Maths

1. Given,

$P(E)=0.6$

$P(F)=0.3$

$P(E\cap F)=0.2$

$P(E/F)=\frac{P(E\cap F)}{P(F)}=\frac{0.2}{0.3}=\frac{2}{3}$

$P(F/E)=\frac{P(F\cap E)}{P(E)}=\frac{0.2}{0.6}=\frac{2}{6}=\frac{1}{3}$

36. For the given inequality, x> –3.

The equation of the line is x= –3.

This line divides the xy-plane into planer I and II. We take a point (0,0) to check the correctness of the inequality.

So, 0> –3 which is true.

So, the solution of the region is I which includes the origin.

The dotted line indicates that any point on the line does not satisfy the inequality.

35. For the given equation inequality y< 2,  the equation of line is y= 2

This line devides the xy-plane into two planer I and II. We take a point (0,0) to check the correctness of the inequality.

So, 0< 2

0< 2 which is false.

So, the solution of the region is II which does not include the origin.

The dotted line indicates that any point on the line does not satisfy the inequality.

34. For the inequality 3y – 5x<30, the equation of line is 3y 5x=30. We consider the table below to plot 3y 5x =30.

$\begin{array}{l}x\\ y\end{array}|\begin{array}{l}0\\ 10\end{array}|\begin{array}{l}-6\\ 0\end{array}|$

This line divides the xy-plane into two planer I and II. We select a point (0,0) and check the correctness the inequality.

3 * 0 – 5 * 0 < 30

0 < 30 which is true.

So, the solution region is I which includes the origin. The dotted line indicates that any point on the line will not satisfy the given inequality.

33.For the inequality –3x+2y≥ –6 the equation of line is – 3x+2y=6.

We consider the table below to plot – 3x+2y= –6.

$\begin{array}{l}x\\ y\end{array}|\begin{array}{l}0\\ -3\end{array}|\begin{array}{l}2\\ 0\end{array}|$

This line divides the xy-plane into two planer I and II. We select a point (0,0) and check the correctness the inequality,

–3 * 0+2 * 0 ≥ –6

0 ≥ –6 which is true.

So, the solution region is I which includes the origin. The continuous line also indicates that any point on the line also satisfy the given inequality.

32. For the inequality, 2x – 3y>6 the eqⁿ is 2x – 3y=6. We consider the table below to plot 2x – 3y=6.

$\begin{array}{l}x\\ y\end{array}|\begin{array}{l}0\\ -2\end{array}|\begin{array}{l}3\\ 0\end{array}|$

This line divides the xy-plane into half planer I and II. We select point (0,0) and check the correctness of the inequality.

2 * 0 – 3 * 0>6

0>6 which is false.

So, the solution region is II which does not includes the origin (0,0).

The dotted line indicates that the any point on the line does not satisfy the given inequality.

31. For the inequality x – y≤ 2 so the equation of line is x – y=2. We consider the table below to plat x – y=2

$\begin{array}{l}x\\ y\end{array}|\begin{array}{l}0\\ -2\end{array}|\begin{array}{l}2\\ 0\end{array}|$

This line divides the xy-plane into half planer I and II. We select point (0,0) and check the correctness of the inequality.

0 – 0 ≤ 2

0 ≤ 2 which is true.

So, the solution region is I which includes the origin (0,0). The continuous line also indicates that any point on the line also satisfy the given inequality.

27. For inequality, x+y < 5, the equation of line is x+y = 5.

We consider the tableto plot x+y=5.

Graph of x+y = 5 is given as dotted line in fig 1. This lines divides by plane in two half planes I and II. We select a point not on the line, say (0,0) which lies in region I.

Since, 0+0 < 5

0 < 5 is true.

The solution region is I. (where origin (0,0) is included)

The dotted line indicates that any point on the line does not satisfy the given inequality.

26. Let 'x' cm be the length of the shortest piece.

Then, the two other remaining length is.

(x + 3) cm and (2x). cm.

So, given that,

2x - (x + 3) ≥ 5 cm.

2x- x - 3 ≥ 5.

x ≥ 5 + 3

x ≥ 8.

And. total length ≤ 91 cm

x + (x + 3) + 2x ≤ 91

4x + 3 ≤ 91

4x + 3 ≤ 91 - 3

$\Rightarrow x\le \frac{88}{4}$

x ≤ 22.

? 8 ≤ x ≤ 22.

Hence the length of the shortest board is greater than or equal to 8 cm and less than or equal to 22 cm.