Locus and Straight Line(p-1) — Maths (commerce) STD 11 Commerce / Arts — Question

\begin{aligned}
& f(x)=\frac{5^x-e^x}{2 x} \text { for } x \neq 0 \\
& =\frac{1}{2}(\log 5-1) \text { for } x=0 \text { at } x=0
\end{aligned}

A card is drawn from a pack of 52 cards. What is the probability that, card is either red or black?

By using determinant, show that the following points are collinear.
P(5, 0), Q(10, -3), R(-5, 6)

Following data gives the age in years and marks obtained by 30 students in an intelligence test.

Age	16	17	22	19	21	16
Marks	16	19	39	50	48	41
Age	21	20	20	23	22	19
Marks	59	44	42	62	37	67

Age	23	20	22	22	23	22
Marks	45	57	35	37	38	56
Age	17	18	16	21	19	20
Marks	54	61	47	67	49	56
Age	17	18	23	21	20	16
Marks	51	42	65	56	52	48

Prepare a bivariate frequency distribution by taking class intervals 16 – 18, 18 – 20,…,etc. for age and 10 – 20, 20 – 30,…, etc. for marks.
Find
(i) marginal frequency distributions.
(ii) conditional frequency distribution of marks obtained when age of students is between 20 – 22.

Find the derivatives of the following w.r.t. x. : \begin{equation}
\frac{x \mathrm{e}^x}{x+\mathrm{e}^x}
\end{equation}