Grade – X Mathematics RW Compound interest 1. A person invests Rs. 31250 at 4% pa. C.I., compounded annually. Calculate i. the interest for the first year ii. the amount at the end of 2 years iii. the interest for the 3rd year 2. A person invests 10000 for three years at a certain rate of interest, compounded annually. At the end of

one year it amounted to Rs. 11200. Calculate i. the rate of interest per annum, ii. the interest accrued in the second year iii. the amount at the end of 3rd year 3. Ramesh invests Rs12800 for three years at the rate of 10% per annum, CI. Find i. the sum due to Ramesh at the end of the first year. ii. the interest he earns for the second year. iii. the total amount due to him at the end of the third year 4. A certain sum amounts to Rs. 5292 in 2 years to Rs. 5556.60 in 3 years at compound interest. Find the

rate and the sum. 5. A certain sum amounts to Rs. 57600 in 2 years and to Rs. 65536 in 4 years at compound interest. Find

the rate and the sum. 6. A sum of money lent at CI amounts to 2809 in 2 yearsand to 2977.54 in 3 years. Find the sum and the

rate of interest. 7. The difference between CI and SI on a certain sim of money for 3 years at 5% p.a. is Rs. 122. Find the

sum lent out. 8. On a certain sum of money , the difference between the CI for a year, payable half yearly, and the simple

interest for a year is Rs.180. Find the sum lent out, if the rate of interest in both the cases is 10%. 9. What sum of money will amount to Rs.3630 in 2 years at 10% per annum compound interest? 10. Find the rate of interest per annum when a sum of Rs.30000 amounts to Rs. 39930 om 1 ½ years. The

interest being compounded semi annually. 11. The simple interest on a sum of money for 2 years at 4% per annum is Rs 340. Find

(i) the sum of money and (ii) the compound interest on this sum for one year payable half yearly at the same rate.

Sales tax and VAT 1. Vikram bought a watch for Rs. 825 which includes 10% sales tax on the list price. What was the cost of

the watch? 2. The price of a TV including sales tax is Rs.13080 Find the marked price. How much more does the

consumer pay for the TV if the sales tax is increased to 13%? 3. Dinesh bought an article for Rs. 374, which includes a discount of 15% on marked price and a sales tax

of 10% on the reduced price. Find the marked price of the arcticle. 4. Ms. Chawla goes to a shop to buy a leather coat which costs Rs.735. The rate of sales tax is 5%. She tells

the shopkeeper to reduce the price to such an extent that she has to pay Rs.735, inclusive of sales tax. Find the reduction needed in the price of the coat.

5. The catalogue price of a colour TV is Rs.2400. The shopkeeper gives a discount of 8% on the listed price. He gives a further off-season discount of 5% on the balance. But sales tax at the rate of 10% is charged on the remaining amount. Find : i. the sales tax amount a customer has to pay. ii. the final price he has to pay for the colour TV.

6. Rohit buys a computer for Rs. 38400 which includes 10% discount and then 6% sales tax is charged on it. Find the listed price so the computer.

7. A man wants to purchase an article of marked price Rs.4000, which is heing sold at a discount. If the sales tax is 10% on the price charged, he can buy the article at Rs.4200. Find the rate of discount allowed.

8. A shop keeper buys a camera at a discount of 20% from the wholesaler,the printed price of the camera being Rs1600 and the rate of sales tax is 6%. The shopkeeper sells it to a buyer at the printed price and charges the tax at the same rate. Find the price ate which the camera is bought. And the VAT paid by the shopkeeper.

9. A manufacturer lists the price of his goods at Rs. 2400 per article. The wholesaler gets a discount of 25% on the goods from the manufacturer. The retailers are allowed a discount of 15% on the listed price by the wholesaler. The prescribed rate of sales tax is 8%. A consumer buys the article form the retailer at

the listed price. Find eh VAT paid by the wholesaler and the retailer. Also find the price paid by the consumer for the article.

10. A manufacturer sells a washing machine to a wholesaler for Rs. 15,000. The wholesaler sells it to a trader at a profit of Rs.1,200 and the trader in turn sells it to a consumer at a profit of Rs.1,800. If the rate of VAT is 8%, find (i) The amount of VAT received by the state government on the sale of machine from the manufacturer and the wholesaler and (ii) the amount that the consumer pays for the machine (4) 11. The list price of an article is Rs 3000.A shopkeeper sells the article to a consumer at the

least price and charges sales tax at the prescribed rate of 8%.If the shopkeeper pays a VAT of Rs 32 to the State Govt. at what price inclusive of sales tax did the shopkeeper buy the article from the wholesaler?

12. Manufacturer sells a washing machine to a trader B for Rs 12500. Trader B sells it to a trader C at profit of Rs 800 and trader C sells it to a consumer at a profit of Rs 1300. If the rate of VAT is 8%, find (i) The amount of tax (under VAT) received by the State Govt. on the sale of this machine. (ii) The amount that the consumer pays for the machine.

Shares and dividends 1. Ajay owns 560 shares of a company. The face value of each share is Rs. 25. The company declares a

dividend of 9%. Calculate : i. The dividend that Ajay will get. ii. The rate of interest on his investment if Ajay had paid Rs. 30 for each share.

2. Aman invests Rs.29929 in shares of par value Rs.26 at 10% premium. The dividend is 15% per annum. Calculate : i. the number of shares ii. the dividend received by him annually iii. the rate of interest he gets on his money.

3. A company with 10000 shares of Rs.100 each, declares an annual dividend of 5%. i. What is the total amount of dividend paid by the company ? ii. What would be the annual income of a man, who has 72 shares, in the company ? iii. If he received only 4% on his investment, find the price he paid for each share.

4. Mr. Lohia invests Rs 26680 in buying Rs 50 shares at a discount of 8%. He sells shares worth Rs15000 at a premium of 6% and the rest at a discount of 10%. Find his total gain or loss from the transaction.

5. Which is better investment : 7% Rs.100 shares at Rs.120 or 8% Rs.10 shares at Rs.13.50 ?

6. A man sold certain Rs. 100 shares paying 10% dividend at a discount of 25% and invested the proceeds in Rs. 100 shares paying 16% dividend quoted at Rs. 80 and thus increased his income by Rs.2000. Find the number of shares sold by him

7. A man invested Rs.45000 in 15% Rs.100 shares quoted at Rs.125. When the market value of these shares rose to Rs.140, he sold some shares, just enough to raise Rs.8400. Calculate : (i) the number of shares he still holds. (ii) the dividend due to him on these shares.

8. By investing Rs. 7500 in a company paying 10% dividend, an income of Rs. 500 is received. What price is paid for each

Rs.100 share?

9. What sum should a person invest in Rs. 25 shares, selling at Rs. 36, to obtain an income of Rs. 720, if the dividend declared is 12%. Also find (i) the no. of shares bought by the person. (ii) The percentage return on his investment.

10. Mr. Sharma has 60 shares of nominal value Rs.100 and he decides to sell them when they are at premium of 60%. He invests the proceeds in shares of nominal value of Rs.50, quoted at 4% discount, paying 18% dividend annually. Calculate

i. The sale proceeds. ii. The number of shares he buys iii. His annual dividend from these shares

11. . A man bought 360 ten – rupees shares paying 12% premium. He sold them when the price rose to Rs.21 and invested the proceeds in five – rupees shares paying 4.5% per annum at Rs.3.5 per share. Find the annual change in his income

12. A person invests Rs.4,368 and buys certain hundred rupee shares at Rs. 91. He sells out shares worth Rs. 2,400 when they have risen to Rs.95 and the remainder when they have fallen to Rs.85. Find the gain or loss on the total transaction.

13. A man invests Rs. 2,688 in buying shares of nominal values Rs. 24 and selling at 12% premium. The dividend on the shares is 15% per annum.

i. Calculate the number of shares he buys ii. Calculate the dividend he receives annually.

BANKING 1. A page from the saving bank account passbook of Mr. Sen is given below. Find the interest Mr. Sen will get from Jan 2006

to June 2006 at the rate of 5% per annum.

Date 2006 Particular Withdrawals (in Rs.) Deposits (in Rs.) Balance (in Rs.)

4 Jan B/F 1500.00

20 Jan By cash 1200.00 2700.00

15 Mar By cheque 2700.00 5400.00

20 Mar To cheque 1000.00 4400.00

6 June By cheque 4200.00 8600.00

16 June To cheque 1200.00 7400.00

2. A page from Saving Bank Passbook of Mrs. Reva is given below. Calculate the interest received by Mrs. Reva from Jan

2006 to June 2006 at the rate of 5% p.a.

Date 2006 Particular Withdrawals (in Rs.) Deposits (in Rs.) Balance (in Rs.)

1 Jan By Balance 1500.00

5 Jan By cash 1200.00 2700.00

20 Jan By cash 1000.00 3700.00

25 Jan To cheque 800.00 2900.00

15 Mar By cash 2500.00 5400.00

20 Mar To cheque 1000.00 4400.00

6 June By cash 3000.00 7400.00

25 June By cash 1200.00 8600.00

3. A page from the saving bank account is given below :

Date 2005 Particular Amount withdrawn(in Rs.)

Deposits (in Rs.) Balance (in Rs.)

01- 07 – 05 B/F 5,000.00

11 - 07 – 05 By cheque 7,000.00 12,000.00

25 - 08 – 05 By cheque 20,000.00

10 - 10 – 05 To cheque 10,000.00 8,000.00 10,000.00

15 - 12 – 05 By cash 12,000.00 22,000.00

29 - 12 – 05 To cheque 19,000.00 3,000.00

The account is closed on 2nd Jan 2006. Find the amount received if the rate of interest is 5% per annum

4. Amar opens a recurring deposit account in a bank. He deposits Rs.500 every month for one year. The interest paid by the bank is 5% per annum. Find the amount he receives at the time of maturity.

5. Shankar deposits Rs.300 per month in a recurring deposit account for 12 months. Find the amount he will receive at the time of maturity at the rate of 5% p.a.

6. Rajaram wants Rs.34575 at the end of 5 years by depositing a certain sum of money on a monthly basis in a bank paying 6% simple interest p.a. What should be the monthly instalment.

7. Rachna gets Rs.38, 805.75 at the end of 6 years at the rate of 6.5% p.a. in a recurring deposit account. Find the monthly instalment.

8. Mrs. Mamun invests Rs.250 every month for 24 months in a bank and collects Rs.6312.50 at the end of the term. Find the rate of simple interest paid by the bank on this recurring deposit.

9. Which is better investment Rs.40000 in a saving deposit with bank for 3 years the interest being compounded half – yearly at the of 6% or Rs.1200 p.m. in a recurring deposit with a bank paying simple interest at 6% p.a. for 3 years

LINEAR INEQUATIONS

1. Solve the inequations and graph the solution set.

2. Solve : and draw graph of the solution set.

3. P is solution set of is the solution set of where . Find the set

4. Find the range of values of x which satisfies. Graph these values of x on the number line.

5. Find the range of values of x which satisfy

Graph these values of x on the real number line.

6. Solve the following inequation, and graph the solution on the number line :

QUADRATIC EQUATIONS

1. 15x2 - 28 = x 2. 3x2 - 5x - 12 = 0 3. 21x2 - 8x - 4 = 0

4. x(2x + 1) = 6 5. abx2 + (b2 - ac) x - bc = 0 6.

7. 8.

9. 10. 11.

12. (x + 4) (x + 5) = 3 (x + 1) (x + 2) + 2x 13.

14.

15. Find the value of p for which the given equations has equal roots :

i. 4x2 - 5x + p = 0 ii. px2 - 8x + 4 = 0 iii. 9x2 + 3px + 4 = 0 iv. (p - 12) x2 + 2 (p - 12) x + 2 = 0

16. Solve the equation 2x - 1/x = 7. Write your answer correct to two decimal places.

17. Find the values of k so that the equation kx(x-2) +6=0 gas two equal roots. Also find the root in each case.

PROBLEMS ON QUADRATIC EQUATIONS

1. A trader buys x articles for a total cost of Rs.600. (i) Write down the cost of one article in terms of x. If the cost per article

were Rs.5 more, the number of articles that can be bought for Rs.600 would be four less. (ii) Write down the equation in x for the above situation and solve it to find x.

2. A shopkeeper buys a certain number of books for Rs.720. If the cost per book was Rs.5 less, the number of books that could be bought for Rs.720 would be 2 more. Taking the original cost of each book to be Rs. x, write an equation in x and solve it.

3. An express train makes a rum of 240 km at a certain speed. Another train, whose speed is 12 km/hr less, takes an hour longer to make the same trip. Find the speed of the express train in km/hr.

4. A train covers a distance of 90km at a uniform speed. Had the speed been 15 km/per hour more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

5. An aeroplane traveled a distance of 400 km at an average speed of x km/hr, on the return the speed was increased by 40 km/4r. Write down an expression for the time taken for (i) The on ward journey (ii) The return journey. If the return journey took 30 minutes less than the onward journey, write down an equation in x and find its value.

6. A rectangle of are a 105 cm2 has its length equal to x cm. Write down its breadth in terms of x. Given that its perimeter is 44cm, write down an equation in x and solve it to determine the dimensions of the rectangle.

7. The perimeter of a rectangular plot is 180m, and its area is 1800m2. Take the length of the plot as x ‘m’. Use the perimeter 180m to write the value of the breadth in terms of ‘x’. Use the value of length, breadth and the area to write an equation in ‘x’. Solve the equation to calculate the length and breadth of the plot.

8. Two squares have sides x cm and (x + 4) cm. The sum of their area is 656 cm2. Express this as an algebraic equation in x and solve the equation to find sides of the squares.

9. The hotel bill for a number of people for over night stay is Rs.4,800. If there were 4 more, the bill each person had to pay would have reduced by Rs.200. Find the number of people.

10. The sum of ages (in years) of a son and his father is 35 and the product of their ages is 150. Find their ages.

11. A two digit number is such that the product of its digits is 8. When 63 is subtracted from the number the digits interchange their places. Find the number.

12. In a auditorium, seats are arranged in rows and columns. The number of rows are equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 10, the total number of seats increased by 300.Find: (i)the number of rows in the original arrangement

(ii) the number of seats in the auditorium after rearrangement

RATIO AND PROPORTIONS

1. How much copper must be mixed with 144 g of zinc in order to make an alloy in which the ratio by weight of zinc to copper

is 9 : 15.

2. There are 40 members in a students council in a school and the ratio of the number of boys to the number of girls in 3 : 1. How many more girls be added to the council to make the ratio of boys to girls 3 : 2?

3. The ratio of the monthly incomes of A and B is 6 : 7 and the ratio of their expenditures is 4 : 5. If the amounts of their monthly savings are Rs.2400 and Rs.2000 respectively. Find their respective monthly income.

4. If a : b = c : d, show that

i. ii. (4a + 9b) : (4a - 9b) = (4c + 9d) : (4c - 9d) iii. (3a - 5b) : (3a + 5b) = (3c - 5d) : (3c + 5d)

5. . If x/a = y/b = z/c, show that

i. ii.

iii.

6.

7. If b is the mean proportional between a and c, show that b (a + c) is the mean proportional between (a2 + b2) and (b2 + c2).

8. If a, b, c are in continued proportion, prove that

i. a : c = (a2 + b2) : (b2 + c2) ii. (a + b) : (b + c) : : a2(b - c) : b2(a - b)

9. What must be added to the numbers 4, 8, 16, 26 so that they become proportional.

10. Using the properties of proportions, solve the following for x :

i. (x3 + 3x) / (3x2 + 1) = 341/91 ii. iii.

11. Solve the following equation using properties of proportion √13 + x + √13 – x = 5 (3) √13 +x - √13 - x FACTOR THEOREM 1. Determine the value of ‘a’ such that (x - 4) is a factor of 2x3 + ax2 + 27x - 28.

2. Find the value of k so that (x + 1) is a factor of k2x2 - 2kx - 3.

3. Show that 2x - 7 is a factor of 2x3 + x2 - 22x - 21. Hence factorise the given expression completely, using factor theorem.

4. If (x - 4) is a factor of x3 + ax2 + 2bx - 24 and a - b = 8, find the value of a and b

5. Find the value of a and b so that the polynomial x3 + ax2 + bx - 6 has a factor x2 - 4x +3.

6. Factorise x3 – 7x2 + 14x – 8 using remainder theorem

MATRICES

1. If , show that 2. Find x and y if

3. Find M

4.

5.

6. Compute A (B + C) and (B + C) A.

7. find the value of x if A2 = B

8. find the value of a , b and c.

9. find A2 - A + BC

10. Compute (AB) C and (CB) A. Is (AB) C = (CB) A.

11.

REFLECTION

1. Use a graph paper to answer the following question i. Plot A (4, 4), B (4, - 6) and C (8, 0), the vertices of a triangle ABC ii. Reflect ABC on the y axis and name it as A’B’C’ iii. Write coordinates of the images A’, B’ and C’ iv. Give a geometrical name for the figure AA’C’B’BC

2. A triangle ABC where A (- 2, 3), B (4, - 4) C (6, - 7) is reflected in the x- axis onto and then

is reflected in the origin onto Write down the co-ordinates of (i) A`, B` C` and (ii) A’’ B’’ C’’.

1 2 2 1 2 7,

3 4 3 2 5 1A B and C

AB AC

3. A (5, 4) is reflected in the x-axis to a point A`.

1. Write down the coordinates of A`. 2. What type of triangle is the figure OAA`, where O is the origin? Give reason. Draw a diagram to represent it 3. State, with reason, whether the triangle OAA` has any line of symmetry. 4. Find the coordinates of A’’, the reflection of A in the y-axis followed by the reflection in the origin. 5. Compare the coordinates of A` and A.

4. A triangle ABC where A (1, 2), B (4, 8) and C (6, 8) is reflected through origin to triangle A, B, C,. Triangle A, B, C, is then reflected in the x - axis to triangle A2, B2, C2. Write down the coordinates of A2, B2, C2. Write down a single transformation.

5. (i) Point P (a, b) is reflected in x-axis to P’ (4, –3). Write down the valuesof a and b. (ii) P’’ is the image of P when reflected in y-axis. Write down the co-ordinates of P’’.

(iii) Name a single transformation that maps P′ to P’’. 6. Use a graph paper to answer the following question (i) Plot A (4, 4), B (4, - 6) and C (8, 0), the vertices of a triangle ABC (ii) Reflect ABC on the y axis and name it as A’B’C’ (iii) Write coordinates of the images A’, B’ and C’ (iv) Give a geometrical name for the figure AA’C’B’BC (v) Find perimeter of the figure AA’C’B’BC (vi) Identify the line of symmetry of AA’C’B’BC 7. Use graph paper for the question .Use 1 cm =1 unit on both axes.

(i) Plot the points A (3,5) and B(-2,-4) (ii) A’ is the image of A when reflected in the x axis.Write down the coordinates of A’ and plot it on

the graph paper. (iii) B’ is the images of B when reflected on the Y-axis, followed by the reflection on the origin.

Write down the co-ordinates of B’and plot it in the graph. (iv) Write down the geometric name of the figure AA’BB’. (v) Name two invariant points under reflection on the x-axis.

8. Points (3,0) and (-1,0)are invariant points under reflection in the line L1;points(0,-3) and (0,1)are invariant point in the line L1. (i) Name the line L1 and L2 . (ii) Write down the images of the points P (3, 4) and Q (-5,-2) on reflection inL1. Name the images as P’ and Q’ respectively. (iii) Write down the images of P and Q on reflection in L2.Name the images as P” and Q”. (iv) State or describe a single transformation that maps P’ and P”.

EQUATIONS OF STRIAGHT LINES 1. A line joining A(4, 5) and B(1, 2) is parallel to the joining C(1, -2) and D(0, k). Find the value of k.

2. Find the equation of the line passing through the intersection of 2x – y = 1 and 3x + 2y + 9 = 0 having slope equal to 1.

3. T he co-ordinates of two points A and B are (0, 4) and (3, 7) respectively. Find

i. The gradiant of AB. ii. The equation of AB. iii. The co-ordinates of the point where the line AB intersects the x-axis.

4. Write down the equation of the line parallel to x – 2y + 8 = 0 and passing through the point (1, 2).

5. Find the equation of the perpendicular dropped from (-1, 2) on the line joined (1, 4) and (2, 3).

6. A (2, -4), B (3, 3) and C (-1, 5) are the vertices of the triangle ABC. Find the equation of

i. The median of the triangle through A. ii. The attitude of the triangle through B.

7. Find the equation of a line, which has y-intercept 4 and which is parallel to the line 2x – 3y = 7. Find the co-ordinates of the point, where it cuts the x-axis.

8. Find the equation of the straight line which passes through the point of intersection of the two lines

2x – y + 5 = 0 ; 5x + 3y – 4 = 0 and is perpendicular to the line x – 3y + 21 = 0

9. A(1, 4), B(3, 2) and C(7, 5) are the vertices of a triangle ABC. Find

i. The co-ordinates of the centroid G of ii. The equation of a line through G and parallel to AB

10. In the figure alongside, the lines are represented. Write down the angles that the lines make with the position direction of the x-axis. Hence determine angle .

DISTANCE AND SECTION FORMULAE 1. The two vertices of a triangle are (– 1, 4) and (5, 2). If the centroid is (0, – 3), find the third

vertex. 2. Three consecutive vertices of a parallelogram ABCD are A(10,-6), B(2,-6) and

C(-4,-2),find the fourth vertex D.

3. A (10, 5), B (6, -3) and C (2, 1) are the vertices of a triangle ABC. L is the mid-point of AB and M is the mid-point of AC.

Write down the co-ordinates of L and M. Show that LM = ½ BC.

4. Two vertices of a triangle are (2, 3), (1, 2) and the centroid is (-1, -2). Find the third vertex.

5. Find the co-ordinates of the centroid of a triangle whose vertices are A (-1, 3), B (1, -1) and C (5, 1).

6. Prove that the points A (-5, 4), B (-1, -2) and C (5, 2) are the vertices of an isosceles right angled triangle.

Find the co-ordinates of D, so that ABCD is a square.

7. The line segment joining A(2, 3) and B(4, -5) is intersected by the x-axis at a point K. Write down the ordinate of the point. Hence find the ratio in which K divides AB.

8. Calculate the ratio in which the line segment joining (3, 4) and (-2, 1) is divided by the y-axis

9. The center O of a circle has the co-ordinates (4, 5) and one point on the circumference is (8, 10). Find the co-ordinates of the other and of the diameter of the circle through this point.

10. If the mid-point of the sides of a triangle are (-2, -5), (3, -2) and (3, -1); find its vertices

11. The co-ordinates of A and B are (2, a) and (-2, a+4). The mid-point of AB is (0, 1). Find the value of a.

12. find the points of trisection of ( - 2, 1) and (1, 4)

13. Find the circumcentre of the triangle formed by the points A (10, 5), B (6, -3) and C (2, 1)

14. Calculate the ratio in which the line joining. A (6, 5) and B (4, -3) is divided by the line y = 2.

SIMILARITY OF TRIANGLES 1. In a triangle ABC, DE is parallel to BC. If AE = 2cm, EC = 4cm and BC = 12cm, find the length of DE

2. In a triangle ABC, DE is parallel to BC. If AD = 4cm, DB = 2cm and the area of

i. Find the area of

ii.

3. In ΔEF , GHǁF such that LP=2 cm and PM=6cm. Also MN=20 cm.

4. . In the diagram alongside, DE is parallel to BC and Prove that

i.

ii.

5. In triangle ABC, DE is parallel to BC. If AD : DB = 2 : 3 and

area of is 8cm2, find the area of BCED.

6. In the figure PQ RS is a parallelogram; PQ = 16cm; QR = 10cm,

L is a point on PR such that RL : LP = 2 : 3, QL produced meets

RS at M and PS produced at N.

i. Prove that triangles RLQ and PLN. are similar. Find the length of PN ii. Find a triangle similar to RLM. Evaluate RM.

7. In triangle PQR, PB bisects and A is the mid-point of PQ.

RC is parallel to PQ and meets AB produced at C. Find QB : BR in two

different ways and show that PR = 2RC

SIMILARITY AS SIZE TRANSFORMATION

1. . On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD has the following measurements. AB = 12cm and

BC = 16cm. Angle A, B, C, and D are all 900 each. Calculate

i. The diagonal distance of the plot in km ii. The are of the plot in sq. km.

2. The scale of map 1 : 2,00,000. A plot of land of area 25 km2 is to be represented on the map. Find

i. The number of kilometers on the ground which is represented by 1 cm on the map ii. The area in km2 that can be represented by 1 cm2 iii. The area on the map that represents the plot of land.

3. A model of a ship is made to a scale of 1 : 200.

i. The length of the model is 5m. Calculate the length of the ship.

ii. The area of the deck of the ship is 1,60,000m2. Find the area of the deck of the model. iii. The volume of the model is 200 litres. Calculate the volume of the ship in m3.

4. A model of a rectangular swimming pool was made to a scale of 1 : 500.

i. The length and width of the actual swimming pool are 100m and 50m respectively. Calculate the length and width on the model in cm.

ii. If the model has the capacity to hold 240cm3 water, how much water can be stored in the actual swimming pool.

CHORD PROPERTIES OF CIRCLES

1. AB and CD are two parallel chords of a circle on the same side of the centre. If AB = 6cm, CD = 12cm and distance

between then is 3cm. Find the radius of the circle.

2. In a circle of radius 13cm, AB and CD are two parallel chords of length 24cm and 10cm respectively. Calculate the distance between the chords, if they are on

i. The same side of the centre ii. Opposite side of the centre

3. In the adjoining fig; CD is a diameter which meets chord AB at E such

that AE = BE = 4cm.If CE = 3cm, find the radius of the circle.

ANGLE PROPERTIES OF CIRCLES 1. In the figure alongside, O is the centre of the circle.If chords

AC and BD intersects at right angles at E and <OAB=35⁰, calculate <EBC.

2. In the adjoining figure ,XY is a diameter of the circle,

PQ is a tangent to the circle at Y.Given that <AXB=50° and <ABX=70°, calculate <BAY and <APY.

3. In the given figure, AB is a diameter and AC is a

chord of a circle such that <BAC=30°.The tangent at C intersects AB produced at D. Prove that BC=BD.

4. In the given figure, PQ is a diameter of the circle whose centre is O.

Given Calculate

5. . In the given diagram a chord ED is parallel to the

diameter AC of the circle ABCDE. If

Calculate .

6. In the given figure, Calculate

i.

ii.

7. . In the adjoining figure, AD is diameter of the circle with centre O. OB is parallel to DC and . Calculate

the numerical value of

i.

ii. and

iii.

8. In the given diagram, Calculate

i.

ii.

iii.

9. In the given diagram AD is a diameter of the circle ABD.

Chords AB and CD meet at P, Find the angle

10. In the figure given, ABCD is a cyclic quadrilateral

in which

Calculate

i.

ii.

iii.

11. In the adjoining diagram, AB is a diameter of the

circle whose centre is O. Given that

calculate

i.

ii.

12. In the adjoining diagram, O is the centre of the circle.

If , find the value of p, q and r.

13. ABCD is a cyclic quadrilateral in which

Find

i.

ii.

iii.

14. In this figure, AD is the diameter of the circle.

If , calculate

i. ii.

15. In the figure, O is the centre of the circle.

and XY is parallel to TZ.

Find giving reason

i.

ii.

iii.

16. ABCD is a cyclic quadrilateral;

AD = DC. Calculate

i.

ii.

iii.

TANGENT PROPERTIES OF CIRCLES

1. In the given figure, ABC is a right-angled with AB = 6cm and BC = 8cm

A circle with centre O has been inscribed inside the triangle.

Calculate the value of x, the radius of the inscribed circle.

2. . In the given diagram SA and SB are tangents to a circle

with centres O and

Calculate the value of p, q, r and s.

3. PQ is a tangent at A to circle with centre O and BD as a diameter.

, calculate:

i.

ii.

4. In the given figure, PQ = RQ, , PC and QC are tangents

to the circle with centre O. Calculate

i. The angle subtended by the chord PQ at the centre,

ii.

5. A, B, C are three points on a circle. The tangent at C meets BA produced at T. Given

that , calculate the angle subtended by AB at the centre.

6. . In the given figure, PQ the tangent to the circle at P.

Write down the values of x, y and z.

7. In the given diagram, the diameter AB is produced to P and PT is a tangent. . Find

i. in terms of x ii. the value of x

8. In the given figure, , calculate

i.

ii.

iii.

9. TAS is at A tangent at A to a circle. If

10. In the given figure alongside,

11. In the adjoining diagram, XT is tangent and XA is secant. If XB = 4cm, XT = 6cm; find AB.

12. In the given figure, PT is a tangent to the circle. If AT = 16cm and AB = 12cm, find PT.

CONSTRUCTIONS AND LOCI

1. Construct a triangle ABC is in which BC = 6cm, AB = 9cm and

i. Construct the locus of all points inside triangle ABC, which are equidistant from B and C. ii. Construct the locus of the vertices of the triangle with BC as base, which are equal in area to triangle ABC. iii. Marks the point Q in your construction which would make triangle QBC equal in area to triangle ABC and isosceles. iv. Measure length CQ

2. . Construct a triangle ABC with AB = 6cm, and BP = 5cm. Complete the rectangle ABCD such that

i. P is equidistant from AB and BC, and ii. P is equidistant from A and D

3. Construct a with CB = 5cm, and BP = 4cm. Complete the rectangle ABCD such that

i. P is equidistant from AB and BC, and ii. P is equidistant from C and D

4. . Using a ruler and compass only

i. Construct a triangle ABC with BC = 6cm, and AB = 3.5cm ii. In the above figure, draw a circle with BC as diameter. Find a point ‘P’ on the circumference of the circle which is

equidistant from AB and BC Measure .

5. Ruler and compass only may be used in this question. All construction lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment.

1. Construct , in which BC= 8cm, AB = 5cm and 2. Construct the locus of points inside the triangle which are equidistant from BA and BC 3. Construct the locus of points inside the triangle which are equidistant from B and C 4. Mark as P, the point which is equidistant from AB, BC and also equidistant from B and C 5. Measure and record the length of PB.

AREAS RELATED TO CIRCLES 1. The length of a wire which is tied as a boundary of a semi-circular park is 108m. Find the radius of the semi-circular park

and area of the park.

2. The wheel of a cart is making 5 revolution per second. If the diameter of the wheel is 84cm, find its speed in Km/hr. Give your answer, correct to nearest km.

3. A bucket is raised from a well by means of a rope which is bound round a wheel of diameter 77cm. Given that the bucket ascends in 1 minute 28 seconds with a uniform speed of 1.1m/sec, calculate the number of complete revolutions the

wheel makes in raising the bucket

4. 11. AC and BD are two perpendicular diameters of a circle ABCD. Given that the area of the shaded portion is 308cm2,

calculate

i. The length of AC and

ii. The circumference of the circle. ( )

5. . In the figure, O is the centre of the bigger circle and AC is its diameter.

Another circle with AB as diameter is drawn.

If AC = 54cm and BC = 10cm, find the area of the shaded region.

6. In the figure, AB is the diameter of the circle with centre O and OA = 7cm.

Find the area of the shaded region.

7. A horse is tethered to one corner of a field which is the shape of an equlateral triangle of side 12m. If the length of the rope

is 7m, find the area of the field which the horse cannot graze

8. From a trapezium ABCD, in which AB || DC and ,

quarter circle BFEC is removed.

Given AB = BC = 3.5, DE = 2cm. Calculate the remaining area.

9. Calculate the area of the shaded portion.

The quadrant shown in the figure are each of radius 7cm

SURFACE AREA AND VOLUME 1. The height of a cons is 40 cm.A small cone is cut off at the top of a plane parallel to its base .If its

volume be 1/64 of the volume of the given cone,at what height above the base is the section cut?

2. The surface area of a solid metallic sphere is 616 cm2. It is melted and recast into smaller spheres of diameter 3.5 cm.

How many such spheres can be obtained ?

3. A metallic sphere of density 2 gm per cubic cm is 42 cm in diameter. Can a person who can lift 90 kg, lift this sphere ?

(Use π = 22/7).

4. A solid metal sphere is cut through its centre into 2 equal parts. If the diameter of the sphere is 3.5 cm, find the total surface of each part correct to two decimal places. (Use π = 22/7).

5. A hollow spherical shell is made of a metal of density 4.9 gm/cm3. If its internal and external diameters are 10 cm and 12 cm respectively, find the weight of the shell. (Take π = 3.1416).

6. Eight metallic spheres, each of radius 2 mm are melted and cast into a single sphere. Calculate the radius of the new (single) sphere. (Use π = 22/7).

7. A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. If the total height of the toy is 15.5 cm, find its total surface area. (Use π = 22/7).

8. A vessel is in the form of a inverted cone. Its height is 8 cm and the radius of its top, which is open is 5 cm. It is filled with water up to the rim. When lead shots each of which is a sphere of radius 0.5 cm are dropped in to the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

9. A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top which is open, is 2.5 cm. It is filled with water up to the rim. When lead shots, each of which is a sphere of radius 0.25 cm are dropped in to the vessel, 2/5 of the water flows out. Find the number of lead shots dropped into the vessel.

10. A vessel in the form of an inverted cone is filled with water to the brim. Its height is 20 cm and diameter is 16.8 cm. Two equal solid cones are dropped in it so that they are fully submerged. As a result, one third of the water in the original cone overflows. What is the volume of each of the solid cones submerged ?

11. A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained.

12. The surface area of a solid metallic sphere is 1256 cm2. It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate : 1. The radius of the solid sphere. II. The number of cones recast. (Take π = 3.14).

13. A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heapof the sand. If the height of the conical heap is 24 cm. Find :

i. Its radius and ii. Its slant height. (Leave your answer in square root form).

14. . Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Find the time in which the level of water in the tank will rise by 21 cm.

15. A well of diameter 3m, is dug 14 m deep. The earth taken out of it has been spread evenly all around it to a width of 4 m, to form an embankment. Find the height of the embankment.(Use π = 22/7).

16. A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and slant height of the conical portion is 53 m, find the area of the canvas needed to make the tent. ( Use p = 22/7).

TRIGONOMETRIC IDENTITIES 1. Without using trigonometric table, evaluate : 7(sin 27˚/cos 63˚) + 3(cos 21˚/sin 69˚) – 7(tan 36˚/cot 54˚) 2. Prove the identity: Sin3A - Cos3A = SinA – CosA 1 – SinA CosA 3. Prove the following identities :

cos A/(1 – tan A) + sin A/(1 – cot A) = sin A + cos A 4. Without using trigonometric tables, evaluate the following :

0 02 0 2 0

0 0

sec17 tan 68cos 44 cos 46

cos 73 cot 22ec

5. (sinA + secA)2 + (cosA + cosecA)2=(1 + secA cosecA)2

6. (1 + 1/tan2A)(1 + 1/cot2A)=1/(sin2A – cos2A)

7. cosecA/(cosecA – 1) + cosecA/(cosecA + 1)=2 + 2tan2A.

8. tanA/(1 – cotA ) + cotA/(1 – tanA)=1 + secA cosecA = 1 + tanA + cotA.

9. (sin225º + sin265º) + √3(tan5ºdtan15ºtan60ºtan75ºtan85º).

10. (cos220º + cos270º)/(sec250º – cot240º) + 2cose258º – 2cot58ºtan32º – 4tan13ºtan37ºtan45ºtan53ºtan77º.

11. Prove that sin ɵ +cos ɵ

sin ɵ − cos ɵ +

sin ɵ − cos ɵ

sin ɵ + cos ɵ =

2

1−2cos2 ɵ

HEIGHT AND DISTANCES 1. Rita was observing a stationary balloon in the morning at an angle of elevation 45°.The shadow of

balloon which was on the western sky fell on the ground at a distance of 100√3 meters from Rita. If the altitude of the sum be 60°, find the height of the balloon from the ground and also the distance of the balloon from Rita.

2. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 15 sec, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of 1500√3 meters,find the speed of the jet plane.

3. A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60º. When he

moves 50 m, away from the bank, he finds the angle of elevation to be 30º. Calculate :

i. The width of the river and ii. The height of the tree.

4. Two poles of equal heights are standing opposite to each other on either side of a road, which is 100 m wide. From a point between them on the road, the angle of elevation of their tops are 30º and 60º. Find the position of the point and also the height of the poles. [Use √3 = 1.732]

5. A man on the top of a vertical tower observes a car moving at uniform speed towards the tower. If it takes 12 minutes for the angle of depression to change from 30º to 45º, how soon, after this, will the car reach the tower ?

6. The angle of elevation of the top of a hill at the foot of the tower is 60º and the angle of elevation of the top of the tower from the foot of the hill is 30º. If the tower is 50 m high, find the height of the hill.

7. From a window 15 m high above the ground in a street, the angle of elevation and depression of the top and foot of another house on the opposite side of the street are 30º and 45º respectively. Show that the height of the opposite house is 23.66 m.(Take √3 = 1.732).

8. The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is 30º. On advancing 150 m towards the foot of the tower, the angle of elevation becomes 60º. Show that the height of the tower is 129.9m.

(Use √3 = 1.732).

9. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 5 m. From a point on the plane the angles of elevation of the bottom and top of the flagstaff are respectively 30º and 60º. Find the height of the tower.

10. As observed from the top of a light-house, 100 m high above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30º to 60º. Determine the distance traveled by the ship during the period of observation.

(Use √3 = 1.732).

STATISTICS 1. The marks obtained by a set of students in an examination are given below:

Marks 5 10 15 20 25 30

No. of students 6 4 6 12 x 4

2. Use graph paper for this question.

The table given below shows the monthly wages of some factory workers. (i) Using the table, calculate the cumulative frequencies of workers. (ii) Draw the cumulative frequency curve. Draw the cumulative frequency curve. Use 2 cm=Rs 500, starting the origin at Rs 6500 on x-axis, and 2 cm=10 workers on y-axis. (iii) Use your graph to write down the median wage in Rs. Wages 6500-7000 7000-7500 7500-8000 8000-8500 8500-9000 9000-9500 9500-10000 Frequency 10 18 22 25 17 10 8

3. The table below shows the distribution of the scores obtained by 120 shooters in a shooting competition. Using a graph

sheet, draw an ogive for the distribution.

Scores obtained Number of shooters

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90

90-100

5 9 16 22 26 18 11 6 4 3

Use your ogive to estimate : 1. The median 2. The inner quartile range 3. the number of shooters who obtained more than 75% scores.

4. The daily wages of 160 workers in a building project are given below :

Wages (in Rs.) 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80

No. of workers 12 20 30 38 24 16 12 8

Using a graph paper, draw an Ogive for the above distribution. Use your Ogive to estimate : (i) The median wage of the workers. (ii) The upper quartile wage of the workers. (iii) The lower quartile wage of the workers. (iv) The percentage of workers who earn more than Rs. 45 a day.

5. Using a graph paper, draw an Ogive for the following distribution which shows a record of the weight in kilograms of 200 students.

Wages (in Rs.) 40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80

No. of workers 5 17 22 45 51 31 20 9

Use your Ogive to estimate the following : (i) The percentage of students weighing 55 kg or more. (ii) The weight above which the heaviest 30% of the students fall. (iii) The number of students who are : (1) under-weight and (2) over-weight, if 55.70 kg is considered as standard weight. 6. For the following frequency distribution draw a histogram. Hence calculate the mode.

Class 0-5 5-10 10-15 15-20 20-25 25-30

Frequency 2 7 18 10 8 5

7. The following table shows the distribution of the heights of a group of a factory workers :

Height (cm) 150-155 155-160 160-165 165-170 170-175 175-180 180-185

No. of workers 6 12 18 20 13 8 6

i. Determine the cumulative frequencies. ii. Draw the cumulative frequency curve on a graph paper. iii. From your graph, write down the median height in cm.

8. Find the mean of the following distribution :

Class Interval 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80

Frequency 10 6 8 12 5 9

9. The marks obtained by a set of students in an examination are given below :

Marks 5 10 15 20 25 30

No. of students 6 4 6 12 x 4

Given that the mean mark of the set is 18, calculate the numerical value of x. 10. The mean of the following frequency distribution is 57.6 and the sum of the observation is 50. Find the missing frequency f1 and f2.

Class 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120

Frequency 7 f1 12 f2 8 5

PROBABILITY 1. An integer is chosen at random form 1 to 100. Find the probability that the number is :

(i) is divisible by 5 (ii) is prime number (iii) is perfect cube 2. A dice is thrown once. What is the probability that the

(i) number is even and (ii) number is greater than 2 3. Two different dice are thrown simultaneously. What is the probability that the sum of two numbers

appearing on the top of disc is (i)9 (ii)atleast10 (iii)13 4. A box contains 17 cards numbered 1,2,3…………..17 and are mixed thoroughly. A card is drawn

atrandom from the box. Find the probability that the number of card is (i) Prime , (ii)divisible by 2 and 3 both , (iii) divisible by 2 or 3.

5. Ankita and Nagma are friends.They were both born in 1990.What is the probability that theyhave (i) same birthday? (ii) different birthdays?

6. A bag contains 1 balls out of which x are black. (i) If a ball at random, what is the probability that it will be a black ball? (ii) If 6 more balls are put in the bag, the probability of drawing a black ball will be double than that of (i).

Find the value of x.