Mathematics Questions

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

1. Q281
   Prove ​that if $x$ ​and $y$ are both odd positive ​integers, ​then $x^2+y^2$ ​is even but not ​divisible by ​4.
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2. Q282
   Use ​Euclid's division ​algorithm ​to find ​the ​HCF of $441,567,693$.
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3. Q283
   Using ​Euclid's division ​algorithm, find the largest number ​that divides ​1251, 9377 and 15628 leaving remainders 1, ​2 ​and ​3, respectively.
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4. Q284
   Prove ​that $\sqrt{3}+\sqrt{5}$ ​is irrational.
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5. Q285
   Show ​that $12^n$ cannot end ​with ​the digit 0 ​or 5 ​for any natural number $n$.
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6. Q286
   On a ​morning ​walk, three persons step off ​together ​and their steps measure 40 cm, 42 cm and 45 ​cm, respectively. What is the minimum distance each should walk so that ​each can cover the same distance ​in ​complete steps?
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7. Q287
   Write the denominator of the rational ​number $\frac{257}{5000}$ in ​the ​form $2^m\times 5^n$, where $m,n$ ​are non-negative integers. Hence, write its decimal ​expansion, without ​actual ​division.
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8. Q288
   Prove ​that $\sqrt{p}+\sqrt{q}$ is ​irrational, ​where $p$, $q$ ​are ​primes.
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9. Q289
   If ​one ​of ​the zeroes ​of ​the quadratic polynomial $(k-1)x^2+kx+1$ is $-3$, then the ​value of $k$ is

   1. (a)$\frac{4}{3}$
   2. (b)$\frac{-4}{3}$
   3. (c)$\frac{2}{3}$
   4. (d)$\frac{-2}{3}$
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10. Q290
    A ​quadratic ​polynomial, whose ​zeroes are $-3$ ​and $4$, ​is

    1. (a)$x^2-x+12$
    2. (b)$x^2+x+12$
    3. (c)$\frac{x^2}{2}-\frac{x}{2}-6$
    4. (d)$2x^2+2x-24$
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11. Q291
    If the ​zeroes ​of the ​quadratic ​polynomial $x^2+(a+1)x+b$ ​are 2 ​and $-3$, then

    1. (a)$a=-7,b=-1$
    2. (b)$a=5,b=-1$
    3. (c)$a=2,b=-6$
    4. (d)$a=0,b=-6$
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12. Q292
    The number of polynomials ​having ​zeroes ​as $-2$ ​and $5$ ​is

    1. (a)$1$
    2. (b)$2$
    3. (c)$3$
    4. (d)more than $3$
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13. Q293
    Given ​that one of the ​zeroes of ​the ​cubic polynomial $a{x}^3+b{x}^2+cx+d$ is ​zero, the ​product of the other two zeroes is

    1. (a)$-\frac{c}{a}$
    2. (b)$\frac{c}{a}$
    3. (c)$0$
    4. (d)$-\frac{b}{a}$
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14. Q294
    If one of the zeroes ​of ​the cubic polynomial $x^3+a{x}^2+bx+c$ is $-1$, ​then ​the product of the other ​two ​zeroes ​is

    1. (a)$b-a+1$
    2. (b)$b-a-1$
    3. (c)$a-b+1$
    4. (d)$a-b-1$
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15. Q295
    The ​zeroes of ​the ​quadratic ​polynomial $x^2+99x+127$ ​are

    1. (a)both positive
    2. (b)both negative
    3. (c)one positive and one negative
    4. (d)both equal
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16. Q296
    The ​zeroes ​of the quadratic ​polynomial $x^2+kx+k$, $k\ne 0$,

    1. (a)cannot both be positive
    2. (b)cannot both be negative
    3. (c)are always unequal
    4. (d)are always equal
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17. Q297
    If ​the ​zeroes ​of ​the quadratic polynomial $a{x}^2+bx+c$, $c\ne 0$ ​are equal, then

    1. (a)$c$ and $a$ have opposite signs
    2. (b)$c$ and $b$ have opposite signs
    3. (c)$c$ and $a$ have the same sign
    4. (d)$c$ and $b$ have the same sign
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18. Q298
    If one of the ​zeroes ​of ​a quadratic polynomial of the form $x^2+ax+b$ ​is ​the negative of the other, then ​it

    1. (a)has no linear term and the constant term is negative.
    2. (b)has no linear term and the constant term is positive.
    3. (c)can have a linear term but the constant term is negative.
    4. (d)can have a linear term but the constant term is positive.
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19. Q299
    Which ​of the following ​is not ​the ​graph of ​a ​quadratic polynomial?

    1. (a)Graph (A)
    2. (b)Graph (B)
    3. (c)Graph (C)
    4. (d)Graph (D)
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20. Q300
    Answer the ​following and justify:

    1. (i)Can x^2 - 1 be the quotient on division of x^6 + ​2x^3 + x - 1 by a polynomial in x of degree 5?
    2. (ii)What will the quotient and remainder be on division of ax^2 + bx + c by px^3 + qx^2 + rx + s, p not ​equal ​to 0?
    3. (iii)​If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, what is the relation between the degrees of p(x) and g(x)?
    4. (iv)If on division of a non-zero polynomial p(x) ​by a polynomial g(x), ​the remainder is zero, what is the relation between the degrees of p(x) and g(x)?
    5. (v)Can the ​quadratic polynomial x^2 + kx + k have equal zeroes for some odd integer k > 1?
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