CLASS 10 Maths Guess Questions Chapter Arithmetic ...

CBSE MATH STUDY: CLASS 10 Maths Guess Questions Chapter Arithmetic ...1) For what value of p, are 2p-1, 7 and 3p three consecutive terms of an A.P? (P=3)

2) Find the value of k, so that 3k + 7, 2k +5, 2k + 7 are in A.P (k= -4)

3) Find the 15^th term from the end of the A.P: 3, 5, 7,………, 201(173)

4) Find the 11^th term from the end of the A.P: 10, 7, 4,……, - 62 (-32)

5) If Sn_, the sum of first n terms of an A.P is given by Sn = 3n² – 4n, then find its nthterm

(6n – 7)

6) The sum of n terms of an A.P. is 3n² + 5n. Find the A.P. Hence, find its 16^th term

(6n + 2, 98)

7) In the following A.P. find the missing term -, 38, -, - , - , -22

8) Find the sum of all natural numbers less than 100 which are divisible by 6 (816)

9) Find the sum of 3 digit numbers which are not divisible by 7 (424214)

10) Find the sum of all three digit numbers which leave the remainder 3 when divided by 5 (99090)

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CBSE10th Arithmetic Progression Study material

1. An 

1.Arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term.
2. The difference between the two successive terms of an A.P. is called the common difference.
3. Each of the number in the list of arithmetic progression is called a term of an A.P.
4. The arithmetic progression having finite number of terms is called a finite arithmetic progression.
5. The arithmetic progression having infinite number of terms is called an infinite arithmetic progression.

X Maths Co-Ordinates geometry Formula

X Surface area and Volume Excel exercise

1. Find the edge of a cube of volume equal to the volume of a cuboid of dimensions 63 cm × 56 cm × 21 cm.
2. Find the number of 5 cm cubes that can be cut out of a 15 cm cube. 3. Three cubes of metals whose edges are 3, 4 and 5 cm respectively are melted and formed into a single cube. If there is no waste in the process, find the edge of the new cube so formed.
4. A school room is to be built to accommodate 70 children, so as to allow 2.2 sq m of floor area and 11 cu m of space for each child. If the room is to be 14 m long, what must be its breadth and height ?
5. How many bricks 20 cm × 10 cm × 7.5 cm be carried by a truck whose capacity to carry load is 6 metric tons ? One cubic meter of bricks weighs 2000 kg. [1 metric ton = 1000 kg]

6. A field is 200 m long and 75 m broad; and a tank 40 meter long, 20 meter broad and
10 meter deep is dug in the field, and the earth taken out of it is spread evenly over the
field. How much is the level of field raised ?
7. Four cubes each of sides 5 cm are joined end to end. Find the surface area of the resulting cuboid.
8. The sides of an open box are 0.5 cm thick and bottom is 1 cm thick. If the internal length, breadth and depth are respectively 14 cm, 10 cm ad 8 cm, find the quantity of material used in the construction of the box.
9. Find the whole surface area of a hollow cylinder open at the ends, if its length is 10 cm, the internal diameter is 8 cm and the thickness is 1 cm [Use π = 3.14]
10. A cubic cm of gold is drawn into a wire 1/5 mm in diameter; find the length of wire. ( π = 3.14)
11. A well with 8.4 meter inside diameter is dug 10 meter deep. Earth taken out of it has been spread all around it to a width of 4 meters to form an embankment. Find the height of the embankment.
12. Find the slant height of a cone whose volume is equal to 12936 cubic meters and the diameter of whose base is 42 meters.
13. The volume of a cone is 616 cubic meters. If the height of cone is 27 meters, find the radius of its base.
14. A conical vessel of internal radius 14 cm and height 36 cm is full of water. If this water is poured into a cylinder with internal radius 21 cm, find the height to which the water rises in the cylinder.
15. Find the diameter of a sphere whose volume is 606.375 cubic meter.
16. A room 12 meters long, 4 meters broad and 3 meters high has two windows 2 m × 1 m and a door 2.5 m × 2 m. Find the cost of papering the walls with paper 50 cm wide at Rs 20 per meter.
17. A hall, whose length is 15 m and breadth is twice its height, takes 250 meters of paper 2 meters wide for its four walls. Find the area of the floor.
18. The length of a room is 1.5 times its breadth. The cost of carpeting it at Rs 150 per square meter is Rs 14400 and the cost of white washing the four walls at Rs 5 per square meter is Rs 625. Find the dimensions of the room.

IX Maths Surface Areas and Volumes Concepts Chapter 13 

Surface area and volume for class 10th
10th surface area and volume Cuboid and a Cube Cylinder Cone 

CBSE MATHEMATICS (Class-9,10) ch-Surface Area and Volumes

X maths Circle,cylinder,square volume surface area

Arithmetic Progression CBSE Class X Self Evaluation Tests For Mathematics

1. 0, –4, –8, –12, ............

2.. 1, 2, 4, 8, 16, ...............

3. 6, 6, 6, 6, 6, ............

4. 5, 5 +Ö 3, 5  2 Ö3, 5+3Ö3,...........

7. Find the sequence whose of nth term is given by :

(a) 2n – 7 (b) –3 + 5n² (c) 4/3 - 6n  Also, determine which of these sequences are A.P’s.

8. Find the A.P. whose nth term is given by :

(a) 9 – 5n (b) –n + 6 (c) 2n + 7

9. Find the 8th term of an A.P. whose 15th term is 47 and the common difference is 4.

10. The 10th term of an A.P. is 52 and 16th term is 82. Find the 32nd term and the general term.

11. The 7th term of an A.P. is – 4 and its 13th term is –16. Find the A.P. [CBSE 2004]

12. Which term of the A.P. 3, 10, 17, .... will be 84 more than its 13th term? [CBSE 2004]

13. Find the A.P. whose third term is 16 and the seventh term exceeds its fifth term by 12.

14. For what value of n is the nth term of the following A.P.’s the same?

1, 7, 13, 19, ..... and 69, 68, 67............... [CBSE 2006C]

15. If the 10th term of an A.P. is 52 and 17th term is 20 more than the 13th term, find the A.P.

16. If seven times the seventh term of an A.P. is equal to nine times the ninth term, show that 16th term is zero.

17. (a) Which term of the A.P. 2, 6, 10, 14, ....... is 78?

(b) Which term of the A.P. 5, 9, 13, 17, ....... is 125?

(a) Which term of the A.P. –4, –1, 2, 5, ....... is 119?

(a) Which term of the A.P. Ö2, Ö18, Ö50 ....... is 21 Ö2 ?

18. How many terms are there in each of the following finite A.P.’s?

(a) –3, –4, –5, –6, ....., – 107

(b) 7, 13, 19, ........, 211

(c) 8, 13, 18, ......., 208

19. (a) Is 216 a term of A.P. 3, 8, 13, 18, ......?

(b) Is 271 a term of A.P. 1, 4, 7, 10, .....?

(c) Is –52 a term of A.P. 10, 7, 4 ......?

(d) Is –190 a term of A.P. –3, –8, –13, ......?

20. Find the 8th term from the end of the A.P. 7, 10, 13, ..... , 184. [CBSE 2005]

21. Find the 20th term from the end of the A.P. 3, 8, 13, ....., 253.

22. Find the value of k so that k – 3, 4k – 11 and 3k – 7 are three consecutive terms of an A.P.

23. Find the value of x so that 3x + 2, 7x – 1 and 6x + 6 are three consecutive terms of an A.P.

24. The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the

first term and the common difference.

25. The 9th term of an A.P. is equal to 7 times the 2nd term and 12th term exceeds 5 times the 3rd term by 2. Find

the first term and the common difference.

26. Find the middle term of an A.P. with 17 terms whose 5th term is 23 and the common difference is –2.

27. For what value of n, the nth terms of the two A.P.’s are same? 2, –3, –8, –13, ...... and –26, –27, –28, –29, .......

28. For what value of n, the nth terms of the two A.P.’s are same? 3, 10, 17, ...... and 63, 65, 67, .....

29. Find the number of integers between 200 and 500 which are divisible by 7.

30. How many numbers of three digits are exactly divisible by 11?

31. The angles of a triangle are in A.P. If the greatest angle equals the sum of the other two, find the angles.

32. Three numbers are in A.P. If the sum of these numbers is 27 and the product is 648, find the numbers.

33. If m times the mth term of an A.P. is equal to n times its nth term, show that the (m + n)th term of the A.P. is

zero.

34. A sum of Rs. 1000 is invested at 8% simple interest per annum. Calculate the interest at the end of 1, 2, 3,

......years. Is the sequence of interest an A.P.? Find the interest at the end of 30 years.

35. Two A.P.’s have the same common difference. The difference between their 100th terms is 111222333.

X Mathematics Sample Paper March _2014

Science Sample Paper Links -2014                                      

Section A

1. If the equation kx2+ 4x+ k=0 has two equal roots, then

(a) k = ±2 (b) k = 0 (c) k = 2 (d) none of these

2. If the sum of n terms of an A.P., is 2n2 +4n, then its nth term is

(a) 4n + 2 (b) 4n + 5 (c) 10n - 4 (d) none of these

3. If tangents TA and TB from a point T to a circle with center O are inclined to each other at an angle of 70° then <TOA is equal to

(a)80° (b) 70° (c) 55° (d) 50°

4. To divide a line segment AB in the ratio 5:3, first a ray AX is drawn so that < BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is

(a) 12 (b) 11 (c) 10 (d) 8

5. If the circumference and area of a circle are numerically equal, then diameter of the circle is 

(a)2x (b) 2 (c) 4 (d) 4x

6. The number of solid spheres, each of diameter 2 cm that could be moulded to form a solid metal cylinder of height 45 cm and diameter 4 cm is

(a)14 (b) 15 (c) 135 (d) none of these

7. From the top of a cliff 20 m high the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is

(a)40 m (b) 60 m (c) 90 m (d) none of these

8. If the distance between the points (5, r) and (1, 0) is 5, then r is

(a)5 (b) -5 (c) 0 (d)  + - 3

9. The area of a triangle with vertices A(4, 0), B(7,0) and C(9, 5) is

a)14 sq. units (b) 28 sq. units (c) 17.5 sq. units (d) none

10. If A(5, -1),B(-3, -2) and C(-1,8) are the vertices of triangle ABC, then the length of the median through A is

(a) √50 units (b) √60 units (c) √62 units (d) √65 units

                                                            Section B

11. Does there exist a equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

12. The nth term of an A.P., cannot be 2n² +1. Justify your answer.

13. AB is a chord of the circle and AOC is its diameter such that <ACB = 50°. If AT is the tangent to the circle at the point A, then <BAT is equal to 50°. Justify your answer.

14. Write True or False and give reason for your answer for the following: A pair of tangents can be constructed from a point P to a circle of radius 6 cm situated at a distance of 5 cm from the centre.

15. Is it true that the distance travelled by a circle wheel of diameter x cm in one revolution is 2px cm? Why?

16. A circle is inscribed in a square of side x cm and another circle is circumscribing the square. Is it true to say that area of the outer circle is two times the area of the inner circle? Give reasons for your answer.

17. Write True or False and justify you answer for the following:

A spherical steel ball is melted to make 10 new identical balls. Then, the radius of each new ball is 1/10th radius of the original ball.

18. A hemisphere is cut out from the top of the cylinder with radius equal to the radius of cylinder. Taking radius as r and height of cylinder as h. find total surface area of solid?

                                                         Section C

19. 50 circular plates, each of radius 7 cm and thickness 1/2 cm are placed one above another to form a Solid right circular cylinder. Find the total surface area and the volume of the cylinder so formed

20. A bag contains cards which are numbered from 2 to 100. A card is drawn at random from the bag. Find the probability that it bears (i) a two digit number (ii) a number which is a perfect square

21. In an AP, the sum of first ten terms is -150 and the sum of its next ten terms is -550.Find the AP.22. Construct a triangle ABC in which BC=9 cm, <B=60° and AB=6 cm. then construct another triangle whose sides are 2/3 of the corresponding sides of tri. ABC.

23. If (1,2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find the values of x and y.

24. Two tangents PA and PB are drawn to a circle with centre O from an external point P. prove that <APB=2 <OAB.

25. A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 60° and angle of depressionof point A from the top of the tower is 45°.Finf the height of the tower. [Take p =1.73].26. A coin is tossed 3 times. List the possible outcomes. Find the probability of getting at least 2 heads.27. ABC is a triangle right angled at A. Semicircles are drawn on AB, AC and BC as diameters. Find the area of the shaded region.28. A canal is 300 cm wide and 120 cm deep. The water in the canal is following with a speed of 20 km/h. How much area will it irrigate in 20 minutes if 8 cm of standing water is desired?

                                                             Section D

29. The interior of building is in the form of a cylinder of diameter 4 cm and height 3.5 m, surmounted by a cone of the same base with vertical angle as a right angle. Find the surface area (curved) and volume of the interior of the building.

30. An AP consists of 37 terms. The sum of three middlemost terms is 225 and sum of last three terms is 429. Find AP.

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

32. The angles of depression of the top and bottom of a building 60 metres high as observed from the top of a tower are 30° and 60°, respectively. Find the height of the tower and also the horizontal distance between the building and the tower.

33. A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, what is its original average speed?

34. The mid-points D, E, F of the sides AB, BC and CA respectively of a triangle ABC are (3, 4), (8, 9), and (6, 7). Find the coordinates of the vertices of the triangle.

CBSE  SUMMATIVE  ASSESSMENT-II  SAMPLE   PAPER [Links]

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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

10th probability illustrative solved examples

ILLUSTRATIVE EXAMPLES

Example 1. An unbiased die is thrown. What is the probability of getting :

(i) an odd number (ii) a multiple of 3 (iii) a perfect square number (iv) a number less than 4.

Solution.

An unbiased die is thrown we may get 1, 2, 3,4,5,6

So,  total number of all possible outcomes= 6

(i) Favourable outcomes for an odd number are 1, 3, 5.

So, no. of favourable outcomes = 3

P (an odd number) = No. of favourable outcomes/ Total no. of possible outcome= 3/6=1/2

(ii) favourable outcomes for a multiple of 3  are 3 and 6.

So, no. of favourable outcomes = 2

P (a multiple of 3) = 2/6= 1/3

(iii) favourable outcomes for ) a perfect square number  are 1 and 4.

So, no. of favourable outcomes = 2

 P (a perfect square number)  =2/6= 1/3

(iv) favourable outcomes for a number less than 4. are 1, 2 and 3.

So, no. of favourable outcomes = 3

P (a number less than 4) = 3/6 =1/2

2. Three unbiased coins are tossed together. Find the probability of getting :

(i) all heads (ii) two heads (iii) one head (iv) at least two heads

Solution. When three unbiased coins are tossed together, possible outcomes are

HHH, HHT, HTH, HTT, THH, THT, TTH and  TTT.       
So, total no. of possible outcomes = 8

(i) favourable outcome = HHH

So, No. of favourable outcome = 1

P (all heads)  =no. of favourable outcomes/Total no. possible outcomes= 1/8

(ii) favourable outcomes are HHT, THH and HTH.

So, no. of favourable outcomes = 3

P (two heads) = 3/8

(iii) favourable outcomes are HTT, THT and TTH.

 So, no. of favourable outcomes = 3       P (one head) =  3/8

(iv) favourable outcomes are HHH, HHT, HTH and THH.

So, no. of favourable outcomes = 4

P (at least two heads) 4/8 =1/2

Example 3. Find the probability that a leap year selected at random will contain 53 Sundays.

Solution. In a leap year, there are 366 days. But  366 days = 52 weeks + 2 days.

Thus, a leap year has always 52 sundays.

The remaining 2 days can be :

(i) Sunday and Monday

(ii) Monday and Tuesday

(iii) Tuesday and Wednesday

(iv) Wednesday and Thursday

(v) Thursday and Friday

(vi) Friday and Saturday

(vii) Saturday and Sunday

Clearly, there are seven elementary events associated with this random experiment.

Let E be the event that a leap year has 53 sundays. Clearly, the event E will happer if the last two

days of the leap year are either Sunday and Monday or Saturday and Sunday.

Favourable no. of elementary events = 2

Hence, required probability = 2/7 

Example 4. One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn.

Find the probability that the card drawn is :

(i) an ace (ii) either red or king  (iii) a face card (iv) a red face card

Solution. here, total no. of possible outcomes = 52.

(i) There are 4 ace cards in a pack of 52 cards. One ace can be chosen in 4 ways.

So, favourable no. of outcomes = 4

P (an ace) no. = of favourable outcomes/Total no. of possible outcomes= 4/52=1/13

(ii) There are 26 red cards, including 2 red kings. Also, there are 4 kings, two red and two black.

card drawn will be a red card or a king if it is any one of 28 cards (26 red cards and 2 black kings)

So, favourable no. of outcomes = 28

        P(either red or king) =28/52= 7/13

(iii) Kings, queens and jacks are the face cards.

So, favourable no. of outcomes = 3 × 4 = 12

      P(a face card) = 12/52 =3/13

(iv)  There are 6 red face cards, 3 each from diamonds and hearts.

So, favourable no. of outcomes = 6

                      P(a red face card) = 6/52= 3/26

Example 5. Cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number of the card is :

(i) an even number                                     (ii) a number less than 14 

(iii) a number which is a perfect square (iv) a prime number less than 20.

Solution.

From 2 to 101, these are (101–2) + 1 = 100 numbers.

So, total no. of possible outcomes = 100.

(i) From 2 to 101, the even numbers are 2, 4, 6, ...., 100 which are 50 in number.

So, number of favourable outcomes = 50

     P(an even number) = no. of favourable outcomes/Total no. of possible outcomes

                                       = 50/100 = 1/2

(ii) From 2 to 101, the numbers less than 14 are 2, 3, ...., 13 which are 12 in number.

So, no. of favourable outcomes    = 12

              P(a number less than 14) = 12/100 = 3/25

(iii) From 2 to 101, the perfect squares are 4, 9, 16, ..... 100, which are 9 in number.

              So, no. of favourable outcomes = 9

 P (a number which is a perfect square) = 9/100

(iv) From 2 to 101, the prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17 and 19 which are 8 in

number.

So, no. of favourable  outcomes  = 8

        P (a prime no. less than 20) = 8/100=2/25

Example 6. A bag contains 3 red balls and 5 black balls.A ball is drawn at random from a bag. What is the

probability that the ball drawn is :  (i) red (ii) not red

Solution. Total number of balls = 3 + 5 = 8

(i) P (red ball) = no. of red balls /Total no.of balls=3/8

(ii) P (not red ball) = 1 – P(red ball) =1 – 3/8 =5/8

Example  7. A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3 . Find the number of blue marbles in the jar.

Solution. Total number of elementary events = 24.

Let there be x green marbles.

P (green marbles is drawn) = x/24

but, P(green marbles is drawn)= 2/3 (given)

So, x / 24  = 2/3         x=24x2/3                       x   =   16

 Number of green marbles = 16

Number of blue marbles = 24 – 16 = 8 Ans.

Example 8. Suppose you drop a die at random on the rectangular region shown in the figure. What is the probability that it will land inside the circle with diameter 1 m?

 Solution. Total area of rectangular region = 3 m × 2 m = 6 m2

Area of the circle = p r²  = p (1/2)² m²        =  p/ 4 m²

P (die to land inside the circle) = p/ 4 m² ÷  6 = p/24 

Example 9. A bag contains 12 balls out of which x are white.

(i) If one ball is drawn at random, what is the probability that it will be a white ball?

(ii) If 6 more white balls are put in the bag, the probability of drawing a white ball will be

double than that in (i). Find x.

Solution. (i) Total number elementary events = 12.

There are x white balls out of which one can be chosen in x ways.

So, favourable number of elementary events = x

 p1 = P (white ball) = no. of favourable outcomes / Total no. of possible outcomes = x /12

 (ii) If 6 more white balls are put in the bag, then total number of balls in the bag =12 + 6 = 18

and, no. of white balls in the bag = (x + 6)

P₂ = P (getting a white ball) = ( x + 6 )/ 18

It is given that, p₂  =  2 p₁

( x + 6 )/ 18 = 2 ( x / 2)

 x = 3

Example 10. Two dice are thrown simultaneously. Find the probability of getting :

(i) a doublet i.e. same number on both dice. (ii) the sum as a prime number.

Solution. Possible outcomes associated to the random experiment of throwing two dice are :

(1, 1), (1, 2), ......., (1, 6)

(2, 1), (2, 2), ......., (2, 6)

.......................................

.......................................

(6,1) , (6, 2), ......., (6, 6)

Total number of possible outcomes = 6 × 6 = 36

(i) The favourable outcomes are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6).

Total no. of favourable outcomes = 6

So, P(a doublet) = no. of favourable outcomes/Total no. of possible outcomes = 6/36= 1/6

(ii) Here, favourable sum (as a prime number) are 2, 3, 5, 7 and 11.

So, favourable outcomes are (1, 1), (1, 2), (2, 1), (1, 4), (4, 1), (2, 3), (3, 2), (1, 6), (6, 1), (2, 5), (5, 2),

(3, 4), (4,3), (6, 5) and (5, 6).

no. of favourable outcomes = 15

P (the sum as a prime number) = 15/36 = 5/12