CBSE+NCERT+10th- Math-Chapter-10-Circles-test-paper

1. Prove that the parallelogram circumscribing a circle is rhombus.

Ans Given : ABCD is a parallelogram circumscribing a circle.

To prove : - ABCD is a rhombus

or

AB=BC=CD=DA

Proof: Since the length of tangents from external are equal in length

AS = AR …..(1)

BQ = BR …..(2)

QC = PC …..(3)

SD = DP …..(4)

Adding (1), (2), (3) & (4).

AS + SD + BQ + QC = AR + BR + PC + DP

AD + BC = AB + DC

AD + AD = AB + AB

Since BC = AD & DC = AB (opposite sides of a parallelogram are equal)

2AD = 2AB

AD = AB …..(5)

BC = AD (opposite sides of a parallelogram)

DC = AB …..(6)

From (5) and (6)

AB = BC = CD = DA

2. A circle touches the side BC of a triangle ABC at P and touches AB and AC when

produced at Q and R respectively as shown in figure. Show that AQ= 1/2(perimeter of triangle ABC)

3. XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that XA+AR=XB+BR

4.  In figure, the incircle of triangle ABC touches the sides BC, CA, and AB at D, E, and F respectively. Show that AF+BD+CE=AE+BF+CD= 1/2(perimeter of triangle ABC),

5. 5. A circle touches the sides of a quadrilateral ABCD at P, Q, R and S respectively. Show that the angles subtended at the centre by a pair of opposite sides are supplementary.

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ASSIGNMENT AREAS RELATED TO CIRCLES CLASS X

AREAS RELATED TO CIRCLES CLASS X
1. The radius of the circle is 3 m. What is the circumference of another circle, whose area is 49 times that of the first?

2. Two circles touch externally. The sum of their areas is 130 p sq. cm and the distance between their centres is 14 cm. Find the radii of the circles.

3. A wire when bent in the form of an equilateral triangle encloses an area of 121 √3 cm² . If the same wire is bent in the form of a circle, find the area of the circle.

4. The area enclosed between the two concentric circles is 770 cm2. If the radius of the outer circle is 21 cm, calculate the radius of the inner circle.

5. A wheel of diameter 42 cm, makes 240 revolutions per minute. Find :

(i) the total distance covered by the wheel in one minute. (ii) the speed of the wheel in km/hr.

6. An arc of length 20pcm subtends an angle of 144° at the centre of the circle. Find radius of circle.

7. The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.

8. In the given figure, the length of the minor arc is 7/24 of the circumference of the circle. Find : 
(i) <AOB

(ii) If it is given that the circumference of the circle is 132 cm, find the length of the minor arc AB and the radius of the circle.

9. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find :

(i) Area of the minor sector (ii) Area of the minor segment

(iii) Area of major sector (iv) Area of major segment ( use p = 3.14 )

10. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. [use p = 3.14, √3 = 1.73]

11. In the given figure, ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of the shaded region.

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12. The following figure shows a rectangle ABCD inscribed in a circle.

(i) If AB = 8 cm and BC = 6 cm, find the area of the circle not included in the rectangle.

(ii) If diameter of the circle is 25 cm and BC = 15 cm, find the area of the circle not included in the given rectangle.

13. A paper is in the form of a rectangle ABCD in which AB = 20 cm and BC = 14 cm. A semi circular portion with BC as diameter is cut off. Find the area of the remaining part 

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X Maths Comprehensive Test chapter Probability

X Maths Comprehensive Test chapter Probability
Q1. A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that

(i) She will buy it? (ii) She will not buy it?

Q2. An unbiased die is thrown. What is the probability of getting:
(i) an even number or a multiple of 3
(ii) an even number and multiple of 3
(iii) a number 3 or 4.

Q3. Two unbiased coins are tossed simultaneously. Find the probability of getting
(i) at least one head. (ii) at most one head. (iii) No head.

Q4. Three unbiased coins are tossed together. Find the probability of getting:
(i) all heads (ii) at least two heads

Q5. Two dice are thrown simultaneously. Find the probability of getting :
(i) the sum as a prime number
(ii) a total of at least 10
(iii) a doublet of even number
(iv) a multiple of 2 on one dice and a multiple of 3 on the other.

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CBSE TEST PAPER MATHEMATICS(Class-10)Circle

I. Fill in the blanks.
a. The word ‘tangent’ comes from the Latin word ------------ 
b. A tangent to a circle intersects it in ----------- point (s).
c. A line intersecting a circle in two points is called a --------
d. A circle can have -----------parallel tangents at the most.
e. The common point of a tangent to a circle and the circle is called ----------


2. Solve these questions (any five) 4X5=20

1. Prove that The tangent at any point of a circle is perpendicular to the radius through the point of contact

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

3. Two tangents TP and TQ are drawn to a circle with centre O from an external point T.(see fig. 1) Prove that < PTQ = 2 < OPQ.

4. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Fig. 2) Find the length TP. 




4. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Fig. 2) Find the length TP. 

4. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Fig. 2) Find the length TP.

5. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.




6. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

7. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

8. A triangle PQR is drawn to circumscribe a circle of radius 4cm. The circle touches QR at D such that   QD = 6 cm and RD = 8 cm. Find PQ and PR.

9. The tangent at a point C of a circle and a diameter AB when extended intersect at P. If <PCA = 110⁰ , find < CBA.


10. In the figure. X.Y. are two parallel tangents to a circle with Center O and another tangent AB with point of contact C intersecting XY at A and X.Y. at B. Prove that <AOB = 90⁰.
10th Maths SA-2 Chapter wise Test Papers Links

10th Maths SA-2 Chapter wise Test Papers Links
Quadratic Equations	Circles	Co-ordinate Geometry
Arithmetic Progressions	Area Related to Circles	Probability
Height and Distance	Surface Areas and Volumes	Sample papers

10th maths Test paper Co-Ordinate Geometry

1. Calculate the distance between the points P(2, 2), Q(5, 4) correct to three significant figures. (Do not consult tables).

2. A is a point on the y-axis whose ordinate is 5 and B is the point (-3, 1). Calculate the length of AB.

3. The distance between A(1, 3) and B(x, 7) is 5. Find the possible values of x.

4. P and Q have co-ordinates (-1, 2) and (6, 3) respectively. Reflect P in the x-axis to P'. Find the length

of the segment P'Q.

5. Point A(2, -4) is reflected in the origin as A'. Point B(-3, 2) is reflected in x-axis at B'. Write the co-ordinates of A' and B'. Calculate the distance A'B' correct to one decimal place.

6. The center of a circle of radius 13 units is the point (3, 6). P(7, 9) is a point inside the circle. APB is a chord of the circle such that AP = PB. Calculate the length of AB.

7. A and B have co-ordinates (4, 3) and (0, 1) respectively. Find (i) the image A' of A under reflection in the y-axis.

(ii) the image B' of B under reflection in the line AA'.

(iii) the length of A'B'.

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CBSE ADDA: CBSE TEST PAPER 10TH MATHEMATICS ...

Assignments class 10 Chapter Co-Ordinate

Q1. Find the distance between the following points:

(a) A(3 , 5) and B(8 , – 7) (b)P( a + b , a – b ) and Q ( a– b , –a – b )

Q2. Find the value of x for which the distance between points A(x, 7) and B (–2, 3) is 4√5 units.

Q3. If the points (3, 2) and (2, –3) are equidistant from points (x, y) show that x + 5y = 0.

Q4. Show that the following points are collinear:
(a) (–5, 6), (–1, 2) and (2, –1)
(b) (4, 3) , (5,1 ) and (1, 9)

Q5. Show that following points are vertices of right triangle. Also, name the right angle.
(a)(4, 4) , ( 3 , 5) , (–1 ,1)
(b)(–2, 3) , ( 8, 3) , ( 6, 7)

Q6. Show that following points are vertices of a rectangle:
(a) (2 , –2) , ( 8, 4) , ( 5, 7 ) , (– 1, 1)
b)(–4 , –1) , (–2 , 4) , ( 4, 0 ) , ( 2, 3 )

Q7. Show that following points are vertices of a square:
a) ( 0 , –1) , ( 2, 1) , ( 0, 3) , (–2, 1)
(b)( 0, 1) , ( 1, 4) , ( 4,3) , ( 3, 0)

Q8. Show that following points are vertices of rhombus:

(a) ( 0, 5 ) , (–2, –2 ) , ( 5 , 0 ) , ( 7, 7 ) (b ) ( 2, –1) , ( 3, 4) , (–-2, 3) , (–3 , –2)

Q9. Show that the points ( a, a ) , (–a, –a ) and (–√3 a , √3 a) form an equilateral triangle.

Q10. Find the co-ordinates of circumcenter of a ∆ ABC where A( 1, 2) ,B ( 3, –4) and C ( 5, –6 ).

Q11. Find radius of the circle, the co-ordinates of the ends of whose diameter are (–1, 2) and (3, –4 ).

Q12. (a) Find the point on x-axis, which is equidistant from points ( 7, 6 ) and ( 9 , 4 ).

(b)Find the point on y-axis, which is equidistant from points ( 5, 2 ) and (–4 , 3 ).

Q13. A point P is at a distance of√10 from the point ( 4, 3). Find the co-ordinates of P, if its ordinate is twice its abscissa.

Q14. A line of length 10 units has (–-2, 3) as one of its end points. If the ordinate of the other end be 9, Show that its abscissa is 6 or–10.

Q15. The opposite angular points of a square be ( 3, 4 ) and ( 1, –1). Find the co-ordinates of the remaining angular points.

Answers

Ans 1 . (a) 13, (b) 2 √a² + b²

Ans 2. 6 or – 10

Ans 10. (11, 2)

Ans11.√13

Ans12. (a) (3, 0)(b ) (0, 15)

Ans13. (3, 6)

Ans15. (9/2, 1/2) and (–1/2 , 5/2 )

CBSE Class 10th maths ASSIGNMENT PROBABILITY

ASSIGNMENT PROBABILITYCLASS X

Q1. A coin is tossed. Find the probability that a head is obtained.

Q2. Find probability of throwing 5 with an ordinary dice.

Q3. Probability of winning a game is 0.4. What is the probability of loosing the game?

Q4. A person is known to hit the target in 3 shots out of 4 shots. Find the probability that the target is not hit.

Q5. Tickets numbered from 1 to 20 are mixed together and a ticket is drawn at random. What is the
probability that the ticket has a number which is multiple of 3 or 7?

Q6. A bag contains 100 identical tokens, on which numbers 1 to 100 are marked. A token is drawn at random. What is the probability that the number on the token is:
(a) an even number
(b) an odd number
(c) a multiple of 3
(d) a multiple of 5
(f) a multiple of 3 and 5
(g) a multiple of 3 or 5
(h) a number less than 20
(i) a number greater than 70(j) a perfect square number(k) a prime number less than 20.

Q7. A card is drawn from a well-shuffled pack of cards. Find the probability that the card drawn is:
(a) a queen
(b) a king bearing diamond sign
(c) a black card
(d) a jack
(e) black and a queen
(f) either black or a queen
(g) a red card
(h) a face card
(i) a diamond or a club
(j) neither heart nor a jack
(k) a 2 of diamond
(l) an ace of hearts
(m) a face card of red color
(n) 10 of a black “suit”

Q8. In a simultaneous toss of two coins, find:
(a) P(2 tails)
(b) P(exactly one tail)
(c) P(no tails)
(d) P(at most one head)
(e) P(one head)

Q9. A coin is tossed successively three times. Find probability of getting exactly one head or two heads.

Q10. Three coins are tossed once. Find probability of: 

(a) 3 heads
(b) exactly 2 heads
(c) atleast 2 heads
(d) atmost 2 heads
(e) no tails
(f) head and tail appear alternatively
(g) atleast one head and one tail

Q11. A dice is thrown once. Find:
(a) P(number 5)
(b) P(number 7)
(c) P(an even number)
(d) P( a number greater than 4)
(e) P( a number less than or equal to 4)
(f) P(a prime number)

Q12. A bag contains 10 white, 6 black and 4 red balls. Find probability of getting:
(a) a white ball
(b) a black ball
(c) not a red ball
(d) a white or a red ball

Q13. Two dice are thrown simultaneously. Find:
(a) P(an odd number as a sum)
(b) P(sum as a prime number)
(c) P(a doublet of odd numbers)
(d) P(a total of atleast 9)
(e) P( a multiple of 2 on one die and a multiple of 3 on other die)
(f) P(a doublet)
(g) P(a multiple of 2 as sum)
(h) P(getting the sum 9)
(i) P(getting a sum greater than 12)
(j) P( a prime number on each die)
(k) P( a multiple of 5 as a sum)

Q14. Find the probability that a leap year at random contains 53 Sundays.

Q15. Two black kings and two black jacks are removed from a pack of 52 cards. Find the probability of getting:
(a) a card of hearts
(b) a black card
(c) either a red card or a king
(d) a red king
(e) neither an ace nor a king
(f) a jack, queen or a king 

Ans
(1)1/2	(2)1/6	(3)0.6	(4)1/4	(5)2
(6) (a)1/2	(b)1/2	(c)33/100	(d)1 /5	(e) 3/50	( f) 47/100
( g) 19/100	(h ) 3/10	(7) (a) 1/13	(b) 1/52	( c) ½	( d) 1/13
( e) 1/26	( f) 7/13	( g)1/2	(h) 4/13	(i) ½	(j) 9/13
( k) 1/52	(l) 1/52	(m) 3/26	(n) 1/26	(8) (a) ¼	(b) ½
( c) ¼	(d)3/5	(e) ½	(9) ¾	(10) (a) 1/8	(b) 3/8
( c) ½	(d ) 7/8	( e) 1/8	( f) ¼	( g) ¾	(11) (a)1/6
(b) 0	( c) ½	(d) 1/3	( e)2/3	( f) 1/2	(12) (a) ½
(b) 3/10	( c) 4/5	( d) 7/10	(13) (a) ½	( b)5/12	( c) 1/12
( d) 5/18	( e) 11/36	( f) 1/6	( g) ½	(h ) 1/9	( i) 0
( j) 1/12	( k) 7/36	(14) 2/7	( 15)