Q. A Father's age is three times the sum of ages of his two children. After $5$ years his age will be twice the sum of ages of two children. Find the age of father.

Q. A die is thrown once. Find the probability of getting a number which
(i) a prime number; (ii) a number lying between $2$ and $6$;

Q. A boat goes $30$ km upstream and $44$ km downstream in $10$ hours. In $13$ hours it can go $40$ km upstream and $55$ km downstream. Determine the speed of the stream and that of the boat in still water

Q.

Two poles of equal heights are standing opposite each other on either side of the road, which is $80m$ wide. From a point between them on the road, the angles of elevation of the top of the poles are ${60}^{\circ }$ and ${30}^{\circ }$, respectively. Find the height of the poles and the distances of the point from the poles.

Q. Prove that : $(\sin \theta +\text{cosec}\theta {)}^2+(\cos \theta +\sec \theta {)}^2=7+{\tan }^2\theta +{\cot }^2\theta$

Q. Find a point on y-axis which is equidistant from the points $(5,-2)$ and $(-3,2)$.

Q. Which term of the AP $3,15,27,39,...$ will be $120$ more than its ${21}^{st}$ term ?

Q. Find the coordinates of a point $A$, where $AB$ is diameter of a circle whose centre is $(2,-3)$ and $B$ is the point $(1,4)$.

Q. Find $c$ if the system of equations $cx+3y+(3-c)=0;12x+cy-c=0$ has infinitely many solutions?

Q. For what values of $k$, the roots of the equation $x^2+4x+k=0$ are real ?

Q. Prove that $\sqrt{2}$ is an irrational number.

Q. Find the mode of the following frequency distribution.

Class	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	8	10	10	16	12	6	7

Q. Find the value of $k$ for which the roots of the equation $3x^2-10x+k=0$ are reciprocal of each other.

Q. Find the value of k such that the polynomial $x^2-(k+6)x+2(2k-1)$ has sum of its zeros equal to half of their product.

Q. Two water taps together can fill a tank in $1\frac{7}{8}$ hours. The tap with longer diameter takes $2$ hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Q. Find $A$ if $\tan 2A=\cot (A-{24}^o)$

Q. Find the value of $({\sin }^2{33}^o+{\sin }^2{57}^o)$

Q. A fraction becomes $\frac{1}{3}$ when $2$ is subtracted from the numerator and it becomes $\frac{1}{2}$ when $1$ is subtracted from the denominator. Find the fraction.

Q. In fig., $DE||BC,AD=1cm$ and $BD=2cm$. What is the ratio of the $ar\left(\Delta ABC\right)$ to the $ar\left(\Delta ADE\right)$ ?

Q. Prove that $\frac{\sin A-\cos A+1}{\sin A+\cos A-1}=\frac{1}{\sec A-\tan A}$.

Q. Find the $HCF$ of $1260$ and $7344$ using Euclid's algorithm.

Q. A man in a boat rowing away from a light house $100m$ high takes $2$ minutes to change the angle of elevation of the top of the light house from ${60}^o$ to ${30}^o$. find the speed of the boat in metres per minute. [Use $\sqrt{3}=1.732$]

Q. Show that every positive odd integer is of the form $(4q+1)$ or $(4q+3)$, where $q$ is some integer.

Q. If $S_n$ the sum of first $n$ terms of an $AP$ is given by $S_n=3n^2-4n$, find the $n^{th}$ term.

Q. $\angle ACB={90}^o$ and $CD\perp AB$, prove that $C{D}^2=BD\times AD$.

Q. A game consists of tossing a coin $3$ times and noting the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game.

Q. Find the area of the shaded region if $ABCD$ is a rectangle with sides $8cm$ and $6cm$ and $O$ is the center of circle.$(Take\pi =3.14)$

Q. Construct a $\Delta ABC$ in which $CA=6cm,AB=5cm$ and $\angle BAC={45}^o$. Then construct a triangle whose sides are $\frac{3}{5}$ of the corresponding sides of $\Delta ABC$.

Q. If the median of the following frequency distribution is $32.5$. find the values of $f_1$ and $f_2$.

Class	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80
Frequency	$f_1$	5	9	12	$f_2$	3	2	40

Q. The marks obtain by $100$ students of a class in an examination are given below.

Marks	No. of Students
0 - 5	2
5 - 10	5
10 - 15	6
15 - 20	8
20 - 25	10
25 - 30	25
30 - 35	20
35 - 40	18
40 - 45	4
45 - 50	2

Draw 'a less than' type cumulative frequency curves (ogive). Hence find median.