The number of elements in the set
\( S=\{x:x\in[0,100] \text{ and } \int_{0}^{x}t^{2}sin(x-t)dt=x^{2}\} \) is:

If the area (in \(sq. units\)) of the region \(\left\{ {\left( {x,y} \right):{y^2} \le 4x,x + y \le 1,x \ge 0,y \ge 0} \right\}\) is \(a\sqrt 2  + b\), then \(a -b\) is equal to

The area of the region enclosed by the curves \(y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|\) and \(y\)-axis is:

If \(\cos \,\alpha  + \cos \,\beta  = \frac{3}{2}\) and \(\sin \,\alpha  + \sin \,\beta  = \frac{1}{2}\) and \(\theta \) is the the arithmetic mean of \(\alpha \) and \(\beta \) , then \(\sin \,2\theta  + \cos \,2\theta \) is equal to 

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero non-coplanar vectors. Let the position vectors of four points \(A, B, \quad C\) and \(D\) be \(\vec{a}-\vec{b}+\vec{c}, \lambda \vec{a}-3 \vec{b}+4 \vec{c}\), \(-\vec{a}+2 \vec{b}-3 \vec{c}\) and \(2 \vec{a}-4 \vec{b}+6 \vec{c}\) respectively. If \(\overrightarrow{A B}\), \(\overline{ AC }\) and \(\overline{ AD }\) are coplanar, then \(\lambda\) is :

Let \(\mathrm{A}(x, y, z)\) be a point in \(x y\)-plane, which is equidistant from three points \((0,3,2),(2,0,3)\) and ( \(0,0,1\) ).
Let \(\mathrm{B}=(1,4,-1)\) and \(\mathrm{C}=(2,0,-2)\). Then among the statements
(S1) : \(\triangle \mathrm{ABC}\) is an isosceles right angled triangle, and
(S2) : the area of \(\triangle \mathrm{ABC}\) is \(\frac{9 \sqrt{2}}{2}\),

Let \(\vec{a}=\hat{i}-3 \hat{j}+7 \hat{k}, \vec{b}=2 \hat{i}-\hat{j}+\hat{k}\) and \(\vec{c}\) be a vector such that \((\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})\). If \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=130\), then \(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}\) is equal to ....................

Let \(\alpha, \beta \in \mathbb{R}\) be such that the system of linear equations
\(x + 2y + z = 5\)
\(2x + y + \alpha z = 5\)
\(8x + 4y + \beta z = 18\)
has no solution. Then \(\dfrac{\beta}{\alpha}\) is equal to :

The ratio of the coefficient of the middle term in the expansion of \((1+x)^{20}\) and the sum of the coefficients of two middle terms in expansion of \((1+x)^{19}\) is \(....\)

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 

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an \(A.P.\) If \(\sum_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4\), then \(4 a_{2}\) is equal to

If the equation of the plane passing through the point \((1,1,2)\) and perpendicular to the line \(x-3 y+2 z-1=04 x-y+z\) is \(Ax + By + Cz =1\), then \(140( C - B + A )\) is equal to \(.........\).

In an examination,\(5\) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is \(..........\).