A wire of length \(20\, \mathrm{~m}\) is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in \(meters\)) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:

If \((1,5,35),(7,5,5),(1, \lambda, 7)\) and \((2 \lambda, 1,2)\) are coplanar, then the sum of all possible values of \(\lambda\) is

Let the sum of the maximum and the minimum values of the function \(f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}\) be \(\frac{m}{n}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\). Then \(\mathrm{m}+\mathrm{n}\) is equal to :

Let \(f(x)\) be a polynomial of degree \(5\), and have extrema at \(x = 1\) and \(x = -1\). If \(\displaystyle\lim_{x \to 0} \left(\dfrac{f(x)}{x^3}\right) = -5\), then \(f(2) - f(-2)\) is equal to:

The position vectors of the vertices \(A, B\) and \(C\) of a triangle are \(2 \hat{i}-3 \hat{j}+3 \hat{k}, \quad 2 \hat{i}+2 \hat{j}+3 \hat{k} \quad\) and \(-\hat{i}+\hat{j}+3 \hat{k}\) respectively. Let \(l\) denotes the length of the angle bisector \(\mathrm{AD}\) of \(\angle \mathrm{BAC}\) where \(\mathrm{D}\) is on the line segment \(\mathrm{BC}\), then \(2 l^2\) equals :

Let \(u=\frac{2 z+i}{z-k i}, z=x+i y\) and \(k>0 .\) If the curve represented by \(\operatorname{Re}( u )+\operatorname{Im}( u )=1\) intersects the \(y\)-axis at the points \(P\) and \(Q\) where \(P Q=5,\) then the value of \(k\) is 

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 ...........

For the curve \(y\, = 3\, sin\,\theta\, cos\,\theta\), \(x\, = e^{\theta}\, sin\,\theta\), \(0 \leq \theta  \leq \pi \) , the tangent is parallel to \(x-\) axis when \(\theta \) is

Consider a hyperbola \(H : x ^{2}-2 y ^{2}=4\). Let the tangent at a point \(P (4, \sqrt{6})\) meet the \(x\) -axis at \(Q\) and latus rectum at \(R \left( x _{1}, y _{1}\right), x _{1}>0 .\) If \(F\) is a focus of \(H\) which is nearer to the point \(P\), then the area of \(\Delta QFR\) is equal to ....... .

Let \(a,b \in R,\left( {a \ne 0} \right)\). if the function \(f\) defined as \(f\left( x \right)\left\{ \begin{array}{l}
\frac{{2{x^2}}}{a}\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,0 \le x < 1\,\,\,\\
a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,1 \le x < \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
\frac{{2{b^2} - 4b}}{{{x^3}}}\,\,\,,\,\,\,\,\,\sqrt 2  \le x < \infty 
\end{array} \right.\,\,\,\,\)  is continuous in the interval \(\left[ {0,\infty } \right)\) , then an ordered pair \((a, b)\) is

For a statistical data \(\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2=371\). He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is

Fifteen football players of a club-team are given \(15\) T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least \(3\) players pick the correct \(T\)-shirt is

If \(a\) and \(b\) are the roots of equation \(x^2-7 x-1=0\), then the value of \(\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}\) is equal to \(........\).