If tangents are drawn from any point on the circle \(x^2+y^2=25\) to the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\), then the angle between the tangents is

If the circle \(S \equiv x^2+y^2+2 g x+4 y+1=0\) bisects the circumference of the circle \(\mathrm{x}^2+\mathrm{y}^2-2 \mathrm{x}-3=0\), then the radius of circle \(\mathrm{S}=0\) is

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three vectors such that \(|\mathbf{a}|=1,|\mathbf{b}|=2,|\mathbf{c}|=3\) and \(2 \mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{c}=\mathbf{c} \cdot \mathbf{a}=2\), Then, \([\mathbf{a} \mathbf{b} \mathbf{c}]^2=\)

Match the following
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=p x+q \\
(p \neq 0), \forall x \in R
\end{array} & \begin{array}{l}
\text {I. } f \text { is neither one-one nor onto }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
f: R \rightarrow R^{+} \cup\{0\} \text { is such that } f(x)=x^2, \forall x \in R
\end{array} & \begin{array}{l}
\text {II. } f \text { is both one-one and onto }
\end{array} \\
\hline \text { (C) } & \begin{array}{l}
f: N \rightarrow N \text { is such that } f(n)=n^2+2 n+3, \forall n \in N
\end{array} & \begin{array}{l}
\text {III. } f \text { is one-one but not onto }
\end{array} \\
\hline \text { (D) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=2\left(\cos ^2 5 x+\sin ^2 5 x\right) \\
\forall x \in R
\end{array} & \begin{array}{l}
\text {IV. } f \text { is onto but not one-one }
\end{array} \\
\hline && V. f \text{ is a constant function and also a bijection} \\
\hline
\end{array}\)
The correct answer is

In a city, 10 accidents take place in a span of 50 days.Assuming that the number of accidents follow the Poisson distribution,the probability that three or more accidents occur in a day, is

For any vector \(\mathbf{r}\)
\(\mathbf{i} \times(\mathbf{r} \times \mathbf{i})+\mathbf{j} \times(\mathbf{r} \times \mathbf{j})+\mathbf{k} \times(\mathbf{r} \times \mathbf{k})\) is equal to

\(\int_0^{2 \pi} \sin ^6 x \cos ^5 x d x\) is equal to

A block of mass \(m\) is lying on a rough inclined plane having an inclination \(\alpha=\tan ^{-1}\left(\frac{1}{5}\right)\). The inclined plane is moving horizontally with a constant acceleration of \(a=2 \mathrm{~ms}^{-2}\) as shown in the figure. The minimum value of coefficient of friction, so that the block remains stationary with respect to the inclined plane is (Take, \(g=10 \mathrm{~ms}^{-2}\) )

The correct decreasing order of priority for the functional group of organic compounds in the IUPAC method of nomenclature is

In a \(\triangle \mathrm{ABC}\), if \(r: \mathrm{R}: r_2=1: 3: 7\) then \(\sin (\mathrm{A}+\mathrm{C})+\sin \mathrm{B}=\)

A man who is running has half the kinetic energy of a boy of half his mass. The man speeds up by \(1 \mathrm{~ms}^{-1}\) and then has the same kinetic energy as the boy. The initial speed of the boy is

If the point \((2, \lambda)\) lies inside the circles \(x^2+y^2=13\) and \(x^2+y^2+x-2 y=14, \lambda\) lies in the set

The ten's digit in \(1 !+4 !+7 !+10 !+12 !+13 !+15 !+16 !+17 !\) is divisible by