The mean of the following distribution is $49$. Find the missing frequency $\text{'}a\text{'}$.

Class	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$
Frequency	$15$	$20$	$30$	$a$	$10$

 

Prove the following identity.

$(\sin A+cosec A{)}^2+(\cos A+\sec A{)}^2=5+{\sec }^2 A·cosec{}^2 A$

Using properties of proportion, find $x:y$, if 

$\frac{x^3+12x}{6x^2+8}=\frac{y^3+27}{9y^2+27}$

The following table shows the distribution of marks in Mathematics

With the help of a graph paper, taking $2 \mathrm{cm}=10 \mathrm{units}$along one axis and  $2 \mathrm{cm}=20 \mathrm{units}$along the other axis, plot an ogive for the above distribution and use it to find the median.

The following table shows the distribution of marks in Mathematics

With the help of a graph paper, taking $2 \mathrm{cm}=10 \mathrm{units}$along one axis and  $2 \mathrm{cm}=20 \mathrm{units}$along the other axis, plot an ogive for the above distribution and use it to find the number of students who scored distinction marks $\left(75\% \mathrm{and} \mathrm{above}\right)$.

The following table shows the distribution of marks in Mathematics

With the help of a graph paper, taking $2 \mathrm{cm}=10 \mathrm{units}$along one axis and  $2 \mathrm{cm}=20 \mathrm{units}$along the other axis, plot an ogive for the above distribution and use it to find the number of students, who passed the examination if pass marks is $35\%$.