probability of drawing red ball in the second draw is:

If \(P(A \cap B)=0.4, P(\bar{A} \cap \bar{B})=0.3\), then the value of \(P(\bar{A})+P(\bar{B})\) is:

The mean of the number of heads in a simultaneous toss of three coins is :

For the given \(5\) values, \(15, 18, 21, 27, 39;\) the three year moving averages are:

"Answer the question on basis of passage given below:
\(G = \frac{\kappa A}{l}\) =k(both \(A\) and \(l\) are unity in their appropriate units in m or cm)
Molar conductivity of a solution at a given concentration is the conductance of the volume \(V\) of solution containing one mole of electrolyte kept between two electrodes with area of cross section \(A\) and distance of unit length (\(l\)). Therefore,
\(\Lambda_m = \frac{\kappa A}{l}\)
Since \(l = 1\) and \(A = V\) (volume containing 1 mole of electrolyte),
\(\Lambda_m = \kappa V\)
Calculate \(\Lambda_m\) for aluminium sulphate. When \(\Lambda_m\) for aluminium chloride, sulphuric acid and hydrochloric acid are 236.2, 459.8 and 121.4 S cm\(^2\) mol\(^{-1}\) respectively:"

Read the passage carefully and answer the Questions
Carboxylic acids are the earliest organic compounds isolated from nature. These are also known as organic acids but are weaker acids as compared to mineral acids. However, they are more acidic than alcohols and phenols because of greater resonance stabilization of the parent acid and its conjugate base. The acidic strength is determined by the \(K _a\), which is called the acidity constant. The greater the value of \(K _a\), the greater the tendency of the acid to ionize and hence the stronger the acid. The acidic strength of the saturated aliphatic carboxylic acid depends upon the inductive effect and its position with respect to - COOH groups. Electron-donating substituents tend to decrease whereas electron -withdrawing substituents tend to increase the acidic strength. The acidic strength of the aromatic carboxylic acid, on the other hand depends upon both the inductive effect and the resonance effect of the substituents.
How can we express the (acidity constant) \(K_{ a }\) of RCOOH ?

Match List-l with List-II

List-l (Enzymes)	List-II (Products)
(A) RNA polymerase I	(I) hnRNA
(B) RNA polymerase III	(II) Polymerization of deoxynucleotides
(C) RNA polymerase II	(III) rRNAs
(D) DNA polymerase	(IV) tRNA, 5srRNA and snRNAS

Choose the correct answer from the options given below:

Let \(f(x)=x|x|\). Then \(f(x)\) is:

Number of solutions of the equation \(\left|\begin{array}{ccc}-1 & 0 & \sin \theta \\ \sin \theta & -1 & 0 \\ 0 & \sin \theta & -1\end{array}\right|=0\) in \((0, \pi)\) is: