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Let $P\left(S\right)$ denote the power set of a set $S$. Which of the following is always true ?

Find the circumradius of a right-angled triangle, whose two adjacent sides containing right angle are 7 cm and 24 cm long?

In the given figure, $\Delta ODC\sim \Delta OBA,\angle BOC={115}^{o}and\angle CDO={70}^o$.

Find (i) $\angle DOC$

(ii) $\angle DCO$

(iii) $\angle OAB$

(iv) $\angle OBA$.

If the circumcenter lies on the exterior of a triangle then the triangle is ______ triangle.

The area of a sector (in $c{m}^2$ )of a circle with radius 6 cm if angle of the sector is ${70}^{\circ }$ will be __ $c{m}^2$. ( Take $\pi$ = $\frac{22}{7}$)

In the figure, AB || DC, then the measure of $\angle D$ is equal to

A line which cuts a circle at two distinct points is called the __ of a circle.

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is 170. If there are at least 6 houses in that row and a is the number of the sixth house, then

Prove that n²-n is divisible by 2 for every positive integer n

The roots of the given equation $x+5=\frac{2x+10}{x-6}$ are

Question 2

For some integer q. every odd integer is of the form



A) q

B) q + 1

C) 2q

D) 2q + 1

Find the sum of the first 22 terms of the A.P. : 8, 3, -2, .......... .

In the given figure, if AC : CB = 5 : 3, then

Express 35 as product of their primes

A sector of a circle of radius 4 cm contains an angle of ${30}^{\circ }$. Find the area of the sector.

In the figure, ABCD is a trapezium in which, AB || DC and $\frac{AO}{OC}=\frac{BO}{OD}=\frac{1}{2}andAB=5cm$. Find the value of CD .

$LetA=\left[\begin{array}{cc}0& \alpha \\ 0& 0\end{array}\right]and(A+I{)}^{50}-50A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],Then,thevalueofa+b+c+dis$

The innder diameter of a glass is 7 cm and it has a raised portion in the bottom in the shape of a hemisphere, as shown in the figure. If the height of the glass is 16 cm, find the apparent capacity and the actual capacity of the glass.

The number ${(1+\sqrt{3})}^2$ is a/an ____________.

Question 5

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Anita buys a new salt cellar in the shape of a cylinder topped by a hemisphere as shown below. The cylinder has a diameter of 6 cm and a height of 10 cm. She pours the salt into the salt cellar, so that it takes up half the total volume of the pot. Find the depth of the salt, marked with x in the diagram.

The point (7,3) falls on which of the following lines?

A solid right circular cone of height 120 cm and radius 60 cm is placed in right circular cylinder full of water of height 180 cm such it touches the bottom.Find the volume of water left in the cylinder, if the radius of the cylinder is equal to the radius of the cone.

$\text{The diameter of a sphere is 6 cm.It is melted and drawn into a wire of diameter 2 mm.The length of the wire is}\left(a\right)12m\left(b\right)18m\left(c\right)36m\left(d\right)66m$

Hussain has 5 red, 3 blue and 2 green socks. If a sock is drawn at random, match the following probabilities.

If the equation $x^2$ - m(2x - 8) - 15 = 0 has equal roots then m =

If $x=\sqrt{a}sin\alpha$ and $y=\sqrt{a}cos\alpha$. The value of $y^2$ + $x^2$ is ___

Question 2 (v)

Write whether the given statement is true or false. Justify your answer.

If the coefficient of $x^2$ and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.