Let \(A(6,8), B(10 \cos \alpha,-10 \sin \alpha)\) and \(C(-10 \sin \alpha, 10 \cos \alpha)\), be the vertices of a triangle. If \(L(a, 9)\) and \(G(h, k)\) be its orthocenter and centroid respectively, then \((5 a-3 h+6 k+100 \sin 2 \alpha)\) is equal to ______ -.

Let a line pass through two distinct points \(P(-2,-1,3)\) and \(Q\), and be parallel to the vector \(3 \hat{i}+2 \hat{j}+2 \hat{k}\). If the distance of the point Q from the point \(\mathrm{R}(1,3,3)\) is 5 , then the square of the area of \(\triangle P Q R\) is equal to :

Let \(\vec{c}\) be the projection vector of \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda\gt0\), on the vector \(\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}\). If \(|\vec{a}+\vec{c}|=7\), then the area of the parallelogram formed by the vectors \(\vec{b}\) and \(\vec{c}\) is ________

Let \(\frac{\sin \mathrm{A}}{\sin \mathrm{B}}=\frac{\sin (\mathrm{A}-\mathrm{C})}{\sin (\mathrm{C}-\mathrm{B})}\), where \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are angles f a triangle \(\mathrm{ABC}\). If the lengths of the sides pposite these angles are \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) respectively, then :

Let \(\mathrm{P}(\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{Q}(1,6,4)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\). Then \(2 \alpha+\beta+\gamma\) is equal to ..............

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \(2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5}\), is .........

The domain of the function \(f(x)=\sin ^{-1}\left(\frac{3 x^{2}+x-1}{(x-1)^{2}}\right)+\cos ^{-1}\left(\frac{x-1}{x+1}\right)\) is :

If the points \(P\) and \(Q\) are respectively the circumcentre and the orthocentre of a \(\triangle ABC\), then \(\overrightarrow{ PA }+\overrightarrow{ PB }+\overrightarrow{ PC }\) is equal to

Let \( a_{1}=1 \) and for \( n\ge1 \), \( a_{n+1}\)
= \(\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}} \). Then \( |\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})| \) is equal to ........... .

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

In a certain town, \(25\%\) of the families own a phone and \(15\%\) own a car; \(65\%\) families own neither a phone nor a car and \(2,000\) families own both a car and a phone. Consider the following three statements \((A)\,\,\,5\%\) families own both a car and a phone
\((B)\,\,\,35\%\) families own either a car or a phone
\((C)\,\,\,40,000\) families live in the town
Then,

Given : \(f(x)=\left\{\begin{array}{ccc}{x} & {,} & {0 \leq x < \frac{1}{2}} \\ {\frac{1}{2}} & {,} & {x=\frac{1}{2}} \\ {1-x} & {,} & {\frac{1}{2} < x \leq 1}\end{array}\right.\) and \(g(x)=\left(x-\frac{1}{2}\right)^{2}, x \in R .\) Then the area (in sq. units) of the region bounded by the curves, \(y=f(x)\) and \(y=g(x)\) between the lines, \(2 \mathrm{x}=1\) and \(2 \mathrm{x}=\sqrt{3},\) is

Let \(\mathrm{ABC}\) be a triangle of area \(15 \sqrt{2}\) and the vectors \(\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \quad \overrightarrow{B C}=a \hat{i}+b \hat{j}+c \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=6 \hat{\mathrm{i}}+\mathrm{d} \hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \mathrm{d}>0\). Then the square of the length of the largest side of the triangle \(\mathrm{ABC}\) is ...........