Sample Paper X Mathematics CBSE SA 2 2014

SAMPLE PAPER X SUBJECT MATHEMATICS CBSE SA – 2  March 2014

SECTION-A
1. The circumference of two circles are in the ratio 2: 3 then the ratio of the areas is:
(a) 2 : 4 (b) 2 : 9 (c) 4 : 9 (d) 4 : 6

2. A silver rod of diameter 2 cm and length 12 cm is drawn into a thin wire of length 24 m of uniform thickness, and then the thickness of the wire is:
(a) 0. 183 (b) 0.173 (c) 0.186 (d) 0.175

3. In two concentric circles, the length of tangent to inner circle is 8cm. Find the radius of outer circle, if the radius of inner circle is 3 cm.
(a) 5 cm (b) 4 cm (c) 3 cm (d) 2 cm

4. A point P is 13 cm from the centre of a circle. Find the length of the tangent drawn to the circle from the point P, if the radius of the circle is 5 cm.
(a) 12 cm (b) 10 cm (c) 8cm (d) 6 cm

5. If PA and PB are tangents from a point playing outside the circle such that PA = 10cm and angle APB, then the length of chord AB is:
(a) 5 cm (b) 4 cm (c) 3 cm (d) 2 cm

6. If 17th term of an A.P. exceeds its 9th term by 64, then the difference is:
(a) 8 (b) 6 (c) 4 (d) 12

7. One coin is tossed three times. The probability of getting 2 heads and 1 heads and 1 tail is:
(a) 1/8 (b) 2/5 (c) 3/8 (d) ¼

8. A vertical stick 20 m long casts a shadow 16 m long. At the same time a tower casts a shadow 48 m long. Then the height of the tower is:
(a)40 m (b) 32 m (c) 96 m (d) 60 m

9. A cone is divided into two parts by drawing a plane through mid – point of its axis, parallel to its base. The ratio of volumes of two parts is:
(a) 2: 3 (b) 1: 2 (c) 1: 3 (d) 1: 7

10. From a point Q , the length of the tangent to a circle is 24cm and the distance of Q from the centre is 25cm. The radius of the circle is:
(a) 7 cm (b) 12 cm (c) 15 cm (d) 24.5 cm

SECTION – B

11. For what value of P, are 2p -1, 7 and 3p three consecutive terms of an A.P.?

12. The length of the minute hand of a clock is 14 cm. Find the area swept out by the minute hand in 1 hour.

13. Find the roots of the quadratic equation 3x – 8/x = 2 ; x does not equal 0

14. If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

15. A letter is drawn at random form the word ‘MATHEMATICS’. Find the probability of drawing each of the different letters in the given word.

16. How many balls each of radius 1 cm can be made from a solid sphere of lead of radius 8cm?

17. It is known that a box of 500 electric tubes contains 15defective electric tubes. One tube is taken out at this box. What is the probability that is a non – defective electric tube?

18. Find the coordinates of the points P,Q and R which divided the line segment joining A (5 , 4) and B (11 , 6) into four equal parts.

SECTION – C

19. The sum of two natural numbers is 8. Determine the numbers, if sum of their reciprocal is 8/15.

20. Draw a right triangle ABC in which AC = AB = 4.5 cm and angle = 90 degree. Draw a triangle similar to triangle to ABC with its sides equal to 5/4th of the corresponding sides of angle ABC.

21. Prove that the tangents drawn at the ends of a chord of circle make equal angles with the chord.

22. In an A.P. the sum of first ten is – 150 and the sum of its next ten terms is – 550.

23. PA and PB are two tangents from an exterior point P to a circle of radius 5 m. If length of the chord AB is 8 cm, then find the length of the tangent.

24. Three cows are tethered with 10 m long rope at the three corners of a triangular field having sides 42 mm 20 m and 34 m. Find the area of the plot which can be grazed by the cows, also find the area of the remaining field (unglazed).

25. The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is ¼. The probability of selecting a blue ball at random from the same jar is 1/3. If this jar contains 10 orange balls, then what is the total number of balls in the jar?

26. If R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a) , then prove that x + y = a + b.

27. The internal and external diameters of a hollow hemispherical shell are 6cm and 10cm respectively. It is melted and recast into a solid cone of base diameter 14 cm. Find the height of the cone so formed.

28. A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle the angle of elevation of the top of the light house from 60degree to 45 degree. Find the speed of the boat.

SELECTION-D

29. If the radii of the ends of a bucket 45 cm high, are 28 cm and 7 cm. Find the capacity of bucket.

30. The side of a square exceeds the side of another square by 4 cm and the sum of the areas of the two squares is 400 sq. cm. Find the dimensions of the squares.

31. The speed of a boat in still water is 11 km/ h. It can go 12 km upstream and return downstream to the original point in 2 hours and 45 minutes. Find the speed of the stream.

32. An iron sphere of radius ‘a’ unites is immerse completely in water contained in a right circular cone of semi – vertical angle 30 degree , water is drained off from the cone till its surface touches the sphere. Find the volume of water remaining in the cone.

33. The sum of first 8terms of an arithmetic progression is 156. The ratio of its 12th tern to its 68th is 1: 5 Calculate the first term and the fifteenth term.

34. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Link for more downloadable CBSE BOARD SAMPLE PAPER-2012

Assignment SA-2 Class X Topic : Height And Distance

1.The angle of elevation of a ladder leaning against a wall is 60^o and the foot of the ladder is 9.5 meter away from  the wall. Find the length of the ladder. [ 19m ]

2. If the length of the shadow cast by a pole be  times the length of the pole, find the angle of elevation of the sun. [ 30^o ]

3.   A tree is broken by the wind. The top stuck the ground at an angle of 30^o and at a distance of 30 m from the root.   Find the total height of the tree.

4.     A circus artist is climbing from the ground along a rope stretched from the top of vertical pole and tied at the ground level 30^o. Calculate the distance covered by the artist in climbing to the top of the pole. [ 24 m ]

5. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60^o. When he was 40 m away from the bank he finds that the angle of elevation to be 30^o.  Find: - 

(i) The height of the tee, 

(ii) The width of the river, correct up to two decimal places.[(i)34.64m (ii) 20m]

6.  An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the  angles of elevations of two planes to a same point on the ground are 60^o and 45^o respectively.  Find the vertical distance between the aeroplanes at that at that instant. [ 1693.34 m ]

7.  The angle of elevation of the top of the hill at the foot of a tower is 60^o and the angle of elevation of the top of tower from the foot of hill is 30^o. If the tower is 50 m high, what is the height of the hill. [ 150 m ]

8.    There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. Let P and Q be       points directly opposite each other on the two banks, and in line with the tree. If the angles of elevation of the top     the tree from P and Q respectively are 30^o and 45^o, find the height of the tree. 

9.        Two pillars of equal heights are on either sides of a roadway, which is 150 m wide. The angles of elevation of       the top of pillars are 60^o  and 30^o at a point on the roadway between the pillars. Find the position of the point    between the pillars and the height of each pillar.       (64.95m)

10.        At the foot of mountain, the elevation of its peak is 45^o. After ascending 1 km towards the mountain up an inclination of 30^o, the elevation changes to 60^o. Find the height of mountain. (1.366 km)

11.        From the top of the building 15m high, the angle of elevation of the top of a tower is found to be 30^o. From the       bottom of the same building, the angle of elevation of the top of tower is found to be 30^o. Find the height of the     tower and the distance between the tower and the building. (22.5m, 12.975m)

12.        A fire in a building B is reported on the telephone to two fire stations P and Q, 10 km apart from each other on a    straight road. P observes that the fire is at angle of 60^o to the road and Q observe that it is an angle of 45^o to the       road. Which station should send its team and how much this team has to travel? (P, 7.32km)

13.        The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sunrays meet the ground  at an angle of 60^o. Find the angle between the sunrays and the ground at the time of long shadow. ( 30^o)

14.        From a point in the cricket ground, the angle of elevation of a vertical tower is found to be θ at a distance of  200m from the tower. On walking 125 m towards the tower the angle of elevation becomes 2θ. Find the height of   tower. (100m)

15.        A boy standing on the ground and flying a kite with 75 m of string at an elevation of 45^o. Another boy is standing on the roof of 25 m high building and is flying his kite at an elevation of 30^o. Both the boys are on the opposite side of the two kites. Find the length of the string that the second boy must have, so that the kites meet.(56.05 m)

16.        As observed from the top of light house, 100m high above the sea level, the angle of depression of a ship, sailing directly towards it, changes from 30^o to 45^o. Determine the distance traveled by the ship during the period of observation.  ( 73.2m)

17.        An aeroplane at an altitude of 200 m observes the angle of depression of opposite points on two banks of a river  to be 45^o and 60^o. Find the width of the river.  ( 315.4m)

18.        From the top of a cliff 150m high, the angles of depression of two boats are 60^o and 30^o. Find the distance  between the boats, if the  boats are (i) on the side of cliff. (ii) on the opposite sides of the cliff.     [ (i) 173.2m (ii) 346.4m ]

19.        A man standing on the deck of a ship, which is 10m above the water level, observe the angle of elevation of the  top of a hill as 60^o and the angle of depression of the base of the hill as 30^o.Calculate the distance of the hill from the ship and the height of the hill.   [17.3m, 40m].

20.        The angle of elevation and depression of the top and the bottom of a light house from the top of the building, 60m high, are 30^o and 60^o respectively. Find (i) The difference between the heights of the light house and the  building (ii) Distance between the light house and the building. [ (i) 20m, (ii) 34.64m]

21.A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from Point ‘A’ on the ground is 60^o and the angle of depression of the point ‘A’ from the top of tower is 45^o. Find the height of tower.  [ 6.83m]

22.        Man on a cliff observes a boat at an angle of depression of 30^o which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is  found to be 60^o. Find the time taken by the boat to reach the shore. [ 9 minutes]

23.  A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly       towards it. If it takes 12 minutes for the angle of depression to change from 30^o to 45^o, how soon after this will       the car reach the observation tower. Give your answer correct to nearest seconds.  [16 min. 24 sec.]

                                                     JSUNIL TUTORIAL CBSE MATHS & SCIENCE

CBSE10th Class Solved Paper - Arithmetic Progression

1. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.

Solution: 12 = a + 2d

106 = a + 49d

So, 106-12 = 47d

Or, 94 = 47d

Or, d = 2

Hence, a = 8

And, n₂₉ = 8 + 28x2 = 64

2. If the 3rd and the 9th terms of an AP are 4 and -8 respectively, which term of this AP is zero?

Solution: -8 = a + 8d

4 = a + 2d

Or, -8 – 4 = 6d

Or, -12 = 6d

Or, d = -2

Hence, a = -8 + 16 = 8

0 = 8 + -2(n-1)

Or, 8 = 2(n-1)

Or, n-1 = 4

Or, n = 5

3. The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

Solution: n₇ = a + 6d

And, n₁₀ = a + 9d

Or, a + 9d – a – 6d = 7

Or, 3d = 7

Or, d = 7/3

4. Which term of the AP: 3. 15, 27, 39, … will be 132 more than its 54th term?

Solution: d = 12,

132/12 = 11

So, 54 + 11 = 65th term will be 132 more than the 54th term.

5.  How many three digit numbers are divisible by 7?

Solution: Smallest three digit number divisible by 7 is 105

Greatest three digit number divisible by 7 is 994

Number of terms

= {(last term – first term )/common difference }+1

= {(994-105)/7}+1

= (889/7)+1=127+1=128

6. How many multiples of 4 lie between 10 and 250?

Solution: Smallest number divisible by 4 after 10 is 12,

The greatest number below 250 which is divisible by 4 is 248

Number of terms: {(248-12)/4}+1

{236/4}+1 = 59+1 = 60

7. For what value of n, are the nth terms of two APs: 63, 65, 67,… and 3, 10, 17,… equal?

Solution: In the first AP       a = 63 and d = 2

In the second AP                 a = 3 and d = 7

As per question,

63+2(n-1) = 2+ 7(n-1)

Or, 61 = 5 (n-1)

Or, n-1 = 61/5

As the result is not an integer so there wont be a term with equal values for both APs.

8. Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.

Solution: As the 7th term exceeds the 5th term by 12, so the 5th term will exceed the 3rd term by 12 as well

So, n₃ = 16

n₅ = 28

n₇ = 40

n₄ or n₆ can be calculated by taking average of the preceding and next term

So, n₄ = (28+16)/2 = 22

This gives the d = 6

AP: 4, 10, 16, 22, 28, 34, 40, 46, ……..

9. Find the 20th term from the last term of the AP: 3, 8, 13, ……, 253.

Solution: a = 3, d = 5

253 = 3 + 5(n-1)

Or, 5(n-1) = 250

Or, n-1 = 50

Or, n = 51

So, the 20th term from the last term = 51 – 19 = 32nd term

Now, n₃₂ = 3 + 5x31 = 158

10. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and the 10th terms is 44. Find the first three terms of the AP.

Solution: a + 3d + a + 7d = 24

Or, 2a + 10d = 24

Similarly, 2a + 14d = 44

So, 44 – 24 = 4d

Or, d = 5

2a + 10x5 = 24

Or, a + 25 = 12

Or, a = -13

So, first three terms of AP: -13, -8, -3,

11. Subba Rao started work in 1995 at an annual salary of Rs. 5000 and received an increment of Rs. 200 each year. In which year did his income reached Rs. 7000.?

Solution: 7000 = 5000 + 200(n-1)

Or, 200(n-1) = 2000

Or, n-1 = 10

Or, n = 11

12. Ramkali saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her savings become Rs. 20.75, find n.

Solution: 20.75 = 5 + 1.75(n-1)

Or, 1.75(n-1) = 15.75

Or, n-1 = 9

Or, n = 10

For more papers visit  CBSE: 10th Class Sample Test Paper- Arithmetic Progression