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Chapter 1: Linear Equation Chapter-1: Linear equation Level – I (QUESTIONS EXACTLY COPIED FROM NCERT) 1. The cost of a notebook is twice the cost of a pen. The linear equation in two variables that represent this statement. (a) x = y (b) x − y = 0 (c) 2x − y = 0 (d) 2x + y = 0 2. Which one of the following options is true for y = 3x + 5. It will have I. A unique solution II. Only two solutions III. Infinitely many solutions (a) Only I (b) Only II (c) Only III (d) II and III 3. Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k. (a) 5 (b) 6 (c) 7 (d) –7 4. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, then the value of a is 2 5 4 3 (a) (b) (c) (d) 3 3 5 2 5. The equation of the graph shown below is. (a) x + y = 0 (b) y = 2 x (c) y = x (d) y = 2 x + 1 6. The equation of the graph shown below is. (a) x + y = 0 (b) y = 2 x (c) y = 2x + 4 (d) y = x − 4 7. The equation of the graph shown below is. (a) x + y = 0 (b) y = 2 x (c) y = 2 x + 1 (d) y = 2 x − 4 8. The equation of the graph shown below is. (a) y = x (b) x + y = 0 (c) y = 2 x (d) 2 + 3 y = 7 x 9. The graph of equation, y = − x + 2 is (a) (b) (c) (d) Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 10. The graph of Equation, 3x − 4 y − 12 = 0 is (a) (b) (c) (d) 11. The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs. 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is 300. Represent the situation algebraically if x and y are respectively the prices (in Rs. per kg) of apples and grapes. 2 x − y = 160 2 x + 4 = 160 2 x + y = 160 (a) (b) (c) (d) None 4 x − 2 y = 300 4 x − 2 y = 300 4 x + 2 y = 300 12. 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the sum of cost of one pencil and that of one pen solved graphically is (a) 3 (b) 5 (c) 8 (d) 2 13. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden. (a) l = 16 m, b = 10 m (b) l = 10 m, b = 16 m (c) l = 20 m, b = 16 m (d) l = 16 m, b = 20 m 14. The difference between two numbers is 26 and one number is three times the other. Find the sum of these numbers. (a) 39 (b) 13 (c) 26 (d) 52 15. The larger of two supplementary angles exceeds the smaller by 18 degrees, then the difference of these angles is (a) 99 (b) 81 (c) 18 (d) –18 16. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages respectively? (a) 10, 40 (b) 40, 10 (c) 20, 50 (d) 50, 20 9 17. A fraction becomes , if 2 is added to both the numerator and the denominator. If, 3 is added to both 11 5 the numerator and the denominator it becomes . Find the fraction. 6 (a) 7/9 (b) 8/9 (c) 10/9 (d) 11/9 18. By Solving 2 x + 3 y = 11 and 2 x – 4 y = –24 and hence find the value of ‘m’ for which y = mx + 3 . (a) 1 (b) –1 (c) 2 (d) –2 x 2y y 19. Solve the following pair of linear equations by cross multiplication method. + = − 1 and x − = 3 2 3 3 (a) x = 2,y = 3 (b) x = 2,y = −3 (c) x = −2,y = 3 (d) x = −2,y = −3 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 20. The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number. (a) 16 (b) 32 (c) 18 (d) 81 21. Meena went to a bank to withdraw 2000. She asked the cashier to give her Rs. 50 and Rs. 100 notes only. Meena got 25 notes in all. Find how many notes of Rs. 50 she received. (a) 10 (b) 15 (c) 20 (d) 5 22. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs. 27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the charge for each extra day. (a) 15 (b) 3 (c) 12 (d) –3 23. For which values of a and b does the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a – b) x + (a + b) y = 3a + b – 2 (a) a = −5, b = −1 (b) a = 5, b = −1 (c) a = 5, b = 1 (d) a = −5, b = 1 24. For which value of k will the following pair of linear equations have no solution? 3x + y = 1, (2k – 1) x + (k – 1) y = 2k + 1 (a) 1 (b) 2 (c) 3 (d) 4 1 1 25. A fraction becomes when 1 is subtracted from the numerator and it becomes when 8 is added to 3 4 its denominator, then the fraction is 12 13 12 5 (a) (b) (c) (d) 5 12 13 12 26. The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle. (a) l = 17, b = 9 (b) l = 9, b = 17 (c) l = 17, b = 10 (d) b = 10, l = 17 27. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water respectively (a) 8 km/h, 3 km/h (b) 3 km/h, 8 km/h (c) 2 km/h, 4 km/h (d) 4 km/h, 2 km/h 28. On Solving the following pairs of equations by reducing them to a pair of linear equations 7x − 2 y =5 xy 8x + 7 y = 15 xy (a) x = 1, y = −1 (b) x = −1, y = 1 (c) x = 1, y = 1 (d) x = −1, y = −1 29. On Solving the following pairs of equations by reducing them to a pair of linear equations: 1 1 3 + = 3x + y 3x − y 4 1 1 −1 − = 2 ( 3x + y ) 2 ( 3x − y ) 8 (a) x = 1, y = −1 (b) x = −1, y = 1 (c) x = −1, y = −1 (d) x = 1, y = 1 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 30. Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water? (a) 4 km/h (b) 5 km/h (c) 6 km/h (d) 7 km/h 31. 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work. (a) 18 days (b) 36 days (c) 24 days (d) 12 days 32. The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the sum of ages of Ani and Biju. (a) 23 year (b) 36 years (c) 39 years (d) 35 years 33. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. (a) 200 km (b) 400 km (c) 600 km (d) 800 km 34. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class. (a) 30 (b) 32 (c) 34 (d) 36 35. In a Δ ABC, C = 3B = 2 (A + B), then B = ? (a) 20o (b) 40o (c) 120o (d) 60o 36. The Solution of pair of linear equations: x y − =0 a b ax + by = a 2 + b2 (a) x = a,y = b (b) x = b,y = a (c) x = −a,y = b (d) x = a,y = −b LEVEL – II (QUESTIONS BASED ON NCERT) 1. The value of k for which the equation 2 x + y = 1,3x + ky = −1 has no solution is. (a) 3/2 (b) 2/3 (c) –3/2 (d) –2/3 2. The system of equation px + 2 y = 5 and 3x + y = 1 have a unique solution if: (a) p  3 (b) p  4 (c) p  5 (d) p  6 3. The equation 3x – y – 2 = 0 and 4 x + 7 y = 1 has. (a) Infinite solution (b) No solution (c) Two solution (d) Unique solution 1 1 5 3 4. The solution of the pair of equations + = 0, − = 4 is. x y x y (a) (2,2) (b) (2, -2) (c) (-2, 2) (d) (-2, -2) Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 5. The graph of the system of equations x + 7 y = 22 and x – 3 y = –18 is: (a) parallel (b) intersects at one point (c) coincident (d) None of these 6. If 22 x+ y = 4x – y –3 = 1, then ( x, y ) is: (a) (–1, –2) (b) (–1, 2) (c) (1, 2) (d) (1, –2) 7. For what values of a and b does the following pair of linear equations have an infinite number of solutions: 2x + 3 y = 7 (a – b) x + (a + b) y = 3 a + b – 2 (a) 5, –1 (b) –5, 1 (c) 5, 1 (d) –5,–l 8. Consider the following statements: A: A linear equation in two variables has infinitely many solutions B: The graph of x = a is a straight line parallel to x–axis What is your opinion? (a) only A is true (b) only B is true (c) Both A and B are true (d) Both A and B are false 9. For what value of k will the following pair of linear equation have no solution: x + 3y = 1 ( k –1) x + ( 2k –1) y = 2k + 1 (a) 0 (b) 1 (c) 2 (d) 3 2x + 4 y 10. If 2 x + y = 2 xy and = 5 then x = ? xy (a) 0 (b) l (c) 2 (d) 2 . ... x =2 x 11. Find x if xx 1 (a) 0 (b) 2 (c) 2 (d) 2 12. Which of the equation given below represent a line parallel to the line represented by the equation 2 y = 5x – 3 ? (a) y = 2 x − 3 (b) 3 y = 7.5x + 8.5 (c) y = 5x + 3 (d) 5 y = 2 x − 3 13. The solution of the equation 2 x + 3 y = 0 and 3 x − 8 y = 0 is (a) x = 0, y = 0 (b) x = 3 , y = 2 (c) x = 0 , y = 2 (d) x = 3 , y = 2 14. The value of k, for which line 4 x + 6 y + k = 0 coincides with the line 2 x + 3 y = 5 (a) 10 (b) – 10 (c) 1/10 (d) 5 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 15. Line a1 x + b1 y + c1 = 0, is inconsistent with line a2 x + b2 y + c2 = 0 , then. a1 b1 a1 b1 c1 a1 b1 c1 (a)  (b) =  (c) = = (d) None a2 b2 a2 b2 c2 a2 b2 c2 16. The value of k for which the system of equation 2 x + 3 y + 10 = 0,x + ky + 5 = 0 , has no solution. 3 3 (a) (b) (c) all k except 3/2 (d) There is no value of k 2 2 3x − y + 1 2 x + y + 2 3x + 2 y + 1 17. The solution of the equation. = = is given by. 3 5 6 (a) x = 2, y = 1 (b) x = 1, y = 1 (c) x = −1, y = −1 (d) x = −1, y = 2 18. If the lines given by a1 x + b1 y = c1 and a2 x + b2 y = c2 intersect at ( X o ,Yo ) then X o and Yo are equal to. c1b1 − b1c1 a1c2 − a2c1 c1b2 + b1c2 a1c2 + a2c1 (a) , respectively (b) , respectively a1b1 − a2b1 a1b2 − b1a2 a1b2 + a2b1 a1b2 + b1a2 b1c2 − c1b2 c1a2 − a1c2 (c) , respectively (d) None of these a1b2 − a2b1 a1b2 − b1a2 19. Solution of the following system of equation is 3 ( 2u + v ) = 7uv; 3 ( u + 3v ) = 11uv. (a) u = 3 / 2,v = 1 (b) u = 1,v = 2 / 3 (c) u = 0,v = 0 and u = 1,v = 3 / 2 (d) u = 0,v = 1 and u = 1 / 2,v = 3 / 2 20. One woman and 2 men can finish some work in 4 days. Three women and two men can finish the same work in 2 days. In how many days one woman alone can finish the work ? (a) 6 (b) 8 (c) 12 (d) 16 21. Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it: (a) There is no such temperature (b) –100 (c) –40 (d) 0 22. A train travels 360 km at a uniform speed. If the speed had been 5km/h more, it would have taken 1 hour less for the same journey. The speed of the train is: (a) 35 km/hour (b) 30 km/hour (c) 20 km/hour (d) 40 km/hour 23. Sonu went to a bank to withdraw Rs. 2000. He asked the cashier to give him Rs. 50 and Rs. 100 notes only. Sonu got 25 notes in all. The number of notes of Rs. 50 received by him are: (a) 8 (b) 12 (c) 10 (d) 15 24. 2 men and 3 women perform a work in 8 days, 6 women and 8 children perform the same work in 4 days and 1 man and 2 children perform the same work in 16 days. Find the number of days 2 men, 3 women and 8 children shall take to perform the same work: (a) 2 (b) 3 (c) 4 (d) 6 25. If 5 men or 7 women can perform a work in 14 days then find the number of days which shall be taken by 9 men and 7 women to perform the same work ? (a) 3 days (b) 4 days (c) 5 days (d) 6 days Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 26. Rs. 49 was divided among 150 children. Each girl got 50 paise and each boy 25 paise. The number of boys was: (a) 101 (b) 102 (c) 103 (d) 104 27. A goods train leaves a station at a certain time at a fixed speed. After 6 hours, an express train leaves the same station and moves in the same direction at a uniform speed of 90 km/hr. This train catches the goods train in 4 hours. The speed of the goods trains is: (a) 36 km/hr (b) 40 km/hr (c) 42 km/hr (d) 45 km/hr 28. A two digit-number has 3 at its unit place. The sum of its digit is one seventh of the number itself. Find the number. (a) 73 (b) 63 (c) 53 (d) 50 29. The sum of the digits of a two digit number is 8. The number obtained by interchanging the two digit exceeds the given number by 36. Then the number is. (a) 26 (b) 62 (c) 71 (d) 17 30. Father age is three times the sum of ages of his two children. After 5 years, his age will be twice the sum of age of two children. The age of father is. (a) 27 (b) 72 (c) 54 (d) 45 31. X can do a piece of work in 20 days and Y can do it in 30 days. They start work together and after some days X leaves and the remaining work is completed by Y in 5 days. Find out the number of days after which X left the job. (a) 15 days (b) 10 days (c) 20 days (d) None of these 32. If from twice the greater of the two numbers, 20 is subtracted, the result is the other number. If from twice the smaller number, 5 is subtracted, the result is the first number. The largest number is. (a) 12 (b) 18 (c) 15 (d) 25 33. If two articles are such that, the cost of two 1st article is Re 1 more than the cost of three 2nd article and cost of three 2nd article is Re 1 more than cost of single article of 1st kind. To find their costs which of the equation you have to solve. (a) 3 y − 2 x = 1,x − 3 y = 1 (b) 2 x − 3 y = 1,3 y − x = 1 (c) 2 x − 3 y = 1,x − 3 y = 1 (d) 2 x + 3 y = 1,3 y − x = −1 34. A year ago, a father was four times son’s age. In six years his age will be 9 more than twice his son’s age. What is the present age of the son? (a) 10 years (b) 9 years (c) 20 years (d) None 35. One kg of tea and one kg of sugar together cost Rs. 95. If the price of tea falls by 10% and that of sugar rises by 20%, then the price of one kg of each combined comes to Rs. 90. The original price of tea in Rs. per kg is. (a) Rs. 72 (b) Rs. 55 (c) Rs. 60 (d) Rs. 80 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation 36. If in a rectangle the length is increased and the breadth is reduced by 2 units each, the area is reduced by 8 square units. If the length is reduced by 1 unit and breadth is increased by 2 units, the area increases by 8 units. Then length and breadth respectively are. (a) 8,6 unit (b) 6, 8 unit (c) 12, 10 unit (d) 8, 10 units 37. The cost of 2 pens is equal to the cost of 5 pencils. If 5 books, 3 pens and 10 pencils together cost Rs. 185, whereas 3 books and 2 pens together cost Rs. 100, then the total cost of 1 book, 1 pen and 2 pencils is. (a) Rs. 35 (b) Rs. 39 (c) Rs. 40 (d) Rs. 32 38. A person can row 8 km up stream and 24 km downstream is 4 hours. He can row 12 km downstream and 12 km upstream in 4 hrs. Then speed of person in still water and the speed of the current is. (a) 8 km/hr, 4 km.hr (b) 4 km/hr, 8 km/hr (c) 10 km/hr, 4 km/hr (d) 10km/hr, 6 km/hr 39. A two-digit number is obtained by both multiplying sum of the digits by 8 and adding 1 or by multiplying the difference of the digits by 13 and adding 2. The number is. (a) 52 (b) 63 (c) 41 (d) 82 40. In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The numbers of cows are. (a) 5 (b) 7 (c) 10 (d) None of these 41. A car travelled from a town P to a town Q at a speed 130 km/h in the return Journey it traveled at a speed of 70 km/h. The average speed of the car (in km/h) is. (a) 91 (b) 94 (c) 90 (d) 100 42. A father’s age is equal to the age of 5 children. In fifteen years, his age will be only half of their united age then, present ages of father is. (a) 54 year (b) 34 year (c) 45 year (d) 43 year 43. A man reaches his office 40 minutes late if he walks 3 km/h but 30 minutes early if he walks 4 km/h his office is at a distance (in km). (a) 12 (b) 15 (c) 14 (d) 84 44. A boat goes 50 km upstream in 8 hours and a distance of 36 km downstream in 6 hours. The speed of the boat (in km/h) is still water is. (a) 5 (b) 6.125 (c) 6.5 (d) 10 45. Five year ago I was three times as old as my son and ten years later I shall be twice as old as my son. How old am I now ? (a) 45 years (b) 50 years (c) 55 years (d) 60 years 46. The sum of two digits of a two-digit number is 9. Also nine times this number is twice the number obtained by reversing the order of the digits, then the number is. (a) 276 (b) 36 (c) 63 (d) 18 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Linear Equation ANSWERS LEVEL – I 1. c 2. c 3. c 4. b 5.b 6. c 7.d 8. b 9. d 10. d 11. c 12. c 13. c 14. d 15. c 16. b 17. a 18. b 19. b 20. c 21.a 22. b 23. c 24. b 25.d 26. a 27. b 28. c 29. d 30. c 31. a 32. d 33. c 34. d 35. b 36. a LEVEL – II 1. a 2. d 3. d 4. b 5. b 6. d 7. c 8. a 9. c 10. b 11. b 12. c 13. a 14. b 15. b 16. d 17. b 18. c 19. c 20. b 21. c 22. d 23. c 24. c 25. c 26. d 27. a 28. b 29. a 30. d 31. b 32. c 33. b 34. d 35. d 36. a 37. b 38. a 39. c 40. b 41. a 42. c 43. c 44. b 45. b 46. d Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458