For \(x \in R\) , \(f\left( x \right) = \left| {\log 2 - \sin x} \right|\) and \(g\left( x \right) = f\left( {f\left( x \right)} \right)\) then ..  .

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

If \((\vec{a}+3 \vec{b})\) is perpendicular to \((7 \vec{a}-5 \vec{b})\) and \((\vec{a}-4 \vec{b})\) is perpendicular to \((7 \vec{a}-2 \vec{b})\), then the angle between \(\vec{a}\) and \(\vec{b}\) (in degrees) is \(......\)

On which of the following lines lies the point of intersection of the line, \(\frac{{x - 4}}{2} = \frac{{y - 5}}{2} = \frac{{z - 3}}{1}\) and the plane, \(x + y + z = 2\) ?

If \(tan\, A\) and \(tan\, B\) are the roots of the quadratic equation, \(3x^2 - 10x - 25 = 0\) then the value of \(3\, sin^2\, (A +B)- 10\, sin\,(A +B). cos\,(A+ B)- 25\, cos^2\, (A+B)\) is

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

In a triangle, the sum of lengths of two sides is \(x\) and the product of the lengths of the same  two sides is \(y.\) If \(x^2 - c^2 = y ,\) where \(c\) is the length of the third side of the triangle, then the circumradius of the triangle is

A line in the \(3-\) dimensional space makes an angle \(\theta \left( {0 < \theta  \le \frac{\pi }{2}} \right)\) with both the \(x\) and \(y\) axes. Then the set ofall values of \(\theta \) is the interval

Let \(p_n\) denote the total number of triangles formed by joining the vertices of an \(n\)-side regular polygon. If \(p_{n+1} - p_n = 66\), then the sum of all distinct prime divisors of \(n\) is:

If \( (\frac{1}{^{15}C_{0}}+\frac{1}{^{15}C_{1}})(\frac{1}{^{15}C_{1}}+\frac{1}{^{15}C_{2}})...(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}) = \frac{a^{13}}{^{14}C_{0}^{14}C_{1}...^{14}C_{12}} \) then 30a is equal to :

\(\int \limits_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^x+6} d x\)

Consider a curve \(y=y(x)\) in the first quadrant as shown in the figure. Let the area \(A_{1}\) is twice the area \(A _{2}\). Then the normal to the curve perpendicular to the line \(2 x -12 y =15\) does NOT pass through the point.

Let \(P=\{z \in \mathbb{C}:|z+2-3 i| \leq 1\}\) and \(Q=\{z \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}\). Let in \(\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|\) be maximum and minimum at \(z_1\) and \(z_2\) respectively. If \(\left|z_1\right|^2+2|z|^2=\alpha+\beta \sqrt{2}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) equals ...........