Q. Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

Q. The table shows the salaries of $280$ persons.

Salary(In thousand)	No. of Persons
$5-10$	$49$
$10-15$	$133$
$15-20$	$63$
$20-25$	$15$
$25-30$	$6$
$30-35$	$7$
$35-40$	$4$
$40-45$	$2$
$45-50$	$1$

Calculate the median salary of the data.

Q. The sum of four consecutive numbers in an $AP$ is $32$ and the ratio of the product of the first and the last term to the product of two middle terms is $7:15$. Find the numbers.

Q. Given that $\sqrt{2}$ is irrational, prove that $(5+3\sqrt{2})$ is an irrational number.

Q. Find $HCF$ and $LCM$ of $404$ and $96$ and verify that $HCF\times LCM=$ Product of the two given numbers.

Q. Find the sum of first $8$ multiples of $3$.

Q. Find the ratio in which $P(4,m)$ divides the line segment joining the points $A(2,3)$ and $B(6,-3)$. Hence find $m$.

Q. Find all zeroes of the polynomial $2x^4-9x^3+5x^2+3x-1$ if two of its zeroes are $2+\sqrt{3}$ and $2-\sqrt{3}$.

Q. Two different dice are tossed together. Find the probability:
(i) of getting a doublet
(ii) of getting a sum $10$, of the number on the two dice.

Q. Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.

Q. What is the value of $({\cos }^2{67}^o-{\sin }^2{23}^o)$?

Q. Find the area of the shaded region in Figure, where arcs drawn with centres $A,B,C$ and $D$ intersect in pairs at mid-point $P,Q,R$ and $S$ of the sides $AB,BC,CD$ and $DA$ respectively of a square $ABCD$ of side $12cm$ [Use $\pi =3.14$].

Q. The following distribution gives the daily income of $50$ workers of a factory:

Daily Income(In)	$100-120$	$120-140$	$140-160$	$160-180$	$180-200$
Number of workers	$12$	$14$	$8$	$6$	$10$

Convert the distribution to a less than type cumulative frequency distribution and draw its ogive.

Q. If $4\tan \theta =3$, evaluate $\left(\frac{4\sin \theta -\cos \theta +1}{4\sin \theta +\cos \theta -1}\right)$.

Q. If $x=3$ is one root of the quadratic equation $x^2-2kx-6=0$, then find the value of $k$.

Q. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Figure. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, find the total surface area of the article.

Q. In an $AP$, if the common difference $\left(d\right)$ is $-4$, and the seventh term $\left(a_7\right)$ is $4$, then find the first term.

Q. Find the distance of a point $P(x,y)$ from the origin.

Q. Prove that the lengths of tangents drawn from an external point to a circle are equal.

Q. A plane left $30$ minutes late than its scheduled time and in order to reach the destination $1500km$ away in time, it had to increase its speed by $100km/h$ from the usual speed. Find its usual speed.

Q. In Fig. $1,ABCD$ is a rectangle. Find the values of $x$ and $y$.

Q. If $A(-2,1),B(a,0),C(4,b)$ and $D(1,2)$ are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides.

Q. Draw a triangle ABC with $BC=6$cm, AB$=5$cm and $\angle$ABC $={60}^o$. Then construct a triangle whose sides are $\frac{3}{4}$ of the corresponding sides of the $\Delta$ ABC.

Q. If $\tan 2A=\cot (A-{18}^o)$, where $2A$ is an acute angle, find the value of A.

Q. In an equilateral $\Delta$ ABC, D is a point on side BC such that BD$=\frac{1}{3}$BC. Prove that $9(AD{)}^2=7(AB{)}^2$.

Q. If $A(-5,7),B(-4,-5),C(-1,-6)$ and $D(4,5)$ are the vertices of a quadrilateral, find the area of the quadrilateral $ABCD$.

Q. A heap of rice is in the form of a cone of base diameter $24$, and height $3.5m$. Find the volume of the rice. How much canvas cloth is required to just cover the heap?

Q. An integer is chosen at random between $1$ and $100$. The probability that it is divisible by $8$ is $\frac{m}{25}$. Find $m$

Q. Given $\Delta$ ABC $\sim$ $\Delta$ PQR, if $\frac{AB}{PQ}=\frac{1}{3}$, then find $\frac{ar\Delta ABC}{ar\Delta PQR}$.

Q. If the area of two similar triangles are equal, prove that they are congruent.