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ICSE Prelims 2017 : Mathematics Must Know Questions (Chatrabhuj Narsee Memorial School (CNM), Mumbai)

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Shri Vile Parle Kelavani Mandal s C.N.M. School & N.D. Parekh Pre-Primary School LAST REVISION WORKSHEET Class : X (MUST KNOW QUESTIONS) 2016-17 MATHEMATICS 1. COMPOUND INTEREST (1) Find the amount and the compound interest on Rs 10,000 at 8 per cent per annum and in 2 years. (Ans: - A = Rs 11,664; I = Rs 1664). (2) Find the amount and the compound interest on Rs 10,000 at 8 per cent per annum in 1 year; Interest being compounded half-yearly. (Ans: - A = 10,816; I = Rs 816). (3) A man borrows Rs 8,000 at 10% compound interest if he repays Rs 1, 500 at the end of the first and Rs 3,200 at the end of second year, find the amount of loan outstanding at the beginning of the third year (Ans:- Rs 4,830). (4) Calculate the compound interest accrued on Rs 16,000 in 3 years, when the rates of interest for successive years are 10% , 12% and 15% respectively. (Ans:- C.I. = Rs 6,668 80). (5) A man borrows Rs 8,000 at 10% compound interest payable every six months . He repays Rs 2,500 at the end of every six months . Calculate the third payment he has to make at the end of 18 months in order to clear the entire loan. (Ans:- Rs 3,879 75). (6) On a certain sum of money , invested at the rate of 5% per annum compounded annually , the difference between the interest of the first year and the interest of the third year is Rs 61 50 . Find the sum. (Ans:- Sum = Rs 12,000). (7) Mrs. Kapoor invested Rs 6,000 every year at the beginning of the year , at 10% per annum compound interest . Calculate the amount of her total savings : (i) upto the end of the second year . (ii) at the beginning of the third year. (Ans:- Amount of 2nd year = Rs 13,860 ; Amount of 3rd year = Rs 19,860). (8) During every financial year , the value of a machine depreciates by 10% . Find the original value (cost) of a machine which depreciates by Rs 2,250 during the second year. (Ans:- Original cost = Rs 25,000). (9) Calculate the difference between the compound interest and the simple interest on Rs 4,000 at 8 per cent per annum and in 2 years . (Ans:- C.I. = Rs 25 60). (10) Ashok borrowed Rs 16,000 at 10% simple interest . He immediately invested this money at 10% compounded interest compounded half-yearly . Calculate Ashok s gain in 18 months. (Ans:- Ashok s gain in 18 months = Rs 122). (11) A sum of money is invested at C.I. payable annually . The amounts of interest in two Successive years are Rs 225 and Rs 240. Fin the rate of interest. (Ans:- rate of interest = 6 ). (12) A person invests Rs 10,000for three years at a certain rate of interest compounded annually. At the end of one year this sum amounts to Rs 11,200. Calculate : (i) the rate of interest per annum . (ii) the amount at the end of the second year. (iii) the amount at the end of the third year. (Ans:- rate of interest = 12% ; amount = Rs 12,544 ; third year amount = Rs 14,049 28). (13) A certain sum of money , placed out at compound interest , amounts to Rs 6,272 in 2 years and to Rs 7,024 64 in 3 years . Find the rate of interest and the sum of money . (Ans:- rate of interest = 12% ; sum = Rs 5,000). (14) A sum of Rs 9,600 is invested for 3 years at 10% per annum of compound interest . (i) What is the sum due at the end of the first year ? (ii) What is the sum due at the end of the second year . (iiii) Find the difference between the answer in (ii) and (i) and find the interest on this sum (difference ) for one year. (iv) Hence , write down the compound interest for the third year. (Ans:- sum of first year = Rs 10,560 ; sum of second year = Rs 11,616 ; difference = Rs 1,056 ; one year interest = Rs 105 60 ; C.I. Rs 1,161 60). (15) 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 . (Ans:- sum of money = Rs 4,250 ; C.I. = Rs 171 70). (16) Calculate the amount and the compound interest on Rs 7,500 in 2 years and at 6% compounded annually . (Ans:- A = Rs 8,427 ; C.I. = Rs 927). (17) Calculate the amount and the compound interest on Rs 12,000 in 3 years when the rate of interests for successive years are 8% , 10% and 15% respectively. (Ans:- A= 16,394 40 ; C.I. = Rs 4,394 40). (18) Calculate the compound interest on Rs 18,000 in 2 years at 15% per annum . (Ans:- C.I. = Rs 5,805). (19) What sum of money will amount to Rs 9,261 in 3 years at 5% per annum compound interest ? (Ans:- Sum = Rs 8,000). (20) On a certain sum , the compound interest in 3 years and at 10 per cent per annum amounts to Rs 2,317 . Find the sum. (Ans:- Sum = Rs 7,000). (21) On a certain sum , the compound interest in two years amounts to Rs 2,256 . If the rates of interest for successive years are 8% and 10% respectively , find the sum. (Ans:- Sum = Rs 12,000). (22) Ramesh lends Rs 15,000 for 2 years at a certain rate of compound interest . If after 2 years , it amounts to Rs 16,224 ; find the rate of interest . (Ans:- Rate = 4%). (23) In what period of time will Rs 12,000 yield Rs 3,972 as compound interest at 10 per cent , if compounded on an yearly bases. (Ans:- time = 3 years). (24) Divide Rs 36,465 between A and B so that when their shares are lent out at 10 per cent compound interest per year , the amount that A receives in 7 years is the same as what B receives in 5 years . (Ans:- A s shares = Rs 16,500 ; B s share = Rs 19,965). (25) Find the amount when Rs 10,000 is invested for 2 years at 10% interest compounded yearly. (Ans:- A= Rs 12,705). (26) John borrowed Rs 20,000 for 4 years under the following conditions : 10% simple interest for the first 2 years . 10% C.I. for the remaining one and a half years on the amount due after 2 years , the interest being compounded half-yearly. Find the total amount to be paid at the end of the four years. (Ans:- Amount paid by the John at the end of 4 years = Rs 28,940 63). (27) A sum of Rs 6,400 earns a compound interest of Rs 1,008 80 in 18 months , when the interest is reckoned half-yearly . Find the rate of interest. (Ans:- Rate = 10%) (28) 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. (Ans:- C.I. = Rs 171 70). (29) The cost of a machine depreciates by 10% every year . if its present worth is Rs 18,000 ; what will be its value after years ? (Ans:- Value after 3 years = Rs 13,122). (30) The population of a town in China increases by 20% every year . if its present population is 2,16,000 , find : (i) its population after 2 years , (ii) its population 2 years ago. (Ans:- population after 2 years = 3,11,040 ; population 2 years ago = 1,50,000). (31) A certain sum of money , lent out at compound interest , amounts to Rs 14,520 in 2 years and to Rs 17,569 20 in 4 years . Fin the rate of interest per annum and the sum. (Ans:- Rate of interest = 10% ; Sum = Rs 12,000). (32) The difference between the compound interest and the simple interest on Rs 9,500 for 2 years is Rs 95 at the same rate of interest per year . Find the rate of interest. (Ans:- Rate of interest = 10%). (33) A sum of money lent out at C.I. at a certain rate per annum doubles itself in 5 years . Find in how many years will the money become eight times of itself at the same rate of interest. (Ans:- time required = 15 years). (34) A loan , taken at 5% C.I. per annum , was repaid in 2 years by paying Rs 1,764 at the end of each year . How much loan was taken ? (Ans:- The sum borrowed = Rs 3,280). (35) A sum of Rs 5,040 is borrowed at 10% C.I. compounded yearly and is paid back in two years by making two equal and annual payments . Calculate the amount of each yearly payment made at the end of each year. (Ans:- x = Rs 2,904). 2. SALES TAX AND VALUE ADDED TAX (VAT) (1) Mr. Gupta purchased an article for Rs 702 including Sales tax . If the rate of Sales Tax is 8%, find the sale price of the article. (Ans:- sale price of the article = Rs 650). (2) Geeta purchased face-cream for Rs 79 10 including Sales Tax . if the printed price of the face-cream is Rs 70 ; find the rate of Sales Tax. (Ans:- The rate of sales tax = 13%). (3) Mrs. Sharma purchased confectionery goods costing Rs 165 on which the rate of Sales Tax is 6% and some tooth-paste , shaving-cream , soap , etc., costing Rs 230 on which the rate of Sales Tax is 10% . if she gives a five-hundred rupee note to the shopkeeper , what money will he return to Mrs. Sharma ? (Ans:- The shopkeeper will return = Rs 72 10). (4) Smith buys a radio-set for Rs 1,696 . The rate of Sales Tax is 6% . He asks the shopkeeper to reduce the price of the radio-set to such an extent that he does not have to pay anything more than Rs 1,696 including Sales Tax . Calculate the reduction as percent , needed in the cost price of the radio-set. (Ans:- reduction as percent of cost price = 6%). (5) The price of an article inclusive of Sales tax of 12% is Rs 2,016. Find its marked price. If the Sales Tax is reduced to 7% , how much less does the customer pay for the article ? (Ans:- marked price = Rs 1,800 ; customer will pay for the article = Rs 90 less). (6) A trader from Meerut buys an article for Rs 3,600 (inclusive of all taxes) from Kanpur . he spends Rs 1,200 on travelling , transportation do the article , etc . If he desires a profit of 15 per cent , how much will a customer pay for the article ? The rate of Sales Tax paid by the customer is 8%. (Ans:- money paid by customer = Rs 5,961 60). (7) A shopkeeper buys an article for Rs 1,500 and spends 20% of the cost on its packing , transportation , etc. Then he marks the article at a certain price. If he sells the article for Rs 2,452 50 including 9% Sales Tax on the price marked , find his profit as per cent . (Ans:- Profit% = 25%). (8) The catalogue price of a computer set is Rs 45,000 . The shopkeeper gives a discount of 7% on the listed price . he gives a further off-season discount of 4% on the balance . However , Sales Tax at 8% is charged on the remaining amount . Find: (i) the amount of Sales Tax a customer has to pay, (ii) the final price he has to pay for the computer set. (Ans:- The amount of Sales Tax = Rs 3,211 68 ; the final price = Rs 43,387 68). (9) Dinesh bought an article for Rs 374 , which included a discount of 15% on the marked price and a sales-tax of 10% on the reduced price . Find the marked price of the article . (Ans:- marked price = Rs 400). (10) A shopkeeper buys an article at a rebate of 20% on the printed price . He spends Rs 40 on transportation of the article . After charging a sales tax of 7% on the printed price , he sells the article for Rs 1,070. Find his gain as per cent. (Ans:- Profit % = 19 %). (11) A shopkeeper buys an article at a discount of 30% from the wholesaler (the printed price of the article being Rs 2,000) and paid sales-tax at the rate of 8% . The shopkeeper sells the article to a buyer at the printed price and charges tax at the same rate . Find the VAT (Value added tax) paid by the shopkeeper . (Ans:- VAT paid by the shopkeeper = Rs 48). (12) A shopkeeper sells an article at its marked price (Rs 7,500) and charges sales-tax at the rate of 12% from the customer . If the shopkeeper pays a VAT of Rs 180 ; calculate the price (inclusive of tax) paid by the shopkeeper . (Ans:- the price paid = Rs 6,720). (13) 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 this machine from the manufacturer and the wholesaler. (ii) the amount that the consumer pays for the machine. (Ans:- VAT received = Rs 1,200 ; VAT received from wholesaler = Rs 96 ; A = Rs 19,440. (14) During a financial year , a shopkeeper purchased goods worth Rs 4,15,000 and paid a total tax of Rs 38,000 . His sales during this period consisted of a taxable turnover of Rs 50,000 for goods taxable at 5% and Rs 3,20,000 for goods taxable at 12% . He also sold tax exempted goods worth Rs 45,000 during this period . Calculate his tax liability (under VAT) for financial year. (Ans:- tax liability (under VAT) = Rs 2,900). (15) An article was bought by a distributor for Rs 15,000 (excluding tax) . He sold it to a trader for Rs 20,000 . The trader sold the article to a retailer for Rs 22,000(excluding tax). Find the VAT paid by the distributor and by the trader if the tax rate was 10 per cent . (Ans:- VAT paid by distributor = Rs 500). (16) A manufacturing company sold a commodity to its distributor for Rs 22,000 including VAT. The distributor sold the commodity to a retailer for Rs 25,000 plus tax (under VAT). If the rate of tax at each stage is 10% , what was the : (i) sale price of the commodity for the manufacturer ? (ii) the amount of VAT paid by the retailer ? (Ans:- the sale price = Rs 20,000 ; Amount of VAT = Rs 300). 3. BANKING (1) Mr. Sharma has a savings bank account with Bank of Baroda . A part of the page of his pass- book is shown below : Date Particulars Amount Amount Balance (In Rs) Withdrawal Deposited (In Rs) (In Rs ) July 1 , 98 B/F --1,500 00 July 8 , 98 By Cheque -1,200 00 2,700 00 July 23 , 98 By Cash -800 00 3,500 00 Aug. , 17 , 98 To Cheque 1,600 00 -1,900 00 Aug . , 27 , 98 By Cash -600 00 2,500 00 Find the amounts on which he will get interest for the months of July , 98 and Aug ., 98. (Ans:- interest for month of July = Rs 2,700 00 ; interest for the month of Aug = Rs 1,900 00). (2) Mr. Dhoni has an account in the Union Bank of India . The following entries are from his passbook: Date Particulars Withdrawal Deposits (In Rs) Balance (In Rs) (In Rs) Jan 3 ,07 B/F --2642 00 Jan 16 To Self 640 00 -2002 00 March 5 By Cash -850 00 2852 00 April 10 To Self 1130 00 -1722 00 April 25 By Cheque -650 00 2372 00 June 15 By Cash 577 00 -1795 00 Calculate the interest from January 2007 to June 2007 at the rate of 4% per annum. (Ans:- Interest = Rs 42 48). (3) Divya opened a savings bank account in a bank on 16th October . Her passbook has the following entries : Date Particulars Amount Amount Balance (In Rs) Withdrawal Deposited (In Rs) (In Rs) Oct ., 18 By Cash -700 00 700 00 Oct ., 25 By Cheque -800 00 1,500 00 Nov ., 5 To Cheque 300 00 -1,200 00 Nov ., 10 By Cash -1,300 00 2,5000 00 Nov ., 18 To Cash 900 00 -1,600 00 Dec ., 3 To Cash 400 00 --1,200 00 Dec ., 21 By Cheque -1,500 00 2,700 00 Jan ., 5 By Cash -300 00 3,000 00 Divya closes the account on 18th January . Calculate the interest earned by her at 5% per annum. (Ans:- Interest = Rs 11 67). (4) Given below are the entries in a Saving Bank A/c passbook:Date Particulars Withdrawal Deposit Balance Feb ., 8 B/F --Rs 8,500 Feb ., 18 To Self Rs 4,000 -Rs 4500 April , 12 By Cash -Rs 2,238 Rs 6738 June , 15 To Self Rs 5,000 -Rs 1738 July , 8 By Cash -Rs 6,000 Rs 7738 Calculate the interest for the six months , February to July , at 4 % p.a. on the minimum balance on or after the 10th day of each month .(Ans:- Rs 111 45). (5) Mr. Shiv Kumar has a Savings Bank Account in the Punjab National Bank. His passbook has the following entries : Date Particulars Withdrawal Deposits (In Rs) Balance (In Rs) (In Rs) April 1, 1997 B/F --3220 00 April 15 By Transfer -2010 00 5230 00 May 8 To Cheque No. 355 298 00 -4932 00 July 15 By Clearing -4628 00 9560 00 July 29 By cash -5440 00 15000 00 Sept 10 To Self 6980 00 -8020 00 Jan 10 , 1998 By Cash -8000 00 16020 00 Calculate the interest due to him at the end of the financial year (March 31st 1998) at the rate of 6% per annum. (Ans:- Interest = 565 78). (6) Given the following details , calculate simple interest at the rate of 6% per annum upto June 30. Date Debit Rs Credit Rs Balance Jan. 1 Jan. 20 Jan. 29 March 15 April 3 May 6 May 8 (Ans:- 1,075 27). -5,000 00 ---3,040 00 -- 24,000 00 -10,000 00 8,000 00 7,653 00 -5,087 00 24,000 00 19,000 00 29,000 00 37,000 00 44,653 00 41,613 00 46,700 00 (7) Mr. Ashok has an account in the Central Bank of India . The following entries are from his passbook:Date Particular Withdrawal Deposits (In Rs) Balance (In Rs) (In Rs) 01.01.05 B/F --1,200 00 07.01.05 By Cash -500 00 1,700 00 17.01.05 To Cheque 400 00 -1,300 00 10.02.05 By Cash -800 00 2,100 00 25.02.05 To Cheque 500 00 -1,600 00 20.09.05 By Cash -700 00 2,300 00 21.11.05 To Cheque 600 00 -1,700 00 05.12.05 By Cash -300 00 2,000 00 If Mr. Ashok gets Rs 83 75 as interest at the end of the year , where the interest is compounded annually , calculate the rate of interest paid by the bank in his Savings Bank Account on 31st December , 2005. (Ans:- principal for 1 month = Rs 200,100). (8) Rajesh deposits Rs 300 per month in a Recurring Deposit Account for 2 years . If the rate of interest is 10% per year ; calculate the amount that Rajesh will receive at the end of 2 years i.e. at the time of maturity. (Ans:- amount that Rajesh will get = Rs 7,950). (9) Mr. R.K. Nair gets Rs 6,455 at the end of one year at the rate of 14% per annum in a Recurring Deposit Account . Find the monthly installment . (Ans:- Installment = Rs 500). (10) Ahmed has a recurring deposit account in a bank . He deposits Rs 2,500 per month for 2 years . If he gets Rs 66,250 at the time of maturity , find: (i) the interest paid by the bank (ii) the rate of interest . (11) Monica has a C.D. Account in the Union Bank of India and deposited Rs 600 per month . If the maturity value of this account is Rs 24,930 and the rate of interest is 10% per annum ; find the time (in years) for which the account was held. (Ans:- the time = 3 years). 4. SHARES AND DIVIDENDS (1) Ravi invested Rs 6,250 in shares of a company paying 6% dividend per annum . If he bought Rs 25 shares for Rs 31 25 ach , find his income from the investment. (Ans:- income = Rs 300). (2) Manoj buys Rs 100 shares at Rs 20 premium in a company paying 15% dividend . Find : (i) the market value of 200shares ; (ii) his annual income ; (iii) his percentage income. (Ans:- market value = Rs 24,000 ; annual income = Rs 3,000 ; percentage income = 12 5%). (3) Find the dividend due at the end of a year on 250 shares of Rs 50 each , if the half-yearly dividend is 4% of the value of the share. (Ans:- dividend at the end of the year = Rs 1,000). (4) A man bought 500 shares , each of face value Rs 10 , of a certain business concern and during the first year , after purchase , receives Rs 400 as dividend on his shares . Find the rate of dividend on the shares. (Ans:- Rate of dividend = 8%). (5) Mukul invests Rs 9,000 in a company paying a dividend of 6% per annum when a share of face value Rs 100 stands at Rs 150 . What is his annual income ? If he sells 50% of his shares when the price rises to Rs 200 , What is his gain in this transaction ? (Ans:- annual income = Rs 360 ; Mukluk gain in this transaction = Rs 1,500). (6) A man wants to buy 62 shares available at Rs 132 (par value being Rs 100). (i) How much he will have to invest ? (ii) If the dividend is 7 5% , what will be his annual income ? (iii) If he wants to increase his annual income by Rs 150 , how many extra shares should he buy ? (Ans:- he have to invest = Rs 8,184 ; annual income = Rs 465 ; extra share = 20). (7) A company with 4000 shares of nominal value of Rs 110 each declares an annual dividend of 15% . Calculate : (i) The total amount of dividend paid by the company . (ii) The annual income of Salman who holds 88 shares in the company . (iii) If he received only 10% on his investment , find the price of Salman paid for each share. (Ans:- the amount of dividend = Rs 66,000 ; annual income = 1,452 ; paid share = Rs 165). (8) A man buys a Rs 40 shares in a company , which he pays 10% dividend . He buys the share at such a price that his profit is 16% on his investment . At what price did he buy the share ? (Ans:- M.V. = Rs 25). (9) 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 . (Ans:- dividend that Ajay will get = Rs 1,260 ; rate of interest = 7 5%). (10) A dividend of 9% was declared on Rs 100 share selling at a certain price . If the rate of return is 7 5% , calculate : (i) the market value of the share ; (ii) the amount to be invested to obtain an annual dividend of Rs 630. (Ans:- M.V. of a share = Rs 120 ; the amount to be invested = Rs 8,400). (11) Which is better investment : 12% at 120 or 8% at 90 ? (Ans:- The first investment is better). (12) A man sells 60 , Rs 15 shares of a company paying 12 per cent dividend , at Rs 21 each and invests the proceeds in Rs 6 shares of another company at Rs 9 each . Find the change in income, if the second company pays a dividend of 8 per cent. (Ans:- change in income = Rs 40 80 less). (13) Mr. Ram Gopal invested Rs 8,000 in 7% Rs 100 shares at Rs 80 . After a year he sold these shares at Rs 75 each and invested the proceeds (including his dividend) in 18% , Rs 25 shares at Rs 41 . Find: (i) his dividend for the first year (ii) his annual income in the second year (iii) the percentage increase in his return on his original investment . (14) Ashok and Sandeep invests Rs 18,000 each in buying shares of two different companies . Ashok buys 7 5% Rs 100 shares at a discount of 20% , whereas Sandeep buys Rs 0 shares at a premium of 20% . If both receive equal dividend at the end of the year , find the rate of dividend receive by Sandeep. (Ans:- Rate of dividend received by Sandeep = 11 25%). (15) John had 1,000 shares of a company with a face value on Rs 40 and paying 8% dividend . He sold some of these shares at a discount of 10% and invested the proceeds in Rs 20 shares at a premium of 50% and paying 12% dividend . If the change in his income is Rs 192 , find the number of shares sold by John. (Ans:- No. of shares sold by John = Rs 600). (16) Divide Rs 40,608 into two parts such that if one part is invested in 8% Rs 100 shares at 8% discount and the other part is invested in 9% R s100 shares at 8% premium , the annual incomes , from both the investment , are equal . (Ans:- the two parts are Rs 19.872 & Rs 20,736). (17) A man has choice to invest in hundred-rupee shares of two companies A and B . Shares of company A are available at a premium of 20% and it pays 8% dividend whereas shares of company B are available at a discount of 10% and it pays 7% dividend , If the man invests equally in both the companies and the sum of the return from them is Rs 936 , find how much , in all , does he invest ? (Ans:- The man invests in all = Rs 12,960). 5. LINEAR INEQUATIONS (1) If the replacement set is the set of natural numbers (N) , find the solution set of : (i) 3 x+4 < 16 (ii) 8 x 4 x 2. (Ans:- (i) solution set = {1, 2, 3} ; (ii) solution set {2, 3, 4, 5, 6, .}). (2) If the replacement set is the set of whole number (W) , find the solution set of : (i) 5 x+4 24 (ii) 4 x 2 < 2 x +10 . (Ans:- (i) solution set = {0, 1, 2, 3, 4} ; (ii) solution set = {0, 1, 2, 3, 4, 5}). (3) If the replacement set is the set of integers , (I or Z) , between -6 and 8 , find the solution set of : (i) 6 x 1 9+x (ii) 15 3 x > x 3 . (Ans:- (i) solution set = {2, 3, 4, 5, 6, 7} ; (ii) solution set = ( 5, 4, 3, 2, 1, 0, 1, 2, 3, 4}). (4) If the replacement set is the set of real numbers (R) , find the solution set of : (i) 5 x 3 x < 11 (ii) 8+3 x 28 2 x . x > 2 and x R} ; (Ans:- (i) Solution set = { x : (ii) Solution set = { x : set = {1, 3, 5}). (5) Solve : 5 x 4 and x R}). 4 , where x is a positive odd integer. (Ans:- Solution (6) Solve the following inequation : 2 y 3 < y+1 4 y+7 ; if : (i) y {Integers} (ii) y R (real numbers). (Ans:- (i) When y {Integers} solution set = { 2, 1, 0, 1, 2, 3}; 2 y (ii) When y R (real numbers) solution set = { y: (7) Simplify : < 4 and y R}). 1 < : x R. Graph the value of x on the real number line. (Ans:- solution = 2 x < 3). (8) List the solution set of 50 3 ( 2 x 5 ) < 25 , given that x W. Also represent the solution set obtained on a number line. (Ans:- required solution set = {7, 8, 9, }. (9) Solve and graph the solution set of 2 < 2x 6 or 2x (Ans:- Graph of solution set of x > 2 or x 4). + 5 13; where x R. 5 < 2x 1 11, x R} (10) Given : P = { x : Q = { x : 1 3 + 4 x < 23, x I} Where R = { real numbers} and I = {integers}. Represent P and Q on two different number lines . Write down the elements P (Ans:- P = 3 < x 6; x R ; Q = 1 x < 5 ; where x I). Q. (11) Find three smallest consecutive whole numbers such that the difference between one-fourth of the largest and one-fifth of the smallest is atleast 3. (Ans:- Required smallest consecutive whole numbers are : 50, 51 and 52). 6. QUADRATIC EQUATIONS (1) Solve: + = 2 . (Ans:- x=2 or x= 1 ). (2) Find the quadratic equation whose solution set is { 2, (3) Use the substitution x = 3 y+1 to solve for (Ans:- y= 1 , or y = ). (4) Without solving the quadratic equation 3 x solution (root) of this equation or not. (Ans:- 2 3}. (Ans:- x y , if 5( 2 x 6 = 0). 3 y+1 )2 + 6( 3 y+1 ) 8 = 0. 2x 1 = 0 , find whether x=1 is a solution of the given 3 x x=1 is a 2 2x 1 = 0). (5) Without solving equation x 2 x + 1 = 0 ; find whether x = 1 is a root of this equation or not . (Ans:- x = 1 is not a root of the given equation x 2 x + 1 = 0). (6) Find the value of k 0. (Ans:- 4 k for which = 1 and k = x=2 is root (solution) of equation kx 2 + 2x 3 = ). (7) If x=2 and x=3 are roots of the equation 3 x values of m and n . (Ans:- m = 7 5 and n = 9). (8) If one root of the quadratic equation 2x Also , find the other root . (Ans:- the other root = 2 mx + 2n = 0 ; find the 6 = 0 is 2 , find the value of a . + ax 2 2 ). (9) Solve each of the following equations by using the formula: (i) 6 5x 2 x 2 3=0 (ii) x 2 = 18 x 77 (iii) 3 and 3 3 ). x 2 + 11 x + 3 = 0. (Ans:- (i) 1 and ; (ii) 11 and 7 ; (iii) (10) Without solving , examine the mature of the roots of the equations: (i) 5 x 2 6x + 7 = 0 (ii) x 2 + 6x +9=0 5x + 2 = 0 . (iii) 2x + 6 x + 3 = 0 (iv) 3 x (Ans:- (i) the roots are imaginary. ; (ii) The roots are rational (real) and equal (iii) The roots are irrational and unequal ; (iv) The roots are rational and unequal). 2 2 (11) Find the value of p , if the roots of the following quadratic equation are equal ( p 3) x 2 + 6 x + 9 = 0. (Ans:- p = 4). (12) Find the value of m , if the roots of the following quadratic equation are equal (4 + m ) x 2 + ( m + 1) x + 1 = 0 . (Ans:- m = 5 or m = 3 ). (13) Solve each of the following equations for x and give , in each case , your answer correct to 2 decimal places . 10 x + 6 = 0 . (i) x 2 (ii) 3 x 2 + 5 x (Ans:- (i) 9 36 and 0 64 ; (ii) 1 09 and 2 76 . 9=0. (14) Solve the following equation : x = 6 . Give your answer correct to two significant figures. (Ans:- 8 2 or 2 2). (15) Solve : (i) 2x x R. 4 5x 2 +3=0 (ii) ( x (Ans:- Required solution = 1, 1 , + (16) Solve : + 2 = (Ans:- Required solution is : , 2 , + 3 x )2 , x 0 and x ( x 2 + 3x ) 6=0, . 1. ). (17) Find the solution set of the equation 3 x 2 8 x 3 = 0 ; when : (i) x Z (integers) (ii) x Q (rational numbers). (Ans:- When x Z , the solution set = {3} ; When x Q , the solution set = {3, }). (18) Solve : ( 2x 3)2 = 25 . (Ans:- x = 4 or x = 1 ). (19) Solve for x : 4( x , 2, (20) Solve : 2 x + ) + 8( x 0 . (Ans:- Solution = ) = 29 . ). (Ans:- x = + a+b , where a+b = x = ). or 0 , ab 0 . 7. SOLVING PROBLEMS (BASED ON QUADRATIC EQUATIONS) (1) Find two natural numbers which differ by 3 and the sum of whose squares is 117. (Ans:- Number are 6 and 9). (2) Five times a certain whole number is equal to three less than twice the square of the number . Find the number. (Ans:- Required whole number is 3). (3) Divide 8 into two parts such that the sum of their reciprocals is . (Ans:- Required parts are 3 and 5). (4) For the same amount of work , A takes 6 hours less than B . If together they complete the work in 13 hours 20 minutes ; find how much time will B alone take to complete the work. (Ans:- B alone will take 30 hrs. to complete the work). (5) The hypotenuse of a right triangle is 13 cm and the difference between the other two sides is 7 cm . Taking x as the length of the shorter of the two sides , write an equation in x that represents the above statement and also solve the equation to find the two unknown sides of the triangle . (Ans:- One side of the triangle = 5 cm ; other side of the triangle = 12 cm). (6) The length of a verandah is 3 m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter. (i) Talking x as the breadth of the verandah , write an equation in x that represents the above statement . (ii) Solve the equation obtained in (i) above and hence find the dimensions of the verandah. x (Ans:- (i) x 2 6 = 0 ; (ii) the length of verandah = 6 m ; its breadth = 3 m). (7) By increasing the speed of a car by 10 km/hr , the time of journey for a distance of 72 km is reduced by 36 minutes . Find the original speed of the car . (Ans:- The original speed of the car = 30 km/hr . (8) Car A travels x km for every litre of petrol , while car B travels ( x + 5) km for every litre of petrol. (i) Write down the number of litres of petrol used by car A nd car B in covering a distance of 400 km. (ii) If car A uses 4 litres of petrol more than car B in covering the 400 km , write down an equation in x and solve it to determine the number of litres of petrol used by car B for the journey. (Ans:- No. of litres of petrol used by car A = litre ; No. of litres of petrol used by car B = litre ; No. of litres of petrol used by car B = 16 litres) . (9) By selling an article for Rs 24 , a trader loses as much per cent as the cost price of the article . Calculate the cost price. (Ans:- C.P of the article is Rs 60 or Rs 40). (10) The sum S of first n natural numbers is given by the relations ; S = n ( n+1 ). Find n , if the sum is 276. (Ans:- n = 23). (11) A two-digit number is such that the product of its digits is 12 . When 36 is added to this number ; the digits interchange their places . Find the number. (Ans:- The required two digit number = 26). (12) Five years ago , a woman s age was the square of her son s age . Ten years hence her age will be twice that of her son s age . Find: (i) the age of the son five year ago. (ii) the present age of the woman. (Ans:- (i) The age of the son 5 years ago = 5 years ; (ii) The present age of the woman = 30 years). (13) A motor-boat , whose speed is 9 km/h in still water , goes 12 km downstream and comes back in a total time of 3 hours . Find the speed of the stream. (Ans:- The speed of the stream = 3 km/hr). (14) A piece of cloth costs Rs 200 . If the piece was 5 m longer and each meter of cloth costs Rs 2 less ; the cost of the piece would have remained unchanged . How long is the piece and what is the original rate per meter ? (Ans:- length of the cloth = 20 m ; rate per m = Rs 10). (15) Some students planned a picnic . The budget for the food was Rs 480 . As eight of them failed to join the party , the cost of the food for each member increased by Rs 10 . Find how many students went for the picnic. (Ans:- No. of students who went for picnic = 16). 8. REMAINDER AND FACTOR THEOREMS (1) In each case. Find the remainder when x (Ans:- Remainder = 8 ). 2 (2) Find the value of k if ( x 2 ) is a factor of determine 8 x + 4 is divided by 2x + 1 x 3 + 2x 2 kx + 10 . Hence, whether (x + 5) is a also a factor. (Ans:- k = 13 ; ( x + 5) is a factor). (3) Find the value of a if the division of ax a remainder of 5. (Ans:- a = 2). (4) When the polynomial 2x 2, the remainder 3 kx 2 3 ax 2 + 9x + ( 5k is 14. Find the value of k . (Ans:- k (5) The polynomials 3 x the same remainder 3 2 + 4x 3) x 10 by x + 3 leaves 8 is divided by x = 2). + 5x 13 and ( a + 1) x 2 7 x + 5 leave when divided by x 3 . Find the value of a . (Ans:- a=5 ). (6) When f ( x ) = x zero and when 3 + ax 2 8 is divided by x 2 , the remainder is bx divided by x+1 , the remainder is 30 . Find the values of a and b . (Ans:- a = 7 and b = 14 ). (7) What number should be added to 2x polynomial is 3 x 3 2 + x so that when the resulting divided by x 2 , the remainder is 3 ? (Ans:- the required number to be added = 3 ). (8) Determine whether x 1 + 1 or not ? (Ans:- x 1 (9) If is a factor of x x 5 + x 4 + x 3 x 2 x is not a factor of the given polynomial). x 2 is a factor of x 2 7 x + 2a , find the value of a . (Ans:- a = 5). (10) Given that x + 2 and x 3 a 6 are factors of x 3 + ax + b ; calculate the values of and b . (Ans:- a and b = 6 ). = 7 (11) Polynomial x 3 x 1 and x 2 is ax 2 + bx 6 leaves remainder 8 when divided by a factor of it. Find the value of a and b . (Ans:- a = 2 and b = 5 ). (12) Using the Factor Theorem, show that ( x 2 ) is a factor of 3 x Hence, factorise the given expression . (Ans:- 3x 2 (13) Show that 2x +7 is a factor of 2x given 3 5x + 5x 2 2 5x 2. 2 = ( x 2 ) ( 3 x+1 ). 11 x 14 . Hence factorise the expression completely, using the factor theorem. (Ans:- ( 2 x +7 ) ( x 2 ) ( x+1 ). (14) Using the Remainder Theorem, factorise the expression 2x completely. 3 + x 3 + ax 2 2x 1 (Ans:- ( x 1 ) ( x+1 ) ( 2 x+1 ). (15) Find the value of a and b so that the polynomial x 2 + bx 45 has ( x 1 ) and ( x+5 ) as its factors. For the values of a and b , as obtained above , factorise the given polynomial completely. (Ans:- a = 13 and b = 31 ; ( x 1 ) ( x+9 ) ( x+5 ). px 2 x 2 (16) If ( x 2 ) is a factor of 2x 3 (i) find the value of p . (ii) with the value of p , factorise the above expression completely. (Ans:- P = 5 ; ( x 2 ) ( x+1 ) ( 2 x+1 ). 9. RATIO AND PROPORTION + 3 y : 3 x + 5 y = 18 : 29, find x : y . (ii) If x : y = 2 : 3, find the value of 3 x + 2 y : 2 x + 5 y . (Ans:- 3 : 4). (1) (i) If 2x (2) If a : b = 5 : 3 , find ( 5a + 8b ) : ( 6 a 7 b ). (Ans:- 49 : 9). (3) Two numbers are in the ratio 3 : 5 . If 8 is added to each number, the ratio becomes 2 : 3 . find the numbers. (Ans:- 24 and 40). (4) (i) What quantity must be added to each term of the ratio 8 : 15 so that it becomes equal to 3:5? (ii) What quantity must be subtrx - 3acted from each term of the ratio a :b becomes c : d ? (Ans:- (i) 2 ; (ii) ). so that it (5) The work done by ( x 3 ) men in ( 2 x+1 ) days and the work done by ( 2 x+1 ) men in ( x+4 ) days are in the ratio 3 : 10. Find the value of x . (Ans:- x = 6). (6) When the fare of a certain journey by an airliner was increased in the ratio 5 : 7 the cost of the ticket for the journey became Rs 1,421. Find the increase in the fare. (Ans:- The increase in the fare = Rs 406). (7) In a regiment, the ratio of number of officers to the number of soldiers was 3 : 31 before a battle. In the battle 6 officers and 22 soldiers were killed. The ratio between the number of officers and the number of soldiers now is 1 : 13. Find number of officers and soldiers in the regiment before the battle. (Ans:- The no. of officers = 21 ; no. of soldiers = 217). (8) If = = and a+b +c = 0 ; show that each given ratio is equal to 1 (Ans:- each of the given ratios is 1 ). (9) Find the compound ratio of : 3 a :2 b , 2m : n and 4 x :3 y (ii) a b : a+b , ( a+b )2 : a 2 + b (i) (Ans:- (i) required compound ratio = 4 amx 2 and a : bny 4 b 4 :( a 2 b 2 )2. ; (ii) required compound ratio = 1 : 1). (10) Find the ratio compounded of the duplicate ratio 5 : 6, the reciprocal ratio of 25 : 42 and the sub-triplicate ratio of 216 : 343. (Ans:- The required compounded ratio = 1 : 1 ). (11) Find : (i) the fourth proportional to 3, 6 and 4 5. (ii) the mean proportional between 6 25 and 0 16. (iii) the third proportional to 1 2 and 1 8. (Ans:- (i) x = 9; (ii) x = 1; (iii) x = 2 7). (12) Quantities a , 2, 10 and b are in continued proportion; find the values of b . a and b = 50). (Ans:- a = 0 4 and (13) What number should be subtracted from each of the numbers 23, 30, 57 and 78; so that the remainders are in proportion. (Ans:- 14 x x = 84 and = 6). (14) What should be added to each of the numbers 13, 17 and 22 so that the resulting numbers are in continued proportion. (Ans:- required number = 3). (15) If ( a 2 + c a, b, c and d (16) If ), ( ab+ cd ) and ( b 2 2 b, are in proportion. (Ans:- a, p : q :: q : (17) If a b mean , prove that r + d 2 ) are in continued proportion; prove that and d c p : = r p 2 and a : b is the duplicate ratio of a+ c are in proportion). : q 2 . (Ans:- p : r ). and b+c , prove that c is the proportional between a and b . (Ans:- c is the mean proportional between a and b ). (18) If = mb a+ c and + a, , prove that = b, c and d are in proportion. b, (Ans:- a, and d c are in proportion). (19) If q is the mean proportional between p and r , prove that : + p 2 q 2+ r 2 = q 4 . ( (20) If (i) (21) If a,b,c (i) = , prove that each given ratio ( (ii) and d = ) 2 2 2 2 ( (iii) and ) is equal to 3 3 1 3 3 3 ) are in proportion , prove that : 2 2 2 2 (ii) 2 2 2 2 = 4 4 4 4 . (22) 6 is the mean proportion between two numbers x and y and 48 is third proportion of x and y . Find the numbers. (Ans:- the required nos. are 3 and 12). (23) If = (24) If a :b y . (Ans:- x , find x : (27) If a :b ( a 2 : 3c 2d . (Ans:- 3 c+2 d , prove that = . , find the values of + . (Ans = 2) = (26) If P = = 13 : 1). = c : d , show that : 3 a+2 b : 3 a 2b = 3c 2d (25) If y : = c: d + ac +c : 3c 2d ). prove that : ) : ( a 2 ac+ c 2 ) = ( b 2 2 + bd +d ) : ( b 2 2 bd+ d (28) Using the properties of proportion, solve the following equation for x : 2 3 (i) 2 = (ii) 2 =5 . (Ans:- (i) x = 11 ; (ii) 1). (29) If x = , prove that : bx 2 3 ax +b=0 . 10. MATRICES (1) Find the values of x , y , [ x 2 y a /2 b+1 ] [ a [ ] 0 3 1 5 = ] and b , if : y . (Ans:- x = 2 , [ ] [ = 3 , a = 2 and b = 4). ] 5 4 3 0 1 3 , B= and C = , find : 3 2 1 4 0 2 (i) A + B and B + A (ii) (A + B) + C and A + (B + C) (ii) Is A + B = B + A ? (iv) Is (A+B) + C = A + (B + C) ? In each case, write the conclusion (if any) that you can draw. (2) Let A = (Ans:- A + B = B+A= [ ] [ ] [ ] 2 4 4 2 ; 2 4 4 2 ; (A + B) + C = 3 1 4 4 ; 2 ). [ ] 3 1 4 4 A + (B + C) = (3) If A = [ 5 4 3 1 (i) A + C ] [ (Ans:- A + C = B A+B [ (4) If matrix A = 2 1 0 4 A 2 6 4 1 ] ] [ ] C= 10 3 2 3 ] [ ] 2 4 1 3 3 2 (Ans:- A = 6 5 5 8 (5) If A = [ 8 6 2 4 ] (6) Given A = [ 1 2 2 3 (Ans:- A + 2B A= ] [ [ [ 3 2 7 4 3 7 2 4 [ 3 5 1 0 ] ; find transpose matrices At and Bt. If ] ; A + At is not possible and B + Bt is possible [ ] ; then solve for 2 2 matrix X such that : B=A. 11 1 3 4 , B = 3C = [ ). (ii) X (Ans:- (i) X = B C. (ii) B + Bt . and B = (i) A + X = B ; find : ). and Bt = [ ] B + Bt = ] ; and B = possible , find : (i) A + At t 3 2 1 0 ; 3 3 3 5 2 1 3 4 3 2 [ and C = (iii) A + B [ A= [ ] ; B= (ii) B ). ] ; 2 1 1 2 3 9 6 10 ] ] ). [ (ii) X = A + B = and C = [ 0 3 2 1 ] 5 11 1 4 ] ). , find : A + 2B 3C. [ 35 ] (7) Given, matrix A = [ 35] (Ans:- X = (8) If A = [ 2 3 4 1 ] [ 17] and matrix B = ; find matrix X such that : A + 2X = B. ). [ ] 1 2 3 5 and B = ; find : (i) AB (ii) BA. (iii) is AB = BA ? (iv) Write the conclusion that you draw from the result obtained above in (iii). [ ] 7 11 7 13 (Ans:- AB = (9) If A = [ 2 1 1 3 (Ans:- A2 (10) Let A = , evaluate A2 [ 3A + 2I = [ ] 3 5 4 2 ] [ ; BA = 6 5 14 14 ). 3A + 2I, where I is a unit matrix of order 2. 1 2 2 3 ] [ 24] and B = ] ). , is the product AB possible ? Give a reason. If yes , find AB. (Ans:- AB = (11) Let A = [260 ] [ ] 3 2 0 5 (i) (A + B) (A (Ans:- (A + B) (A (A + B) (A ). [ ] 1 0 1 2 and B = (ii) A2 B) B) = B) A2 [ B2 Is (A + B) (A 6 14 5 23 B2). ; find : ] ; A2 B2 = B) = A2 [ 8 16 3 21 ] ; B2 ? [ (12) Given : y = 3 8 9 4 ] [] x y [ 28] = , find x and y . (Ans:- x = and ). [ ] [ ] 1 3 2 0 (13) If B and C are two matrices such that B = 2 4 5 1 the matrix A so that BA = C. (Ans:- A = (14) Find the matrix M, such that M (15) Given : [ 8 2 1 4 ] .X= [1012] [ 3 6 2 8 (Ans:- the order of the matrix X = a b [ xy] = = [ 17 7 4 8 ] , find ). [ 2 16 ] . (Ans:- M = [ 4 ; Write down : (i) the order of the matrix X the matrix X = ] and C = [22] (ii) the matrix X. =2 1 . (16) State with reason . Whether the following are true or false A, B, C, are matrices of order 2 2. (i) A . B = B . A (ii) A . (A . C) = (A . B) . C (iii) (A + B)2 = A 2 + 2AB + B2 (iv) A . ( B + C) = A . B + A . C (Ans:- (i) False , as matrix multiplication is not commutative. (ii) True , as matrix multiplication is always associative . (iii) False , as laws of algebra are not applicable to matrices . (iv) True , as in the case of matrices the multiplication is always distributive over addition). 11. CO-ORDINATE GEOMETRY : REFLECTION 5]). (1) The triangle A(1, 2), B(4, 4) and C(3, 7) is first reflected in the line y = 0 onto triangle A`B`C` and then triangle A`B`C` is reflected in the origin onto triangle A``B``C``. Write down the co-ordinates of : (i) A`, B` and C` (ii) A``, B`` and C``. (Ans:- A` = (1, 2 ) ; B` = (4, 4 ) ; C` = (3, 7 ) ; A``= ( 1 , 2) ; B`` = ( 4 , 4) ; C`` = ( 3 , 7). (2) A point P is reflected in the x -axis . Co-ordinates of its image are (8, 6 ). (i) Find the co-ordinates of P . (ii) Find the co-ordinates of the image of P under reflection in the y -axis. (Ans:- P = (8, 6) ; Co-ordinates of the image of P under reflection in the y -axis = ( 8 , 6). (3) Points ( 5, 0) and (4, 0) are invariant points under reflection in the line L1 ; points (0, 6) and (0, 5) are invariant on reflection in the line L2 . (a) Name or write equations for the lines L1 and L2 . (b) Write down the images of P(2, 6) and Q( 8, 3) on reflection in L1 . Name the images as P` and Q` respectively . (c) Write down the images of P and Q on reflection in L2 . Name the images as P`` and Q`` respectively. (d) State or describe a single transformation that maps Q` onto Q`` . (Ans:- (a) The line L1 is x -axis , whose equation is y = 0 ; The line L2 is y -axis , whose equation is x = 0 ; (b) P` = (2, 6 ); Q` = ( 8, 3) ; (c) P`` = ( 2 , 6) ; Q`` = (8, 3 ). (4) (i) Find the reflection of the point P( 1, 3) in the line x = 2. (ii) Find the reflection of the point Q(2, 1) in the line y + 3 = 0. (Ans:- (i) P` (5, 3) ; (ii) Q` (2, 7) (5) The points P(5, 1) and Q( 2 , 2 ) are reflected in line x = 2 . Use graph paper to find the images P` and Q` of points P and Q respectively in line x = 2. Take 2 cm equal to 2 units. (Ans:- the co-ordinates of P` = ( 1 , 1). (6) Use a graph paper for this question. (Take two division = 1 unit on both the axes). Plot the points P (3, 2) and Q ( 3, 2) . From P and Q, draw perpendiculars PM and QN on the x -axis. (a) Write the co-ordinates of points M and N. (b) Name the image of P on reflection in the origin. (c) Assign the special name to geometrical figure PMQN and find its area. (d) Write the co-ordinates of the point to which M is mapped on reflection in : (i) x -axis, (ii) y -axis, (iii) origin . (Ans:- Co-ordinates of M = (3, 0) and Co-ordinates of N = ( 3, 0) (7) Use graph paper for this question. The points A(2, 3), B(4, 5) and C(7, 2) are the vertices of ABC. (i) Write down the coordinates of A`, B`, C` if A` B` C` is the image ABC, when reflected in the origin. (ii) Write down the co-ordinates of A``, B``, C`` if A`` B`` C`` is the image of ABC, when reflected in the x -axis. (iii) Mention the special name of the quadrilateral BCC`` B`` and find its area. (Ans:- A` = ( 2, 3), B` = ( 4, 5) and C` = ( 7, 2); A``= (2, 3), B`` = (4, 5) and C`` = (7, 2) ; BCC`` B`` is an isosceles trapezium ; area of quadrilateral BCC`` B`` = 21 sq- unit). 12. DISTANCE AND SECTION FORMULAE (1) Find the distance between the points (3, 6) and (0, 2). (Ans = 5). (2) Find the distance between the origin and the point : (i) ( 12, 5) (ii) (15, 8). (Ans:- (i) 13 ; (ii) 17). (3) Find the co-ordinates of points on the x -axis which are at a distance of 5 units from the point (6, 3). (Ans:- required points on the x -axis. Are (2, 0) and (10, 0). (4) KM is a straight line of 13 units . If K has the co-ordinates (2, 5) and M has the co-ordinates (2, 5) and M has the co-ordinates ( x , 7), find the value of x . (Ans:- x = 7 or x = 3). (5) Which point on the y -axis is equidistant from the points (12, 3) and ( 5, 10). (Ans = (0, 2). (6) Use the distance formula to show that the points A(1, 1), B(6, 4) and C(4, 2) are collinear. (Ans:- Given points A, B and C are collinear). (7) Show that the points A (8, 3), B (0, 9) and C (14 , 11) are the vertices of an isosceles right-angled triangle. (Ans:- the triangle ABC isosceles right-angled triangle). (8) Show that the quadrilateral ABCD with A (3, 1), B (0, 2) , C (1, 1) and D (4, 4) is a parallelogram. (Ans:- opposite sides of the quadrilateral ABCD are equal , it is a parallelogram). (9) Find the area of a circle, whose centre is (5, 3) and which passes through the point ( 7, 2). (Ans:- Area of the circle = 530 66 sq. units). (10) Find the points on the x -axis whose distances from the points A(7, 6) and B( 3, 4) are in the ratio 1 : 2. (Ans:- Required points on x -axis are : (9, 0) and ( , 0). (11) Point P( x , y ) is equidistance from the points A( 2, 0) and B(3, 4) , Prove that : 10 x 8 y = 21 . (Ans:- Hence proved that = 10 x 8 y = 21 ). (12) Find the co-ordinates of the circumcentre of the triangle ABC; whose vertices A, B and C are (4, 6) , (0, 4) and (6, 2) respectively. (Ans:- circumcentre of the given triangle = (3, 3). (13) Find the co-ordinates of point P which divides the joint of A (4, 5) and B(6, 3) in the ratio 2 : 5. (Ans:- P = , ( ) . (14) Find the ratio in which the point (5, 4) divides the line joining points (2, 1) and (7, 6). (Ans:- the required ratio is 3 : 2). (15) In what ratio is the line joining the points (4, 2) and (3, 5) divided by the x -axis ? Also, find the co-ordinates of the point of intersection. (Ans:- The ratio = 2 ; 5 and the required point of intersection = ( , 0) . (16) Calculate the ratio in which the line joining the points (4, 6) and ( 5, 4) is divided by the line y= 3 . Also, find the co-ordinates of the point of intersection. (Ans:- the required point of intersection = ( , 3). (17) The origin O, B ( 6, 9) and C (12, 3) are vertices of triangle OBC. Point P divides OB in the ratio 1 : 2 . Find the co-ordinates of points P and Q . Also, show that PQ = (Ans:- point P = ( 2, 3) ; point Q = (4, 1) ; point PQ = 1 3 1 3 BC. BC). (18) Show that P (3, m 5 ) is a point of trisection of the line segment joining the points A (4, 2) and B (1, 4) . Hence , find the value of m . (Ans:- m = 5). (19) Find the co-ordinates of the mid-point of the line segment joining the points P (4, 6) and Q ( 2, 4) . (Ans:- Mid-point = (1, 1). (20) A (14, 2), B (6, 2) and D (8, 2) are three vertices of a parallelogram ABCD. Find the co-ordinates of the fourth vertex C. (Ans:- the vertex C = (0, 2). (21) The mid-point of the line segment joining (3m, 6) and ( 4, 3n ) is (1, 2m 1). Find the values of m and n . (Ans:- m = 2 and n = 0). (22) In triangle ABC, P ( 2, 5) is mid-point of AB , Q (2, 4) is mid-point of BC and R ( 1, 2) is mid-point of AC . Calculate the co-ordinates of vertices A, B and C. (A = ( 5, 3) ; B = (1, 7) ; C = (3, 1). (23) ABC is a triangle and G(4, 3) is the centroid of the triangle . If A = (1, 3) , B = (4, b ) and C = ( a , 1), find a and b . Find the length of side BC. (Ans:- a = 7 ; b = 5 ; BC = 5 units). 13. EQUATION OF A LINE (1) The line , represented by the equation 3 x 8 y=2 , passes through the point ( k , 2). k . (Ans:- k = 6). Find the value of (2) Does the line 3 x= y +1 bisect the line segment joining A ( 2, 3) and B (4, 1) ? (Ans:- the given line bisects the join of A and B). (3) Find the slope of the line segment whose inclination is : (i) 60 o (ii) 52 o . (Ans:- slope 3 ; slope = 1 2799). (4) Find the slope of the line passing through the points A ( 2, 3) and B (2, 7) . Also find : (i) the inclination of the line AB, (ii) slope of the line parallel to AB, (iii) slope of the line perpendicular to AB. (Ans:- (i) 1 ; (ii) 1 ; (iii) 1). (5) The line joining A ( 3, 4) and B (2, 1) is parallel to the line joining C (1, 2) and D (0, x ). Find x . (Ans:- 1). (6) Given the points A (2, 3) , B ( 5, 0) and C ( 2, a ) are collinear . Find a . (Ans:1 ). (7) Find the equation of a line : (i) whose inclination is 45 o and y -intercept is 5. (ii) with inclination = 60 o and passing through ( 2, 5). (iii) passing through the points ( 3, 1) and (1, 5). (Ans:- (i) x+5 ; (ii) 3 x + 2 3 + 5 ; (iii) x+4 ). (8) Find the equation of the line whose x -intercept is 8 and y -intercept is 12. (Ans:- 3 x 24 ). (9) Find the equation of the line whose slope is 3 and x -intercept is also 3. (Ans:- 3 x+ y +9=0 ). (10) Find the equation of the line which pases through (2, 7) and whose y -intercept is 3. (Ans:- 2 x +3 ). (11) The equation of a line is 3 x 4 y+12=0 . It meets the x -axis at a point A and the -axis at point B. Find : (i) the co-ordinates of point A and B ; (ii) the length of intercept AB, cut by the line within the co-ordinates axes. (Ans:- A = ( 4, 0) ; B = (0, 3) ; AB = 5). (12) Write down the equation of the line whose gradient is and which passes through P, where P divides the line segment joining A ( 2, 6) and B (3, 4) in the ratio 2 : 3. (Ans:- 3 x 2 y+4=0 ). y (13) Find the equation of the lines which pas through the point ( 2, 3) and are equally inclined x+5 ; CD = x+ y=1 ). to the co-ordinates axes. (Ans:- AB = (14) Find the slope and (c) y -intercept of the line 2 x 3 y 4=0 . (Ans:- ( m )= ; ). (15) Given two straight lines 3 x 2 y=5 and 2 x +ky+7=0 . Find the value of k for which the given lines are : (i) parallel to each other (Ans:- (i) ; (ii) perpendicular to each other . (ii) 3 ). (16) Find the equation of the line passing through (2, 1) and parallel to the lines 2 x y=4 . (Ans:- 2 x 5 ). (17) Find the equation of the line which passes through the point ( 2, 3) and is perpendicular to the line 2 x +3 y +4=0 . (Ans:- 3 x+12 ). (18) Given two points A ( 5, 2) and B (1, 4) , find : (i) mid-point of AB ; (ii) slope of AB ; (iii) slope of perpendicular to AB; (iv) equation of the perpendicular bisector of AB. (Ans:- (i) ( 2, 1) ; (ii) 1 ; (iii) 1 ; (iv) x+1 ). (19) ABCD is a rhombus . The co-ordinates of A and C are (3, 6) and ( 1, 2) respectively . Write down the equation of BD. (Ans:- x+ y=5 ). 14. SYMMETRY (1) Construct a triangle ABC, in which AB = AC = 5cm and BC = 3 2cm . Using the ruler and compasses only draw the reflection A`BC of triangle ABC in BC . Draw the lines of symmetry of the figure ABA`C. (2) Use a ruler and compass in this question. (i) Construct the quadrilateral ABCD in which AB = 5 cm , BC = 7cm and angle ABC = 120 o, given that AC is its only line of symmetry. (ii) Write down the geometrical name of the quadrilateral . (iii) Measure and record the length of BD in cm. (3) Us egraph paper for this question . Plot the points A (8, 2) and B (6,4) . These points are the vertices of a figure which is symmetrical about x=6 and y=2 . Complete the figure on the graph . Write down the geometrical name of the figure . (4) Use graph paper for this question . use 2cm = 1 unit on both the axes . Plot the points A ( 2, 4) , B (2, 1) and C ( 6, 1). (i) Draw the line of symmetry of triangle ABCD. (ii) Mark the point D if the line in (i) and the line BC are both lines of symmetry of the quadrilateral ABCD . Write down the co-ordinates of point D. (iii) What kind of quadrilateral is figure ABCD ? (iv) Write down the equations of BC and the line of symmetry named in (i). (5) Use graph paper for this question A(0, 3) , B(3, 2) and O(0, 0) are the vertices of triangle ABO (i) Plot the triangle on a graph sheet taking 2cm = 1 unit on both the axes . (ii) Plot D the reflection of B in the Y axis , and write its co-ordinates . (iii) Give the geometrical name of the figure ABOD. (iv) Write the equation of the line of symmetry of the figure ABOD. 15. SIMILARITY (With Applications to Maps and Models) (1) In the given figure , AB and DE are perpendicular to BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm , calculate AD. (Ans:- 16 cm). (2) In the adjoining figure , ABC is a triangle right-angled at vertex A and AD is altitude. (i) Prove that : ABD is similar to CAD. (ii) If BD = 3 6 cm and CD = 6 4 cm ; find the length of AD. (Ans:- AD = 4 8 cm). (3) In the given figure , DE // BC. (i) Prove that ADE and ABC are similar (ii) Given that AD = Also, find BC, calculate DE, if BC = 4 5 cm. Ar .( ADE) Ar .( ABC ) and Ar .( ADE) Ar .(trapezium BCED) . (Ans:- DE =1 5 cm). (4) In the figure, given alongside, PB and QA are perpendiculars to the line segment AB. If PO = 6 cm , QO = 9 cm and area of POB = 120 cm2, find the area of QOA. (Ans:- 270 cm2). (5) In ABC , angle B = 90 , AB = 12 cm and AC = 15 cm . D and E are points on AB and AC respectively such that angle AED = 90 and DE = 3 cm . Calculate the area of ABC and then the area of ADE. (Ans:- area = 54 cm2 ; Arc = 6 cm2). (6) The scale of map is 1 : 50,000. In the map , a triangular plot ABC of land has the following dimensions : AB = 2 cm , BC = 3 5 cm and angle ABC = 90 . Calculate : (i) the actual length of side BC, in km , of the land . (ii) the area of the plot in sq. km. (Ans:- (i) 1 75 km ; (ii) 0 875 sq. km). (7) A rectangle tank has length = 4 m , width = 3 m and capacity = 30 m3 . A small model of the tank is made with capacity 240 cm3 . Find : (i) the dimensions of the model . (ii) the ratio between the total surface area of the tank and its model. (Ans:- (i) 8cm 6 cm 5 cm ; (ii) 2500 : 1). (8) In the figure, given alongside, Angle QPS = angle RPT and angle PRQ = angle PTS . (i) prove that triangles PQR and PST are similar. (ii) If PT : ST = 3 : 4 ; find the ratio between QR : PR. 16. LOCI (LOCUS AND ITS CONSTRUCTION) (1) Construct a triangle ABC in which AB = 6 cm , BC = 7cm and CA = 6 5 cm. Find the point P equidistance from B and C ; and also equidistance from AB and BC. (2) Construct a quadrilateral ABCD , having given AB = 2 6cm ; BC = 4 0 cm , CD = 3 2cm . AD = 2 cm and diagonal BD = 3 6cm. Mark a point P on diagonal AC, equidistance from B and C. (3) Use ruler and compasses only for this question (i) Construct ABC , where AB = 3 5 cm , BC = 6cm and ABC = 60 o. (ii) Construct the locus of points inside the triangle which are equidistance from BA and BC. (iii) Construct the locus of points inside the triangle which are equidistance from B and C. (iv) Mark the point P which is equidistance from AB, BC and also equidistance form B and C Measure and record the length of PB. (4) Construct a triangle ABC with AB = 7cm, BC = 8 cm and angle ABC = 60 o. Locate by construction the point P such that : (i) P is equidistance from B and C . (ii) P is equidistance from AB and BC. Measure and record the length of PB. (5) A straight line AB is 8cm long . Draw and describe the locus of a point which is : (i) always 4cm from the line AB. (ii) equidistance from A and B. Mark the two points X and Y, which are 4 cm from AB and equidistance from A and B. Describe the figure AXBY. (6) Use graph paper for this question . Take 2cm = 1 unit on both the axes. (i) Plot the points A (1, 1), b (5, 3) and C (2, 7). (ii) Construct the locus of points equidistance from A and B (iii) Construct the locus of points equidistance from AB and AC. (iv) Locate the point P such that PA = PB and P is equidistance from AB and AC. (v) Measure and record the length PA in cm. (7) Construct a triangle BCP given BC = 5cm, BP = 4 cm and angle PBC = 45 o. (i) Complete the rectangle ABCD such that : (a) P is equidistance from AB and BC. (b) P is equidistance from C and D. (ii) Measure and record the length of AB. 17. CIRCLES (1) Chords AB and CD of a circle with centre O, intersects at at a point E. If OE bisects angle AED, prove that chord AB = chord CD. (Ans:- Chord AB = chord CD). (2) In the figure, given alongside, CD is a diameter which meets the chord AB at E, such that AB at E, such that AE = BE = 4 cm .If CE is 3 cm , find the radius of the circle. (Ans:- r = 4 cm). (3) A chord of length 48 cm is at distance of 10 cm from the centre of the circle. Another chord of length 20 cm is drawn in the same circle , find its distance from the centre of the circle. (Ans:- 24 cm). (4) The line segment joining the mid-points of two parallel chords of a circle passes through the centre . Prove it. (5) Chords AB and CD of a circle area parallel to each other and lie an opposite sides of the centre of the circle . If AB = 36 cm , CD = 48 cm and the distance between the chords is 42 cm ; find the radius of the circle. (Ans:- R = 30 cm). (6) AB and CD are two equal chords of a circle with centre O . If AB and CD, on being produced , meet at a point P outside the circle prove that : (a) PA = PC (b) PB = PD . (7) Two circles with centre A and B intersect each other at point P and Q . Prove that the centreline AB bisects the common chord PQ perpendicularly. (Ans:- AB bisects PQ). (8) Out of two unequal chords of a circle, the bigger chord is closer to the centre of the circle . Prove it. (9) In the adjoining figure ; angle AOC = 110 ; calculate : (i) ADC (ii) ABC (Ans:- (i) 55 ; (ii) 125 (iii) OAC. ; (iii) 35 (10) In the adjoining figure , PQ = PR and PRQ = 70 . Find QAR. (Ans:- 40 ). (11) The given figure shows a circle through the points A, B, C, and D . If BAC = 67 , find: DBC + DCB. (Ans:- 113 ). (12) In the adjoining figure ; AC is a diameter of the circle. AB = BC and AED = 118 . Calculate : (i) DEC ). (ii) DAB . (Ans:- (i) 28 ; (ii) 73 (13) In the given figure , ABD = 55 BAD = 80 . and BDC = 45 . Find : (i) BCD (ii) ADB Hence , show that AC is a diameter. (Ans:- (i) 100 ; (ii) 45 ). (14) In the figure, given alongside, O is the centre of the circle and angle AOC = 160 . Prove that : 3 y 2 x=140 . (15) In the given figure , I is the incentre of triangle ABC . AI produced meets the circumcirle of the triangle ABC at point D. If angle BAC = 50 and angle ABC = 70 , find : (i) angle BCD (ii) angle ICD (iii) angle BIC (Ans:- (i) 25 ; (ii) 55 ; (iii) 115 ). (16) If two sides of a cyclic quadrilateral are parallel , prove that the other two sides are equal. 18. TANGENTS AND INTERSECTING CHORDS (1) ABC is a right angled triangle with AB = 6 cm and BC = 8 cm . A circle with center O has been inscribed inside the triangle . Calculate the value of x , the radius of the inscribed circle. (Ans:- 2 cm). (2) A, B and C are three points on a circle . The tangent at C meets BA produced at T . Given that angle ATC = 36 and that angle ACT = 48 , calculate the angle subtended by AB at centre of the circle . (Ans:- 96 ). (3) Two circles with radii 25 cm and 9 cm touch each other externally . Find the length of the direct common tangent . (Ans:- 30 cm). (4) The centers of two circles with radii 6 cm and 2 cm are 10 cm apart . Calculate the length of the transverse common tangent. (Ans:- 6 cm). (5) In the figure given alongside , PQ = QR , angle RQP = 68 , PC and QC are tangents to the circle with centre O , calculate the values of (i) angle QOP (ii) angle QCP. (Ans:- (i)112 ; (ii) 68 ). (6) In the given figure , AB is the diameter and AC is the Chord of a circle such that angle BAC = 30 . The tangent At C intersects AB produced at D. Prove that : BC = BD (7) In the given figure , PT touches a circle with centre O at R. Diameter SQ when produced meets PT at P. If angle SPR = x x + 2 y and angle QRP = = 90 . y , show that (8) In the given figure , PM is a tangent to the circle and PA = AM. Prove that : (i) PMB is isosceles (ii) PA PB = MB2 (9) In a right triangle ABC, a circle with AB as diameter is drawn to intersects the hypotenuse AC in P . Prove that the tangent at P, bisects the side BC. (10) ABC is an isosceles triangle with AB = AC . A circle through B touches side Ac at it middle point D and intersects side AB in point P . Show that : AB = 4 AP. (11) The given figure shows an isosceles triangle ABC inscribed In a circle such that AB = AC . If DAE is a tangent to the circle at point A , prove that DE is parallel to BC. (Ans:- hence proved). (12) AB is the diameter of a circle with centre O. A line PQ touches the given circle at point R and cuts the tangents to the circle through A and B at points P and Q respectively . Prove that :angle POQ = 90 . (13) From an exterior point P , a tangent PT and a secant PAB are drawn to a given circle. If points A nad B lie on the circumference of the circle , prove that: (i) triangles PAT and PBT are similar. (ii) PA PB = PT2. 19. CONSTRUCTIONS (Circles) (1) Draw a circle of diameter 12 cm. Mark a point at a distance of 10 cm from the centre of the circle . Draw tangents to the given circle from this exterior point . Measure the length of each tangent . (2) Draw a circle of radius 4 5 cm. Draw two tangents to this circle so that the angle between the tangents is 60 o. (3) Construct an equilateral triangle ABC with side 6 cm . Draw a circle circumscribing the triangle ABC. (4) Construct a circle inscribing an equilateral triangle with side 5 6 cm. (5) Draw an inscribing circle of a regular hexagon of side 5 8 cm. (6) Construct a regular hexagon of side 4 cm . Construct a circle circumscribing the hexagon . (7) Draw a circle of radius 3 5 cm . Mark a point P outside the circle at a distance of 6 cm from the centre . Construct two tangents from P to the given circle . Measure and write down the length of one tangent. 20. CIRCUMFERENCE AND AREA OF A CIRCLE (1) The circumference of a circle is 123 2 cm . Calculate : (i) the radius of the circle; (ii) the area of the circle, correct to the nearest cm2; (iii) the effect on the area of the circle , if its radius is doubled. (Ans:- (i) 19 6 cm ; (ii) 1207 cm2 ; (iii) 4 times the original area). (2) The area enclosed by the circumference of two concentric circles is 423 5 sq.cm. If the circumference of the outer circle is 132 cm, calculate the radius of the inner circle. (Ans:- r = 17 5 cm). (3) The diameter of a cycle wheel is 28 cm . How many revolutions will it make in moving 13 2 km ? (Ans:- 15,000). (4) A sheet is 11 cm long and 2 cm wide . Circular pieces 0 5 cm in diameter are cut from it to prepare discs . Calculate the number of discs that can be prepared. (Ans:- 88). (5) In the given figure, AB is the diameter of a circle with centre O and OA = 7 cm . Find the area of the shaded region (Ans:- 66 5 cm2). (6) In the adjoining figure; PS is a diameter of a circle and is of length 6 cm . Q and R are point on the diameter such that PQ, QR and Rs are equal . Semicircles are drawn with PQ and QS as diameters . Find the perimeter of the shaded portion. Also, find the area of the shaded portion. (Ans:- perimeter = 18 cm ; area = 9 sq. cm). (7) A circle circumscribe a rectangle with sides 12 cm and 16 cm . Calculate the circumference of the circle. (Take = 3 14). (Ans:- 62 8 cm). (8) The radius of a circle is 21 cm . Calculate the area and perimeter of : (i) its sector with angle 120 at the centre. (ii) its quadrant. (Ans:- (i) 462 sq. cm ; (ii) 346 5 cm2). (9) AC and BD are two perpendiculars of a circle ABCD. Given that the area of the shaded portion is 308 cm2. Calculate : (i) the length of AC and (ii) the circumference of the circle . (Take = ). (Ans:- (i) 28 cm ; (ii) 88 cm). (10) The given figure shows a square ABCD of side 20cm . Semi-circles are drawn with each side of the square as diameter . Find the area of the shaded portion. (Take = 3 14). (Ans:- 228 cm2). 21. CONE AND SPHERE (Surface Area and Volume) (1) The area of the curved surface of a cylinder is 4,400 cm2 and the circumference of its base is 110 cm . Find : (i) the height of the cylinder (ii) the volume of the cylinder. (Ans:- H = 40 cm ; volume = 38,500 cm3). (2) The barrel of a fountain-pen , cylindrical in shape , is 7 cm long and 5 mm in diameter . A full barrel of ink in the pen will be used up when writing 310 words on an average. How many words would use up a bottel of ink containing on-fifth of a litre ? (Asnwer correct to the nearest 100 words) . (Ans:- 45,100 words). (3) A cylindrical tube open at both the ends is made of metal . the internal diameterof the tube is 11 2 cm and its length is 21cm. The metal everywhere is 0 4 cm. Calculate the volumne of the metal in the tube , correct to one place of decimal. (Ans:- 306 2 cm3). (4) The radius of the base and the height of a right circular cone are 7 cm and 24 cm respectively Find the volume and the total surface area of the cone . (Ans:- volume = 1232 cm3 ; area = 704 cm2). (5) Find what length of canvas 2 m in width is required to make a conical tent 12 m in diameter and 63 m in slant height . Also, find the cost of the canvas at the rate of Rs 15 per meter. (Ans:- length = 594 m ; cost = Rs 8,910). (6) Find the area of the canvas required to make a conical tent 14 m high and 96 m in diameter . Given that : (i) 20% of the canvas is used in folds and stitching. (ii) canvas used in folds and stitching is 20% of the curved surface area of the tent. (Ans:- (i) 9,428 m2 ; (ii) m2 = 9,051 m2. (7) The capacity and the base area of a right circular conical vessel are 9856 cm3 and 616 cm2 respectively . Find the curved surface area of the vessel. (Ans:- 2200 cm2). (8) From a solid cylinder of height 36 cm and radius 14 cm , a conical cavity of radius 7 cm and height 24 cm is drilled out . Find the volume and the total surface area of the remaining solid. (Ans:- volume = 20944 cm2 ; area 4796 cm2). (9) If the surface area of a sphere is 616cm2 , find its volume. (Ans:- 1437 cm3). (10) The internal and external diameters of a hollow hemisphere vessel are 42 cm and 45 5 cm respectively . Find its capacity and also its outer curved surface area. (Ans:- capacity = 19,404 cm3 ; area = 3253 25 cm2). (11) 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 heap of the sand . If the height of the conical heap is 24 cm, find : (i) the radius and (ii) the slant height of the heap . Give your answer correct to one place of decimal. (Ans:- r = 36 cm ; L = 43 3 cm). (12) The radius of the base of a cone and the radius of a sphere are the same , each being 8 cm . Given that the volumes of these two solids are also the ame , calculate the slant height of the cone . (Ans:- l = 32 98 cm). (13) The radius of a sphere is 9 cm .It is melted and drawn into a wire of diameter 2 mm . Find the length of the wire in meters . (Ans:- 972 m). (14) A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness . Find the thickness of the wire. (Ans:- cm). (15) 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 formed . (Ans:- 126). (16) A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm . Find the height of the cone. (17) A toy is in the form of a cone mounted on a hemisphere with the same radius . The diameter of the base of the conical portion is 12 cm and its height is 8 cm . Determine the surface area and the volume of the toy ( = 3 14). (Ans:- area = 414 48 cm2 ; volume = 753 6 cm3). (18) From a solid cylinder , whose height is 8 cm and radius is 6 cm , a conical cavity of height 8 cm and of base radius 6 cm is hollowed out . find the volume of the remaining solid . Also, find the total surface area of the remaining solid. (Ans:- 603 cm3 ; area = 603 cm2). (19) A vessel 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 into the vessel , out . Find the number of lead shots dropped into the vessel. (Ans:- 440). 22. TRIGONOMETRICAL IDENTITIES (1) Prove the identity : tan A + cot A = sec A . cosec A. (2) prove that : (i) cos4 A sin4 A = 2 cos2 A 1 (ii) (1 + cot A)2 + (1 cot A)2 = 2 cosec2 A (iii) tan4 A + tan2 A = sec4 A sec2 A . (3) Prove that : (i) sin A 1+cos A + (ii) 1+cos A 1 cos A = (cosec A + cot A )2 (iii) cos A cot A 1 sin A (4) Prove that : sec A tan A cosec A+ cot A (5) Prove that : (i) 1 sin A 1+ sin A 1+cos A sin A = 2 cosec A = 1 + cosec A . = cosec A cot A sec A+ tan A = sec A tan A of the water flows (ii) (6) Prove that :- (i) tan A + sec A 1 tan A sec A+1 cos A 1 tan A = (8) If x=a 2 2 + 2 (9) Evaluate : (i) = cos A + sin A 1 ) = tan 2 A 1 sin A sin 4 A 2 and tan A sin A = n ; Prove that : m sec A cos B, 2 2 . sin A sin A cos A + (ii) (1 + tan2 A) + (1 + (7) If tan A + sin A = m 1+sin A cos A y 2 2 = b sec A sin B and z 2 n2 = 4 mn sin28 cos 62 = c tan A ; show that : cot 56 tan 34 (ii) (iii) cosec50 sec 40 (ii) tan2 25 cot2 27 cot2 27 . (11) prove that : (i) cos 55 sin 35 + sin 55 cos 35 tan 72 cot18 cot 72 tan 18 (iii) sec 70 sin 20 (12) Without using table , evaluate : + cosec 70 2 tan53 cot 37 cot ( 90 A ) cos A cot A = 1 = 0 cos 20 =2 . cot 80 tan 10 (13) Prove that : (i) sin (90 A) cos ( 90 A) = (ii) . = 1. (10) Evaluate : (i) cos 58 sin 32 (ii) . tan A 2 1+ tan A 1 = cos2 A. (iii) sec2 63 (14) Without using mathematical tables , find the value of x if. cos x = cos 60 cos 30 + sin 60 sin 30 . (15) Given cos 38 sec (90 2A) = 1 ; find the value of angle A. 23. HEIGHTS AND DISTANCES (1) The length of the shadow of a vertical tower is 3 times its height . Find the angle of elevation on the sun. (Ans:- 30 . (2) The angle of elevation of the top of a tower at a distance of 120m from its foot on a horizontal plane is found to be 30 . Find the height of the tower. (Ans:- 69 28m). (3) A guard observes an enemy boat , from an observation tower at a height of 180 m above sea level , to be at an angle of depression of 29 . (i) Calculate , to the nearest meter , the distance of the boat from the foot of the observation tower. (ii) After some time , it is observed that the boat is 200 m from the foot of the observation tower . Calculate the new angle of depression. (Ans:- (i) 325 m ; (ii) 41 59` (4) Two people standing on the same side of a tower in a straight line with it measure the angles of elevation of the top of the tower as 25 and 50 respectively. If the height of the tower is 70 m , find the distance between the two people. (Ans:- 91 38m). (5) The length of the shadow of a vertical tower on level ground increases by 10 m, when the altitude of the sun changes from 45 to 30 . Calculate the height of the tower , correct to two decimal places . (Ans:- 13 66m). (6) An observer on the top of a cliff ; 200 m above the sea-level, observes the angles of depression of the two ships to be 45 and 30 respectively . Find the distance between the ships , if the ships are : (i) on the same side of the cliff , (ii) on the opposite sides of the cliff. (Ans:- (i) 146 4 m ; (ii) 546 4 m). (7) 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 to 45 , how soon after this will the car reach the observation tower ? (Ans:16 39). (8) The angle of elevation of a stationary cloud from point 25 m above a lake is 30 and the angle of depression of its reflection in the lake is 60 . What is the height of the cloud above the lake-level. (Ans:- 50 m). (9) From a point on the ground , the angle of elevation of the top of a vertical tower is found to be such that its tangent is . On walking 50 m towards the tower , the tangent of the new angle of elevation of the top of the tower is found to be . Find the height of the tower . (Ans:- 120m). (10) A vertical pole and a vertical tower are on the same level ground . From the top of the pole the angle of elevation of the top of the tower is 60 and the angle of depression of the foot of the tower is 30 . Find the height of the tower if the height of the pole is 20 m.(Ans: 80m) 24. MEASURES OF CENTRAL TENDENCY (MEAN, MEDIANS, QUARTILES AND MODE) (1) The weights (in kilogram) of 5 persons are : 37, 35, 71, 57 and 45. Find the arithmetic mean of their weights. (Ans:- 61 kg). (2) Find the mean of : x 5 4 6 5 7 6 8 3 9 2 f (Ans:- 6 85). (3) Find the mean of : Class interval Frequency 0 10 10 (Ans:- Mean = 24 ). 10 20 6 20 30 8 30 40 12 40 50 5 50 55 5 55 60 3 (4) Find mean of the following distribution using short-cut method : C.I. 25 40 7 f 40 45 6 45 50 9 (Ans:- mean = 46). (5) Find the mean of the following distribution: Class interval 20 30 Frequency 10 (Ans:- Mean = 49 6). 30 40 6 40 50 8 50 60 12 60 70 5 70 80 9 (6) If the mean of the following distribution is 7 5, find the missing frequency: Variable: Frequency : (Ans:- f 5 20 = 16). 6 17 7 f 8 10 9 8 10 6 11 7 12 6 (7) The total number of observations in the following distribution table is 120and their mean is 50. Find the values of missing frequencies f 1 and f 2 . Class: 0 20 Frequency : 17 f (Ans:- 1 = 28 and f2 20 40 f1 = 24). 40 60 32 60 80 f2 80 100 19 (8) Find the median of : 7, 12, 15, 6, 20, 8, 4 and 10. (Ans:- Median = 9). (9) The weights of 45 children in a class were recorded, to the nearest kg, as follows : Wt. (in nearest kg) 46 48 50 52 No. of children 7 5 8 12 Calculate the median weight. (Ans:- Median weight = 52 kg). 53 10 54 2 55 1 (10) Find the median for the following distribution : C.I. 0 10 Frequency 5 (Ans:- median is 26). 10 20 7 20 30 10 30 40 8 40 50 5 (11) The marks obtained by 200 students in an examination are given below :Marks No. of student s 0 10 05 10 20 10 20 30 11 30 40 20 40 50 27 50 60 38 60 70 40 70 80 29 80 90 14 90 100 06 Using a graph paper , draw an Ogive for the above distribution . Use your Ogive to estimate : (i) the median ; (ii) the number of students who obtained more than 80% marks in the examination and (iii) the number of students who did not pass , if the pass percentage was 35. Use the scale as 2cm = 10 marks on one axis and 2cm = 20 students on the other axis. (Ans:- Median = 57 ; no. of students scored more than 80% = 20 ; no. of student not pass = 34). (12) 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 No. of workers 12 20 30 38 24 Using the graph paper, draw an ogive for the above distribution. Use your Ogive to estimate : (i) the median wage of the workers (ii) the percentage of the workers who earn more than Rs 45 a day. 50 60 16 60 70 12 70 80 8 (Ans:- median = 35 ; the percent of workers who earn more than Rs 45 a day = 30%). (13) Find the lower quartile, upper quartile and inter quartile range for the data : 9, 11, 15, 19, 17, 13, 7. (Ans:- Lower quartile = 9 ; upper quartile = 17 ; inter quartile range = 8). (14) From the following frequency distribution table, find : (i) Lower quartile (ii) Upper quartile (iii) Inter-quartile range. C.I. 5 10 10 - 15 15 20 20 25 25 30 Frequency 3 4 6 9 7 (Ans:- Lower quartile = 15 5 ; Upper quartile = 25 5 ; Inter-quartile range = 10). 30 35 1 (15) Using a graph paper, draw an ogive for the following distribution which shows the marks obtained in the General Knowledge paper by 100 students. Marks 0 10 10 20 20 30 30 40 40 50 50 60 60 70 No. of students 5 10 20 25 15 12 9 (Ans;- Median = 36 ; No. of students scoring marks above 65 = 8). (16) 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 . Score obtained Number of shooters 0 10 5 10 20 9 20 30 16 30 40 22 40 50 26 50 60 18 60 70 11 70 80 6 80 90 4 90 100 3 Using your ogive to estimate : (i) the median (ii) the interquartile range. (iii) The number of shooters who obtained more than 75% scores. (Ans:- Median = 43 ; inter-quartile = 27 ; 10). (17) Find the mode of following data : 4, 7, 4, 3, 2, 7, 7, 6, 4, 7, 8 . (Ans:- Mode = 7). (18) Find the mode from the following frequency distribution : Number 8 9 10 11 12 13 Frequenc 3 8 12 15 14 17 14 12 15 8 16 6 70 80 4 y (Ans:- Mode = 13). (18) Find the mode of the following frequency distribution : (Using histogram) Class 20 30 30 40 40 50 50 60 60 70 Frequency 4 7 9 11 6 (Ans:- Mode = 53). 70 80 2 25. PROBABILITY (1) A bag contains a black ball, a red ball and a green ball, all the balls are identical in shape and size . Mohit takes out a ball from the bag, without looking into it . What is the probabilty that the ball drawn is: (i) red ball ? (ii) balck ball ? (iii) green ball ? (Ans:- P(red ball) = ; P(black ball) = ; P(green ball) = ). (2) In a single throw of a die, find the probability of getting a number: (i) greater than 2 (ii) less than or equal to 2. (iii) not grater than 2. (Ans:- P(greater than 2) = ; (less than or equal to 2) = ; P(not greater than 2) = ). (3) From a well-shuffled deck of 52 cards , one card is drawn . Find the probability that the card drawn will: (i) be a face card (ii) not be a face card . (Ans:- P(face card) = ; P(not a face card) = ). (4) In a badminton match between Rajesh and Joseph, the probability of winning of Rajesh is 0 58 . Find the probability of : (i) not wining of Rajesh ; (ii) wining of Joseph. (Ans:- P(not wining of Rajesh) = 0 42 ; P(wining of Joseph) = 0 42). (5) In a single throw of a die, find the probability of getting : (i) 7 (Ans:- P(getting a number 7) = 0 ; P(number less than 7) = 1). (ii) a number less than 7. (6) A die is thrown once. Find the probability of getting :(i) an odd number (ii) a number greater than 4 (iii) a number between 2 and 6. (Ans:- P(an odd number) = ; P(no. greater than 4) = ; P(no. between 2 & 6) = ). (7) Two dice are thrown simultaneously . Find the probability that : (i) both the dice shown the same number. (ii) the first dice shows 6. (iii) the total (sum) of the numbers on the dice is 9. (iv) the product of the numbers on the dice is 8. (v) the total of the numbers on the dice is greater than 9. (Ans:- P(dice show the same number) = ; P(first dice shows 6) = ; P(total no. on the dice is 9) = ; P(total product of the number on the dice is 8) = ; P(total no. on the dice is greater than 9) = ). (8) A card is drawn from a pack of 100 cards numbered 1 to 100 . Find probability of drawing a number which is a perfect square . (Ans:- required probability = ). (9) Three identical coins are tossed together . What is the probability of obtaining : (i) all heads (ii) exactly two heads (iii) exactly one head (iv) at least one head (v) at least two heads (vi) all tails. (Ans:- (i) P (all heads) ; (ii) P (exactly two heads) = ; (iii) P (exactly one head) = ; (iv) P (at least one head) = ; (v) P (at least two heads) = ; (iv) P (all tails) = ). (10) Two dice are rolled simultaneously. Find the probability of : (i) obtaining a total of at least 9. (ii) getting a multiple of 2 on one die and a multiple of 3 on the other die. (iii) getting a multiple of 3 as the sum. (Ans:- (i) required probability = ; (ii) required probability = ; (iii) required probability = ). (11) A box contains some black balls and 30 while balls . If the probability of drawing a black ball is two-fifths of a while ball; find the number of black balls in the box. (Ans = 12). (12) From a pack of 52 playing cards all cards whose number are multiples of 3 are removed . A card is now drawn at random . What is the probability that the card drawn is (i) a face card (ii) an even numbered red card ? (Ans: (i) ; (ii) ) . X-X-X-X-X-X-X (All The Best)

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