NCERT CBSE 10th Class Maths Test paper Real Numbers

1. Express 140 as a product of its prime factors

2. Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.

3. Find the LCM and HCF of 6 and 20 by the prime factorization method.

4. State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.

5. State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating decimal.

6. Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.

7. Use Euclid’s division algorithm to find the HCF of 135 and 225

8. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m

9. Prove that √3 is irrational.

10. Show that 5 – √3 is irrational

11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Maths class X : introduction to trigonometry notes

1. In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine

a. sin A, cos A

b. sin C, cos C
2.    Given 15 cot A = 8. Find sin A and sec A 

3.    If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

4.    In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

5.     State whether the following are true or false. Justify your answer. 

CBSE MATHS X Number system Questions Bank

1. Express 140 as a product of its prime factors

2. Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.

3. Find the LCM and HCF of 6 and 20 by the prime factorization method.

4. State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.

5. State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating decimal.

6. Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.

7. Use Euclid’s division algorithm to find the HCF of 135 and 225

8. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m

9. Prove that √3 is irrational.

10. Show that 5 – √3 is irrational

11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

12. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

13. Express 156 as a product of its prime factors.

14. Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.

15. Find the HCF and LCM of 12, 36 and 160, using the prime factorization method.

16. State whether 6/15 will have a terminating decimal expansion or a non-terminating repeating decimal.

17. State whether35/50 will have a terminating decimal expansion or a non-terminating repeating 18. decimal.

19. Find the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two numbers.

20. Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Prove that ( √n -1 + √n +1 ) is irrational, for every nÎN

21. Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer.

CBSE NCERT 10th Linear equation in two variables Test Paper

MULTIPLE CHOICE QUESTIONS

1. Every linear equation in two variables has ___ solution(s).

(a) no (b) one (c) two (d) infinitely many

2. a1/a2 = b1/b2=c1/c2 is the condition for

(a) intersecting lines (b) parallel lines (c) coincident lines (d) none

3. For a pair to be consistent and dependent the pair must have

(a) no solution (b) unique solution (c) infinitely many solutions (d) none of these

4. Graph of every linear equation in two variables represent a ___

(a) point (b) straight line (c) curve (d) triangle

5. Each point on the graph of pair of two lines is a common solution of he lines in case of

(a) Infinitely many solutions (b) only one solution (c) no solution (d) none of these

6. Which of he following is the solution of the pair of linear equations

3x – 2y = 0, 5y – x = 0

(a) (5, 1) (b) (2, 3) (c) (1, 5) (d) (0, 0)

7. One of the common solution of ax + by = c and y-axis is _____

(a) (0, c/b) (b) (0,b/c ) (c) , 0 , (c/ b ) (d) (0, c/ b)

8. If the value of x in the equation 2x – 8y = 12 is 2 then the corresponding value of y will be (a) –1 (b) +1 (c) 0 (d) 2

9. The pair of linear equations is said to be inconsistent if they have

(a) only one solution (b) no solution (c) infinitely many solutions. (d) both a and c

10. On representing x = a and y = b graphically we get ____

(a) parallel lines (b) coincident lines (c) intersecting lines at (a, b) (d) intersecting lines at (b, a)

11. How many real solutions of 2x + 3y = 5 are possible

(a) no (b) one (c) two (d) infinitely many

12. The value of k for which the system of equation 3x + 2y = – 5, x – ky = 2 has a unique solutions.

(a) K = 2/ 3 (b) K ¹ 2/3 (c) K = -2 /3 (d) K ¹ - 2/3

13. If the lines represented by the pair of linear equations 2x + 5y = 3, 2(k + 2) y + (k + 1) x = 2k

are coincident then the value of k is ____

(a) –3 (b) 3 (c) 1 (d) –2

14. The coordinates of the point where x-axis and the line represented by x/2 + 4/3 = 1 intersect, are

(a) (0, 3) (b) (3, 0) (c) (2, 0) (d) (0, 2)

15. Graphically x – 2 = 0 represents a line

(a) parallel to x-axis at a distance 2 units from x-axis.

(b) parallel to y-axis at a distance 2 units from it.

(c) parallel to x-axis at a distance 2 units from y-axis.

(d) parallel to y-axis at a distance 2 units from x-axis.

16. If ax + by = c and lx + my = n has unique solution then the relation between the coefficients will

be ____

(a) am ¹ lb (b) am = lb (c) ab = lm (d) ab¹ lm

SHORT ANSWER TYPE QUESTIONS
17. Form a pair of linear equations for : The sum of the numerator and denominator of fraction is 3

less than twice the denominator. If the numerator and denominator both are decreased by 1, the

numerator becomes half the denominator.

18. Amar gives Rs. 9000 to some athletes of a school as scholarship every month. Had there been 20 more athletes each would have got Rs. 160 less. Form a pair of linear equations for this.

19. Find the value of k so that the equations x + 2y = – 7, 2x + ky + 14 = 0 will represent concident lines.

20. Give linear equations which is coincident with 2 x + 3y - 4 = 0

21. What is the value of a for which (3, a) lies on 2x – 3y = 5

22. The sum of two natural nos. is 25 of their difference is 7. Find the nos.

23. Dinesh in walking along the line joining (1, 4) and (0, 6), Naresh is walking along the line joining (3, 4,) and (1,0). Represent on graph and find the point where both of them cross each other.

24. Solve the pair or linear equations

x – y = 2 and x + y = 2. Also find p if p = 2x + 3

25. For what value of K the following system of equation are parallel.

2x + Ky = 10 3x + (k + 3) y = 12

26. For m a pair of linear equations for the following situation assuming speed of boat in still water

as ‘x’ and speed of stream ‘y’ : A boat covers 32 km upstream and 36 km downstream in 7 hours/

It also covers 40 km upstream and 48 km downstream in 9 hours.

27. Check graphically whether the pair of linear equations 3x + 5y = 15, x – y = 5 is consistent. Also

check whether the pair is dependent.

28. For what value of p the pair of linear equations

(P + 2) x – (2 p + 1)y = 3 (2p – 1) , 2x – 3y = 7 has unique solution.

29. Find the value of K so that the pair of linear equations :

(3 K + 1) x + 3y – 2 = 0 (K2 + 1) x + (k–2)y – 5 = 0 is inconsistent.

30. Given the linear equation x + 3y = 4, write another linear equation in two variables such that the

geometrical representation of the pair so formed is (i) intersected lines (ii) parallel lines (iii) coincident lines.

31. Solve x – y = 4, x + y = 10 and hence find the value of p when y = 3 x –p

32. Determine the value of K for which the given system of o linear equations has infinitely many solutions: Kx + 3y = K – 3 , 12x + Ky = K

33. Find the values of and for which and following system of linear equations has infinite no of solutions : 2x + 3y = 7 2 x + ( + )y = 28.

34. Solve for x and y : [ x +1 ]/ 2 + [y-1 ]/3 = 8 , [ x +1 ]/ 3 + [y-1 ]/2 = 8

35. Solve for x and y : 2x + 3y = 17 2 x + 2 – 3 y+1 = 5.

36. Solve for x and y

139x + 56y = 641 , 56x + 139y = 724

37. Solve for x and y , 5 / [x + y ] + 1/ [x – y ] =2 , 15 / [x + y ] - 5/ [x – y ] = -2

38. Solve for x and y

37x + 43y = 123 43x + 37y = 117

39. Check graphically whether the pair of eq. 3x + 2y – 4 = 0 and 2x – y – 2 = 0 is consistent. Also find the coordinates of the points where the graphs of the lines of equations meet the y-axis.

LONG ANSWER TYPE QUESTIONS
40. Solve for x and y

1/ 2[2x+3y] + 12/7[3x-2y] = ½ , 7/ [2x+3y] + 4/[3x-2y] = 2 for 2x + 3y ¹0 and 3x – 2y¹ 0

41. Solve for p and q [p+q ] / pq =2 and [p-q ] / pq =6

42. Solve for x and y , 2/[3x+2y] + 3/[3x-2y]=17/5 ; 5/[3x+2y] + 1/[3x-2y]=2

43. Solve for x and y . 6 /[x+y] = 7/ [x-y] + 3 ; 1/ 2[x+y] = 1/ 3[x-y]

44. . Solve for x and y ; 2/√x + 3/√y =2 ; 4/√x - 9/√y =2 ;

45. ax + by = 1 ; bx+ay = [(a+b)2 ]/ [a2 +b2] - 1

46. If from twice the greater of two nos., 20 is subtracted, the result is the other no. If from twice the smaller no., 5 is subtracted, the result is the greater no. Find the nos.

47. 27 pencils and 31 rubbers together costs Rs. 85 while 31 pencils and 27 rubbers together costs Rs. 89. Find the cost of 2 pencils and 1 rubber.

48. The area of a rectangle remain the same if its length is increased by 7 cm and the breadth is decreased by 3 cm. The area remains unaffected if length is decreased by 7 cm and the breadth is increased by 5 cm. Find length and breadth.

49. A two digit no. is obtained by either multiplying the sum of the digits by 8 and adding 1; or by multiplying the difference of the digits by 13 and adding 2. Find the no. How many such nos.are there. 50. A no. consists of three digits whose sum is 17. The middle one exceeds the sum of other two  by 1. If the digits are reversed, the no. is diminished by 396. Find the no.

51. A boatman rows his boat 35 km upstream and 55 km down stream in 12 hours. He can row 30 km. upstream and 44 km downstream in 10 hours. Find the speed of he stream and that of the boat in still water. Hence find the total time taken by the boat man to row 50 cm upstream and 77 km downstream

52. Ashok covers 60 km in 1½ hours with the wind and 2 hours against the wind. Find the speed of the Ashok and speed of the wind.

53. The distance between school and metro station is 300 m. Kartikay starts running from school towards metro station, while Ashu starts running from metro station to school. They meet after 4 minutes. Had Kartikay doubled his speed and Ashu reduced his speed to third of the original they would have met one minute earlier. Find their speeds.

54. Puru chase Vinayak who is 5 km ahead. Vinayak is travelling at a speed of 80 km/h and Puru chase at an average speed of 90 km/h. After how much time Puru met Vinayak.

55. In a unit-test the no. of hose that passed and the no. of these that failed were in the ratio 3:1. Had 8 more appeared and 6 less passed, the ratio of passes to failures would have been 2:1. Find how many appeared?

56. In a function if 10 guests are sent from room A to B, the no. of guests in room A and B are same. If 20 guests are sent from B to A, the no. of guests in A is double the no. of guests in B. Find no. of guests in both the rooms in the beginning.

57. In a function Madhu wished to give Rs. 21 to each person present and found that she fell short of Rs. 4 so she distributed Rs. 20 to each and found that Rs. 1 were left over. How much money did she gave and how many persons were there.

58. A mobile company charges a fixed amount as monthly rental which includes 100 minutes free per month and charges a fixed amount these after for every additional minute. Abhishek paid
Rs. 433 for 370 minutes and Ashish paid Rs. 398 for 300 minutes. Find the bill amount under the same plain, if Usha use for 400 minutes.

10th chapter: Pair of Linear Equations in two Variables

1. Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if she travels 2 km by rickshaw, and the remaining distance by bus. On the other and, if she travels 4 km by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and of the bus.

2. A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of the stream. A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.

3. A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.


4. A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the station A to B costs Rs 2530. Also, one reserved first class ticket and one reserved first class half ticket from A to B costs Rs 3810. Find the full first class fare from station A to B, and also the reservation charges for a ticket.

5. A shopkeeper sells a saree at 8% profit and a sweater at 10% discount, thereby, getting a sum Rs 1008. If she had sold the saree at 10% profit and the sweater at 8% discount, she would have got Rs 1028. Find the cost price of the saree and the list price (price before discount) of the sweater.

6. Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received Rs 1860 as annual interest. However, had she interchanged the amount of investments in the two schemes, she would have received Rs 20 more as annual interest. How much money did she invest in each scheme?

7. Vijay had some bananas, and he divided them into two lots A and B. He sold the first lot at the rate of Rs 2 for 3 bananas and the second lot at the rate of Re 1 per banana, and got a total of Rs 400. If he had sold the first lot at the rate of Re 1 per banana, and the second lot at the rate of Rs 4 for 5 bananas, his total collection would have been Rs 460. Find the total number of bananas he had .

8. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled.How long would it take for each pipe to fill the pool separately?

9. Jamila sold a table and a chair for Rs 1050, thereby making a profit of 10% on the table and 25% on the chair. If she had taken a profit of 25% on the table and 10% on the chair she would have got Rs 1065. Find the cost price of each.

10. Had Ajita scored 10 more marks in her mathematics test out of 30 marks, 9 times these marks would have been the square of her actual marks. How many marks did she get in the test?

11. For which values of p and q, will the following pair of linear equations have infinitely many solutions?
              4x + 5y = 2 ; (2p + 7q) x + (p + 8q) y = 2q – p + 1.

12. Solve the following pair of linear equations: 21x + 47y = 110; 47x + 21y = 162

13. Draw the graphs of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.

14. For which value(s) of λ , do the pair of linear equations  λx + y = λ2 and x + λy = 1 have
(i) no solution?      (ii) infinitely many solutions? (iii) a unique solution?

15. For which value(s) of k will the pair of equations kx + 3y = k – 3 ; 12x + ky = k have no solution?