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ICSE Class X Prelims 2025 : Mathematics

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Pratyush KU Mahapatro (Ronit)
St. Joseph's Convent School, Ganjam
10th

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Maths last minute revision 1 STD : 10TH MATHS LMR BOOK GOODS AND SERVICES TAX (1) When a chartered accountant (CA) provided his services to Mr. Rao in filing his income returns, his bill for services was 8260 inclusive of 18% GST. What is the original amount of the bill? How much is the GST paid to the State Government by the CA? Ans: 630 (2) Sachin visits a department stores and buys the following articles: Medicine costing 1250 , GST @ 5%; a packet of sweets 450 , GST @ 5%, soaps, hair oil 500, GST @ 18%, a chess board 300 with a discount of 10%, GST @ 12% calculate: (1) the total amount of GST paid (2) the total bill amount including GST paid by Sachin. Ans: (i) 207.40 (3) When Mr.Mukharjee stayed in a hotel 2 days he had to payed 7080 including 18%GST. What is the tariff of the hotel for the unit of accommodation? (ii) 2677.40 Ans: 3000 (4) A shopkeeper buys certain quantity of cashew nuts for 7200 and sells it to a consumer at a profit of 25%. If the rate of GST is 5%, find the GST paid by the shopkeeper to the Central and State Governments. Ans: CGST = 45 (5) A manufacturer sells a sewing machine for 5500. the wholesaler sells it to the retailer SGST = 45 for 7500. the shopkeeper makes a profit of 1000 by selling it to the consumer. if the GST charged at each stage is 12% , find: (a) the amt. of GST paid by the manufacturer to the central government. (b) the amt. of GST received by the state govt. 2 (c). the final price paid by the consumer Ans: (i) 330 (ii) 510 (iii) 9520 (6) A shopkeeper buys a printer at a discount of 30% on the marked price of Rs. 8000. He sells the printer to a customer at marked price. GST charged at each stage is 18%. If the sales are intra-state ,Find: (i) GST paid by the shopkeeper to the Central Government. (ii) The price paid by the shopkeeper for the article inclusive of tax. (iii) The cost to the customer inclusive of tax. (iv) the amount of tax received by the State Government Ans: (i) 216 (ii) 6608 (iii) 9440 (iv) 720 (7) A manufacturer marks a mobile for 6000. He sells it to a wholesaler at 25% discount. The wholesaler sells it to a retailer at 20% discount on market price. If the retailer sells it at market price and GST charged is 12% at every stage. find : (1) the GST paid to the Central government by the wholesaler. (2) the GST paid to the State government by the retailer (3) the amount which the consumer pays. Ans: (i) 18 (ii) 72 (8) The printed price of an air conditioner is 40000. The wholesaler allows a discount of 10% (iii) 6720 on it to the shopkeeper.The shopkeeper sells the AC to a customer at a discount of 5% on the marked price. GST is charged at the rate of 28%. The sales are intra-state. Find: (1) the price inclusive of tax paid by the shopkeeper. (ii) the tax paid by the wholesaler to the State Government. (iii) the GST paid by the shopkeeper to the State Government. (iv) the tax received by the Central Government. (v) the total amount paid by the customer inclusive of tax. Ans: (i) 46080 (ii) 5040 3 (iii) 280 (iv) 5320 (v) 48640 (9) A manufacturer sells an article to a wholesaler with marked price 2000 to a wholesaler at a discount of 20% on the marked price. The wholesaler sells it to a retailer at a discount of 10% on the marked price. The retailer sells the article at the marked price.If the GST paid by the wholesaler is 24, find : (i) the rate of GST (ii) the GST paid by the retailer (iii) the price paid by the customer. Ans: (i)12% (ii) 24 (10) A shopkeeper sells some edible oil for 7,200 at its marked price. The shopkeeper pays (iii) 2240 GST of 120 to the Government. If the GST charged throughout is 5%, calculate the price paid by the shopkeeper for the oil inclusive of tax. Ans: 5040 (11) The marked price of an article is 6000 and rate of GST is 12%. a shopkeeper buys it at a discount and sells it at its marked price. if the sales are intra-state and the shopkeeper paid 36 under GST to the State Government , find (i) the amount (inclusive of GST ) paid by the shopkeeper and (ii) the % of discount received by him. Ans: (i) 6048 (ii) 10% (12) A wholesaler buys a TV from a manufacturer for 25000.He marks the price of the TV 20% above his cost price and sells it to a retailer at a 10% discount on the marked price. If the rate of GST is 28% find: (i) The marked price (ii) The retailer s cost price inclusive of tax. (iii) The GST paid by the wholesaler to the State Government. Ans: (i) 30000 (ii) 34560 (iii) 280 (13) The price of an article is Rs.5120 inclusive of GST, at the rate of 28% on its listed price. A customer asks the dealer for a discount on the listed price so that after charging the GST, 4 the selling price will be same as the listed price. What is the amount of discount that the dealer must allow for the deal? Ans: 875 (14) A manufacturer supplies some blankets worth Rs. 40000 to a dealer at a profit of 15%. The dealer sells these to a shopkeeper at a profit of Rs. 12000. If the rate of GST is 12%, calculate: (i) the input GST of the dealer (ii) the GST paid by the dealer to the government (iii) the price paid by the shopkeeper inclusive of GST Ans: (i) 5520 (ii) 1440 (15) The price of an article is Rs. 4410 inclusive of GST, at the rate of 5% on its listed price. (iii) 64960 A customer asks the dealer for a discount on the listed price so that after charging the GST, the selling price will be same as the listed price. (i) what is the list price? (ii) What is the amount of discount that the dealer must allow for the deal? Ans: (i) 4200 (ii) 200 (16) Ayush purchased a computer for 28320 which included 20% discount on the list price and 18% tax under GST on the remaining price. find the list price of the computer. (17) Mr.Bedi visits the market and buys the following articles: Ans: 30000 Medicines costing 950, GST @5%; A pair of shoes costing 3000, GST @18% A Laptop bag costing 1000 with the discount of 30%, GST @18% Calculate: (i) The total amount of GST paid. (ii) The total bill amount including GST paid by Mr. Bedi. Ans: (i) 713.50 (ii) 5363.50 5 BANKING (1) Sanjana opens a recurring deposit scheme and deposits 800 per month for a period of 2 years . if the rate of interest is 9% p.a. find the amount payable at the end of 2 years. Ans: 21000 (2) Shivangi deposits 500 every month in a recurring deposit scheme and receives 16550 at the end of 2 1 years. calculate the rate of interest given by the bank. 2 Ans: 8% (3) Fateh deposits a certain sum of money every month in a recurring deposit account for 2 years. If the bank pays interest at 10% p.a and Fateh receives Rs 66,250 as the maturity value of the account, what sum of money did he pay every month. Ans: 2500 (4) Puneet has a recurring deposit account in a bank and deposits 400 per month. If he receives 10100 at the time of maturity, find the time for which the account is held if the rate of interest is 5% p.a. Ans: 2 years (5) Archana deposited 400 per month for 3 years in a bank s recurring deposit account. If the bank pays interest at the rate of 11% p.a., find the amount she gets on maturity. Ans: 16842 (6) Joseph has an account in recurring deposit scheme for 2 years. He deposits 1500 per month. If the rate of interest 8% p.a., calculate the amount he would received at the time of maturity. Ans : 39000 (7) Rajesh deposited 1000 every month in a recurring deposit account for 3 years. calculate the rate of interest if the matured value is 40,440. Ans: 8% (8) Vineeta deposits 800 per month in a cumulative deposit account for 3 years. if the amount payable at the time of maturity is 31464; calculate the rate of interest. Ans: 6% (9) Zaheeda deposits a certain sum of money,every month in a recurring deposit account for 2 years. If she receives 37875 at the time of maturity and the rate of interest is 5% , find the monthly deposit. Ans: 1500 (10) Srinidhee deposits a certain sum of money every month in the recurring deposit scheme for 5 years at 6% p.a. If the amount payable to her at the time of maturity of the account is 55320, find the monthly instalment. Ans: 800 6 (11) Shriya opened a cumulative deposit scheme with a bank for 3 years. If the rate of interest is 8% p.a. and the bank pays 1776 as interest at the time of maturity. Find the monthly deposit and the maturity value of the investment. Ans: 400 , 16176 (12) Sonya deposits 300 per month at 8% p.a. in a bank and receives 4740 on maturity of the account. Find out the total time for which the recurring deposit account is held. Ans: 15 months (13) Arnav deposits 500 every month at 12% p.a.in a bank in a recurring deposit scheme. The bank pays 5275 on maturity. Find the time for which the account is held. Ans:10 months (14) Ritika deposits 200 every month in a recurring deposit scheme at 8%p.a. If she gets 1648 as the maturity amount ,find the period for which the account is held. Ans:8 months (15) Mr Motasha opens a recurring deposit account of 600 per month at 12% p.a. If he is paid 37668 as maturity amount, how many instalments does he need to pay? Ans: 12 7 Linear Inequations (1) IF x R, Solve the following inequations and represent the solution set on the number line. 3x - 3 < 27 - 2x 3x + 7 Ans: { x : 4 x < 6, x R} (2) Solve the following inequations and represent the solution set on the number line. where x I: -2 1 6 11<5 3 6 6 Ans: {-3,-2,-1,0,1,2,3,4,5} (3) Given P= {x : 5 < 2x 1 11, x R} and Q = { x : -1 4x + 3 < 23 , x I }. Represent P and Q on number lines. write down the elements of P Q. Ans: (i) P {x | 3 < x 6, x R} (ii) Q ={-1,0,1,2,3,4} (4) Solve the following inequations and represent the solution set on the number line. 1 25 2 < 5 40, Z Ans: {10, 11, 12} (5) Solve the following inequations and represent the solution set on the number line. 8 7 2(2 ), Ans: {0, 1, 2, 3} (6) Solve the following inequations and represent the solution set on the number line. 11 15 2 < 23 , Z Ans:{-3, -2, -1, 0, 1, 2} 8 (7) Solve the following inequations and represent the solution set on the number line. 12 2 5 > 2( 2) , y I Ans: {-2, -1, 0} (8) Solve the following inequations and represent the solution set on the number line. 2 < + , R Ans: { : < , } (9) Solve the following inequations and represent the solution set on the number line. 5 < 2 + 1 5 , R Ans:{x : 3 < x 2, x R} (10) Solve the following inequations and represent the solution set on the number line. 5 < N 2 7 2 , Ans: x ={1, 2, 3} (11) Solve the following inequations and represent the solution set on the number line. + <4 , Ans : x = {0, 1, 2, 3} (12) Solve the following inequations and represent the solution set on the number line. 2 , Ans:{1, 2, 3, 4, 5} 9 (13) Solve the following inequations and represent the solution set on the number line. +1< , Ans:{x : 3 x < 2, x R} (14) Solve for x and write the solution set for the following inequations and represent it on the number line. x - 2 3 (x - 2) + 5 < 8 - x, x R Ans: {x: - x<2 , x R} (15) Solve for x and write the solution set for the following inequations and represent it on the number line. 4x - 3 2x + 7 < 6x - 1 , x Z Ans: {3, 4, 5} (16) Solve for x and write the solution set for the following inequations and represent it on the number line. x-3 + 2 2x + , x I Ans: {-3,-2,-1,0,1,2,3,4} (17) Solve for x and write the solution set for the following inequations and represent it on the number line. 13x - 5 < 15x + 4 < 7x + 12 , x R 1 Ans: { : 4 < 2 < 1, } 10 (18) Find P Q and represent it on the number line. P = {x : 8x - 1 > 5x + 2, x N} and Q = {x : 7x - 2 3(x + 6) , x N} Ans. Q = {5, 6, 7, .}; P Q = Q (19) Solve the following inequations and represent the solution set on the number line. 4x - 19 < -2 x- ,x R Ans. {x : 4 x < 5, x R} (20) Given A = { x : -1 < 2x - 5 < 11, x R}, B= { x : -11 3x - 2 10, x R} (i) represent A and B on the number lines. (ii) Also represent A B on the number line. Ans: (21)Find the values of x, which satisfy the inequations -3 < - 2 , x I. graph the solution set on the number line. Ans.{ 2, 1, 0, 1, 2} (22) A = {x : 11x 5 > 7x + 3, x R} and B = {x : 18x 9 15 + 12x, x R}. Find the range of set A B and represent it on a number line. Ans. {x : x 4, x R} 11 (23) Find the values of x, which satisfy the inequations 2 < 5 1 6 2 2 3 2, x W. Graph the solution set on the number line. ANS:{0, 1, 2, 3, 4} (24) Solve the following inequations and represent the solution set on the number line: 81 < 1 4 2 2 7 1 , x I. 2 Ans:{ -2,-1,0,1} (25) Solve the following inequations and represent the solution set on the number line: 3 5 + 2 < x + 4 + 5, x R 2 Ans:{x : -5 < x 2, x R} 12 QUADRATIC EQUATIONS (1) If one root of the quadratic equation mx2 - 9x 10 = 0 is 2, find the value of m, and also find the other root. Ans: 2, 5 7 (2) Solve the following equations and give your answer correct to 2 decimal places. 5x2- 3x 4 = 0 Ans: -0.64 (3) Solve the following equations and give your answer correct to 3 significant figures x - 18 = 6 (4) Find the nature of roots of the following equations without solving: Ans: 8.20 or -2.20 (i) 5x2 - 8x 12 = 0 (ii) 3x + 48 = 24 (iii) 2x2 - 9x + 13 = 0 (iv) 7x2 - 7x 42 = 0 Ans: (i) irrational and unequal (ii) real and equal (iii) imaginary (iv) rational and unequal (5) Without solving the following quadratic equation, find the value of m for which the given equation has real and equal roots. x2 + 2(m - 1)x + (m + 5) = 0 (6) Solve for x. give your answer correct to 2 decimal places. Ans: 4 or -1 (i) x2 - 9x 12 = 0 (ii) x + 1 = 3 (iii) 2x - 2 = 7 (iv) x2 - 11x + 25 = 0 Ans: (i) 10.18,-1.18 (ii) 2.62,0.38 (iii) 3.77,-0.27 (7) Solve and give your answer correct to 3 significant figures (iv) 7.79,3.21 (i) 2x - 1 = 7 13 (ii) 5x(x + 2) = 3 (iii) (x - 1)2 - 3x + 4 = 0 (iv) (x - 4)2 - 5x 3 = 0 Ans: (i) 3.64,-0.138 (ii) 0.265,-2.27 (iii) 3.62,1.38 (iv) 11.9,1.09 (8) Without solving the following equations, find the value of p, for which the given equation has equal roots. (i) x2 + (p - 1)x + (p + 2) = 0 (ii)(p + 6)x2 + (p + 3)x + 1 = 0 (iii)x2 + (p + 3)x + (3p + 1) = 0 Ans: (i) 7 or -1 (ii) 3,-5 (iii) 1 or 5 (9) If x = 2 is one root of the equation (k - 3)x kx 8 = 0 ,find the value of k. also, find the other 2 root of the equation. Ans: 10, - 4 (10) If x = 4 is one root of (k + 2)x - (5k + 2)x 4 = 0, find the value of k. also, find the other 2 root of the equation. 7 Ans : 5, - 1 (11) The sum of two natural numbers is 14 and the sum of their reciprocals is numbers. 7 24 . find the 7 Ans: 8 and 6 (12) The difference of squares of two natural numbers is 180.The square of the smaller number is 8 times the larger number. Find the two numbers. Ans: 18 and 12 (13) A two-digit positive numbers id such that the product of its digits is 8. if 18 is added to the number , the digits interchange their places. find the number. Ans: 24 (14) IN a two-digit number, the unit s digit exceeds its ten s digit by 1, and the product of the given number and its ten s digit is 280. find the number. Ans: 56 (15) The sum of areas of two squares is 225 m . if the difference of their perimeters is 12 m, 2 find the length of sided of the squares. 14 Ans: 9 m and 12 m (16) The length of a rectangle exceeds the breadth by 5 m. if the length was decreased by 4 m and breadth was doubled, then the area would be increased by 40 m2. find the length. Ans: 13 m (17) A wire of length 60cm is bent to form a right-angled triangle and its hypotenuse is 26 cm. find the other two sided of the triangle. Ans: 10 cm and 24 cm (18) The age of grandmother is square of her granddaughter s age. four years ago, she was 15 times the child s age. find their present ages. Ans: 7 years , 49 years or 8 years , 64 years (19) Five years ago, a women s age was the square of her son s age. four years hence, her age will be thrice that of her son s age find: (i) The age of the son five years ago (ii) The present age of the women. Ans: (i) 6 years (ii) 41 years (20) A plane travels at a distance of 2400 km at a certain speed. But on return trip due to bad weather, it reduces its speed by 50 km/h and covers the same distance in 12 minutes more than that of onward journey .Find the original speed of the plane. Ans: 800 km/h (21) In winter, a train travels a distance of 264 km at a certain speed. in summer, it travels 8 km/h faster than in winter and takes 22 minutes less than in winter. find its speed in winter. Ans: 72 km/h (22) A boat goes 12 km downstream and returns moving upstream to the same spot after 4 1 hours. the speed of the current is 2 km/h. find the speed of the boat in still water. 2 Ans: 6 km/h (23) One pipe can fill a tank in 3 hours less than the other. the two pipes can fill the tank in 3 hours 36 minutes. find the time each pipe would take to fill the tank. Ans; 9 hours and 6 hours (24) Amrita bought some pens for Rs. 360. when the price of each was reduced by Rs.3, she could buy 6 more pens for the same cost Rs.360. find the original cost of the pen. Ans: Rs.15 (25) Some glass flower vases were bought for Rs. 6000. Ten were damaged during transporting. The remaining were sold for a total profit of 1200 by selling each for Rs. 60 more than 15 what was paid. Find the number of vases bought. Ans:50 (26) In an auditorium , the number of rows was equal to the number of seats in each row . If the number of rows is doubled and the number of seats in each row is reduced by 12 , then the total number of seats is increased by 1300 . How many rows were there ? how many seats were there? (27) Find the value of k for which the following equation has equal roots x2 + 4kx + (k2 - k + 2) = 0 Ans: no. of rows=50 no. of seats=2500 Ans: k = 2 , -1 3 16 RATIO AND PROPORTION 2+ (1) If 2 2 2 = find the value of 17 8 (i) x : y (ii) 3+ 3 3 3 Ans: (i) 5:3 (ii) 76 (2) Two numbers are in the ratio 3 : 4. if 4 is subtracted from each term, then the ratio becomes 5 : 7. find the original numbers. (3) The incomes of A and B are in the ratio 5 : 8. each saves Rs. 1000 and the ratio of their expenditure is 8 : 13. find their incomes. (4) If a, b,c ,d are proportional, prove that (5) If a + c = 2b and (6) If 3 +4 3 +4 =3 1 4 3 4 5 +7 5 7 (8) If (9) If 6 3 (10) If (11) If 3+3 2 Ans: Rs.25000,Rs.40000 = 5 ++7 . 5 7 , then prove that = 2 2+ +1 +1 = 14( +1) 13( +1) = 5 ,using properties of proportion solve for a. 3 + = 172,find 171 3 +1 + x+1 3x+1 x1 3+48 12 2+64 = Ans: 24 and 32 + 1 = 2 , then prove that a:b = c:d (7) Use properties of proportion to solve for x: 4+9 49 a:b = 4 ,use properties of proportion and solve for x. 3+75 15 2+125 Ans: x=3 Ans: 1 Ans: 4:3 Ans: 8 ,calculate x : y. (12) If a,b,c are in continued proportion, prove that Ans: 4:5 (i) (a + b + c) (a b + c) = a2 + b2 + c2 17 (ii) 2+ 2+ (iii) (iv) ( (v) + 2+ = + + 2+ + + + + + 2+ 2+ + + = ) = abc =a b+c (13) If x,y, and z are in continued proportion prove that , (14) If x = 6 + , find the value of +3 3 = +3 (16) Given 3 4 =3 +4 +3 +4 3 +4 2 2+ = y4 3 (15) Using properties of proportion, solve for x : 3 4 +3 4 2 2+ 2+ 3 3 =4 2+3 2 3 2 , show that a,b,c ,d are in proportion. Ans: 2 Ans: 0 or 10 (17) Find two numbers such that the mean proportion between them is 24 and third proportional to them is 192. (18) If x = +2 + a 2b a+2b a 2b Ans: 12 and 48 , using properties of proportion show that bx2 ax + b = 0 (19) If a,b,c,d are in continued proportion , prove that (i) 3 7 7 +3 = 3 3 7 7 3+3 (ii) (a + b + c )(b + c + d ) = (ab + bc + cd)2 (20) What same number should be subtracted from 23,30,57 and 78 so that the remainder are in proportion? Ans: 6 (21) Three numbers are in continued proportion . If the middle number is 18 and the sum of first and last is 39, find the numbers. (22) What number must be added to each of the numbers 6,15,20 and 43 to make them proportional? Ans: 12,18 and 27 Ans = 3 (23) What least number must be added to each of the numbers 5,11,19 and 37 so that they are in proportional? Ans: 2 18 (24) Given that 3+3 3+3 = 63. using componendo and dividendo , find a : b. 62 (25) If x,y,z are in continued proportion, prove that ( + ) ( + ) (26) If b is the mean proportion between a and c show that (27) If 7 +2 7 2 Ans: 3 : 2 = 4+ 2 2+ 4+ 2 2+ = = 5 , use properties of proportion to find the value of 3 (i) m : n (ii) 2+ 2 Ans: (i) 8:7 (ii) 113 15 (28) Using properties of proportion , solve for x. Given that x is positive: 2 + 4x2 1 2x 4x2 1 =4 Ans: (29) If x= 2 +1 + 2a 1 2a+1 2a 1 , prove that x - 4ax + 1 = 0. (30) Using properties of proportion find x:y , given : 2+2 2 +4 = 5 8 2+3 3 +9 Ans: 2:3 19 FACTORISATION OF POLYNOMIAL (1) Find the remainder when 4x3 + 6x2 - 8x - 10 is divided by (2x + 1). Ans: -5 (2) If the remainder is 7 when 2x3 - 3x2 + ax - 5 is divided by (2x - 3), find the value of a. Ans: a = 8 (3) When 3x + ax + bx - 6 is divided by (x + 2) ,the remainder is 20 and (x - 2) is a factor of the 3 2 polynomial. find a and b. Ans: a = 4 and b = -17 (4) What should be subtracted from the polynomial 2x3 + 5x2 - 11x - 10 so that (2x + 7) is a factor? (5) prove that (x - 2) is a factor of x3 - 7x + 6. hence, factorise the given expression. (6) Using the Remainder theorem, factorise completely the following polynomial 3x3 + 2x2 - 19x + 6. Ans: a = 4 Ans: (x - 2)(x + 3)(x - 1) Ans: (x - 2)(x + 3)(3x -1) (7) prove that (2x - 1) is a factor of 2x3 + x2 13 + 6. hence, factorise the given polynomial. (8) If (3x + 1) is a factor of 3x3 + 4x2 - 35x - 12, factorise the expression. Ans: (2x - 1)(x + 3)(x - 2) Ans: (3x + 1)(x + 4)(x - 3) (9) When the two polynomials x3 - px2 + x + 6 and 2x3 - x2 - (p + 3)x - 6 are divided by (x - 3), the remainder is same. find the value of p. Ans: p = 1 (10)(x2 + 2x - 15) is a factor of x3 + ax2 + bx - 30. find the values of a and b. (11) Using the Remainder theorem, factorise completely the following polynomial: x3 + 10x2 - 37x + 26. Ans: a = 4 and b = -11 Ans: (x -1)(x + 13)(x - 2) (12) Find the value of k if (x - 2) is a factor of x + 2x kx + 10.hence, determine whether (x + 5) 3 2 is also a factor. Ans: k = 13, yes (13) What should be added to 2x3 + 5x2 - 28x - 18 so that (x - 3) is a factor of the resulting polynomial? (14) Given that (x + 2) is a factor of (3x + 4)3 - (5x + a)3 , find the value of a. ANS: 3 Ans: a = 8 20 (15) Find the value of a ,if (x - a) is a factor of x3 - a2x + x + 2. Ans: -2 (16) Show that 2x + 7 is a factor of 2x3 + 5x2 - 11x - 14. Hence factorise the given expression completely, using the factor theorem. Ans: (x - 2)(x + 1)(2x + 7) (17) If (x - 2) is a factor of 2x - x px 2 3 2 (i) Find the value of p. (ii) with the value of p factorise the above expression completely. Ans: (i) 5 (ii) (x - 2)(x + 1)(2x + 1) (18) Find the a if the two polynomials ax3 + 3x2 - 9 and 2x3 + 4x + a ,leave the same remainder when divided by x + 3. Ans: 3 (19) What must be subtracted from 16x - 8x + 4x + 7,so that the resulting expression has 2x + 1 3 as a factor? Ans: 1 (20)Using the remainder theorem find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2.hence, find k if the sum of two remainders is 1. (21) Use the factor theorem to factorise 6x3 + 17x2 + 4x - 12 completely. Ans = -2 Ans: (x + 2)(x + 1)(2x + 5) (22) What must be added to the polynomial 2x - 3x - 8x, so that it leaves a remainder 10 when 3 divided by 2x + 1? 2 Ans: 7 21 MATRICES 1 3 4 (1) A= [ ] and B = [ 5 6 7 2 ] , find (i) A + B 8 9 (2) If P = [ 3 (3) If [ 2 ]+X= 4 [ 4 7 5 (4) If [ 8 ] , find (i) P 7 6 7 5 2 ] [ 9 3 1 4 3 2 (6) If A=[ 5 ] =[ +4 1 0 0 4 4 2 6 3 (7) Let A = [ ] and B = 5 4 6 [ 2 3 (iii) P P 1 2 2 1] 8 (ii) [ + 10 3 2 2 2 2 ] 15 ] 12 Ans: 1 [ 1 7 1 4 5 6 1 ] ] 4 ] = 3 + 4 3 [ ] 5 1 1 (ii) x = 3,y = 1 or 3,z = 4 ], Find the matrix D such that 3A - 2B + 2D = 0. ] . find : (i) A + A Ans: [ 6 Ans: (i) [ 1 (9) A=[3 7 5] and B =[ 4 2 ] . Is AB possible? If so, find the product. 8 7 ] [ ] 3 4 6 (i) Write the order of matrix X. 4 6 (ii) A A where A is the transpose of A. 7 and B =[ .Is A x B possible? Give reason.If so, find the product. 6 8 1 2 Ans: (i) x = 3,y = 2, z = -7 4 2 14 ] (ii) [ Ans: [ 3 (8) A= [ 5 (10) Given [ 6 9 7 18 Ans: (i) [ ] (ii) [ 8 6 15 0 1 (iii) [ ] 1 0 8 4 4 12 ] + M, Find the matrix M. (5) Find x,y,z in the following: (i) [ Ans: (i) [ 4 ] ,Find the matrix X. 1 2 ] -M=[ 6 7 4 (ii) P + P (ii) A B X= (ii) Find the matrix X. 5 ] 11 1 0 ] (ii) [ 9 12 9 ] 0 53 Ans: [ ] 83 Ans: [41 Ans: (i) 2 x 1 2 (ii) [ ] 3 46] 22 2 3 4 1 6 7 ] ,B = [ ] ,C = [ ] . find A(B + C), AB + AC and draw your 4 5 1 2 3 1 conclusion from the result. (11) If A = [ 4 2 0 (12) If A = [ ] , B=[ 6 3 1 5 (13) If X+Y =[ 11 10 2 2 ] and C=[ 1 1 ] and X-Y = [ 4 9 2 5 14 3 1 ] ] , Find A2 A + BC. ,Find the matrices X and Y. 25 ] 47 2 Ans: [ 3 2 2 3 Ans: X= [ ] ,Y= [ 7 6 4 5 1 2 ] . find k if M - 6M + kI = Null Matrix, Where I is an identity matrix of 1 2 order 2 x 2. (14) Given M = [ 4 Ans: [26 50 2 3 8 and R = [ (15) Let Q= [ ] 5 6 19 (16) If [ 4 3 (17) If A = 5 1 4 [ 3 9 [ [ 2 sin 30 2 cos 0 8 2 ] 4 1 tan 45 30 (20) Evaluate: 2 ]+3 [ ] 6 =[ 15 1 2 ] 1 and B = [ sec 60 3 4 ] [ ] sin 90 4 3 ] 8 ] 9 Ans: k = 9 ]. find the matrix P if PQ = R. Ans: 1 [ 7 ,find x and y. 5 7 , B= and AX = B .Find the (i) order of matrix X, (ii) matrix X. ] [ ] 4 6 (18) Given A= [ (19) Evaluate: ][ 5 15 4 2 ] 1 Ans: x = 3, y = 4 2 Ans: (i) 2 x 1 (ii) [ ] 3 ] and AB = I ,find the values of x and y. 9 Ans: x = -2 and y = -4 Ans: [ 11 10 10 ] 11 4 cos 60 8 ] [ ] 3 tan 30 9 (21) Find the values of x and y if 22 3 [ ] [ 2] = [ ] y 2y 5 36 26 Ans: [ ] 25 23 (22) Given P= [ (23) Given [ 4 7 ] and Q = [ ] .if PX = Q ,find the matrix X. 4 11 3 b 3 2 b] [ 4 3 6 ] = [14 ] ,B= [ 4 5 Ans: a=1 , b= 4 2] and PA=B (ii) the matrix P. 4 2 6 3 ],B=[ 2 (27) Find x and y, if [ (28) Evaluate: 2 (29) If A= [ 1 3 1 3 ][ ] 3y 2 y 4 sin 30 [ sin 90 2 4 ],C= [ 4 5 2 2 0 ], B = (31) Find x and y if : [ 2 (32) A = [ 1 0 2 [ 6 9 ],B=[ 2 Ans : [ 16 3 21 Ans: [13 14 2 and I is the identity matrix of the same order and At is the ] 3 14 ] 13 Ans: 11 [ 16 0 ] Find X such that A + 2X = 2B + C. 3 3 y (ii) [7 4 2 4 ], C = [ 0 0 2 3 2 5 ][ ]=[ ] 4y 1 12 4 3 1 2 ],C= 8] ] Ans: x=2, y=1 2 cos 60 4 5 ][ ] 2 cos 0 5 4 5 4 ], B= [ 3 1 6 3 ] .Find A2 A + BC. Ans: (i) 1 x 2 16 =[ ] transpose of matrix A , and At. B + BI (30) A = [ Ans: a = -2, b = 5 20], find a and b. find: (i) the order of matrix P. (26) A = [ 3 Ans: [ ] 5 5 3 ] [ ] = 7 [3 ] ,find a and b. 2 2b 2 6 3 Ans: x = 2, y = 3 2 (24) Given [a (25) A [ 1 3 2 ] Find A2 + AC 5B. [ 1 4 Ans : 2 [ 3 2 ] 5 ] 1 Ans: x=1, y=2 24 (33) A = [ 2 1 (34) Simplify: 0 1 0 ],I=[ ] and A2 = 9A + MI. Find M. 7 0 1 sin sin A [ cos 3 (35) If A = [ 5 cos cos ] + cos A [ sin sin 0 4 ], and B= [ 1 1 sin ] cos 2 , find A2 2AB + B2 ] 0 Ans: 23 [ 17 3 ] 6 Ans: -14 Ans: 1 0 [ ] 0 1 Ans: [51 54 20 ] 17 25 ARITHMETIC PROGRESSION (1) Write the 17th term of the AP, whose first 2 terms are -2 and 3. Ans: 78 (2) If the third term of an AP is 11 and the seventh term is 27 ,find the fifth term. Ans: 19 (3) Which term of the AP: 23,44,65,86, ., is 212? Ans: 10 (4) Which term of the sequences 30, 29 1 , 28 1 , 27 1 , ., is the first negative term? 2 2 2 (5) Insert 5 numbers between 5 and 14, so the resulting sequences is in AP. Ans: 42nd Ans: 5,6.5,8,9.5,11,12.5,14 (6) The general term of a sequence is given by an = -3n + 5. Is the sequence in an AP? If so find the 11th term and common difference. (7) The n term of two APs is same.find the value of n and the n term. th AP1 : 57,54,51, th AP2 : 1,5,9, (8) Find the number of integers between 50 and 300 which are divisible by 7. (9) (3x + k), (2x + 9) and (x + 13) are 3 consecutive terms of an AP, find k. Ans: -28 Ans: n = 9 and 33 Ans: 35 Ans: 5 (10) Find the 20th term of an AP whose third term is 7 and the seventh term exceeds 3 times the third term by 2. also, find the nth term. Ans: 75, 4n-5 (11) An AP has 21 terms .The sum of 10 ,11 and 12 terms is 129 and the sum of the last 3 terms th th is 237.Find the AP. (12) Find the 6th term from the end of the A.P. 19,14,9, .(-46). (13) Find the 6 term from the end of AP. th 17,14,11, .., which has 30 terms. th Ans: 3,7,11, .75,79,83 Ans: -21 Ans: -55 (14) The sum of three numbers in an AP is -12 and their product is 36. Find the numbers. Ans: -9,-4,+1 26 (15) Find 4 numbers in AP whose sum is -4 and the sum of whose square is 84. (16) Find the 5 numbers in AP, whose sum is 12 1 2 and ratio of first to last term is 2:3. Ans:2, 9 Ans: -7,-3,1,5 , 5 , 11 ,3 or 4 2 4 1 2,2 , 21 , 23 4 2 4 ,3 (17) Divide 22 into 4 parts which are in AP.such that the ratio of the product of extremes to the product of means is 5:14. (18) Find the sum of 17 terms of an AP, whose middle term is 40. (19) In an AP,18th term is 48 and 32nd term is 104. find Ans: 1,4,7,10 Ans: 680 (i) the first term and the common difference. (ii) the sum of first 50 terms. Ans: (i) -20 (ii) 3900 (20) Solve for x: 1 + 4 + 7+ .+ x = 247 Ans: 37 (21) The sum of first 6 terms of an AP is 63. The ratio of its 10th term to its 20th term is 1:2. Calculate the first and 15th terms. (22) The ratio of 6th and 14th terms of an AP is 2:5. Find the ratio of Ans: 45 (i) 4th and 19th terms, (ii) the sum of first 5 terms and the sum of first 12 terms. Ans: (i) 2:11 (ii) 1:6 (23) An AP consists of 29 terms. The sum of 3 middle most terms is 375 and the sum of the last 3 terms is 531. find the AP. (24) How many terms of the AP 7,10,13, are needed to get a sum of 710? Ans: 69,73,77,81, Ans: 20, - (25)(i) find the sum of all odd numbers between 100 and 300 (ii) find the sum of all multiples of 7 lying between 500 and 800. (26) The sum of first n terms of an AP is 3 2+5 2 . find the 25th term. 71 3 not possible Ans: (i) 20000 (ii) 27993 Ans: 76 27 (27) The sum of first n terms of an AP is given by n + 8n . find its 12th term. Ans: 31 (28) Find 4 consecutive terms in an AP whose sum is 22 and the sum of 3 , and 4 term is 21. rd th Ans: -2,3,8,13 (29) The sum of first 15 terms is 105 in an A.P. the sum of next 15 terms is 780. find the first 3 terms. Ans: -14,-11,-8, 28 (1) Plot A(2,3) and B(6,3) REFLECTION (i) Reflect A in the origin to get the image D. (ii) Reflect A in the x-axis to get the image C. (iii) write the coordinates of C and D. (iv) What kind of figure is ABCD? find its area. (v) What is the reflection of C in the y-axis? (vi) Name two points from the figure which are invariant on reflection in y-axis. Ans: (iii) C(2,-3),D(-2,-3) (IV) parallelogram, 24 sq.units (v) D(-2,-3) (vi) (0,0),(0,-3) (2) Plot P(5,3) on a graph paper. The point P (5, 3) was reflected in the origin to get the image P . (i) Write down the co-ordinates of P . (ii) If M is the foot of the perpendicular from P to the x-axis, find the co-ordinates of M. (iii) If N is the foot of the perpendicular from P to the x-axis, find the co-ordinates of N. (iv) Name the figure PMP N. (v) Find the area of the figure PMP N. Ans: (iv) parallelogram (v) 30 sq.units (3) Plot P(2,4), Q(-2,1),R(5,0) . Reflect points P and Q in x-axis to get P and Q . (i) Write the coordinates. (ii) Give a geometrical name to the figure formed by joining the points PQQ P R. Find its area and perimeter. (iii) Name its axis of symmetry and write its equation. (iv) Name two points from the figure which are invariant on reflection in x-axis. Ans: (ii) pentagon, 32 sq.units,22 (iii) x axis, y=0 (iv) (5,0), (-2,0) (4) Plot P(6,3) and Q(3,0). refletct P in the x-axis to get P . Write the co-ordinates of P .O is the origin. give the geometrical name of POP Q. Write the equation of the line of symmetry for the figure POP Q. find its area. (5)(i) Plot the points A(4,6) and B(1,2) on the graph paper. Ans: arrowhead, x-axis, y = 0,9 sq.units (ii) A is the image of A when reflected in x-axis. 29 (iii) B is the image of A when reflected in the line AA (iv) Give a geometrical name to the figure ABA B . Ans: (ii) A =(4,-6) (iii) B =(7,2) (iv) kite (6)(i) Plot A(0,5),B(2,5),C(5,2),D(5,-2),E(2,-5) AND F(0,-5). (II) Reflect B,C,D AND E on the y-axis and name these B ,C ,D and E res. and write their coordinates. (iii) Name the figure formed by BCDEE D C B . Ans: (iii) octagon 30 SECTION FORMULA (1) P divides the line segment joining A(1,-6) and B(6,4) in the ratio 2:3. find the coordinates of P. Ans: (3,-2) (2) Find the ratio in which (4,b) divided the line joining the points (6,-2) and (-3,16).also, find b. Ans: 2:7, b = 2 (3) In what ratio is the line joining the points (4,3) and (-10,-4) divided by the x-axis? also, find the coordinates of the point of intersection. Ans: 3:4, (-2,0) (4) In what ratio does the line y=3 divide the line joining the points A(2,6) and B( -12,-1)? find the coordinates of the point of intersection. (5) G(2,5) is the centroid of ABC where A= (4,1). find the midpoint of side BC. Ans: 3:4, (-4,3) Ans: (1,7) (6) In the adjoining figure, P(-4,3) divides AB in the ratio 3:2. find the coordinates of A and B. (7) If A(-6,0), and B(0,6) , find the points of trisection of the line segment AB. (8) If A =(-3,7) and B=(1,9) , find the coordinates of the midpoint of AB. Ans: A=(-10,0), B(0,5) Ans: (-4,2) or (-2,4) Ans: (-1,8) (9) If A=(3,-4) and midpoint of the line segment AB is (5,1) ,find the coordinates of B. Ans: (7,6) (10) In the given figure, M(3,-2) is the midpoint of AB. if A and B are on x-axis and y-axis res. find the coordinates of A and B. Ans: A(6,0) , B (0,-4) (11) If ABCD is a parallelogram and A= (3,-1) ,B = (5,6),C = (7,3) , find the coordinates of D. Ans: (5,-4) 31 (12) The centroid of ABC is (-1,4). if A= (5,-6), B = (-2,3), find the coordinates of C. Ans: (-6,15) (13) The line segment joining A(2,3), B(6,-5) is intercepted by the X axis at the point K. write the coordinate of the K. Hence ,find the ratio in which K divides AB. (14) find the coordinates of the centroid of a triangle whose vertices are: A(-1,3), B(1,-1) , C(5,1) Ans: 0, 3:5 Ans: (5 , 1) 3 (15) The midpoint of the line segment joining (2a,4) and (-2,2b) is (1,2a + 1). find the values of a and b. (16) If The line joining the points A(4,-5), B(4,5) is divided by point P so that coordinates of P. Ans: a = 2, b = 3 = 2 . Find the 5 Ans: (4,-1) (17) Given a line segment AB joining the points A(-4,6) B(8,-3) .find: (i) the ratio in which AB is divided by the y-axis. (ii) find the coordinates of the point of intersection. Ans: (i) 1:2 (ii) (0,3) (18) In the given figure, the line segment AB meets X-axis at A and Y-axis at B. The point P(-3,4) on AB divides it in the ratio 2:3. Find the coordinates of A and B. Ans: A(-5,0), B(0,10) (19) Calculate the ratio in which the line joining A(-4,2), B(3,6) is divided by point P(x,3). Also find x. Ans: 1:3, 9 (20) P (1,-2) is a point on the line segment A(3,-6), B(x,y) such that AP:BP is equal to 2:3. Find the coordinates of B. (21) M and N are two points on x-axis respectively. P(3,2) divides the line segment MN 4 Ans: B(-2,4) in the ratio 2:3 find: (i) the coordinates of M and N. 32 (ii) slope of the line MN. Ans: (i) M(5,0), N(0,5) (22) In which ratio is the line joining P(5,3), Q(-5,3) divided by the y-axis? Also find the coordinates of the point of intersection. (ii) -1 Ans: 1:1 , (0,3) 33 EQUATION OF STRAIGHT LINE (1) Prove that A (6,6) , B (-2,0) and C (2,3) are collinear points. (2) If A (-3,5), B (2,-1), C(a,11) are collinear ,find the value a. Ans: -8 (3) Line 3x ky = 1 is perpendicular to the line AB where A (7, -2) and B (-5,6). Find the value of k. Ans: 2 (4) In the given circle, C(-1,-4) is the centre and A(5,1) is a point on the circle. IF AT is a tangent to the circle, find the value of k if T(k, -2). Ans: 15 2 (5) Given A = (k,4), B = (6,-3),C = (1,8), D = (5,-6).find the value of k if (i) AB || CD (ii) AB CD. Ans: (i) 4 (ii) 61 (6) In the given diagram, equation of AB is 3y = x + 2 3 and equation of AC is 3x y 2 = 0. 2 (i) write down the angles which the lines AB and AC make with the positive direction of x-axis. (ii) hence, find BAC. Ans: (i) ABC = 30 , ACX = 60 (7) If the point (2m , 3m -1) lies on the line 4x - 3y = 1, find the value of m. (ii) BAC = 30 Ans: 2 (8) The line through A(a,6) and B(9,5) is parallel to the line through C( 4,1) and D(7,4). find the value of a. (9) If A(p, -1) ,B(3,-8), and C(5,6) are collinear, find the value of p. (10) A(1,k), B(3,5) and C(6,2) are vertices of ABC. if ABC = 90 , find the value of k. Ans: 10 Ans: 4 Ans: 3 34 (11) The gradient of the line joining P(6,k) and Q (1 - 3k, 3) is 1. (i) find k. (ii) find the midpoint of PQ. 2 Ans: (i) -11 (ii) (20,-4) (12) If lines 4y - 7x = 12 and ay - 8x = 1 are perpendicular to each other, find the value of a. Ans: -14 (13) If the lines through A(a,4) and B(5,-2) is perpendicular to the line 2x - 4y = 1 , find the value of a. Ans: 2 (14) If the line 4x - 3y = 9 is perpendicular to the line through A(1,-2) and B(k,4) ,find the value of k. Ans: -7 (15) Find the measure of angle in the figure. Ans: 45 (16) In the given figure, equation of AB is x - 3 y +1 = 0 and equation of AC is x y 2=0 (i) Write down the angles that AB and AC make with the positive direction of x-axis. (ii) Find BAC. (17) Find the equation of a line through A(4,-3) and B(6,5). (18) Find the equation of a line passing through (-3,2) and parallel to 5x + 4y = 6. Ans: (i) 30 , 45 (ii) 15 Ans: y = 4x-19 Ans: 4y = -5x - 7 or 5x + 4y + 7 = 0 (19) Find the equation of a line passing through (4,-5) and perpendicular to 2x + 3y = 1. Ans: 2y = 3x - 22 (20) Find the equation of a line through the point of intersection of the lines 2x + 5y = 9 and 5x - 2y = 8 and perpendicular to the line 4x + 3y = 7 35 (21) In the triangle A (2,-3), B (6,7), C (-8,5) find the equation of Ans: 4y = 3x - 2 (i) the altitude through A. (ii) the median through A. Ans: (i) y = -7x + 11 (ii) y = -3x + 3 (22) ABCD is a kite, B = (7,-1), and D = (9,-5) form the shorter diagonal. find the equation of the longer diagonal AC. Ans: 2y = x - 14 (23) Find the equation of a line through (-6,7) and with the x-intercept of 5 units. Ans: 7x + 11y = 35 (24) Given equation of line L1 is y = 4. (i) write the slope of the line L2 if L2 is the bisector of angle O. (ii) write the coordinates of point P. (iii) Find the equation of L2 Ans: (i) tan 45 = 1 (25) Find the equation of a lines equally inclined to the co-ordinates axes and passing (ii) P(4,4) (iii) y = x through the point P (5,-1). Ans: (i) y=x-6 (ii) x+y=4 36 (26) In the given figure, find the equation of the line through P (5,4) making an angle of 45 with x-axis. Also, find the coordinates of Q and R. Ans: y = x - 1 (27) Find the equation of a line passing through the point (3,2) and its x-intercept is double of y-intercept. (28) A line passes through the point P (2,9) and cuts off positive intercepts, on the Q(1,0), R(0,-1) Ans: x + 2y = 7 x-axis and y-axis in the ratio 2:3 .Find the equation of the line. Ans: 3x + 2y = 24 (29) ABCD is a rhombus.the coordinates of A and C are (3,6) and (-1,2) res. write down the equation of BD. ANS: x + y = 5 (30) Write down the equation of the line whose gradient is 3 and which passes through P, where 2 P divides the line segment joining A(-2,6) and B(3,-4) in the ratio 2:3. (31) A(1,4), B(3,2),C(7,5) are the vertices of a ABC .find: Ans: 2y = 3x + 4 (i) the coordinates of the centroid G of ABC. (ii) the equation of a line ,through G and parallel to AB. Ans: (i) (11 , 11 ) 3 3 (ii) 3x+3y=22 (32) A straight passes through the points P(-1,4) and Q(5,-2). It intersects the coordinate axes at points A and B. M is the midpoint of the segment AB. Find : (i) the equation of the line . (ii) the coordinates of A and B. 37 (iii) the coordinate of M . Ans: (i) x+y=3 (ii) A(3,0), B(0,3) (iii) (1.5,1.5) (33) Find the value of k for which the lines kx - 5y + 4 = 0 and 4x - 2y + 5 = 0 are perpendicular to each other. Ans: 5 2 (34) P(3,4), Q(7,-2) R(-2,-1) are the vertices of triangle PQR. Write down the equation of the median of the triangle,through R. (35) In the given figure, write Ans: 7y = 2x - 3 (i) the coordinates of A,B, C (ii) the equation of the line through A and parallel to BC. ANS: (i) A(2,3),B(-1,2),C(3,0) (ii) x + 2y = 8 (36) If the lines y = 3x + 7 and 2y + px = 3 are perpendicular to each other,find the value of p. Ans: 2 (37) Find the equation of the line parallel to the line 3x + 2y = 8 and passing through the point (0,1). (38) Points A and B have coordinates (7,-3) and (1,9) res. find: 3 Ans: 3x + 2y = 2 (i) the slope of AB (ii) the equation of the perpendicular bisector of the line segment AB. (iii) the value of p if (-2,p) lies on it. Ans: (i) -2 (ii) 2y=x+2 (iii) 0 38 (39) Find the value of p for which the lines 2x + 3y 7 = 0 and 4y px 12 = 0 are perpendicular to each other. Ans: 6 (40)A and B are two points on the x-axis and y-axis res. P(2,-3) is the midpoint of AB. find the (i) coordinates of A and B (ii) slope of line AB (iii) equation of line AB. Ans: (i) A(4,0), B(0,-6) (ii) 3 (iii) 2y=3x-12 2 (41) The equation of a line is 3x + 4y 7 = 0 .find: (i) the slope of the line (ii) the equation of a line perpendicular to the given line and passing through the intersection of the lines x y + 2 = 0 and 3x + y 10 = 0 . (42) ABCD is a parallelogram where A(x,y), B(5,8),C(4,7),D(2,-4) .find (i) coordinates of A (ii) equation of diagonal BD. Ans: (i) 3 (ii) 4x-3y+4=0 4 Ans: (i) (3,-3) (ii) y=4x-12 (43) The line through A(-2,3) , B(4,b) is perpendicular to the line2x - 4y = 5. find the value of b. (44) In ABC ,A(3,5), B(7,8),C(1,-10).Find the equation of the median through A. Ans: -9 Ans: y + 6x = 23 (45) Find the value of a for which the following points A(a,3) ,B(2,1) and C(5,a) are collinear. hence find the equation of the line AB. Ans: 4 or -1, 2x+3y=7 (46) A line AB meets X-axis at A and Y-axis at B. P(4, 1) divides AB in the ratio 1:2. or y=x-1 (i) Find the Coordinates of A and B. (ii) find the equation of the line through P and perpendicular to AB. 39 Ans: (i) A(6,0), B(0,-3) (48) The slope of a line joining P(6,k) and Q (1 - 3k, 3) is (i) k. 1 2 (ii) 2x + y = 7 .Find (ii) Midpoint of PQ using the value of k found in (i). (49) A(-1,3), B(4,2),C(3,-2) are the vertices of a triangle. Ans: (i) -11 (ii) (20,-4) (i) Find the coordinates of the centroid G of the triangle. (ii) Find the equation of the line through G and parallel to AC. Ans: (i) G(2,1) (ii) 5x + 4y = 14 (50) The vertices of a ABC are A(3,8), B(-1,2), C(6,-6) .Find (i)Slope of BC. (ii) Equation of a line perpendicular to BC and passing through A. Ans: (i) 8 7 (ii) 8y = 7x + 43 (51) Find the value of P if the lines , 5x - 3y + 2 = 0 and 6x py + 7 = 0 are perpendicular to each other. Hence find the equation of a line passing through (-2,-1) and parallel to 6x py + 7 = 0. Ans: -10, 3x + 5y + 11 = 0 40

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