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Multiple Choice Questions (MCQs), 1., , Which of the following is a solution of the 9, The sum of two numbers is 137 and their, + 5y, 3x, 8?, difference is 43. Algebraic representation of the, equation, (b)x, =, 0,, y, =, 1, 0, 1,, y, (a) r, situation can be, (d) x= 0, y = 0, (), 1,y =1, ( a ) x - y = 137, r, +y = 180, =, , =, , 2., , Which of the following is not a solution of, , (b) 2(x+ y) 137, 2(x, (c), +y =137, x-y, (d) +y 43, x-y =, =, , the equation 3x + 2y = 5?, , (a), , =, , 1, y, , x =, , 3, y, , x = 3, y = - 2, , =4 (d) r=1, y, , (c)x=-1, y, 3., , (b), , 1, , =, , = 4 is a, , solu ion, , =, , =, , (b), , =, , (a), , 2x+ 3y -, , (c), , 2x-3y +17 =0, , 4., , 2 , y = l is a solution of the linear equation, =, , 17=0, , (d), , 2y = 5, , The point of intersection of the lines, represented by 3x -2y =6 and the x-axis is, , (b) (0,-3), (d) (0, 3), , c)-2,0), , 5 chairs and 4 tables together cost 2800, while 4 chairs and 3 tables together cost 2170., can be, Algebraic representation of the situation, 6., , (a) 5x-4y, (b) 5x + 4y, (c) 5x + 3y, (d) 5x-3y, , =, =, =, =, , 2800, 4x3y =2170, 2800, 4x+ 3y =2170, 2800, 4x+ 3y 2170, 2800, 4r 3y 2170, , denominator, , =, , (a)r+y =8,, , )-8, 8., , a, , (b) *+y= 8,+1- 2, y, , daughter is, , one-third the age, , of mother is, If the present age, , years, then the age, after 15 years is, , (in years), , X, , of the daughter, , ( b ) T + 1 5, , (c)x+5, , be, , (d) +y=8, y+1 2, , ot her mother., , (a)+15, , Algebraic, , y+1, , The age of, , 3, , d)-15, , (c), , always coincident, , (d), , intersecting or coincident, , 12. The, , pair of equations x, , -3x - 9y, , +2, , (a), , a, , (b) exactly, , 3y +5 =0 and, , 0 has, , two solutions, , infinitely many solutions, no solution, , pair of equations, , 2x+y-6, (a), , =, , +, , unique solution, , 13. The, , 7. The s u m of the numerator and, is increased, of a fraction is 8. If the denominator, , 2, , (a) intersecting, (b) parallel, , (d), , =, , can, , pair oflinear equations in two variables, is inconsistent, then the lines represented by, 11. Ifa, , (c), , =, , by 1, the fraction becomes, representation of the situation, , )-(), , two equations are, , 3x- 4y = 8, , 5., 5., , (a) (2, 0), , 1, , (b), , (a), , (d) 2x+ 3y +17 =0, b) 4x, , (a) 2x+ 7y 11, (c) x-3y =5, , 137, , 10. The value ofa so that the point (3, a) lies, 3y 5, is, on the line represented by 2x, , of the linear, , 3x + 2y, , 43, , =, , ., , equation, 17 = 0, , -y), =43, , a, , 2x, , -, , 3y, , +, , 4, , 0 and, , =, , 0 has, , unique solution, , (b) exactly two solutions, (c) infinitely many solutions, (d) no solution, , pair of equations 9x + 3y + 5 0 and, 6x+2y +7 =0 represents lines which are, (a) parallel, =, , 14. The, , (b) intersecting, (c) coincident, , at one, , (d) perpendicular, 15. The, , to each other, , pair of equations, , (a), , one solution, , (b), , two solutions, , (c) infinitely, (d), , point, , y, , =, , many solutions, , no solution, , 3 and y, , =, , 8 has

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16. The pair of equations* 1 nnd y, , is, , (c) intersecting lines which are porpendicular, (d) intersecting lines but not perpendicular, 17. If am, , (), , (b), , (a), 28., , bl, then the pair of equations ax +, , (d), , 3, , The pair of equations x + 2y +5 = 00 ane, and, , 0 have, , -3x 6y +1, , by = c, lx + my =n, , (a), (b), (c), (d), , nd, 3 and, , 27. The value of y, when, , represents, (n) parallel ines, (b) coincident lines, , (a) a unique solution, (b) exactly two solutions, , has a unique solution, has no solution, has infinitely many solutions, may or may not have a solution, , (), , infinitely many solutions, , (d), , no solution, , 18. The value of k for which the system of, , 29. If a pair of linear equations is consistent,, , equations r + 3y 3, has no, lution, is, , then the lines will be, , (a), , 10 (b), , 0 and 4x + ky + 7 =0, , 7, , (d) 12, (c) 3, 19. Find the value of k for which the system, , of equations kx y, , = 4, 10x - 2y = 3 has no, , solution., (a), , (b), , 6, , 4, , (c)5, , d), , 2, , 20. The point of the intersection of the lines, x - 3 = 0 and y, , - 5, , 0 is, , (a)-3,5), (c), , parallel, , (b) always coincident, (c), , intersecting or coincident, , (d) always intersecting, 30., , The pair of equations y = 0 and y =-7 has, , (a), (b), , one solution, two solutions, , (infinitely many solutions, (d) no solution, , (b) (3, 5), (d) (3, -5), , (0, -5), , (a), , 21. Find the conditions to be satisfied by, , 31. For what value of k, do the equations 3r- y, , coefficients for which the following pair of, , + 8 0 and 6x - ky = -16 represent coincident, 1ines?, , equations ax + by + c = 0, dx + ey +f= 0 represent, , coincident lines., , (a) ab, , ed; bf = ce, , (b), , ae = bd; bc, , ad = bc; bf = ce, , (d), , ae = bd; bf = ce, , (a), , = ef, , 1/2, , (b)-1/2, , (c) 2, , 32. If the lines given by 3x + 2ky, , 22. The difference between twonumbers is 24 and, one number is three times the other number., 36, 12 (b), , 15, 4, , 30, 10 (), , 40, 14, , (d), , 33, 11, , cx-y, , 58°, 32°, , (b) 53, 370, , many solutions is, (a) 3, , (), , 64°, 26°, , (d) 29°, 63, , (c), , -12, , 34., , If r, , 24. In a AABC, if 2 C =50° and 2A exceeds 2B, by 44°, then 2A =, , (a) 43, , (b) 40, (c) 67, (d) 87, If 3Y= 9 and x -2y=6 represent a system, , 25., of the equations, then the value of r + y is, , (b) -6, of, , equations, , ( a ) x = 9, y = 5, , (b), , * = 5, y = 9, , (c)x= 9, y = 9, , (d), , x = 5, y = 5, , (b)-3, (d) no value, , = a, , and y =, , b is the solution of the, , equations x-y = 2 and x+y = 4, then the values, , of, , (a), , and b are,, 3 and 5, , (c), , 3 and 1, , a, , respectively, (b) 5 and 3, (d) -1 and -3, , 35. Aruna has only? 1 and, , (d) None of these, , pair, , 3, , 2 and 6x - 2y = 3 will have infinitely, , (a), , 26. The solution of the, 1 4 and x - y =4 is, , ), , 33. The value of c for which the pair of equations, , 23. The larger of two complementary angles, exceeds the smaller by 16 degrees. Find them., , (a)-2, (c)4, , = 2 and 2x +, , 5y + 1 = 0 are parallel, then the value of k is, , Find them., (a), , (d)-2, , 2 coins with her., , f the total number of coins that she has is 50, r, , +, , y, , and the amount, , then, respectively, , of money with her is 7 75,, , the number of 7 1 and 7 2 coins are,, , (a), , 35 and 15, , (b) 35 and 20, , (), , 15 and 35, , (d) 25 and 25

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Pair of Linear Equations in Two Variables, 36. The, , father's, , age is six times, his son's age., Four years hence, the age of, the, father will be, four times his son's age, The present, , years) of the son is, (a), , 4, , (b), , age, (in, , 5, , (c) 6, 37. The following pair of, linear, x+3y 7- 0, 2x + 9y = 5, (a), , has a unique solution, , (b), , has, , (c), , no solution, , (d), , may, , infinitely, or, , many, , equations:, , system, , of, , (d), =, , -, , b) k, , 2, , (c)k 2, , If u +, , 2, , k* 3, , )k5, , u, , 20, , +, , 2, , k, , (d), , 21 and find the, , =, , 39x +6y, , (c) 12, , pair of linear equations, k, , =, , k = 1, , 13x +, , 4 has infinitely many, (b) k =2 () k= 4, +, , -=Oand b, (a), , (d) 18, , pair, , of, , ky, , =, , k and, , solutions if, (d), , k = 6, , equations, , ab = a +b is, y, , x = a, y = b, , (b), , ( c ) x = -a, y = b, , x = a, y = - b, , (d) None of these, , 6x-4y=, , (d) k-2, , following pair, , of linear equations:, x+3y 7 =0, 2x + 6y = 14 have, (a) One solution, , (a), , x = 2, y =, , (b), , (C)x = 6, y = 7, , 51., , Graphically, , x = 4, y = 4, , (d) All of these, the, , pair, , of, , equations, , 6x-3y + 10 = 0, 2x-y + 9, , lines which are, (a) intersecting, , exactly, (b) intersecting exactly, , (b) Two solutions, (c), , 47., , (b), , 50. Which of the following is/are solutions of, the pair of equations 3x 2y = 4 and, , 40. For what value of k, the, equations 3x- y+, 8 0; 6x - ky =-16, represent, coincident lines?, (a) 0, (b) 1, (c)-1, (d) 2, 41. The, , k1, , (a), , 39. Write the value of k for, which the system, of equations x + 2y, 0, kx y =0 has, unique, solution., , (a) k, , (c), , 49. The solution of the, , (b), , 6, , k =4 (d) k= 3, pair of linear equations 2x + ky-3 =0,, , (a), , 48. The, , 4x+ y= 3 and 8x + 2y = 5k., (a), , 2, , (b), , k =6, , 4u, 27,, value of (u v)., (a) 5, (b) 2, , a, , Determine k, equations has infinite solutions, , 46. The, , 2, , -, , solution, for which the, , 38., , =, , 6x+y+7 = 0 has a unique solution if, , (d) 3, , solutions, , may not have, , 31, , (a) k, , No solution, , (d) Infinitely many solutions, 42. For what value of k, will the following pair, , at, , one, , at two, , (c) coincident, (d) parallel, , of linear equations 2x + 3y = 4 and (k + 2) x +, , 52. It is, , = 0 represent two, , given that there, , is, , point, points, , no, , solution to the, , 6y= 3k +2 have infinitely many solutions, (d) 4, (b) 2, ) 3, , system of equations x + 2y = 3, ax+ by = 4., , (a) 1, , Which one of the following is true?, , 43. What should be the value of m, for the, , (a), , a has a unique value, , given equations 5x + my = 4 and 15x + 3y = 12, , (6), , b has a unique value, , to have infinitely many solutions?, (d), c) 3, (a) 1, (b) 2, , ( c ) acan have more than one value, 4, , (d) a has exactly two different values, , 44. Inthe given figure, ABCDisa parallelogram., , 53. Find the value ofk for which the systemof, , Find the value ofx +y., , equations kx + 2y = 5, 3x + y = 1 has a unique, , solution., (a) k 2, , 6cm, 1, , (a) 2, , (b), , 0c, , 1, , m, , -y, (c) 3, , The pair, , (c) k 6, , (d) k= 6, , 54. The system of equations:x+ 2y = 6, 3x +, , (d), , 6y 18, (a) is inconsistent, (b) has a unique solution, , 4, , of equations 3x + 4y =k and, 12y = 6 has infinitely many solutions if, 45., , (b) k 3, , 9xx, , (c) has infinite number of solutions, (d) None of these

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55. The equations representing the, , given graph, , are, , of k for which the system, 57. Write the value, t, 0, 2 x - y = 0 has, Ry, nique, of equations x, solution., , (a), , (b) k, , 2, , d) k, , (c)k, , (2,2), , 2, , (3,, 2), , x, , -1, -2,-1), -1, 1-1, , 58. 5 books and 7 pens together costs, 79, whereas 7 books and 5 pens together costs, 777. Represent the situation algebraically., (a) 5x + 7y = 77, 7x 5y = 79, 5x-7y =77, 7x + 5y = 79, , x, , 2, , -, , -5,4), , (a), , 7x + 2y = 11; y -2x = 3, , (b), , 2x + 7y = 11; 5x + (35y/2) = 25, , 5x + 7y = 77, 7x + 5y = 79, , (d), , 6x+ 7y = 79, 7x + 5y = 77, , 59. Mr. Mehra investedR, , 60,000 partly, , at 3%, , per annum and the remaining at 4% per annum., the total interest received after one, 1s, , year, , 2,200. Obtain system of linear equations., (a) t y 6000, 3x + 4y 2200, (b) t, y= 60000, 3x + 4y, 220000, , ( ) 3 x - 1y = 10; 8y - 6x = 4, , (d), , (6), (c), , =, , 3x- 4y = 1; 8y 6 x = 4, , =, , =, , 56. Find the value of k for which the, , of equations, x + y - 4 = 0 and 2x + ky, has no solution., , (a) 2, (c) 4, , (b) 3, (d) 5, , system, , 3 =0, , (C) *t y= 60, 3x + 4y = 22, (d) 3x + 4y ==60000, x + y = 220000, 60. If the, system, 4x + ky = 10 has, , find k., (a) 6, , of, , equations, , 2x, , +, , 3y, , ,, , infinitely many solutions, the, , (b), , 5, , (c) 4, , (d) 2

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ANSWERS, 1., , (c): The given equation is 3x+ 5y = 8, , (a), , L.HS. - 3() + 5(0)-3, , (b), , L.HS.,, , (d), , L.HS, 3(0) + 5(0) = 0, , after 15 years, So, age of the daughter, , R.HS., , 9., , 3(0) + 5(1) 5 * R.HS, (c) L.H.S. 3(1) +5(1) =8 R.H.S., -, , -, , (c), , x+y= 137 and, , R.H.S., , (d): The given equation is 3x + 2y, , 10., , (b), , L.H.S., , =3(3) +2(4) 17, , R.H.S., , L.H.S., L.H.S., , x =2, y, , 5., , Now,, , =-, , -, , = 1 i s the solution of t h e equation 2x +7y = 11, , According to, , the, , =6, , a chair be, , question, 5x, , +, , and 6x + 2y + 7 = 0, , x, , 4y, , (d):, , Here, a=9, b =3, c =5, a, , =, , and that of a, , table, , 2800, , Let the numerator and denominator of the, , fraction be x and y respectively., , According to the question,, 8., , x, , +y, , (a) : P r e s e n t a g e o f m o t h e r =, , 8 and, , =, , x, , b2, , a2, , 14. (a): The given pair of equations is 9x +3y + 5=0, , and 4x +3y =2170, 7., , -3, , The given equations has a unique solution., , Required point of intersection is (2, 0)., , y., , Since, 1, , i), , 6, , x=2, Let the cost of, , =1, , =, , -, , be, , =, , -, , Equation of x-axis is y= 0, Putting y =0 in (i), we get 3x 2(0), , (b):, , 1-9,4, , 3y + 4 0 and 2x + y - 6 0, Here, a =2, b =-3, c 4, a2 2, ba=1, c, =-6, , 2x, , 4(2) 2(1) =6 # R.H.S., 1# R.H.S., =2 3(1), 3(2) 4(1) 2# R.H.S., , (a): We have, 3x-2y, , 6., , =and=, , 13. (a): 1The given pair of equations is, , 0., , (b), (c), (d), , =, , 3, b2 =-9, c, = 2, , The given pair of equations has no solution., , is the solution of the equation, , (a): (a) LH.S. 2(2) +7(1) =11 = R.H.S, L.H.S., , Now, 41=, , R.H.S., , =, , 4., , =, , 0., , Here, a =1, b = 3, C= 5, a2, , R.H.S., , 0, , 2(3) -3(4) +17 11, 2(3) +3(4) +17 35, , 3x +2y-17, , 3a = 1, , allel, , -3x 9y +2, , =4, , 5, , =, , (d): The given equations are X * 3y + 5 =0 and, , (b): (a) L.HS, = 2(3) + 3(4)-17 = 1 # R.H.S., , = 3, y, , -y= 43, , tent,, then the lines represented by two equations are paralle, , + R.H.S., , 3., , x, , years, , 1-=, b, 6 2'b,, a2, , =6, b =2, c2 =7., , C_, , 2'c 7, , Since, - 1 1, a2, , b2C2, , The given pair of equations represents parai, , lines., 15. (d): Both the lines represented by the equato, y =3 and y = 8 are parallel to, , axis. So,, , parallel to, , each other., , Present, , y., , 11. (b): When a pair of linear equations is inconsisten, , R.H.S., , x=1,y=-2is not the solution of the given equation., , =, , x, , a = 1/3, , R.H.S., , 12., , =, , years, wh, , R.H.S., , (a), (b), (c), , (c) L.HS., (d) L.H.S., , and y,, , (b): Since, (3, a) lies on the line 2x -3y=5, , 2(3) -3(0), , 5, , x, , 5 6 - 3 a =5, , L.H.S. 3(1) +2(1) =5, L.H.S. =3(3) +2(-2)=5, L.H.S. = 3(-1) + 2(4) 5, (d) L.H.S. = 3(1) + 2(-2) =-1, , x, , 'According to the question,, , r - 1 , y =1 is the solution of given equation., 2., , two numbers be, , Let the, , age of daughter =years, , ne, , given, , pair of equations, , has, , no, , solution.

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16. (c):r -1 represents a line parallel to y-axis at a, , x+y =130, , distance of 1 unit from y-axis and y= 1 represents a line, , Also, ZA - ZB= 44°, , x - y= 44°, , parallel to x -axis at a distance of 1 unit from x-axis., The pair of equations intersect each other at, (1,1), and are perpendicular., , 2x- 174°x = 870 L A = 87o, , 17. (a): Given equations of lines are ax + by - c =0 and, , 25. (b): We have, 3 -V = 9 = 3-V = 32, , l x+my - n = 0., , X-y2, , ), , and x - 2y = 6, , 1i), , From (i), x = y +2, , Since, ambl, The, , pair of equations has a unique solution., (d): The given equations are x + 3y -3 0 and 4x, , 18., , .1), , Adding () and (ii), we get, , .1), , Substituting the value of x in (ii), we get, , y+2-2y =6 y 4, , =, , +ky+7 =0., Here, a, , 1, b, = 3, c - 3 , a, , 4, b, = k, c, , Substituting the value of y in (ii), we get, x= - 4 + 2= - 2, , = 7, , 'Xty =-2, , The system of equations has no solution, if, , +, , (-4), , =, , -6, , 26. (a): The given pair of equations is, x + y = 14, , and, , 19. (c): The given system of equations are, , ..i), , x -y=4, , -i), , kx - y - 4 = 0 and 10x - 2y - 3 =F0., , Adding (i) and (i), we get 2x = 1 8 x =9, , Here, a, = k, b = -1, c = -4 and a, = 10, b, = -2, c, = -3, , Substituting the value of r in (), we gety= 14 - 9 =5, , The given system will have no solution, if, , Hence, x = 9, y = 5., , a2, , 27. (a): The given equations are, , 0, , 3, , >k=, , k=5, , b2, , Hence, the given system will have no solution, if k = 5., , 20. (b): The given lines are, , x-3 0, and y, , 5 =0, , which, , y=5,, , -(11, , X, , Putting =u and =v in (i) and (i), we have, , x=3, which is parallel to y-axis., , -, , 11-7, , y, , is parallelto x-axis., , u+E 3, , Hence, the lines intersect at (3, 5)., , and u, , 21. (d): The given pair of equations is, ax + by + c=0 and dx + ey +f=0, , -, , U =, , .(iii), iv), , 7, , Adding (ii) and (iv), we get 2u, , 10, , u= 55, , For coincident lines,, , - a2, , 22., , a e bd and bf=, =, , ba, , ce, , C2, , (a) : Let the two numbers be x and y, where x>y., , According to the question,, X-y =, and x, , = 3y, , x-, , Subtracting (ii), x, , 23., , ...), (i), , 24, , = 3y = 3, , 3y =, , from, x 12 =, , 0, , (i),, , we, , get 2y, , =, , y, , 24, , =, , (b): Let the two angles be x and y, where x >y., , According to the question,, and x - y, , and, , (il),, , we, , get, , the value of, , Substituting, Let, , x, , x, , 2x, in, , +, , +5, , =, , 0 and -3x 6y +, -, , Now, -, , 2, , 1 =0, , b2, , =, , 106°, , (1),, , we, , x, , ZB, , +, , =, , 53, , 29., , (c) A pair of linear equations is consistent if they, intersecting (one solution) or coincident (infinite, solution)., 30. (d): The given pair of equations, is y 0 and y -7., =, , =, , X'<, , 0, , get, , ZC, , x + y + 50° = 180°, , measures, , of LA and ZB, , s u m property], =180 [By angle, IGiven, Z C = 50°], , C2, , So, the given pair of equations have no solution., , -7, , and y be the, , respectively., Now, 2A, , 2y, , ...), , y=90°- 53° 37, (d):, , We have, x+, , ...(ii), , = 16°, , Adding (i), , (d):, , are, , 36, , X+y = 90°, , 24., , 12, , 28., , From the graph, it is clear that both lines, So, they have no solution., , are, , parallel.

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WtG CBSE Board Term-I Mathematics Class-10, , 38, 31., , 38. (b): The given system of equations is, , (c):Given equations of lines are 3x - y +8 = 0, , 4x+y -3 0 and 8x+ 2y-5k =0, , and 6r - ky + 16 = 0, , =, , For infinite solutions,, , These equations represent coincident lines if, , 32. (c): Given equations of lines are, , 5k=6k=, , 3x +2ky 2 = 0 and 2x +5y + 1 = 0, , (a): The given system of equations is, x+2y 0 and kr -y =0, , 39., , Since the given lines are parallel, , h,, , -k, , Forunique solution, ,, , 33. (d): The given equations of lines are, , k, , (d): The given system of equations is, 3x-y+8= 0 and óx - ky + 16 =0, , 40., , cx- y = 2 and 6x - 2y = 3, , For coincident lines,, , c x - y - 2 = 0 and 6x - 2y - 3 = 0, , For infinitely many solutions,, , 1--1, a2 b2 C2, , and, , 41. (d): The given pair of linear equations is, r+ 3y - 7 = 0 and 2xr + 6y -14 = 0, , c = 3 and c =4, , Here, -5-2.4,9=4, , Since, c has different values, which is not possible., So, there exists no value of c for which given equations, have infinitely many solutions., 34., , --9, , (c):Since x = a and y =b is the solution of the, , equations x, a -b, , y, , -, , =, , 2 and, , y =4,, , x +, , so, , ), .i), , 2, , and a + b = 4, , On adding (i) and (i), we get 2a, , a=3, , 6, , =, , Substituting the value of a in equation (), we get, , The given pair of linear equations has infinitely, many solutions., , (b): Given pair of equations is 2x+3y -4 0, and (k+ 2)x+ 6y- (3k+2)- 0, For infinitely many solutions,, , 42., , 3-b 2 b = 1, , (d):, , 35., , Let the number of R1 coins, , a n d n u m b e r of R2 coins =, , =, , k+2 6-(3k+2), , x, , y, , According to the question, x +y 50, And, (rx 1) + (y 2) 75 x + 2y 75, 50, Subtracting (i) from (i), we get y 75, , ), , =, , x, , =, , From(), r+25, 36., , (c):, , Let the, , .(i), , =, , 50, , x=, , -, , =, , 25, , So, , 43. (a): Given pair of equations is, , 5x+my- 4 and 15x +3y, , 25, , For, , present age of father is, , x, , k+2-4 k-2, , years and the, , 12., , infinitely many solutions,, , present age of son is y years., +, Acording to the question, (r+ 4) 4(y 4), =, , r-4y, , 44., , = 12, , .(i), , and x = 6y, , (d): We know, , bisect each other., , that,, , diagonals of a, , parallelogram, , Substituting the value of x from (ii) in (i), we get, , xty= 10, , 2y 12, , Adding () and (i), we get 2x 16x=8, , y=6, , Present age of son is 6 years., , 37., , (a): The given pair of linear equations is, , x+3y -7, , Here, , =, , 0 and 2x+9y, , -5 =0, , 3, , From, , solution., given pair of equations has a unique, , (), 8 +y= 10 y-2, , Now, x+y=8 +2 =4, 45., , (a): The pair of equations, 4y- k and 9x + 12y- 6, 3x+ 4y-k= 0 and 9x, +12y-6-0, , 3x+, has, , Since,, The, , and x - y = 6, , infinitely many solutions if, C2, , ..)

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U1 T W VdilauCS, , Palr ol LINedr tqUduolS, , 46. (d): The, , of linear equations 2x, , pair, , 6x+y+7=0, , and, , has, , unique, , +, , ky, , -3, , 0, , . h , i - 1-2,3, , solution if, , -and, , a2, 47., , b2, , a2, , (c), , 2a=b and 3b # 8 i.e., bz, , Given equations are, , u+47= 27, and u + 20, , .(1), 11), , 21, , a can have more than one value., , 53. (c): Writing the equations in standard form,, , Subtracting (i) from i), we get, 27 6, , 3x + y, , Putting the value of v in (i), we get, 27u, , 1u+12, , kx +2y, , we get, , v=3, =, , 1, , =0;, , 0, , For unique solution, we must have, , 15, , Now, u - v = 15 - 3, , 5, , 12, , k6, , 48. (b): The pair of linear equations, 13x + ky, , 54., , k = 0 and 39x + 6y - (k +4) = 0 has infinitely, , many solutions if, , (c): The given system of equations is, , x+2y, , =0, 3x + 6y, , 6, , 18, , =, , 0, , Now, 1--2-and, -=, b6, and, , a2, , b2, , C2, , 3, , k=2, , 23, 1 bC, a2, , 2k =4, k+4=3k, k+4, , and, , 49. (a):We have, and, , r=y), , abab=?+b, y, , solutions., 55. (d): Equations in option (d) satisfy the given graph., 56. (a): The condition for system of equations to be, inconsistent(i.e., no solution) is, , (11), a2, , Putting x=y in equation (i), we get, , b2, , k=2, , abb ab 2+b, ay, , C2, , T h e given system ofequations has infinitely many, , k=2, , =0, , b2, , C2, , and k, , k=2, , 57. (b): The required condition i s ,, , y, , 2, k, , b+a) =at +b y=b., , F o r any value, , From ) x=b=a., , ofkother thanthe given system, , has unique solution, , Hence x = a, y = b, , in, (a), (b), (d): All the values of (x, y) given options, 50, and (c), satisfy the given pair of equations., , Option (d) is the correct answer, 1., , (d):, , 6x-3y, , +, , The, , given pair of, , 10, , linear, +9, , 0 and 2x y, , 41-=3,, b1 -=3, 2, , =, , equations, , 1, , i.e., - 1, 2, b2, , 0., , and cost of 7 books and 5 pens is, , 77., , 59. (b): Let the amount invested at 3% be, amount invested at 4% be, Now, x, , 2, , C1, , of equations, , represents, , +y =, , 3x4y, 100, 60., , parallel lines., c ) : The given system of, =0, X*2y 3 0, ax, -, , 79, , x and the, , y., , 60000, , and total interest after one year, , C2, Graphically, given pair, , ne, , 5x+7y =, , +by -4, , given system, , equations is, , of equations has, , of one, , 7x +5y= 77, , and, , 02, , 58. (d) : Let the cost of orne book be x and that, pen be y . Cost of 5 books and 7 pens is 79., , n o solution, , 2200, , 3x + 4y, , 220000, , 100, , (a): 2x +3y = 5 and 4x + ky = 10 has infinitely, , many solutions, so