Let \([ t ]\) denote the greatest integer \(\leq t\). If for some \(\lambda \in R -\{0,1\}, \lim \limits_{x \rightarrow 0}\left|\frac{1-x+|x|}{\lambda-x+[x]}\right|=L,\) then \(L\) is equal to

\(\sum \limits_{ k =0}^6{ }^{51- k } C _3\) is equal to

The odd natural number \(a,\) such that the area of the region bounded by \(y =1, y =3, x =0, x = y ^{a}\) is \(\frac{364}{3}\), equal to.

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

Let \(f : S \rightarrow S\) where \(S =(0, \infty)\) be a twice differentiable function such that \(f ( x +1)= xf ( x )\) If \(g: S \rightarrow R\) be defined as \(g(x)=\log _{e} f(x),\) then the value of \(\mid g "(5)- g "(1) \mid\) is equal to :

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

Let \(\sum\limits_{k = 1}^{10} {f\,(a\, + \,k)} \, = \,16\,({2^{10}}\, - \,1),\) where the function \(f\) satisfies \(f(x + y) = f(x) f(y)\) for all natural numbers \(x, y\) and \(f(1) = 2.\) Then the natural number \(‘ a '\) is

The letters of the word '\(MANKIND\)' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word '\(MANKIND\)' is \(.....\)

The value of \(\frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots+100^2 \times 101}\) is

Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and R be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m} \mathrm{n}\) is equal to :-

Let \(z = a +i b , b \neq 0\) be complex numbers satisfying \(z ^{2}=\overline{ Z } \cdot 2^{1-|z|}\). Then the least value of \(n\) \(\in N\), such that \(z ^{ n }=( z +1)^{ n }\), is equal to.

The coefficients \(a, b, c\) in the quadratic equation \(a x^2+b x+c=0\) are from the set \(\{1,2,3,4,5,6\}\). If the probability of this equation having one real root bigger than the other is \(p\), then \(216\) p equals:

Let \(S\) be the set of all real values of \(k\) for which the system oflinear equations \(x +y + z = 2\) ; \(2x +y - z = 3\) ; \(3x + 2y + kz = 4\) has a unique solution. Then \(S\) is