Appendix 1 · Proofs in Mathematics

NCERT Class 10 Maths Exercise A1.2

All 7 questions of NCERT Class 10 Maths Exercise A1.2 (Proofs in Mathematics), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Given that ​all women are mortal, and suppose ​that A is ​a ​woman, what can ​we ​conclude about A?
   Show answer

   $A$ is mortal.

   Step-by-step solution →
2. Q2
   Given ​that the ​product ​of two rational numbers is ​rational, and suppose $a$ ​and $b$ are ​rationals, what can you ​conclude ​about $ab$?
   Show answer

   $ab$ is a rational number.

   Step-by-step solution →
3. Q3
   Given that ​the decimal expansion of irrational ​numbers ​is ​non-terminating, non-recurring, and $\sqrt{17}$ ​is irrational, ​what can ​we conclude about the decimal expansion of $\sqrt{17}$?
   Show answer

   The decimal expansion of $\sqrt{17}$ is non-terminating and non-recurring.

   Step-by-step solution →
4. Q4
   Given that $y=x^2+6$ ​and $x=-1$, what can we ​conclude about ​the ​value ​of $y$?
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   $y=7$.

   Step-by-step solution →
5. Q5
   Given ​that $ABCD$ is ​a ​parallelogram and $\angle B=80^{\circ }$. ​What can you conclude ​about the ​other angles ​of the parallelogram?
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   $\angle A=\angle C=100^{\circ }$ and $\angle B=\angle D=80^{\circ }$.

   Step-by-step solution →
6. Q6
   Given ​that $PQRS$ ​is a cyclic ​quadrilateral and ​also its diagonals ​bisect ​each other. What can you ​conclude about the quadrilateral?
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   $PQRS$ is a rectangle (all four angles equal $90^{\circ }$).

   Step-by-step solution →
7. Q7
   Given that $\sqrt{p}$ is irrational for all ​primes $p$ and also suppose that 3721 is a prime. Can you conclude that $\sqrt{3721}$ ​is ​an ​irrational number? Is your ​conclusion correct? Why or why ​not?
   Show answer

   Yes - from the two given premises the conclusion "$\sqrt{3721}$ is irrational" **does** follow deductively. But no, the conclusion is not correct, because $3721=61^2$ is not a prime and $\sqrt{3721}=61$ is rational; the premise was wrong, so the conclusion is false.

   Step-by-step solution →