📝 Real Numbers (Questions)
11. The decimal expansion of the rational number `frac{33}{2^2 . 5}` will terminate after
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) more than 3 decimal places
Ans: (b)
12. 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 `r`
must satisfy
(a) 1 < r < b
b) 0 < r < b
(c) 0 `\le` r < b
(d) 0 < r < b
Ans: (c)
13. The ratio of LCM and HCF of the least composite and the least prime numbers is
(a) `1:2`
(b) `2:1`
(c) `1:1`
(d) `1:3`
Ans: (b)
14. If LCM(x, 18) =36 and HCF(x, 18) =2, then x is
(a) 2
(b) 3
(c) 4
(d) 5
Ans: (c)
★★★★★★
Assertion & Reason
(a) If both Assertion and Reason are true and Reason is the correct explanation of Assertion.
(b) If both Assertion and Reason are true and Reason is not the correct explanation of Assertion.
(c) If Assertion is true but Reason is false.
(d) If Assertion is false but Reason is true.
1. Assertion (A): (7 × 13 × 11) + 11 and ( 7 × 6 × 5 × 4 × 3 × 2 × 1) + 3 have exactly composite number.
Reason (R): ( 3 × 12 × 101) + 4 is not a compostive number.
Ans: (c)
2. Assertion (A): The product of two consecutive positive integers is divisible by 2.
Reason (R): 132333343563715 is a composite number.
Ans: (b)
★★★★★★
Case Studies
CASE STUDY 1.
To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B.
1. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B?
a) 144
b) 128
c) 288
d) 272
Ans: (c)
2. If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF (32, 36) is
a) 2
b) 4
c) 6
d) 8
Ans:(b)
3. 36 can be expressed as a product of its primes as
a) `2^2 × 3^2`
b) `2 × 3^3`
c) `2^3 × 3`
d) `2^0 × 3^0`
Ans:(a)
4. `7 × 11 × 13 × 15 + 15`
a) Prime number
b) Composite number
c) Neither prime nor composite
d) None of the above
Ans: (b)
5. If `p` and `q` are positive integers such that `p = ab^2` and `q= a^2b`, where `a, b` are prime numbers, then the LCM `(p, q)` is
a) `ab`
b) `a^2b^2`
c) `a^3b^2`
d) `a^3b^3`
Ans: (b)
CASE STUDY 2:
A Mathematics Exhibition is being conducted in your School and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience.
Observe the following factor tree and answer the following:
1. What will be the value of x?
a) 15005
b) 13915
c) 56920
d) 17429
Ans: (b)
2. What will be the value of y?
a) 23
b) 22
c) 11
d) 19
Ans: (c)
3. What will be the value of z?
a) 22
b) 23
c) 17
d) 19
Ans: (b)
4. According to Fundamental Theorem of Arithmetic 13915 is a
a) Composite number
b) Prime number
c) Neither prime nor composite
d) Even number
Ans: (a)
5. The prime factorisation of 13915 is
a) `5 × 11^3 × 13^2`
b) `5 × 11^3 × 23^2`
c) `5 × 11^2 × 23`
d) `5 × 11^2 × 13^2`
Ans: (c)
CASE STUDY 3:
A seminar is being conducted by an Educational Organisation, where the participants will be educators of different subjects. The number of participants in Hindi, English, and Mathematics are 60, 84, and 108 respectively.
1. In each room the same number of participants are to be seated and all of them being in the same subject, hence maximum number of participants that can be accommodated in each room are
a) 14
b) 12
c) 16
d) 18
Ans: (b)
2. What is the minimum number of rooms required during the event?
a) 11
b) 31
c) 41
d) 21
Ans: (d)
3. The LCM of 60, 84 and 108 is
a) 3780
b) 3680
c) 4780
d) 4680
Ans: (a)
4. The product of HCF and LCM of 60,84 and 108 is
a) 55360
b) 35360
c) 45500
d) 45360
Ans: (d)
5. 108 can be expressed as a product of its primes as
a) `2^3 × 3^2`
b) `2^3 × 3^3`
c) `2^2 × 3^2`
d) `2^2 × 3^3`
Ans: (d)
★★★★★★
Q) A positive integer is of the form `3q + 1`, `q` being a natural number. Can
you write its square in any form other than `3m + 1`, i.e., `3m` or `3m + 2`
for some integer `m`? Justify your answer.
Q) Show that the square of an odd positive integer is of the form `8m + 1`,
for some whole number `m.`
Q) Prove that if `x` and `y` are both odd positive integers, then `x^2 + y^2`
is even but not divisible by 4.
Q) Use Euclid’s division lemma to show that the cube of any positive integer
is of the form `9m, 9m + 1 or 9m + 8.`
Good questions
ReplyDeleteThanks Sir for good question
ReplyDelete