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 \(..........\).

Let \(\left\{a_{n}\right\}_{n=0}^{\infty}\) be a sequence such that \(a _{0}= a _{1}=0\) and \(a _{ n +2}=2 a _{ n +1}- a _{ n }+1\) for all \(n \geq 0\). Then, \(\sum\limits_{ n =2}^{\infty} \frac{ a _{ n }}{7^{ n }}\) is equal to

Let \( y=y(x) \) be the solution curve of the differential equation \( (1+x^{2})dy+(y-\tan^{-1}x)dx=0, \) \( y(0)=1 \). Then the value of \( y(1) \) is:

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let \(f\left( x \right) = \int\limits_0^x {g\left( t \right)dt} \), where \(g\) is a non zero even function. If \(f(x+5) = g(x)\) , then \(\int\limits_0^x {f\left( t \right)dt} \) equals

If one real root of the quadratic equation \(81x^2 + kx  + 256 = 0\) is cube of the other root, then a value of \(k\) is

Let R be a relation defined on the set \( \{1,2,3,4\}\times\{1,2,3,4\} \) by \( R=\{((a,b),(c,d)):2a+3b=3c+4d\} \).
Then the number of elements in R is

Statement \(-1:\) The line \(x - 2y = 2\) meets the parabola, \(y^2 + 2x = 0\) only at the point \((-2, - 2).\) Statement \(-2:\) The line \(y = mx - \frac{1}{{2m}}(m \ne 0)\) is tangent to the parabola, \(y^2 = - 2x\) at the point \(\left( { - \frac{1}{{2{m^2}}}, - \frac{1}{m}} \right).\)

The system of linear equations \(x + \lambda y - z = 0,\lambda x - y - z = 0\;,\;x + y - \lambda z = 0\) has a non-trivial solution for:

Let \( A=\{2,3,5,7,9\} \). Let R be the relation on A defined by \(x\) Ry if and only if \( 2x\le3y \). Let \(l\) be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then \( l+m \) is equal to :

Let \(S=\{4,6,9\}\) and \(T=\{9,10,11, \ldots, 1000\}\). If \(A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in N, a_{1}, a_{2}, a_{3}, \ldots, a_{k} \in S\right\}\) then the sum of all the elements in the set \(T - A\) is equal to \(......\)

If \(S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\dfrac{5\theta}{2} = \cos 7\theta \cos\dfrac{7\theta}{2}\right\}\), then \(n(S)\) is equal to _______.

The area (in sq. units) of the part of the circle \(x ^{2}+ y ^{2}=36\) which is outside the parabola \(y ^{2}=9 x ,\) is