Mathematics Questions

Browse 1,240 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

1. Q241
   Prove that the ​sum ​of ​two consecutive odd ​numbers ​is ​divisible ​by 4.
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2. Q242
   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 $8$.
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3. Q243
   If $p\ge 5$ is a ​prime number, show that $p^2+2$ ​is divisible ​by 3. ​[Hint: ​Use ​Example 11]
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4. Q244
   Let ​x and ​y be rational numbers. Show ​that ​xy ​is a rational ​number.
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5. Q245
   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.]
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6. Q246
   A ​line ​parallel to ​side $BC$ of triangle $ABC$, ​intersects $AB$ and $AC$ ​at $D$ and $E$ ​respectively. ​Prove that $\frac{AD}{DB}=\frac{AE}{EC}$.
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7. Q247
   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.
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8. Q248
   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.
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9. Q249
   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.
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10. Q250
    Write the converses of the following statements. Also, ​decide in each case whether the converse ​is true or false.

    1. (i)​If triangle $ABC$ is isosceles, then its base angles are equal.
    2. (ii)If an integer is odd, then its square is an odd integer.
    3. (iii)If $x^2=1$, then $x=1$.
    4. (iv)If $ABCD$ is a parallelogram, then $AC$ and $BD$ bisect each other. ​
    5. (v)If $a,b$ and $c$ are whole numbers, then $a+(b+c)=(a+b)+c$. ​
    6. (vi)If $x$ and $y$ are two odd numbers, then $x+y$ is an even number.
    7. (vii)If vertices ​of a parallelogram lie on ​a circle, then it is a rectangle.
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11. Q251
    Consider the following situation. ​A problem dating back to the early 13th century, posed by Leonardo Fibonacci, asks how many ​rabbits you would have if you started with one pair and let them reproduce. Assume that a pair of rabbits produces a pair of offspring each month, ​and that each pair of rabbits produces their first offspring at ​the age of $2$ months. Month by month, the number of pairs of rabbits is given by the ​sum ​of the pairs in the two preceding months, except for the $0$th and the $1$st months. The table alongside lists the number of pairs for months $0$ to $16$; after about $16$ months you have nearly $1600$ pairs of rabbits. Clearly state the ​problem and the different stages of mathematical ​modelling in this situation.
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12. Q252
    In each of the problems below, show ​the different stages of mathematical modelling for solving ​the ​problem. An ornithologist ​wants to estimate the number ​of parrots in a large field. She uses a net to catch some, and catches $32$ parrots, which she rings and sets free. The following week she manages to net $40$ parrots, of which $8$ are ringed.

    1. (i)What fraction of her second catch is ringed?
    2. (ii)Find an estimate of the ​total number of ​parrots in the field.
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13. Q253
    Suppose ​the adjoining figure represents ​an aerial photograph of a forest with each dot representing a tree. Your purpose is to find ​the ​number of trees ​there are ​on this tract of land as part of an environmental census.
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14. Q254
    A T.V. can ​be purchased for Rs 24000 cash or for Rs 8000 cashdown payment and six monthly instalments of Rs 2800 each. Ali goes to market to buy a T.V., and he has Rs 8000 with him. He has ​now two options. One is to buy TV under instalment scheme or to make cash payment by taking ​loan from some ​financial society. The society charges simple interest at the rate of 18% ​per annum simple interest. Which option is ​better for Ali?
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15. Q255
    For ​some ​integer $m$, ​every even integer ​is ​of ​the ​form

    1. (a)$m$
    2. (b)$m+1$
    3. (c)$2m$
    4. (d)$2m+1$
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16. Q256
    For ​some integer $q$, ​every odd integer is of ​the ​form

    1. (a)$q$
    2. (b)$q+1$
    3. (c)$2q$
    4. (d)$2q+1$
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17. Q257
    $n^2-1$ ​is ​divisible ​by 8, ​if $n$ ​is

    1. (a)an integer
    2. (b)a natural number
    3. (c)an odd integer
    4. (d)an even integer
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18. Q258
    If the HCF of $65$ ​and $117$ ​is expressible in ​the ​form $65m-117$, ​then the value of $m$ ​is

    1. (a)$4$
    2. (b)$2$
    3. (c)$1$
    4. (d)$3$
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19. Q259
    The largest ​number which divides $70$ ​and $125$, leaving remainders $5$ ​and $8$, ​respectively, ​is

    1. (a)$13$
    2. (b)$65$
    3. (c)$875$
    4. (d)$1750$
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20. Q260
    If ​two positive integers $a$ and $b$ are written as $a=x^3{y}^2$ ​and $b=x{y}^3$; $x$, $y$ ​are prime ​numbers, ​then HCF $(a,b)$ ​is

    1. (a)$xy$
    2. (b)$x{y}^2$
    3. (c)$x^3{y}^3$
    4. (d)$x^2{y}^2$
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