Mathematics Village (10th Mathematics )

One Hour Test

How many two-digit numbers are divisible by 3?
Find the 11th term from the last term of the AP : 10, 7, 4, . . ., – 62.
A sum of Rs 1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of 30 years
In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on.There are 5 rose plants in the last row. How many rows are there in the flower bed?
If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.
A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year Assuming that the production increases uniformly by a fixed number every year, find : the production in the 1st year (ii) the production in the 10th year (iii) the total production in first 7 years
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on In how may rows are the 200 logs placed and how many logs are in the top row?
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . . What is the total length of such a spiral made up of thirteen consecutive semicircles?
If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?

Real Numbers SHORT ANSWER TYPE QUESTIONS

Real Numbers

SHORT ANSWER TYPE QUESTIONS [2 Marks]

1. Show that every positive even integer is of the from 2m, and that every positive odd integer is of the form 2m + 1, where m is some integer.

2. Show that any positive odd integer is of form 4m + 1 or 4m + 3, where m is some integer.

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

4. Find the HCF of 1656 and 4025 by Euclid’s method.

5. Factorise 34650 using factor tree.

6. Find the HCF of 255 and 867 by prime factorisation.

7. Find the largest number which can divide 3528 and 2835.

8. Find the LCM of 2520 and 2268 by prime factorisation.

9. Find the smallest number which is divisible by 85 and 119.

10. Show that 5 - Ö3 is irrational.

11. Show that 3Ö2 is irrational.

12. Show that 1 / Ö2 is irrational.

13. Write the denominator of the rational number 257 / 5000 in the form 2^m × 5ⁿ, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

14. The values of the remainder r, when a positive integer a is divided by 3, are 0 and 1 only. Is it true? Justify your answer.

15. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.

16. Show that the sum and product of two irrational numbers (5 + Ö2) and (5 – Ö2) are rational numbers.

17. Without actually performing the long division, find if 987 / 10500 will have terminating or non-terminating repeating decimal expansion. Give reason for your answer.

18. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

19. Show that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some integer q.

20 Can the number 6n, n being a natural number, end with the digit 5? Give reasons.

21. Use Euclid’s division lemma to show that square of any positive integer is either of form 3m or 31m+ for some integer m.

22. Find the L.C.M. of 120 and 70 by fundamental theorem of Arithmetic.

23. Write 60 in form of factor tree.

24. Without actually performing the long division, state whether the following number has a terminating decimal expansion or non terminating recurring decimal expansion 543 / 225.

25. Use Euclid’s division algorithm to find HCF of 870 and 225.

26. Check whether 6ⁿ can end with the digit 0, for any natural number n.

27. Explain why 11 × 13 × 15 × 17 + 17 is a composite number.

28. Show that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q + 1, where q is some integer.

29. Check whether 15n can end with digit zero for any natural number n.

30. Find the LCM of 336 and 54 by prime factorisation method.

31. Find the LCM and HCF of 120 and 144 by fundamental theorem of arithmetic.

32. Use Euclid’s Lemma to show that square of any positive integer is of form 4m or 41m+ for some integer m.

33. Using fundamental theorem of arithmetic, find the HCF of 26, 51 and 91

34. Find the LCM and HCF of 15, 18, 45 by the prime factorisation method..

35. Find the HCF and LCM of 306 and 54. Verify that HCF × LCM = Product of the two numbers

Real Number MCQs

Real Number

Xth Mathematics

MULTIPLE CHOICE QUESTIONS

1 Euclid’s division algorithm can be applied to :

(a) only positive integers

(b) only negative integers

(d) all integers except 0.

2. For some integer m, every even integer is of the form :

(a) m

(b) m + 1

(d) 2m + 1

3. If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is :

(a) 1

(b) 2

(d) 4

4. If two positive integers p and q can be expressed as p = ab² and q = a³b, a; b being prime numbers, then LCM (p, q) is :

(a) ab

(b) a²b²

(d) a³b³

5. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is :

(a) 10

(b) 100

(d) 2520

6. 7 × 11 × 13 × 15 + 15 is :

(a) composite number

(b) prime number

(d) none of these

7. 1.2348 is :

(a) an integer

(b) an irrational number

(d) none of these

8. 2.35 is :

(a) a terminating decimal

(b) a rational number

(d) both (a) and (c)

9. 3.24636363... is :

(a) a terminating decimal number

(b) a non-terminating repeating decimal number

(d) both (b) and (c)

10. For some integer q, every odd integer is of the form :

(a) 2q

(b) 2q + 1

(d) q + 1

11. If the HCF of 85 and 153 is expressible in the form 85m – 153, then the value of m is :

(a) 1

(b) 4

(d) 2

12. The decimal expansion of the rational number 47 / 2².5. will terminate after :

(a) one decimal place

(b) three decimal places

(d) more than 3 decimal places

13. If two positive integers p and q can be expressed as p = ab² and q = a²b; a, b being prime numbers, then LCM (p, q) is :

(a) ab

(b) a²b²

(b) a³b³

14. Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where :

(a) 0 < r ≤ b

(b) 1 < r < b

(d) 0 ≤ r < b

15. Following are the steps in finding the GCD of 21 and 333 :

333 = 21 × m + 18

21 = 18 × 1 + 3

n = 3 × 6 + 0

The integers m and n are :

(a) m = 15, n = 15

(b) m = 15, n = 18

(d) m = 18, n = 15

16. HCF and LCM of a and b are 19 and 152 respectively. If a = 38, then b is equal to :

(a) 152

(b) 19

(d) 76

17. (n + 1)2 – 1 is divisible by 8, if n is :

(a) an odd integer

(b) an even integer

(d) an integer

18. The largest number which divides 71 and 126, leaving remainders 6 and 9 respectively is :

(a) 1750

(b) 13

(d) 875

19. If two integers a and b are written as a = x³y² and b = xy⁴; x, y are prime numbers, then H.C.F. (a, b) is :

(a) x³y³

(b) x²y²

(d) xy²

20. The decimal expansion of the rational number 145171250 will terminate after :

(a) 4 decimal places

(b) 3 decimal places

(d) 1 decimal place

Back to Mathematics Village

Assignment of Real Numbers

Case Studies

Linear Equation of Two Variable assignment (Graph )

Arithmetic Progression

Real Numbers SHORT ANSWER TYPE QUESTIONS

Real Number MCQs