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MATHS M.C.Q (1 Marks)

Real Numbers · UP Board (UPMSP) STD 10 (UPMSP - English Medium). 30 questions.

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M.C.Q (1 Marks)

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30 questions · timed · auto-graded

MCQ 11 Mark
A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13?
  • A
    0
  • B
    1
  • C
    3
  • ✓
    5
Answer
Correct option: D.
5
Let the number be n.When the number is divided by 143, leaves 31 as remainder.
⇒ The given number is of the form, 143x + 31
⇒ n = 143x + 31, where x is the quotient
⇒ n = 13(11x) + 13(2) + 5
⇒ n = 13(11x + 2) + 5
So, here the remainder will be 5 when divided by 13
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MCQ 21 Mark
$\text{HCF}$ of $\left(2^3 \times 3^2 \times 5\right),\left(2^2 \times 3^3 \times 5^2\right)$ and $\left(2^4 \times^3 \times 5^3 \times 7\right)$ is:
  • A
    $30$
  • B
    $48$
  • ✓
    $60$
  • D
    $105$
Answer
Correct option: C.
$60$

$(2^3 \times 3^2 \times 5), (2^2 \times 3^3 \times 5^2)$ and $(2^4 \times ^3\times 5^3 \times 7)$
$HCF = 2^2 \times 3 \times 5 = 60$

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MCQ 31 Mark
2.13113111311113... is:
  • A
    An integer.
  • B
    A rational number.
  • ✓
    An irrational number.
  • D
    None of these.
Answer
Correct option: C.
An irrational number.
An irrational number is a number that is non-terminating and non-repeating.
2.13113111311113... is neither terminating nor repeating, and hence is an irrational number.
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MCQ 41 Mark
a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a + b) is:
  • ✓
    2
  • B
    3
  • C
    5
  • D
    8
Answer
Correct option: A.
2
Since 3 is the least prime factor of a, and 5 is the least prime factor of b, so, 2 cannot be a factor of either.
$\therefore$ a and b are both odd.
We know that, sum of two odd numbers is alwayas even.
So, a + b is even.
⇒ The least prime factor of (a + b) is 2
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MCQ 51 Mark
What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?
  • ✓
    13
  • B
    9
  • C
    3
  • D
    585
Answer
Correct option: A.
13
70 and 125 are divided by the largest number leaving remainders 5 and 8 respectively.70 - 5 = 65
125 - 8 = 117
So, 65 and 117 are exactly divisible by the required number.
Thus, the required number is the HCF of 65 and 117
HCF(65, 117) = 13
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MCQ 61 Mark
Which of the following has terminating decimal expansion?
  • A
    $\frac{32}{91}$
  • ✓
    $\frac{19}{80}$
  • C
    $\frac{23}{45}$
  • D
    $\frac{25}{42}$
Answer
Correct option: B.
$\frac{19}{80}$
A number is a terminating decimal, if the denominator is of the form $2^m \times 5^n$,
where m and n are non$-$negative integers.
$\frac{32}{91}=\frac{32}{7\times13}$
$\frac{19}{80}=\frac{19}{2^4\times5}$
$\frac{23}{45}=\frac{23}{3^2\times5}$
$\frac{25}{42}=\frac{25}{2\times3\times7}$
Clearly, option $(b)$ is a terminating decimal, since its denominator is of the form $2^m \times 5^n$
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MCQ 71 Mark
$\frac{1}{\sqrt{2}}$ is:
  • A
    A fraction.
  • B
    A rational number.
  • ✓
    An irrational number.
  • D
    None of these.
Answer
Correct option: C.
An irrational number.
An irrational number is a number that is non-terminating and non-repeating.$\frac{1}{\sqrt{2}}=\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}$ ...(Rationalising the denominator)
$=\frac{\sqrt{2}}{2}$
$=\frac{1}2{}\times\sqrt{2}$
Now, $\frac{1}2{}$ is rational but $\sqrt2$ is irrational.
Product of a rational number and an irrational number is irrational.
Hence, $\frac{1}{\sqrt2}$ is an irrational number.
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MCQ 81 Mark
Which of the following rational numbers is expressible as a terminating decimal?
  • A
    $\frac{124}{165}$
  • B
    $\frac{131}{30}$
  • ✓
    $\frac{2027}{625}$
  • D
    $\frac{1625}{462}$
Answer
Correct option: C.
$\frac{2027}{625}$

A number is a terminating decimal, if the denominator is of the form $2^m \times 5^n$,
where m and n are non$-$negative integers.
$\frac{124}{165}=\frac{124}{3\times5\times11}$
$\frac{131}{30}=\frac{131}{2\times3\times5}$
$\frac{2027}{625}=\frac{2027}{5^4}=\frac{2027}{2^0\times5^4}$
$\frac{1625}{462}=\frac{1625}{2\times3\times7\times11}$
Clearly, option (c) is a terminating decimal, since its denominator is of the form $2^m \times 5^n.$

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MCQ 91 Mark
$2.\overline{35}$ is:
  • A
    An integer.
  • ✓
    A rational number.
  • C
    An irrational number.
  • D
    None of these.
Answer
Correct option: B.
A rational number.
$2.\overline{35}=2.35353535\dots$
Which is repeating decimal number, and hence is a rational number.
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MCQ 101 Mark
The number 3.24636363... is:
  • A
    An integer.
  • ✓
    A rational number.
  • C
    An irrational number.
  • D
    None of these.
Answer
Correct option: B.
A rational number.
3.24636363...Which is repeating decimal number, and hence is a rational number.
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MCQ 111 Mark
The decimal expansion of the rational number $\frac{37}{2^2\times5}$ will terminate after:
  • A
    One decimal place.
  • ✓
    Two decimal places.
  • C
    Three decimal places.
  • D
    Four decimal places.
Answer
Correct option: B.
Two decimal places.
The prime factorisation of the denominator is $2^2 \times 5$
Since $2 > 1,$
The decimal expansion will terminate after $2$ decimal places.
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MCQ 121 Mark
$\sqrt2$ is:
  • A
    A rational number.
  • ✓
    An irrational number.
  • C
    A terminating decimal.
  • D
    A non-terminating repeating decimal.
Answer
Correct option: B.
An irrational number.
An irrational number is a number that is non-terminating and non-repeating.
$\sqrt2=1.4142135\dots$ which is neither terminating nor repeating, and hence is an irrational number.
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MCQ 131 Mark
$0.\overline{68}+0.\overline{73}=?$
  • A
    $1.\overline{41}$
  • ✓
    $1.\overline{42}$
  • C
    $0.\overline{141}$
  • D
    None of these.
Answer
Correct option: B.
$1.\overline{42}$
Consider, $\text{x}=0.\overline{68}$$\Rightarrow\text{x}=0.6868\dots\ \ \dots(\text{i})$
Multiply by 100
$\Rightarrow\text{100x}=68.68\dots\ \ \dots(\text{ii})$
Subtracting (i) from (ii), we get
$\text{99x}=68$
$\Rightarrow\text{x}=\frac{68}{99}\dots(\text{A})$
Consider, $\text{x}=0.\overline{73}$
⇒ x = 0.7373... ...(iii)
Multiply by 100
⇒ 100x = 73.73... ...(iv)
Subtracting (iii) from (iv), we get
$\text{99x}=73$
$\Rightarrow\text{x}=\frac{73}{99}\dots(\text{B})$
Adding (A) and (B), gives us
$\frac{68}{99}+\frac{73}{99}=\frac{141}{99}=1.42424\dots$
$\Rightarrow0.\overline{68}+0.\overline{73}=1.42424\dots$
$=1.\overline{42}$
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MCQ 141 Mark
The decimal representation of $\frac{71}{150}$ is:
  • A
    A terminating decimal.
  • ✓
    A non$-$terminating, repeating decimal.
  • C
    A non$-$terminating and non-repeating decimal.
  • D
    None of these.
Answer
Correct option: B.
A non$-$terminating, repeating decimal.

A number is a terminating decimal, if the denominator is of the form $2^m \times 5^n$,
where m and n are non$-$negative integers.
The prime factorisation of the denominator is $2 \times 3 \times 50^2$
So, the denominator will be non$-$ terminating.
Since $\frac{71}{150}$ is a rational number, it will surely be repeating.

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MCQ 151 Mark
What is the largest number that divides each one of 1152 and 1664 exactly?
  • A
    32
  • B
    64
  • ✓
    128
  • D
    256
Answer
Correct option: C.
128
The largest number that divides each one of 1152 and 1664 exactly will be the HCF of the numbers.Using Euclid's Division Algorithm,
1664 = 1152 × 1 + 512
1152 = 512 × 2 + 128
512 = 128 × 4 + 0
So, HCF(1152, 1664) = 128
Hence, the largest number is 128
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MCQ 161 Mark
$\pi$ is :
  • A
    An integer.
  • B
    A rational number.
  • ✓
    An irrational number.
  • D
    None of these.
Answer
Correct option: C.
An irrational number.
An irrational number is a number that is non $-$ terminating and non $-$ repeating.
$\pi=3.1415926\dots$
Which is neither terminating nor repeating, and hence is an irrational number.
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MCQ 171 Mark
What is the least number that divisible by all the natural numbers from $1$ to $10 ($both inclusive$)$?
  • A
    $100$
  • B
    $1260$
  • ✓
    $2520$
  • D
    $5040$
Answer
Correct option: C.
$2520$

To find the least number divisible by all the natural numbers is the $\text{LCM}$ of the numbers from $1$ to $10$
Find the prime factorization of each of the numbers to find the $\text{LCM}$.
$1, 2, 3, 5, 7, 4 = 2^2, 6 = 2 \times 3, 8 = 2^3, 9 = 3^2, 10 = 2 \times 5$
$\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 2520$

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MCQ 181 Mark
Which of the following is a pair of co-primes?
  • A
    (14, 35)
  • ✓
    (18, 25)
  • C
    (31, 93)
  • D
    (32, 62)
Answer
Correct option: B.
(18, 25)
Two numbers are said to be co-primeIf the HCF between them is 1
14 = 2 × 7
35 = 5 × 7
HCF(14, 35) = 7
18 = 2 × 3 × 3
25 = 5 × 5
HCF(18, 25) = 1
31 = 1 × 31
93 = 3 × 31
HCF(31, 93) = 31
32 = 2 × 2 × 2 × 2 × 2
62 = 2 × 31
HCF(32, 62) = 2
Since the HCF(18, 25) is 1, 18 and 25 is the pair of co-primes.
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MCQ 191 Mark
Euclid's division lemma sates that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy:
  • A
    $1<\text{r}<\text{b}$
  • B
    $0<\text{r}\le\text{b}$
  • ✓
    $0\le\text{r}<\text{b}$
  • D
    $0<\text{r}<\text{b}$
Answer
Correct option: C.
$0\le\text{r}<\text{b}$
Euclid's division lemma states that,
For any positive integers a and b, there exist unique integers q and r such that
$\text{a}=\text{bq}+\text{r},$ where $0\le\text{r}<\text{b}$
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MCQ 201 Mark
The number 1.732 is:
  • A
    An irrational number.
  • ✓
    A rational number.
  • C
    An integer.
  • D
    A whole number.
Answer
Correct option: B.
A rational number.
Since the number is a terminating decimal number, it is a rational number.
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MCQ 211 Mark
The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number?
  • A
    36
  • B
    45
  • C
    9
  • ✓
    81
Answer
Correct option: D.
81
Let the two n umbers be a and b.
HCF × LCM = ab
⇒ 27 × 162 = 54 × b
⇒ b = 81
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MCQ 221 Mark
What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?
  • A
    15
  • ✓
    16
  • C
    9
  • D
    5
Answer
Correct option: B.
16
245 and 1029 are divided by the largest number leaving remainders 5 in each case.
245 - 5 = 240
1029 - 5 = 1024
So, 240 and 1024 are exactly divisible by the required number.
Thus, the required number is the HCF of 240 and 1024
HCF(240, 1024) = 16
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MCQ 231 Mark
On dividing a positive integer n by 9, we get 7 as remainder. What will be the remainder if (3n - 1) is divided by 9?
  • A
    1
  • ✓
    2
  • C
    3
  • D
    4
Answer
Correct option: B.
2
On dividing n by 9 the remainder is 7
⇒ n = 9q + 7, where q is the quotient
⇒ 3n = 3(9q + 7)
⇒ 3n = 27q + 21
⇒ 3n - 1 = 27q + 21 - 1
⇒ 3n - 1 = 27q + 20
⇒ 3n - 1 = 27q + 18 + 2
⇒ 3n - 1 = 9(3q + 2) + 2
So, the remainder will be 2
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MCQ 241 Mark
Which of the following is an irrational number?
  • A
    $\frac{22}{7}$
  • B
    $3.1416$
  • C
    $3.\overline{1416}$
  • ✓
    $3.141141114...$
Answer
Correct option: D.
$3.141141114...$
An irrational number is a number that is non-terminating and non-repeating.
Option (a) is a rational number, while option (c) is a repeating decimal number, and so are rational numbers. Option (d) is an irrational number.
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MCQ 251 Mark
The simplest form of $\frac{1095}{1168}$ is:
  • A
    $\frac{17}{26}$
  • B
    $\frac{25}{26}$
  • C
    $\frac{13}{16}$
  • ✓
    $\frac{15}{16}$
Answer
Correct option: D.
$\frac{15}{16}$
$\frac{1095}{1168}=\frac{5\times3\times73}{2\times2\times2\times2\times73}$
$=\frac{5\times3}{2\times2\times2\times2}$
$=\frac{15}{16}$
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MCQ 261 Mark
The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is:
  • A
    8000
  • B
    1600
  • ✓
    320
  • D
    1605
Answer
Correct option: C.
320
Let the two n umbers be a and b.
HCF × LCM = ab
⇒ 5 × LCM = 1600
⇒ LCM = 320
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MCQ 271 Mark
$\text{LCM}$ of $(2^3 \times 3 \times 5)$ and $(2^4 \times 5 \times 7)$ is:
  • A
    $40$
  • B
    $560$
  • C
    $1120$
  • ✓
    $1680$
Answer
Correct option: D.
$1680$
$(2^3 \times 3 \times 5)$ and $(2^4 \times 5 \times 7)$
$\text{LCM} = 2^4 \times 3 \times 5 \times 7 = 1680$
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MCQ 281 Mark
The decimal expansion of the number $\frac{14753}{1250}$ will terminate after:
  • A
    One decimal place.
  • B
    Two decimal places.
  • C
    Three decimal places.
  • ✓
    Four decimal places.
Answer
Correct option: D.
Four decimal places.

The prime factorisation of the denominator is $2 \times 5^2$
Since $4 > 1,$
The decimal expansion will terminate after $4$ decimal places.

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MCQ 291 Mark
$\big(2+\sqrt{2}\big)$ is:
  • A
    An integer.
  • B
    A rational number.
  • ✓
    An irrational number.
  • D
    None of these.
Answer
Correct option: C.
An irrational number.
An irrational number is a number that is non-terminating and non-repeating.
Now, 2 is a rational number and $\sqrt2$ is an irrational number.
Sum of a rational number and an irrational number is irrational.
Hence, $\big(2+\sqrt{2}\big)$ is an irrational number.
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MCQ 301 Mark
If $a =\left(2^2 \times 3^3 \times 5^4\right)$ and $b =\left(2^3 \times 3^2 \times 5\right)$, then $\operatorname{HCF}( a , b )= ?$
  • A
    $90$
  • ✓
    $180$
  • C
    $360$
  • D
    $540$
Answer
Correct option: B.
$180$
$a = 2^2 \times 3^3 \times 5^4$
$b = 2^3 \times 3^2 \times 5$
$\text{HCF} (a, b) = 2^2 \times 3^2 \times 5 = 180$
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