Two poles, \(\mathrm{AB}\) of length \(a\) metres and \(\mathrm{CD}\) of length \(\mathrm{a}+\mathrm{b}(\mathrm{b} \neq \mathrm{a})\) metres are erected at the same horizontal level with bases at \(\mathrm{B}\) and \(\mathrm{D} .\) If \(\mathrm{BD}=\mathrm{x}\) and \(\tan \angle\,ACB=\frac{1}{2}\), then:

The number of the real solutions of the equation : \( x|x+3|+|x-1|-2=0 \) is

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be a G.P. such that \(a_{1}<0\); \(a_{1}+a_{2}=4\) and \(a_{3}+a_{4}=16 .\) If \(\sum\limits_{i=1}^{9} a_{i}=4 \lambda,\) then \(\lambda\) is equal to 

Let \(\alpha, \beta\) be the roots of the equation \(x^2-\sqrt{2} x+2=0\). Then \(\alpha^{14}+\beta^{14}\) is equal to

Let \(f\) be a real valued continuous function defined on the positive real axis such that \(g(x)=\int_0^x \mathrm{t} f(\mathrm{t}) \mathrm{dt}\). If \(\mathrm{g}\left(x^3\right)=x^6+x^7\), then value of \(\sum_{r=1}^{15} f\left(\mathrm{r}^3\right)\) is :

The radius of circle, having minimum area, which touches the cruve  \(y = 4 - {x^2}\) and the lines \(y = \left| x \right|\) is :

Let \(\mathrm{S}=\{1,2,3,4,5,6,9\} .\) Then the number of elements in the set \(\mathrm{T}=\{\mathrm{A} \subseteq \mathrm{S}: \mathrm{A} \neq \phi\) and the sum of all the elements of \(\mathrm{A}\) is not a multiple of 3\(\}\) is ..... .

Two tangents are drawn from the point \(\mathrm{P}(-1,1)\) to the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-6 \mathrm{y}+6=0\). If these tangents touch the circle at points \(A\) and \(B\), and if \(D\) is a point on the circle such that length of the segments \(A B\) and \(A D\) are equal, then the area of the triangle \(A B D\) is eqaul to:

The value of \(\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} d x,\) where \([x]\) is the greatest integer \(\leq x ,\) is

Suppose \(f ( x )\) is a polynomial of degree four, having critical points at \(-1,0,1\) . If \(T =\{ x \in R \mid f ( x )= f (0)\},\) then the sum of squares of all the elements of \(T\) is

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

Which of the following is true for \(y ( x )\) that satisfies the differential equation \(\frac{d y}{d x}=x y-1+x-y ; y(0)=0\)

Among : \(( S 1): \lim _{ n \rightarrow \infty} \frac{1}{ n ^2}(2+4+6+\ldots \ldots \ldots+2 n)=1\) (S2) : \(\lim _{ n \rightarrow \infty} \frac{1}{ n ^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots \ldots .+ n ^{15}\right)=\frac{1}{16}\)