If \(\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k},|\vec{b}|=5\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{6}\), then the area of the triangle formed by these two vectors as two sides is :

\(\tan ^{-1} \frac{1}{\sqrt{3}}-\sin ^{-1} \frac{1}{2}\) is equal to :

If \(P(A)=\frac{3}{5}, P(B)=\frac{1}{2}\) and \(P(A \cap B)=\frac{1}{4}\), then \(P(\bar{A} \mid \bar{B})\) is

The function \(f:[0,1] \rightarrow R\) given by \(f(x)=\alpha x^3\) has:

If \(A=\left[\begin{array}{cc}2 & 1 \\ 1 & 4 \\ -1 & 6 \\ 0 & 7\end{array}\right]\) and \(B=\left[\begin{array}{cccc}1 & 2 & 3 & 0 \\ 2 & -1 & 6 & 7\end{array}\right]\), then

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

If the line drawn from the point (-2, -3, -4) meets a plane at right angle at the point (1, -2, 4), then equation of plane is:

Read the passage carefully and answer the questions based on the passage:
The number of reacting species (atoms, ions or molecules) taking part in an elementary reaction, which must collide simultaneously in order to bring about a chemical reaction is called molecularity of a reaction. In the rate equation Rate \(=k[\mathrm{~A}]^x[\mathrm{~B}]^y\)
\(x\) and \(y\) indicate how sensitive the rate is to the change in concentration of A and B, respectively. Sum of these exponents, i.e., \(x +y\) gives the overall order of a reaction where \(x\) and \(y\) represent the order with respect to the reactants A and B respectively. Hence, the sum of powers of the concentration of the reactants in the rate law expression is called the order of that chemical reaction. For a first order reaction, the concentration of the reactant varies as \([\mathrm{R}]=[\mathrm{R}]_0 e^{-k t}\)
The units of first and zero order rate constant of reactions are respectively:

If \(A=\left[\begin{array}{cc}-2 & 6 \\ -5 & -1\end{array}\right]\) then \(A^{-1}\) is :

The demand function for a certain product is represented by the equation: p = 20 + 5x - \(3 x^2\), where \(x\) is the number of units demanded and p is the price per unit (in Rs.), then the marginal revenue when 2 units are sold, is:

Huygen's principle allows us to understand and determine.

Value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log (\tan x) d x\) is

A person started keeping aside 10,000 each year for his child's college education in a sinking fund. The amount he will receive after 6 years, if the rate of interest is \(10 \%\) p.a., is: [Use \((1.1)^{\wedge} 6=1.771\) ]