Appendix 1 · Proofs in Mathematics

NCERT Class 10 Maths Exercise A1.1

All 4 questions of NCERT Class 10 Maths Exercise A1.1 (Proofs in Mathematics), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   State whether the following statements are always true, always false ​or ambiguous. Justify your answers.

   1. (i)All mathematics textbooks are interesting.
   2. (ii)The distance ​from the Earth to the Sun ​is approximately $1.5\times 10^8$ km.
   3. (iii)All human beings ​grow old.
   4. (iv)​The journey ​from Uttarkashi to Harsil is ​tiring.
   5. (v)The woman saw an elephant through a pair ​of binoculars.
   Show answer

   $\text{(i)}$ Ambiguous, $\text{(ii)}$ Always true, $\text{(iii)}$ Always true, $\text{(iv)}$ Ambiguous, $\text{(v)}$ Ambiguous.

   Step-by-step solution →
2. Q2
   State whether the following statements are true or ​false. Justify ​your answers.

   1. (i)All hexagons are polygons.
   2. (ii)Some ​polygons are pentagons.
   3. (iii)Not all even numbers are ​divisible by 2.
   4. (iv)Some real numbers are ​irrational.
   5. (v)Not ​all real ​numbers are rational.
   Show answer

   $\text{(i)}$ True, $\text{(ii)}$ True, $\text{(iii)}$ False, $\text{(iv)}$ True, $\text{(v)}$ True.

   Step-by-step solution →
3. Q3
   Let $a$ and $b$ be real numbers such that $ab\ne 0$. ​Then which ​of the following ​statements are true? Justify your answers.

   1. (i)Both $a$ and $b$ must be zero.
   2. (ii)Both $a$ and $b$ must be non-zero. ​
   3. (iii)​Either $a$ or $b$ must ​be non-zero.
   Show answer

   (i) False, (ii) True, (iii) False (NCERT reads 'either ... or' as exactly one): only (ii) is true.

   Step-by-step solution →
4. Q4
   Restate the following ​statements with appropriate conditions, so ​that they become true.

   1. (i)​If $a^2>b^2$, ​then $a>b$.
   2. (ii)If $x^2=y^2$, then $x=y$.
   3. (iii)​If $(x+y{)}^2=x^2+y^2$, then $x=0$.
   4. (iv)The ​diagonals of a ​quadrilateral bisect each ​other.
   Show answer

   (i) If $a>0$ and $a^2>b^2$, then $a>b$. (ii) If $xy>0$ and $x^2=y^2$, then $x=y$. (iii) If $(x+y{)}^2=x^2+y^2$ and $y\ne 0$, then $x=0$. (iv) The diagonals of a **parallelogram** bisect each other.

   Step-by-step solution →