How many minimum number of times must a man toss a fair coin so that the probability of having at least one head is more than \(90 \%\) ?

The function \(f ( x )=\log (\sin x )\) is ___________

If \(f(x)=|x|+|x-5|\), then which of the following statements are TRUE?
(A) \(f\) is a continuous function everywhere
(B) \(f\) is a continuous function except x = 5 and x = 0
(C) \(f\) is a continuous function except x = 0 but not differentiable at x = 5
(D) \(f\) is a continuous function everywhere but not differentiable at x = 0 and x = 5
Choose the correct answer from the options given below:

The differential equation whose solution is \(A x^2+B y^2=1\) where \(A\) and \(B\) are arbitrary constant is of:
(A) first order and first degree
(B) second order and first degree
(C) second order and second degree
(D) second order
Choose the correct answer from the options given below :

The function \(f:[-1,1] \rightarrow \mathbb{R}\) (set of real numbers) given by \(f(x)=\frac{x}{x+3}\) is

(A) The degree of differential equation \(\frac{d^2 y}{d x^2}+\sin \left(\frac{d y}{d x}\right)=0\) is 2
(B) The degree of differential equation \(\frac{d}{d x}\left(\frac{d y}{d x}\right)+\left(\frac{d y}{d x}\right)^3+\sin x=0\) is 1
(C) The order of differential equation \(2 x^2 \frac{d^2 y}{d x^2}-3 \frac{d y}{d x}+y=0\) is 2
(D) The number of arbitrary constants in a particular solution of a differential equation of third order are 3
(E) The integrating factor of the differential equation \(\frac{d y}{d x}-y=\cos x\) is \(e^{-x}\)
Choose the correct answer from the options given below:

If \(\vec{a}\) and \(\vec{b}\) are two unit vectors and \(\vec{a}+\vec{b}\) is also a unit vector, the magnitude of \(\vec{a}-\vec{b}\) is

The area (in sq. units) of the region bounded by the curve \(y=x^5\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=1\) is equal to:

Given a linear programming problem max \(z=22 x+18 y\) subject to \(x+y \leq 20\), \(360 x+240 y \leq 5760\) and \(x \geq 0, y \geq 0\). Its corner points are:

Endosperm is completely consumed during embryo development in the seed of -

Read the passage carefully and answer the questions
Replacement of a hydrogen atom in a hydrocarbon by an alkoxy or carboxyl group yields a class of compounds known as ethers. Ethers are classified as symmetrical or unsymmetrical on the basis of groups attached to the oxygen atoms. Diethyl ether, a symmetrical ether, has been widely used as an inhalation anesthetic. Ethers can be prepared by acid catalyzed intermolecular dehydration of alcohols and Williamson's synthesis. Acid catalyzed dehydration of alcohols is not generally preferred as it gives a mixture of elimination and substitution products. In Williamson's synthesis, an alkyl halide is allowed to react with sodium alkoxide. Ethers containing substituted Alkyl groups may also be prepared by this method. The C-O bond in ether is weakly polar and is cleaved under drastic conditions with excess of hydrogen halides. In electrophilic substitution, the alkoxy group deactivates the aromatic ring and directs the incoming group to ortho and para positions.
In Williamson synthesis, the alkoxide ion attacks the alkyl halide via which pathway?

A first order reaction is 50% completed in \(1.26 \times 10^{14}\) s.
How much time would it take for 100% completion?

Light of two different frequencies whose photons have energies 1 eV and 2.5 eV respectively, illuminate a metal of work function 0.5 eV. The ratio of the maximum KE of the emitted electrons will be :