The function \(f: R \rightarrow R\) is defined by \(f(x)=\cos ^2 x+\sin ^4 x\) for \(x \in R\), then \(f(R)\) is equal to

If \(\omega\) is a complex cube root of unity, then \(\sum_{k=1}^6\left(\omega^k+\frac{1}{\omega^k}\right)^2=\)

The volume (in cubic units) of the tetrahedron with edges \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) is

If the chord \(x+y=1\) of the circle \(x^2+y^2=a^2\) subtends a right angle at the origin, then \(a=\)

\(\lim _{n \rightarrow \infty} \sum_{r=1}^n \cot ^{-1}\left(r^2+\frac{3}{4}\right)=\)

The area of the region that is common to the circle \(x^2+y^2=16 a^2\) and the parabola \(y^2=6 a x\) is

If \(x+2 y-3=0,3 x+4 y-7=0,2 x+3 y-4\) \(=0\) and \(4 x+5 y-6=0\) are the equations of four lines, then

A force, \(\overrightarrow{\mathrm{F}}=(4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}) \mathrm{N}\) is acting on a body making an angle \(\theta\) with the horizontal. Then the angle ' \(\theta\) ' is

MATCH THE FOLLOWING:

List - I	List - II
(Metal in drinking water)	(Maximum prescribed concentration in ppm)
A) Mn	I) 0.05
B) Zn	II) 3.0
C) Cd	III) 0.005
D) Cu	IV) 5.0

If a point \(P\) denotes a complex number \(z=x+i y\) in the argand plane and if \(\frac{z+1}{z+i}\) is a purely real number, then the locus of \(P\) is

Two spheres A & B of radii \(4 \mathrm{~cm} \& 6 \mathrm{~cm}\) are given charges of \(80 \mu \mathrm{C} \& 40 \mu \mathrm{C}\) respectively. If they are connected by a fine wire, the amount of charge flowing from one to the other is

Match the following
\(\begin{array}{ll}
\text{List - I} & \text{List - II} \\
\text{A) } \mathrm{Al}_2 \mathrm{O}_3+2 \mathrm{NaOH}+3 \mathrm{H}_2 \mathrm{O} \rightarrow 2 \mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_4\right] & \text{I) Roasting} \\
\text{B) } \mathrm{Ni}(\mathrm{CO})_4 \xrightarrow{230^{\circ} \mathrm{C}} \mathrm{Ni}+4 \mathrm{CO} & \text{II) Calcination} \\
\text{C) } \mathrm{Fe}_2 \mathrm{O}_3 \cdot 3 \mathrm{H}_2 \mathrm{O} \xrightarrow{\Delta} \mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \mathrm{O} & \text{II) Vapour phase refining} \\
\text{D) } 2 \mathrm{PbS}+3 \mathrm{O}_2 \rightarrow 2 \mathrm{PbO}+2 \mathrm{SO}_2 & \text{IV) Electrolysis} \\
& \text{V) Leaching}
\end{array}\)
The correct answer is

The half-life periods of a first order reaction at \(300 \mathrm{~K}\) and \(400 \mathrm{~K}\) are \(50 \mathrm{~s}\) and \(10 \mathrm{~s}\) respectively. The activation energy of the reaction in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is \((\log 5=0.70)\)