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Chapter 1: Real Numbers

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This page provides comprehensive Chapter 1: Real Numbers - HOTS Worksheet - SJMaths. High Order Thinking Skills (HOTS) worksheet for Class 10 Real Numbers.

HOTS (High Order Thinking Skills)

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  1. Question 1: Prove that $\sqrt{2} + \sqrt{3}$ is irrational.
    Solution: Let $x = \sqrt{2} + \sqrt{3}$ be rational. $x^2 = 2 + 3 + 2\sqrt{6} = 5 + 2\sqrt{6}$. $\sqrt{6} = (x^2 - 5)/2$. Since $x$ is rational, RHS is rational, but $\sqrt{6}$ is irrational. Contradiction.
  2. Question 2: Use Euclid's division lemma to show that the square of any positive integer is either of the form $3m$ or $3m+1$ for some integer $m$.
    Solution: Let $a = 3q+r$, $r=0,1,2$. $a^2 = (3q)^2 = 9q^2 = 3(3q^2) = 3m$. Or $a^2 = (3q+1)^2 = 9q^2+6q+1 = 3(3q^2+2q)+1 = 3m+1$. Or $a^2 = (3q+2)^2 = 9q^2+12q+4 = 3(3q^2+4q+1)+1 = 3m+1$.
  3. Question 3: Find the HCF of 65 and 117 and express it in the form $65m + 117n$.
    Solution: HCF is 13. $117 = 65 \times 1 + 52$; $65 = 52 \times 1 + 13$. So $13 = 65 - 52 = 65 - (117 - 65) = 65 \times 2 - 117 \times 1$. Thus $m=2, n=-1$.
  4. Question 4: Find the largest number which divides 2053 and 967 leaving remainders 5 and 7 respectively.
    Solution: Numbers are $2053-5=2048$ and $967-7=960$. HCF(2048, 960) = 64.
  5. Question 5: Prove that one and only one out of $n, n+2$ and $n+4$ is divisible by 3, where $n$ is any positive integer.
    Solution: Let $n = 3q+r$. If $r=0, n$ is divisible. If $r=1, n+2=3q+3$ is divisible. If $r=2, n+4=3q+6$ is divisible.
  6. Question 6: Two tanks contain 504 and 735 liters of milk respectively. Find the maximum capacity of a container which can measure the milk of either tank an exact number of times.
    Solution: HCF(504, 735). $504 = 21 \times 24$, $735 = 21 \times 35$. HCF is 21 liters.
  7. Question 7: Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.
    Solution: LCM(24, 15, 36) = 360. Greatest 6 digit number is 999999. $999999 = 360 \times 2777 + 279$. Number $= 999999 - 279 = 999720$.
  8. Question 8: Prove that $n^2 - n$ is divisible by 2 for every positive integer $n$.
    Solution: $n^2 - n = n(n-1)$. Product of two consecutive integers is always even (divisible by 2).
  9. Question 9: Show that the cube of any positive integer is of the form $9m, 9m+1$ or $9m+8$.
    Solution: Let $a = 3q+r$. $a^3 = (3q+r)^3$. If $r=0, a^3=27q^3=9m$. If $r=1, a^3=27q^3+27q^2+9q+1=9m+1$. If $r=2, a^3=27q^3+54q^2+36q+8=9m+8$.
  10. Question 10: Find the HCF of 180, 252 and 324 using Euclid's division algorithm.
    Solution: HCF(180, 252) = 36. HCF(36, 324) = 36. So HCF is 36.
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