Mathematics Questions

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

1. Q361
   Values ​of $k$ ​for which ​the quadratic equation $2x^2-kx+k=0$ ​has ​equal roots is

   1. (a)$0$ only
   2. (b)$4$
   3. (c)$8$ only
   4. (d)$0,8$
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2. Q362
   Which constant must ​be added and ​subtracted ​to solve the ​quadratic ​equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$ ​by the method ​of completing the square?

   1. (a)$\frac{1}{8}$
   2. (b)$\frac{1}{64}$
   3. (c)$\frac{1}{4}$
   4. (d)$\frac{9}{64}$
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3. Q363
   The quadratic ​equation $2x^2-\sqrt{5}x+1=0$ has

   1. (a)two distinct real roots
   2. (b)two equal real roots
   3. (c)no real roots
   4. (d)more than 2 real roots
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4. Q364
   Which ​of ​the ​following ​equations ​has two distinct real ​roots?

   1. (a)$2x^2-3\sqrt{2}x+\frac{9}{4}=0$
   2. (b)$x^2+x-5=0$
   3. (c)$x^2+3x+2\sqrt{2}=0$
   4. (d)$5x^2-3x+1=0$
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5. Q365
   Which ​of ​the ​following equations ​has ​no real roots?

   1. (a)$x^2-4x+3\sqrt{2}=0$
   2. (b)$x^2+4x-3\sqrt{2}=0$
   3. (c)$x^2-4x-3\sqrt{2}=0$
   4. (d)$3x^2+4\sqrt{3}x+4=0$
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6. Q366
   $(x^2+1{)}^2-x^2=0$ has

   1. (a)four real roots
   2. (b)two real roots
   3. (c)no real roots
   4. (d)one real root
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7. Q367
   State whether ​the ​following ​quadratic ​equations have two distinct real roots. ​Justify your ​answer.

   1. (i)$x^2-3x+4=0$
   2. (ii)$2x^2+x-1=0$
   3. (iii)$2x^2-6x+\frac{9}{2}=0$
   4. (iv)$3x^2-4x+1=0$
   5. (v)$(x+4{)}^2-8x=0$
   6. (vi)$(x-\sqrt{2}{)}^2-2(x+1)=0$
   7. (vii)$\sqrt{2}{x}^2-\frac{3}{\sqrt{2}}x+\frac{1}{\sqrt{2}}=0$
   8. (viii)$x(1-x)-2=0$
   9. (ix)$(x-1)(x+2)+2=0$
   10. (x)$(x+1)(x-2)+x=0$
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8. Q368
   Write whether the following statements are true ​or false. Justify your answers.

   1. (i)​Every ​quadratic ​equation has exactly ​one root.
   2. (ii)​Every quadratic equation has at least one real root.
   3. (iii)Every quadratic equation has at least two roots.
   4. (iv)Every quadratic equation has at most two roots.
   5. (v)If the coefficient of $x^2$ and ​the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
   6. (vi)If the coefficient of $x^2$ and the constant term have the same ​sign and if the coefficient of $x$ term is zero, then the quadratic equation has no real roots.
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9. Q369
   A ​quadratic equation ​with ​integral coefficient has ​integral roots. ​Justify your answer.
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10. Q370
    Does there ​exist ​a quadratic ​equation ​whose coefficients are ​rational but both of its ​roots ​are irrational? Justify your ​answer.
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11. Q371
    Does ​there exist ​a quadratic equation whose coefficients ​are all distinct irrationals ​but both ​the ​roots are ​rationals? Why?
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12. Q372
    Is $0.2$ ​a ​root of ​the equation $x^2-0.4=0$? ​Justify.
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13. Q373
    If $b=0$, $c<0$, ​is it true ​that the ​roots ​of $x^2+bx+c=0$ are ​numerically equal ​and ​opposite in sign? ​Justify.
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14. Q374
    Find the roots of the quadratic equations by using the quadratic ​formula ​in each of the following:

    1. (i)2x^2 - 3x - 5 = 0 ​
    2. (ii)5x^2 + 13x + 8 = 0
    3. (iii)-3x^2 + 5x + ​12 ​= ​0
    4. (iv)-x^2 + 7x - 10 = 0
    5. (v)x^2 ​+ 2*$\sqrt{2}$*x - 6 = 0 ​
    6. (vi)x^2 - 3*$\sqrt{5}$*x + 10 = 0
    7. (vii)(1/2)x^2 - $\sqrt{11}$*x + 1 = 0.
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15. Q375
    Find the ​roots ​of the following quadratic equations ​by ​the factorisation method: ​

    1. (i)$2x^2+\frac{5}{3}x-2=0$
    2. (ii)$\frac{2}{5}{x}^2-x-\frac{3}{5}=0$ ​
    3. (iii)$3\sqrt{2}{x}^2-5x-\sqrt{2}=0$
    4. (iv)$3x^2+5\sqrt{5}x-10=0$
    5. (v)$21x^2-2x+\frac{1}{21}=0$
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16. Q376
    In an ​AP, ​if $d=-4$, $n=7$, $a_n=4$, ​then $a$ is

    1. (a)$6$
    2. (b)$7$
    3. (c)$20$
    4. (d)$28$
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17. Q377
    In ​an ​AP, ​if $a=3.5$, $d=0$, $n=101$, then $a_n$ ​will ​be

    1. (a)$0$
    2. (b)$3.5$
    3. (c)$103.5$
    4. (d)$104.5$
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18. Q378
    The list ​of ​numbers $-10,-6,-2,2,\dots$ ​is

    1. (a)an AP with $d=-16$
    2. (b)an AP with $d=4$
    3. (c)an AP with $d=-4$
    4. (d)not an AP
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19. Q379
    The ​11th term ​of ​the ​AP: $-5,-\frac{5}{2},0,\frac{5}{2},\dots$ is

    1. (a)$-20$
    2. (b)$20$
    3. (c)$-30$
    4. (d)$30$
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20. Q380
    The ​first ​four terms of an AP, ​whose first term is $-2$ ​and the common difference ​is $-2$, are

    1. (a)$-2,0,2,4$
    2. (b)$-2,4,-8,16$
    3. (c)$-2,-4,-6,-8$
    4. (d)$-2,-4,-8,-16$
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