Let \(m\) and \(n\) be the numbers of real roots of the quadratic equations \(x^2-12 x+[x]+31=0\) and \(x ^2-5| x +2|-4=0\) respectively, where \([ x ]\) denotes the greatest integer \(\leq x\). Then \(m ^2+ mn + n ^2\) is equal to \(..............\).

An urn contains \(5\) red and \(2\) green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is

A circle in inscribed in an equilateral triangle of side of length \(12\). If the area and perimeter of any square inscribed in this circle are \(\mathrm{m}\) and \(\mathrm{n}\), respectively, then \(m+n^2\) is equal to

Let \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \quad a_{i}>0, i=1,2,3\) be \(a\) vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of \(\vec{a}\) on the vector \(3 \hat{i}+4 \hat{j}\) be \(7\) . Let \(\vec{b}\) be a vector obtained by rotating \(\vec{a}\) with \(90^{\circ}\). If \(\vec{a}, \vec{b}\) and \(x\)-axis are coplanar, then projection of a vector \(\vec{b}\) on \(3 \hat{i}+4 \hat{j}\) is equal to

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

If the line \(x = y = z\) intersects the line \(x \sin A+ y\) \(\sin B + z \sin C -18=0= x \sin 2 A + y \sin 2 B + z\) \(\sin 2 C -9\), where \(A , B , C\) are the angles of a triangle \(ABC\), then \(80\left(\sin \frac{ A }{2} \sin \frac{ B }{2} \sin \frac{ C }{2}\right)\) is equal to \(..........\).

The number of bijective functions \(f :\{1,3,5, 7, \ldots \ldots . .99\} \rightarrow\{2,4,6,8, \ldots \ldots, 100\}\), such that \(f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \ldots f(99), \quad\) is

If the position vectors of the vertices \(A, B\) and \(C\) of a \( \Delta ABC\) are respectively \(4\hat i + 7\hat j + 8\hat k\,,\,2\hat i + 3\hat j + 4\hat k\) and \(2\hat i + 5\hat j + 7\hat k\) then the position vector of the point, where the bisector of \(\angle A\) meets \(BC\) is

\(\mathop {\lim }\limits_{y \to 0} \frac{{\sqrt {1 + \sqrt {1 + {y^4}} }  - \sqrt 2 }}{{{y^4}}} = \)

If all the six digit numbers \(x_1 x_2 x_3 x_4 x_5 x_6\) with \(0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6\) are arranged in the increasing order, then the sum of the digits in the \(72^{\text {th }}\) number is \(............\).

Five numbers are in \(A.P.\), whose sum is \(25\) and product is \(2520 .\) If one of these five numbers is \(-\frac{1}{2},\) then the greatest number amongst them is

If \(1+\left(1-2^{2} \cdot 1\right)+\left(1-4^{2} \cdot 3\right)+\left(1-6^{2} \cdot 5\right)+\ldots \ldots+\left(1-20^{2} \cdot 19\right)\) \(=\alpha-220 \beta,\) then an ordered pair \((\alpha, \beta)\) is equal to 

Consider a set of \(3 n\) numbers having variance \(4.\) In this set, the mean of first \(2 n\) numbers is \(6\) and the mean of the remaining \(n\) numbers is \(3.\) A new set is constructed by adding \(1\) into each of first \(2 n\) numbers, and subtracting \(1\) from each of the remaining \(n\) numbers. If the variance of the new set is \(k\), then \(9 k\) is equal to .... .