Mathematics Questions

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

1. Q261
   If ​two ​positive integers $p$ and $q$ can ​be expressed as $p=a{b}^2$ ​and $q=a^{3}b$; $a,b$ being prime ​numbers, ​then LCM $(p,q)$ ​is

   1. (a)$ab$
   2. (b)$a^2{b}^2$
   3. (c)$a^3{b}^2$
   4. (d)$a^3{b}^3$
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2. Q262
   The ​product of ​a non-zero rational ​and ​an ​irrational number ​is

   1. (a)always irrational
   2. (b)always rational
   3. (c)rational or irrational
   4. (d)one
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3. Q263
   The ​least ​number that is ​divisible by all the ​numbers from 1 ​to 10 (both inclusive)

   1. (a)$10$
   2. (b)$100$
   3. (c)$504$
   4. (d)$2520$
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4. Q264
   The ​decimal ​expansion ​of ​the rational number $\frac{14587}{1250}$ ​will terminate ​after:

   1. (a)one decimal place
   2. (b)two decimal places
   3. (c)three decimal places
   4. (d)four decimal places
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5. Q265
   Write whether every ​positive integer can be of the ​form ​4q + 2, ​where q ​is ​an ​integer. Justify your ​answer.
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6. Q266
   "The product of two consecutive ​positive integers is divisible ​by 2". Is ​this statement ​true ​or false? Give reasons.
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7. Q267
   "The product ​of three ​consecutive positive integers ​is divisible by ​6." ​Is this statement true or false? Justify your ​answer.
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8. Q268
   Write ​whether the square of ​any ​positive ​integer can ​be ​of the form $3m+2$, where $m$ is a natural number. ​Justify ​your answer.
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9. Q269
   A positive integer is of ​the ​form 3q + ​1, q being ​a natural number. Can you ​write ​its square in any form other than 3m + ​1, i.e., 3m or 3m + 2 for some integer m? Justify your ​answer.
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10. Q270
    The ​numbers 525 ​and 3000 are both divisible ​only ​by 3, 5, 15, 25 ​and ​75. What is HCF (525, 3000)? ​Justify your ​answer.
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11. Q271
    Explain ​why $3\times 5\times 7+7$ is ​a ​composite ​number.
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12. Q272
    Can ​two ​numbers ​have ​18 as their ​HCF and 380 as their LCM? ​Give ​reasons.
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13. Q273
    Without actually performing ​the long ​division, find if $\frac{987}{10500}$ ​will have terminating or ​non-terminating ​(repeating) ​decimal ​expansion. Give reasons for ​your answer.
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14. Q274
    A rational number ​in its decimal expansion is $327.7081$. What can you say ​about the ​prime factors of $q$, ​when ​this number is expressed in ​the form $\frac{p}{q}$? Give ​reasons.
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15. Q275
    Show that the square of any ​positive integer ​is either ​of the form $4q$ ​or $4q+1$ ​for some ​integer $q$.
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16. Q276
    Show that ​cube of any positive integer ​is of ​the form $4m$, $4m+1$ ​or $4m+3$, for ​some integer $m$.
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17. Q277
    Show that the ​square of ​any positive integer ​cannot be of ​the ​form $5q+2$ ​or $5q+3$ for any integer $q$.
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18. Q278
    Show ​that the square of any ​positive ​integer cannot be of ​the ​form $6m+2$ or $6m+5$ for ​any ​integer $m$.
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19. Q279
    Show ​that the ​square of ​any ​odd integer ​is of ​the form $4q+1$, ​for some ​integer $q$.
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20. Q280
    If $n$ is ​an odd ​integer, then ​show that $n^2-1$ ​is ​divisible by ​8.
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