### Arithmetic Progression

Arithmetic Progression

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*n*th terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?

### AREAS RELATED TO CIRCLES

##
**AREAS RELATED TO CIRCLES**

__MCQs__

1 If the perimeter of a semicircular protractor is 36 cm, then its diameter is :

(a) 10 cm (b) 12 cm

(c) 14 cm (d) 15 cm

2 The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 40 cm and 9 cm is

(a) 41 cm (b) 49 cm

(c) 82 cm (d) 62 cm

3 If the diameter of a semicircular protractor is 14 cm, then the perimeter of the protractor is :

(a) 26 cm (b) 14 cm

(c) 28 cm (d) 36 cm

4 The area of a circle whose circumferenc is 44 cm is :

(a) 152 cm

^{2}(b) 153 cm^{2}
(c) 154 cm

^{2}(d) 150 cm^{2}
5 A region in the circle, bounded by an arc and a chord, including the arc and the chord is

(a) sector (b) segment

(c) minor arc (d) major arc

6 If a circle of radius 7 cm is divided into 10 equal parts, then the area of each sector is :

(a) 14.5 cm2 (b) 15.4 cm2

(c) 15.6 cm2 (d) 16.5 cm2

7. If the diameter of the wheel of a cycle is 7cm, then its area is :

(a) 77 cm2 (b) 22 cm2

(c)772 cm2 (d) 770 cm2

8. If the radius of a circle is doubled, then area of the circle becomes :

(a) double (b) triple

(c) four times (d) same

9. The ratio of the area and circumference of a circle of radius 4 cm is :

(a) 4 : 1 (b) 2 : 1

(c) 1 : 2 (d) 8 : 1

10. The diameter of a circle of area 154 cm2 is :

(a) 7 cm (b) 14 cm

(c) 21 cm (d) 7 2 cm

11. If 100 p cm2 is the area of a circle, then its circumference is :

(a) 50p cm b) 20p cm

(c) 10p cm (d) 25p cm

12. The area of a circle of circumference 12 is :

(a) 3p (b) 3/p

(c) 2p (d) 2/p

13. If the area of a sector of a circle of radius 6 cm is 9 p cm, then the angle subtended at the centre of the circle is :

(a) 30° (b) 60°

(c) 90° (d) 120°

14. If a copper wire of length 88 cm is bent in the form of a circle, then the radius of the circle is :

(a) 7 cm (b) 14 cm

(c) 772 cm (d) none of these

15. The area of the circle that can be inscribed in a square of side 6 cm is :

(a) 36p cm2 (b) 18p cm2

(c) 12p cm2 (d) 9p cm2

16. The area of the square that can be inscribed in a circle of radius 8 cm is :

(a) 256 cm2 (b) 128 cm2

(c) 64 2 cm2 (d) 64 cm2

17. The radius of a circle whose circumference is equal to the sum of the circumferences of the two circles of diameters 36 cm and 20 cm is :

(a) 56 cm (b) 42 cm

(c) 28 cm (d) 16 cm

18. The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm is :

(a) 31 cm (b) 25 cm

(c) 62 cm (d) 50 cm

19. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is :

(a) 2 units (b) p units

(c) 4 units (d) 7 units

20. A steel wire, when bent in the form of a square,enclosed an area of 121 cm2. The same wire is bent in the form of a circle. The area of the circle is :

(a) 154 cm2 (b) 145 cm2

(c) 451 cm2 (d) 541 cm2

21. A race track is in the form of a ring whose inner and outer circumferences are 437 m and 503 m respectively. The width of the track is :

(a) 10.5 m (b) 20.5 m

(c) 21 m (d) 30 m

22. Two circles touch internally. The sum of their areas is 116p cm2 and the distance between their centres is 6 cm. The radii of the circles are :

(a) 4 cm and 9 cm (b) 4 cm and 10 cm

(c) 5 cm and 10 cm (d) 4 cm and 20 cm

23. If the sum of the circumferences of two circles with radii R1 and R2 is equal to the circumference of a circle of radius R, then :

(a) R1 + R2 = R

(b) R1 + R2 > R

(c) R1 + R2 < R

(d) nothing definite can be said about the relation among R1, R2 and R.

24 The perimeter of the sector of a circle whose central angle is 45° and radius 7 cm is :

(a) 39 cm (b) 19.5 cm

(c) 35 cm (d) 17.5 cm

25. If the circumference of a circle and the perimeter of a square are equal, then :

(a) Area of the circle = Area of the square

(b) Area of the circle > Area of the square

(c) Area of the circle < Area of the square

(d) Nothing definite can be said about the relation between the areas of the circle and square.

26. The length of a wire in the form of an equilateral triangle is 44 cm. If it is rebent into the form of a circle, then area of the circle is :

(a) 484 cm2 (b) 176 cm2

(c) 154 cm2 (d) 144 cm2

27. If the perimeter of a circle is equal to that of a square, then the ratio of their areas is :

(a) 22 : 7 (b) 14 : 11

(c) 7 : 22 (d) 11 : 14

28. It is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in the locality. The radius of the new park would be :

(a) 10 m (b) 15 m

(c) 20 m (d) 24 m

29. The length of the minute hand of a clock is 6 cm.The area swept by the minute hand in 10 minutes is :

(a) 12p cm2 (b) 36p cm2

(c) 9p cm2 (d) 6p cm2

30. If we decrease the radius of a circle by 20%, then its circumference will be reduced by

(a) 40% (b) 10%

(c) 20% (d) 50%

### 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 2

*m*, and that every positive odd integer is of the form 2

*m*+ 1, where

*m*is some integer.

**2.**Show that any positive odd integer is of form 4

*m*+ 1 or 4

*m*+ 3, where

*m*is some integer.

**3.**Show that any positive odd integer is of the form 6

*m*+ 1, or 6

*m*+ 3, or 6

*m*+ 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

*× 5*

^{m}*, where*

^{n}*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 3

*q*or 3

*q +*1 or 3

*q +*2 for some integer

*q*.

**20**Can the number 6

*n*,

*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 3

*m*or 31

*m*+ 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 6

*can end with the digit 0, for any natural number*

^{n}*n.*

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

**28.**Show that every positive even integer is of the form 2

*q*and that every positive odd integer is of the form 2

*q +*1, where

*q*is some integer.

**29.**Check whether 15

*n*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 4

*m*or 41

*m*+ 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

(c) all integers

(d) all integers except 0.

**2.**For some integer

*m*, every even integer is of the form :

(a)

*m*
(b)

*m*+ 1
(c) 2

*m*
(d) 2

*m*+ 1**3.**If the HCF of 65 and 117 is expressible in the form 65

*m*– 117, then the value of

*m*is :

(a) 1

(b) 2

(c) 3

(d) 4

**4.**If two positive integers

*p*and

*q*can be expressed as

*p = ab*

^{2}and

*q*=

*a*

^{3}

*b*,

*a*;

*b*being prime numbers, then LCM (

*p, q*) is :

(a)

*ab*
(b)

*a*^{2}*b*^{2}
(c)

*a*^{3}*b*^{2}
(d)

*a*^{3}*b*^{3}**5.**The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is :

(a) 10

(b) 100

(c) 504

(d)
2520

**6.**7 × 11 × 13 × 15 + 15 is :

(a) composite number

(b) prime number

(c) neither composite nor
prime

(d) none of these

**7.**1.2348 is :

(a) an integer

(b) an
irrational number

(c) a rational number

(d) none of these

**8.**2.35 is :

(a) a terminating decimal

(b) a rational number

(c) an irrational number

(d) both (a) and (c)

**9.**3.24636363... is :

(a) a terminating decimal
number

(b) a non-terminating
repeating decimal number

(c) a rational number

(d) both (b) and (c)

**10.**For some integer

*q,*every odd integer is of the form :

(a) 2

*q*
(b) 2

*q +*1
(c)

*q*
(d)

*q +*1**11.**If the HCF of 85 and 153 is expressible in the form 85

*m*– 153, then the value of

*m*is :

(a) 1

(b) 4

(c) 3

(d) 2

**12.**The decimal expansion of the rational number 47 / 2

^{2}.5. will terminate after :

(a) one decimal place

(b)
three decimal places

(c) two decimal places

(d) more than 3 decimal places

**13.**If two positive integers

*p*and

*q*can be expressed as

*p = ab*

^{2}and

*q = a*

^{2}

*b*;

*a*,

*b*being prime numbers, then LCM (

*p, q*) is :

(a)

*ab*
(b)

*a*^{2}*b*^{2}
(c)

*a*^{3}*b*^{2}
(b)

*a*^{3}*b*^{3}**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*
(c) 0 <

*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
(c)

*m*= 15,*n =*16
(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

(c) 38

(d) 76

**17.**(

*n +*1)2 – 1 is divisible by 8, if

*n*is :

(a) an odd integer

(b) an
even integer

(c) a natural number

(d) an
integer

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

(a) 1750

(b) 13

(c) 65

(d)
875

**19.**If two integers

*a*and

*b*are written as

*a = x*

^{3}

*y*

^{2}and

*b = xy*

^{4};

*x, y*are prime numbers, then H.C.F. (

*a, b*) is :

(a)

*x*^{3}*y*^{3}
(b)

*x*^{2}*y*^{2}
(c)

*xy*
(d)

*xy*^{2}**20.**The decimal expansion of the rational number 145171250 will terminate after :

(a) 4 decimal places

(b) 3 decimal places

(c) 2 decimal places

(d) 1 decimal
place

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