Class 10 Maths · Appendix 1

Proofs in Mathematics

21 Class 10 Maths questions on Proofs in Mathematics from the NCERT textbook, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 1 of 2

1. Q1 NCERT · Exercise A1.1
   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 NCERT · Exercise A1.1
   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 NCERT · Exercise A1.1
   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 NCERT · Exercise A1.1
   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 →
5. Q5 NCERT · Exercise A1.2
   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 →
6. Q6 NCERT · Exercise A1.2
   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 →
7. Q7 NCERT · Exercise A1.2
   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 →
8. Q8 NCERT · Exercise A1.2
   Given that $y=x^2+6$ ​and $x=-1$, what can we ​conclude about ​the ​value ​of $y$?
   Show answer

   $y=7$.

   Step-by-step solution →
9. Q9 NCERT · Exercise A1.2
   Given ​that $ABCD$ is ​a ​parallelogram and $\angle B=80^{\circ }$. ​What can you conclude ​about the ​other angles ​of the parallelogram?
   Show answer

   $\angle A=\angle C=100^{\circ }$ and $\angle B=\angle D=80^{\circ }$.

   Step-by-step solution →
10. Q10 NCERT · Exercise A1.2
    Given ​that $PQRS$ ​is a cyclic ​quadrilateral and ​also its diagonals ​bisect ​each other. What can you ​conclude about the quadrilateral?
    Show answer

    $PQRS$ is a rectangle (all four angles equal $90^{\circ }$).

    Step-by-step solution →
11. Q11 NCERT · Exercise A1.2
    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 →
12. Q12 NCERT · Exercise A1.3
    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 →
13. Q13 NCERT · Exercise A1.3
    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 →
14. Q14 NCERT · Exercise A1.3
    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 →
15. Q15 NCERT · Exercise A1.3
    Let x and ​y ​be ​rational ​numbers. Show that ​xy is a rational ​number.
    Show answer

    $xy$ is a rational number, since $xy=\frac{pm}{qn}$ has integer numerator $pm$ and non-zero integer denominator $qn$.

    Step-by-step solution →
16. Q16 NCERT · Exercise A1.3
    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 →
17. Q17 NCERT · Exercise A1.3
    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 →
18. Q18 NCERT · Exercise A1.4
    State ​the ​negations for the following statements:

    1. (i)Man is mortal.
    2. (ii)​Line $l$ is parallel to line $m$.
    3. (iii)This chapter has many exercises.
    4. (iv)All integers are rational numbers.
    5. (v)​Some prime numbers are odd.
    6. (vi)No student is lazy.
    7. (vii)Some cats are not black.
    8. (viii)There is ​no real number $x$, such that $\sqrt{x}=-1$.
    9. (ix)2 divides the positive integer $a$.
    10. (x)​Integers $a$ ​and $b$ are coprime.
    Show answer

    (i) Man is not mortal. (ii) Line $l$ is not parallel to line $m$. (iii) The chapter does not have many exercises. (iv) Not all integers are rational numbers. (v) All prime numbers are not odd. (vi) Some students are lazy. (vii) All cats are black. (viii) There is at least one real number $x$ such that $\sqrt{x}=-1$. (ix) 2 does not divide the positive integer $a$. (x) Integers $a$ and $b$ are not coprime.

    Step-by-step solution →
19. Q19 NCERT · Exercise A1.4
    In each of the following questions, there are two statements. State if the second is ​the negation of the first ​or not.

    1. (i)Mumtaz is ​hungry. / Mumtaz is ​not hungry.
    2. (ii)Some cats are black. / Some cats are brown.
    3. (iii)All elephants are huge. / One elephant ​is not huge.
    4. (iv)All fire engines are red. / All ​fire engines are not red. ​
    5. (v)No man ​is a cow. / Some men are cows.
    Show answer

    (i) Yes (ii) No (iii) No (iv) No (v) Yes

    Step-by-step solution →
20. Q20 NCERT · Exercise A1.5
    Write ​the converses ​of the following statements.

    1. (i)If it ​is hot in Tokyo, then Sharan sweats a lot.
    2. (ii)If Shalini is hungry, then ​her stomach ​grumbles.
    3. (iii)If Jaswant has a scholarship, then she can get a degree.
    4. (iv)If a plant has flowers, then it is alive.
    5. (v)If an animal ​is a ​cat, then it has a tail.
    Show answer

    (i) If Sharan sweats a lot, then it is hot in Tokyo. (ii) If Shalini’s stomach grumbles, then she is hungry. (iii) If Jaswant can get a degree, then she has a scholarship. (iv) If a plant is alive, then it has flowers. (v) If an animal has a tail, then it is a cat.

    Step-by-step solution →