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Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above do the following:

ABC is an isosceles triangle in which AB = AC, circumscribed about a circle as shown in the fig. Prove that the base is bisected by the point of contact.

Subtract the sum of 883 and -479 from -76.

Question 69

In the following question, state whether the given statements are true (T) or false (F).

$24.58=2\times 10+4\times 1+5\times 10+8\times 100$

Q. Prove that if x and y are odd positive integers, then x^2+y^2 is even but not divisible by 4.

Evaluate: $\frac{\sin {30}^{\circ }+\tan {45}^{\circ }-\text{cosec}{60}^{\circ }}{\sec {30}^{\circ }+\cos {60}^{\circ }+\cot {45}^{\circ }}$

A man has some hens and cows.If the number of heads be 48 and number of feet be 140,how many cows are there?

Find the missing frequencies in the following frequency distribution whose mean is 50.
$\begin{array}{ccccccc}\text{x}& 10& 30& 50& 70& 90& \text{Total}\\ \text{f}& 17& f_1& 32& f_2& 19& 120\end{array}$

A train travels 180 km at a uniform speed. If the speed had been 9 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.

A solid is composed of a cylinder with hemispherical ends. If the whole length of the solids is $104$ cm and the radius of each of the hemispherical ends is $7$ cm, find the cost of polishing its surface at the rate of Rs.$10$ per $d{m}^2$.

A 32cm tall cylindrical bucket of base diameter 36cm, is filled with sand and this container is emptied on the ground forming a conical heap of sand. If the height of the heap is 24cm. Find the slant height of the heap.

The three angles of a quadrilateral are ${75}^{\circ }$, ${90}^{\circ }$ and ${75}^{\circ }$. What is the fouth angle?

The sum of three numbers in A.P. is –3, and their product is 8. Find the numbers.

In Fig. 12.25, ABCD is a square of side 14 cm. With centers A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region.

Question 6 (ii)

Simplify the expression and find the value if x is equal to 2:
3(x + 2) + 5x - 7

Find the mean of the following data using assume mean method :

Class 0-10 ; F 8

Class 10-20 ; F 12

Class 20-30 ; F 10

Class 30-40 ; F 6

A largest sphere is to be carved out of a right circular cylinder of radius 7 cm and height 14 cm. Find the volume of the sphere. (Answer correct to the nearest integer).

In the given figure, AB is the diameter of the circle. Find the value of $\angle$ ACD

Find the nature of the roots of the following quadratic equations:
(i) $2x^2-8x+5=0$

(ii) $3x^2-2\sqrt{6}x+2=0$
(iii) $5x^2-4x+1=0$

(iv) $5x(x-2)+6=0$
(v) $12x^2-4\sqrt{15}x+5=0$

(vi) $x^2-x+2=0$

Construct an isosceles triangle with base 6cm and height 7cm. Construct another triangle similar to this, with ratio of corresponding sides as 4:3.

The largest positive integer that will divide 398 436 and 542 leaving remainder 7 11 and 15 respectively

The diagonals of a rhombus are 8 cm and 15 cm. Find its side.

A circus artist is climbing a $30m$ long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. The height of the pole, if the angle made by the rope with the ground level is ${30}^{\circ }$ wii be

What's the formula for finding a cubic polynomial given the sum, sum taken to at a time, and the product of its zeroes ?

Find x and y if $\left(\begin{array}{cc}x+3& 4\\ y-4& x+y\end{array}\right)=\left(\begin{array}{cc}8& 4\\ 2& 11\end{array}\right)$

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O is 60 then find the length of OP

Solve the pair of linear equations:

1/2x+1/3y=2

1/3x+1/2y=13/6

Find the equation of the line through (1, 3) and making an intercept of 5 on the y-axis.

A rhombus whose side measures 40 cm has one of its angles as ${60}^{\circ },$, then the length of its diagonals are

Solve the equation $2x-\frac{1}{3}=7$.Write your answer correct to two decimal places.

Factorize $\sqrt{3}$ $x^2$ + 5x + 2$\sqrt{3}$ = 0 by splitting the middle term and find the value of x.

Let R be a relation on the set N be defined by $\left\{\right(x,y)|x,y|N,2x+y=41\}$. Then R is

Figure depicts a racing track whose left and right ends are semi-circular. The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find the area of the track (in $m^2$).

Product of $\frac{7}{6}$ and reciprocal of $\frac{11}{66}$ is __________.

If $△ABC$ is isosceles with AB = AC, prove that the tangent at A to the circumcircle of $△ABC$ is parallel to BC. [2 MARKS]

The points A(-a,b), B(-a,c) and C(a,c) form the vertices of a triangle. Which of the following is true?

The following table gives the marks obtained by the students of a class in an exam out of 50.

$\begin{array}{cc}\text{Marks}& \text{No. of Students}\\ Below10& 5\\ Below20& 10\\ Below30& 23\\ Below40& 31\\ Below50& 40\end{array}$

(i) Convert the cumulative frequency distribution into a regular frequency distribution.

(ii) Find the mean. [3 MARKS]

Question 3 (iii)
Prove that $6+\sqrt{2}$ is irrational.

In the given figure, C and D are points on the semi-circle described on AB as diameter. Given angle BAD = ${70}^0$ and angle DBC= ${30}^0$. Calculate $\angle BDC$

A tree in each year grows 4cm less than it grew in previous year. If it grew 1 metre in the first year, in how many years will it have ceased growing and what will be its height then,