The equation \(\cos ^{-1}(1-x)-2 \cos ^{-1} x=\frac{\pi}{2}\) has

$f\left(x\right)=\left\{\begin{array}{cl}\frac{{72}^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}},& x\ne 0\\ k\log 2\log 3,& x=0\end{array}\right.$ Find the value of $\text{'}k\text{'}$ for which the function $f$ is continuous.

The equation of an ellipse in its standard form, given the distance between its foci is 2 units and the length of its latusrectum is \(\frac{15}{2}\) units, is

Match the following elements of \(\left[\begin{array}{ccc}1 & -1 & 0 \\ 0 & 4 & 2 \\ 3 & -4 & 6\end{array}\right]\) with their co-factors and choose the correct answer.
\(\begin{array}{lcll} & \text { Element } & & \text { Co-factor } \\ \text { A. } & -1 & \text { (1) } & -2 \\ \text { B. } & 1 & (2) & 32 \\ \text { C. } & 3 & (3) & 4 \\ \text { D. } & 6 & \text { (4) } & 6 \\ & & \text { (5) } & -6\end{array}\)

The base of a triangle is 80 and one of the base angles is \(60^{\circ}\). If the sum of the lengths of the other two sides is 90, then the shortest side is of length

If \(\alpha, \beta\) and \(\gamma\) are non-zero vectors such that \(|\beta|=|\gamma|=1\) and \(|\alpha|=10\), then \((\alpha \times(\beta+\gamma) \times(\beta \times \gamma) \cdot(\beta-\gamma)=\)

A team of 5 students is to be selected from 12 students. If two particular students are to be included in that team, then the number of ways that such team can be selected is

One mole of ethanol (l) was completely burnt in oxygen to form \(\mathrm{CO}_2(\mathrm{~g})\) and \(\mathrm{H}_2 \mathrm{O}(l)\). What is the \(\Delta_r \mathrm{H}^{\ominus}\) (in \(\mathrm{kJ} \mathrm{mol}^{-1}\) ) for this reaction?
(The standard enthalpy of formation \(\left(\Delta_f \mathrm{H}^{\ominus}\right)\) of \(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(l), \mathrm{CO}_2(\mathrm{~g})\) and \(\mathrm{H}_2 \mathrm{O}(l)\) is respectively \(-277,-393\) and \(-286 \mathrm{~kJ} \mathrm{~mol}^{-1}\).)

If the circle \(S \equiv x^2+y^2+2 g x+4 y+1=0\) bisects the circumference of the circle \(\mathrm{x}^2+\mathrm{y}^2-2 \mathrm{x}-3=0\), then the radius of circle \(\mathrm{S}=0\) is

A radar system can detect an enemy plane in one out of 10 consecutive scans. The probability that it cannot detect an enemy plane at least two times in four consecutive scans, is

Match List-I with List-II
\(\begin{array}{|c|l|l|l|}\hline & \text{List-I (Reaction)} & & \text{List-II (Enzyme)} \\ \hline \text{A.} & \begin{array}{l} \text{Hydrolysis of starch to} \\ \text{maltose}\end{array} & \text{I} & \text{Diastase} \\ \hline \text{B.} & \begin{array}{l} \text{Conversion of proteins} \\ \text{to peptides} \end{array} & \text{II} & \text{Pepsin} \\ \hline \text{C.} & \begin{array}{l} \text{Hydrolysis of sucrose} \\ \text{to glucose and fructose}\end{array} & \text{III} & \text{Invertase} \\ \hline \text{D.} & \text{Glucose to ethanol} & \text{IV} & \text{Zymase} \\ \hline\end{array}\)
The correct answer is

One end of a steel wire of radius ' \(r\) ' is fixed to a ceiling and a load of \(3 \mathrm{~kg}\) is attached to the free end of the wire. Another wire made of copper of radius ' \(2 r\) ' is attached to the bottom of \(3 \mathrm{~kg}\) load and a \(2 \mathrm{~kg}\) load is attached to the free end of the copper wire. The ratio of longitudinal strains produced in copper and steel wires is
(Young modulus of steel \(=20 \times 10^{10} \mathrm{Nm}^{-2}\) )
(Young modulus of copper \(=12 \times 10^{10} \mathrm{Nm}^{-2}\) )

If the mean of the data $p, 6,6,7,8,11,15,16$, is $3$ times $p$, then the mean deviation of the data from its mean is