Chapter 1 · Real Numbers

NCERT Exemplar Class 10 Maths Exercise 1.2

All 10 questions of NCERT Exemplar Class 10 Maths Exercise 1.2 (Real Numbers), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Write ​whether ​every positive integer can be of the form 4q + ​2, ​where q ​is an integer. Justify ​your answer.
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   No - not every positive integer is of the form $4q+2$. By Euclid's division lemma an integer can be $4q$, $4q+1$, $4q+2$ or $4q+3$; the form $4q+2$ covers only those leaving remainder 2 (e.g. it misses 1, 3, 4, 5).

   Step-by-step solution →
2. Q2
   "The ​product of ​two consecutive ​positive integers ​is ​divisible by 2". Is ​this statement true ​or false? Give ​reasons.
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   True. Of any two consecutive integers one is even, so the product $n(n+1)$ always has an even factor and is divisible by $2$.

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3. Q3
   "The product of three ​consecutive positive integers ​is ​divisible ​by 6." Is this statement true or ​false? ​Justify your ​answer.
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   True. Among three consecutive integers one is a multiple of 3 and at least one is even, so the product is divisible by both 2 and 3, hence by 6.

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4. Q4
   Write whether ​the square ​of any ​positive ​integer can be of ​the form $3m+2$, where $m$ is a natural ​number. Justify your ​answer.
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   No. The square of any positive integer is always of the form $3m$ or $3m+1$, never $3m+2$.

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5. Q5
   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.
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   No - the square of a number of the form $3q+1$ is always of the form $3m+1$. It can never be $3m$ or $3m+2$, because $(3q+1{)}^2=3(3q^2+2q)+1$.

   Step-by-step solution →
6. Q6
   The ​numbers ​525 and 3000 are both divisible only by 3, ​5, 15, 25 ​and 75. ​What ​is HCF ​(525, 3000)? ​Justify your answer.
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   HCF$(525,3000)=75$, since 75 is the greatest of the common divisors $3,5,15,25,75$.

   Step-by-step solution →
7. Q7
   Explain ​why $3\times 5\times 7+7$ is ​a ​composite ​number.
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   $3\times 5\times 7+7=7(3\times 5+1)=7\times 16=112$, a product of two integers greater than 1, so it is composite.

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8. Q8
   Can two numbers have 18 ​as their ​HCF and 380 as ​their ​LCM? Give ​reasons.
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   No. The HCF must divide the LCM, but $18$ does not divide $380$ ($380=18\times 21+2$). So no two numbers can have HCF 18 and LCM 380.

   Step-by-step solution →
9. Q9
   Without actually ​performing the ​long division, find if $\frac{987}{10500}$ will have terminating ​or non-terminating (repeating) ​decimal ​expansion. Give reasons for your ​answer.
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   $\frac{987}{10500}=\frac{47}{500}$ and $500=2^2\times 5^3$, so the decimal expansion is terminating.

   Step-by-step solution →
10. Q10
    A ​rational ​number in ​its decimal ​expansion ​is $327.7081$. What can you say about the prime factors ​of $q$, when this number ​is expressed in the form $\frac{p}{q}$? ​Give reasons.
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    Since 327.7081 has a terminating decimal expansion, $q$ in $\frac{p}{q}$ (lowest terms) has prime factors only of the form $2^m\times 5^n$ -- that is, $q$ is made up only of 2s and 5s.

    Step-by-step solution →