1. A motorboat takes 6 hours to cover 100km down stream and 30km upstream. If the motorboat goes 75km down stream and returns back to its starting point in 8hours, find the speed of the motorboat in still water and the rate of the stream.
2. Solve the following system of linear equations: 3x - 5y = -1, x - y = -1
3. The polynomial x4 +bx³ +59 x² +cx +60 is exactly divisible by x² +4 x +3. Find the values of b and c.
4. If the equation (1 + m2) x2 + 2 mcx + (c2 - a2) = 0 has equal roots, prove that c2 = a2(1 + m2)
5. A polygon has 10 sides .The lengths of the sides starting with the smallest form an AP .If the perimeter of the polygon is 420 Cm and the length of the longest side is twice that of the shortest side
6. A train covers a distance of 90 Km at uniform speed .Had the speed been 15 Km /hr more . it would have taken 30 minutes less for the journey find the original speed of the train
7. A party of tourists booked a room in a hotel for Rs 1200 , three of the members failed to pay as a result others had to pay Rs 20 more (each) . How many tourists were there in the party
8. A passenger train takes 2 hours less for journey of 300 km, if its speed is increased by 5 km/h from its usual speed, find its usual speed.
9. A takes 16 days less than the time taken by B to finish a piece of work. If both A and B together can finish it in 6 days, find the time taken by B to finish the work.
10. Obtain all the zeroes of the polynomial f(x)=3x4+6x3-2x2-10x-5 if two of its zeroes are √5/5and - √5/3 .
11. Solve the following system of linear equations graphically. Shade the region bounded by these lines and y-axis. Also find the area of the shaded region.
x –y =1 ; 2x + y =8
12. A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously, fill the pool in the same time during which the pool is filled by third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately.
13. A piece of cloth costs Rs 200. If the piece was 5 m longer and each metre of cloth costs Rs 2 less the cost of piece would have remained unchanged. How long is the piece and what is the original rate per metre.
14. Prove that 5 – √3 is an irrational number.
15. In a potato race, a bucket is placed at the starting point, which is 5m from the first potato and other potatoes are placed 3m apart in a straight line. There are ten potatoes in a line. A competitor starts from the bucket, picks up the nearest potato, runs back to pick up the next potato, runs to the buckets to drop it in and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
16. Is 1/7 a non-terminating decimal, why?
17. What is the solution of the system of equation a1x+b1y+c1 = 0 and
a2x+ b2y+c2=0 if the graph of the equation intersect each other at point P (a , b).
18. What is the relationship between sum of zeros of a cubic polynomial ax3+bx2+cx+d (where a ¹ 0) and its coefficients?
19. State Euclid’s division algorithm, find the HCF of 18864 and 6075.
20. Solve for x and y: 4/x + 3y =14 ; 3/x - 4y =23 . Where x ¹0
21. For which value(s) of ‘a’ and ‘b’ , the following pair of linear equations have infinite number of solutions 2x + 3y = 7 ; (a – b)x + (a+b)y = 3a + b – 2.
22. If α , β and γ are roots of the equation x3–3 x2 – x + 3 = 0, then find (i) α + β + γ ii) αβγ
23. Determine value(s) of ‘p’ for which the quadratic equation 4x2 -3px + 9 = 0 has real roots.
24. 13. Solve the system of equations: bx/ a -ay/b + a + b=0 ; bx –ay + 2ab = 0
25. Find a zeros of the polynomial x4 + x3 - 9x2 – 3x + 18, if it is given that two of its zero as are –√3 and . √3
26. Show that one and only one out of n , n+2 or n+4 is divisible by 3. Where ‘n’ is any positive integers.
27. Solve the following pair of equations graphically :
x + 2y =5 ; 2x +3y = -4
Also find the points where the lines meet the x-axis.
28. Solve the following pair of linear equation graphically:
3x-2y-1 =0 2x-3y+6=0
29. Solve the following pair of linear equations graphically:
2x+3y=8
x +4y=9
30. Solve the following pair of linear equation graphically:
x-2y=4
x-y =3
31. 4 chairs and 3 tables cost Rs 2100 and 5 chairs and 2 tables cost Rs 1750. Find the cost of 1 chair and 1 table separately.
32. Five years ago, A was thrice as old as B and 10 years hence A shall be twice as old as B. what are the present ages of A and B?
33. In a two digit number, unit’s digit is twice the ten’s digit. If the digits are reversed, new number is 27 more that the original number. Find the number.
34. A and B are friends and A is elder to B by 2 years. A’s father D is twice as old as A and B is twice as old as his sister C. if the ages of D and C differ by 40 years, find the age of A.
35. Ten years ago, father was twelve time as old as his son and ten years hence he will be twice as old as his son will be. Find their present ages.
36. A man has only 20 paisa coins and 25 paisa coins in his purse. If he has 50 coins in all totaling Rs 11.25, how many coins of each kind does she has?
37. The age of two girls are in the ratio 5:7 Eight years ago their ages were in the ratio 7:13. Find their present ages.
38. The taxi charges in a city comprise of a fixed charge together with the charge for the distance covered. For a journey of 10km, the charges paid are Rs 75 and for a journey of 15km, the charge paid are Rs 110. What will a person have to pay for travelling a distance of 25 km?
39. A man travels 600km partly by train and partly by car. If he covers 400km by train and the rest by car, it takes him 6 hours and 30 minutes. But if he travels 200 km by train and rest by car, he takes half an hour longer. Fine the speed of the train and that of the car.
40. The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number.
41. Two years ago a father was five times as old as his son. Two years later, his age will be 8years more than three times the age of the son. Find the present ages of father and son.
42. The monthly incomes of A and B are in the ratio of 9:7 and their monthly expenditures are in the ratio of 4:3. If each saves Rs 1600per month, find the monthly incomes of each.
43. A number consisting of two digits is equal to 7 times the sum of its digits. When 27 is subtracted from the number, the digits interchange their places. Find the number.
2. Solve the following system of linear equations: 3x - 5y = -1, x - y = -1
3. The polynomial x4 +bx³ +59 x² +cx +60 is exactly divisible by x² +4 x +3. Find the values of b and c.
4. If the equation (1 + m2) x2 + 2 mcx + (c2 - a2) = 0 has equal roots, prove that c2 = a2(1 + m2)
5. A polygon has 10 sides .The lengths of the sides starting with the smallest form an AP .If the perimeter of the polygon is 420 Cm and the length of the longest side is twice that of the shortest side
6. A train covers a distance of 90 Km at uniform speed .Had the speed been 15 Km /hr more . it would have taken 30 minutes less for the journey find the original speed of the train
7. A party of tourists booked a room in a hotel for Rs 1200 , three of the members failed to pay as a result others had to pay Rs 20 more (each) . How many tourists were there in the party
8. A passenger train takes 2 hours less for journey of 300 km, if its speed is increased by 5 km/h from its usual speed, find its usual speed.
9. A takes 16 days less than the time taken by B to finish a piece of work. If both A and B together can finish it in 6 days, find the time taken by B to finish the work.
10. Obtain all the zeroes of the polynomial f(x)=3x4+6x3-2x2-10x-5 if two of its zeroes are √5/5and - √5/3 .
11. Solve the following system of linear equations graphically. Shade the region bounded by these lines and y-axis. Also find the area of the shaded region.
x –y =1 ; 2x + y =8
12. A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously, fill the pool in the same time during which the pool is filled by third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately.
13. A piece of cloth costs Rs 200. If the piece was 5 m longer and each metre of cloth costs Rs 2 less the cost of piece would have remained unchanged. How long is the piece and what is the original rate per metre.
14. Prove that 5 – √3 is an irrational number.
15. In a potato race, a bucket is placed at the starting point, which is 5m from the first potato and other potatoes are placed 3m apart in a straight line. There are ten potatoes in a line. A competitor starts from the bucket, picks up the nearest potato, runs back to pick up the next potato, runs to the buckets to drop it in and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
16. Is 1/7 a non-terminating decimal, why?
17. What is the solution of the system of equation a1x+b1y+c1 = 0 and
a2x+ b2y+c2=0 if the graph of the equation intersect each other at point P (a , b).
18. What is the relationship between sum of zeros of a cubic polynomial ax3+bx2+cx+d (where a ¹ 0) and its coefficients?
19. State Euclid’s division algorithm, find the HCF of 18864 and 6075.
20. Solve for x and y: 4/x + 3y =14 ; 3/x - 4y =23 . Where x ¹0
21. For which value(s) of ‘a’ and ‘b’ , the following pair of linear equations have infinite number of solutions 2x + 3y = 7 ; (a – b)x + (a+b)y = 3a + b – 2.
22. If α , β and γ are roots of the equation x3–3 x2 – x + 3 = 0, then find (i) α + β + γ ii) αβγ
23. Determine value(s) of ‘p’ for which the quadratic equation 4x2 -3px + 9 = 0 has real roots.
24. 13. Solve the system of equations: bx/ a -ay/b + a + b=0 ; bx –ay + 2ab = 0
25. Find a zeros of the polynomial x4 + x3 - 9x2 – 3x + 18, if it is given that two of its zero as are –√3 and . √3
26. Show that one and only one out of n , n+2 or n+4 is divisible by 3. Where ‘n’ is any positive integers.
27. Solve the following pair of equations graphically :
x + 2y =5 ; 2x +3y = -4
Also find the points where the lines meet the x-axis.
28. Solve the following pair of linear equation graphically:
3x-2y-1 =0 2x-3y+6=0
29. Solve the following pair of linear equations graphically:
2x+3y=8
x +4y=9
30. Solve the following pair of linear equation graphically:
x-2y=4
x-y =3
31. 4 chairs and 3 tables cost Rs 2100 and 5 chairs and 2 tables cost Rs 1750. Find the cost of 1 chair and 1 table separately.
32. Five years ago, A was thrice as old as B and 10 years hence A shall be twice as old as B. what are the present ages of A and B?
33. In a two digit number, unit’s digit is twice the ten’s digit. If the digits are reversed, new number is 27 more that the original number. Find the number.
34. A and B are friends and A is elder to B by 2 years. A’s father D is twice as old as A and B is twice as old as his sister C. if the ages of D and C differ by 40 years, find the age of A.
35. Ten years ago, father was twelve time as old as his son and ten years hence he will be twice as old as his son will be. Find their present ages.
36. A man has only 20 paisa coins and 25 paisa coins in his purse. If he has 50 coins in all totaling Rs 11.25, how many coins of each kind does she has?
37. The age of two girls are in the ratio 5:7 Eight years ago their ages were in the ratio 7:13. Find their present ages.
38. The taxi charges in a city comprise of a fixed charge together with the charge for the distance covered. For a journey of 10km, the charges paid are Rs 75 and for a journey of 15km, the charge paid are Rs 110. What will a person have to pay for travelling a distance of 25 km?
39. A man travels 600km partly by train and partly by car. If he covers 400km by train and the rest by car, it takes him 6 hours and 30 minutes. But if he travels 200 km by train and rest by car, he takes half an hour longer. Fine the speed of the train and that of the car.
40. The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number.
41. Two years ago a father was five times as old as his son. Two years later, his age will be 8years more than three times the age of the son. Find the present ages of father and son.
42. The monthly incomes of A and B are in the ratio of 9:7 and their monthly expenditures are in the ratio of 4:3. If each saves Rs 1600per month, find the monthly incomes of each.
43. A number consisting of two digits is equal to 7 times the sum of its digits. When 27 is subtracted from the number, the digits interchange their places. Find the number.
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