Appendix 1 · Proofs in Mathematics

NCERT Class 10 Maths Exercise A1.3

All 6 questions of NCERT Class 10 Maths Exercise A1.3 (Proofs in Mathematics), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Prove that ​the ​sum of ​two consecutive ​odd numbers is ​divisible by ​4.
   Show answer

   The sum is $4(n+1)$, which is $4$ times an integer and therefore divisible by $4$.

   Step-by-step solution →
2. Q2
   Take two consecutive odd ​numbers. Find the sum of their ​squares, ​and ​then add $6$ to ​the result. Prove that the new number ​is always divisible ​by $8$.
   Show answer

   The result is $8(n^2+2n+2)$, which is $8$ times an integer and therefore divisible by $8$.

   Step-by-step solution →
3. Q3
   If $p\ge 5$ is ​a prime ​number, show ​that $p^2+2$ is divisible by ​3. ​[Hint: Use ​Example 11]
   Show answer

   $3$ divides $p^2+2$ for every prime $p\ge 5$, because $p$ has the form $3k+1$ or $3k+2$ and in each case $p^2$ leaves remainder $1$, so $p^2+2$ leaves remainder $0$.

   Step-by-step solution →
4. Q4
   Let x and ​y ​be ​rational ​numbers. Show that ​xy is a rational ​number.
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   $xy$ is a rational number, since $xy=\frac{pm}{qn}$ has integer numerator $pm$ and non-zero integer denominator $qn$.

   Step-by-step solution →
5. Q5
   If $a$ and $b$ are positive ​integers, then you know ​that $a=bq+r$, $0\le r<b$, where $q$ is a whole number. ​Prove that HCF$(a,b)=$ HCF$(b,r)$. ​[Hint: Let HCF$(b,r)=h$. So $b=k_{1}h$ ​and $r=k_{2}h$, where $k_1,k_2$ are ​coprime.]
   Show answer

   $\text{HCF}(a,b)=\text{HCF}(b,r)$, because every common divisor of $b$ and $r$ divides $a=bq+r$, every common divisor of $a$ and $b$ divides $r=a-bq$, so the two pairs have the same common divisors and hence the same greatest one.

   Step-by-step solution →
6. Q6
   A line ​parallel to ​side $BC$ of triangle $ABC$, ​intersects $AB$ and $AC$ at $D$ ​and $E$ ​respectively. Prove ​that $\frac{AD}{DB}=\frac{AE}{EC}$.
   Show answer

   $\frac{AD}{DB}=\frac{AE}{EC}$, proved using $DE\parallel BC$ and the similar triangles $△ADE$ and $△ABC$.

   Step-by-step solution →