\(\int_0^{\pi / 2} \frac{\sin ^8 x}{\sin ^8 x+\cos ^8 x} d x\) is equal to

The equation of tangent to the curve \(y=\sqrt{3 x-2}\) which is parallel to the line \(4 x-2 y+5=0\) is :

Which of the following statements is incorrect?

For some constant 'k', if the system of linear equations
\(2 x-y+3 z=1\)
\(x-2 y+z=3\)
\(k x+y-z=0\)
has a unique solution, then

Which of the following statements is/are true?
(A) The vector sum of the three sides of a triangle in order is \(\vec{c}\)
b (B) The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector \(\vec{r}=a \hat{i}+b \hat{j}+c \hat{k}\) are related as \(l=\frac{a}{r}, m=\frac{b}{r}, n=\frac{c}{r}\)
(C) If θ is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta\)
(D) The cross product of two vectors is commutative
Choose the correct answert from the option given below :

Let E and F are events associated with an experiment. If \(P(E)=0.4, P(F)=0.8\) and \(P(F \mid E)=0.6\) then \(P(E \mid F)\) is:

If a line makes angles \(\alpha, \beta\) and \(\gamma\) with positive \(x\)-axis, \(y\)-axis and \(z\)-axis respectively, then the value of \(\sin ^2 \frac{\alpha}{2} \cos ^2 \frac{\alpha}{2}+\sin ^2 \frac{\beta}{2} \cos ^2 \frac{\beta}{2}+\sin ^2 \frac{\gamma}{2} \cos ^2 \frac{\gamma}{2}\) is

The first isolated restriction endonuclease was:

Read the passage carefully and answer the Questions
Werner in 1898, propounded his theory of coordination compounds. In coordination compounds metals show two types of linkages (valences)-primary and secondary. The primary valences are normally ionisable and are satisfied by negative ions. The secondary valences are non-ionisable. These are satisfied by neutral molecules or negative ions. The secondary valence is equal to the coordination number and is fixed for a metal. The ions or group bound by the secondary linkages to the metal have characteristic spatial arrangements corresponding to the different coordination numbers. Both double salts as well as complexes are formed by the combination of two or more stable compounds in stoichiometric ratio. However, they differ in the fact that double salts such as carnallite, Mohr's salt, potash alum etc. dissociate into simple ions completely when dissolved in water. But complex ions do not dissociate completely into simple ions.
Which of the following statement is correct as per Werner theory?

The speed of an electromagnetic wave in a medium of dielectric constant 2.25 and relative permeability 4 is:

In a LPP , if the objective function \(Z=a x+b y\) has the same maximum value at two corner points of the feasible region, then the number of points at which \(Z_{\max }\) occurs is:

The reaction in which an amide is treated with bromine in presence of ethanolic solution of sodium hydroxide is called

Which of the following are the properties of Normal Distribution function f(x) and Normal probability curve:
(A) The probability of success remains the same in each trial and the number of trials is small in number.
(B) The curve is bell shaped and is symmetrical about the mean.
(C) If set of n trials are repeated N times, then frequency f(r) of r successes is given by
f(r) = N.P(r) = \(N e^{-m} \frac{m^r}{r!}\), r = 0, 1, 2...
(D) As x increases numerically, f(x) decreases rapidly and the maximum value of f(x) occurs at x = μ(mean).
Choose the correct answer from the options given below: