If \(\theta \in[0, \pi]\) is the angle between any two non-zero vectors \(\vec{a}\) and \(\vec{b}\), such that \(|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|\), then \(\theta=\)

Two persons A and B throw a die alternately till one of them gets a six and wins the game. If A begins, then the probabilities of winning of A and B respectively are

A particle moves along the curve \(6 y=x^3+2\). The point(s) on the curve at which the \(y\)-coordinate is changing 8 times as fast as the \(x\)-coordinate are:

The general solution of the differential equation \((1+y) d x-2 x d y=0\) is :

The relation R in R (set of real numbers) is defined by \(R=\left\{(a, b): a \leq b^3\right\}\), then \(R\) is

For the differential equation \((x+y) d y+(x-y) d x=0\), which of the following is/are correct?
(A) Differential equation is homogeneous
(B) Order of differential equation is 1
(C) Integrating factor of differential equation is \(e ^{ x }\)
(D) Degree of the equation is not defined
Choose the correct answer from the options given below:

For \(x>0\), the minimum value of \(\frac{x}{\log _e x}\) is

The relationship between \(a\) and \(b\) making \(f(x)\) continuous at \(x=3\), where \(f(x)=\left\{\begin{array}{ll}a x+1, & x \leq 3 \\ b x+3, & x>3\end{array}\right.\) is:

Answer the question on the basis of passage given below :
Formaldehyde is the first member of the aldehydic series. It is present in green leaves of plants where its presence is supposed to be due to the reaction of \(CO_2\) with water in the presence of sunlight and chlorophyll. Commercially, formaldehyde is obtained by the controlled oxidation of methane in the presence of molybdenum oxide (\(MoO\)) acting as a catalyst. Formaldehyde is a constituent of a number of polymers also called resins. These are non volatile solids or semi solids which are obtained directly from certain plants and can be synthesized in the laboratory.
Benzaldehyde is less reactive towards nucleophilic addition reaction than formaldehyde because of:

Which of the following compound is a gem-dihalide?

A cell of emf 2.2 V gives a current of 0.2 A through a resistance of 9 \(\Omega\). The internal resistance of the cell will be:

The differential equation of the family of parabolas \(y^2=4 a x\) where \(a\) is an arbitrary constant is :

Given that two identical coils, one of copper and another of aluminium are rotated with the same angular speed in an external magnetic field, which of the following statements is true?