M.C.Q (1 Marks) - Maths STD 10 Questions - Vidyadip

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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.
$\Rightarrow $ The given number is of the form, $143x + 31$
$\Rightarrow n = 143x + 31$, where x is the quotient
$\Rightarrow n = 13(11x) + 13(2) + 5$
$\Rightarrow n = 13(11x + 2) + 5$
So, here the remainder will be $5$ when divided by $13$

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MCQ 21 Mark

$HCF$ 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.

$60$

$\left(2^3 \times 3^2 \times 5\right),\left(2^2 \times 3^3 \times 5^2\right)$ and $\left(2^4 \times{ }^3 \times 5^3 \times 7\right)$
$HCF = 2^2× 3 × 5 = 60$

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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

$\pi$ 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 51 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.
$\Rightarrow $ The least prime factor of $(a + b)$ is $2$

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MCQ 61 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 71 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 $2^m × 5^n$, 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 $2^m × 5^n$

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MCQ 81 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 91 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 $2^m × 5^n$, 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 $2^m × 5^n$.

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MCQ 101 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 111 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 121 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 $2^2 × 5$
Since $2 > 1,$
The decimal expansion will terminate after 2 decimal places.

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MCQ 131 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 141 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, $\text{x}=0.\overline{73}$
$\Rightarrow x = 0.7373... ...(iii)$
Multiply by $100$
$\Rightarrow 100x = 73.73... ...(iv)$
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 151 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 $2^m× 5^n$, where m and n are non-negative integers.
The prime factorisation of the denominator is $2 × 3 × 50^2$
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 161 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 171 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 181 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 $LCM$ of the numbers from $1$ to $10$
Find the prime factorization of each of the numbers to find the $LCM$.
$ 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 191 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-prime
If the $HCF$ between them is $1$
$14 = 2 \times 7$
$35 = 5 \times 7$
$HCF(14, 35) = 7$
$18 = 2 \times 3 \times 3$
$25 = 5 \times 5$
$HCF(18, 25) = 1$
$31 = 1 \times 31$
$93 = 3 \times 31$
$HCF(31, 93) = 31$
$32 = 2 \times 2 \times 2 \times 2 \times 2$
$62 = 2 \times 31$
$HCF(32, 62) = 2$
Since the $HCF(18, 25)$ is $1, 18$ and $25$ is the pair of co-primes.

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MCQ 201 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 211 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 221 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 $b$.
$HCF \times LCM = ab$
$\Rightarrow 27 \times 162 = 54 \times b$
$\Rightarrow b = 81$

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MCQ 231 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 241 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 $7$
$\Rightarrow n = 9q + 7,$ where q is the quotient
$\Rightarrow 3n = 3(9q + 7)$
$\Rightarrow 3n = 27q + 21$
$\Rightarrow 3n - 1 = 27q + 21 - 1$
$\Rightarrow 3n - 1 = 27q + 20$
$\Rightarrow 3n - 1 = 27q + 18 + 2$
$\Rightarrow 3n - 1 = 9(3q + 2) + 2$
So, the remainder will be $2$

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MCQ 251 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 261 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 271 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 $b$.
$HCF \times LCM = ab$
$\Rightarrow 5 \times LCM = 1600$
$\Rightarrow LCM = 320$

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MCQ 281 Mark

$LCM$ of $ \left(2^3 \times 3 \times 5\right) \text { and }\left(2^4 \times 5 \times 7\right) $ is:

A
$40$
B
$560$
C
$1120$
✓
$1680$

Answer

Correct option: D.

$1680$

$ \left(2^3 \times 3 \times 5\right) \text { and }\left(2^4 \times 5 \times 7\right) $
$\text { LCM }=2^4 \times 3 \times 5 \times 7=1680 $

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MCQ 291 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 $2 × 5^2$
Since $4 > 1,$
The decimal expansion will terminate after 4 decimal places.

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MCQ 301 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 311 Mark

If $\mathrm{a}=\left(2^2 \times 3^3 \times 5^4\right) \text { and } \mathrm{b}=\left(2^3 \times 3^2 \times 5\right)$, then $HCF (a, b) = ?$

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 $
$ \operatorname{HCF}(a, b)=2^2 \times 3^2 \times 5=180$

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M.C.Q (1 Marks) - Maths STD 10 Questions - Vidyadip