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NCERT Exemplar · Class 10 Mathematics Quadratic Equations

28 questions · 28 still being checked

EXERCISE 4.1 1–11 (part 1 of 4)

  1. Choose the correct answer from the given four options in the following questions:

    Exercise 1

    Which of the following is a quadratic equation?
    (A)
    x2+2x+1=(4−x)2+3\displaystyle x^2+2 x+1=(4-x)^2+3
    (B)
    −2x2=(5−x)(2x−25)\displaystyle -2 x^2=(5-x)\left(2 x-\frac{2}{5}\right)
    (C)
    (k+1)x2+32x=7\displaystyle (k+1) x^2+\frac{3}{2} x=7, where k=−1\displaystyle k=-1
    (D)
    x3−x2=(x−1)3\displaystyle x^3-x^2=(x-1)^3

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    NCERT’s answer
    (D)
    (D)\[x^3-x^2=(x-1)^3 \] \[(x-1)^3=x^3-3x^2+3x-1 \] \[x^3-x^2=x^3-3x^2+3x-1 \] \[2x^2-3x+1=0 \](A), (B), (C) each cancel to a linear equation.

  2. Exercise 2

    Which of the following is not a quadratic equation?
    (A)
    2(x−1)2=4x2−2x+1\displaystyle 2(x-1)^2=4 x^2-2 x+1
    (B)
    2x−x2=x2+5\displaystyle 2 x-x^2=x^2+5
    (C)
    (2x+3)2+x2=3x2−5x\displaystyle (\sqrt{2} x+\sqrt{3})^2+x^2=3 x^2-5 x
    (D)
    (x2+2x)2=x4+3+4x3\displaystyle \left(x^2+2 x\right)^2=x^4+3+4 x^3

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    NCERT’s answer
    (C)
    (C)\[(\sqrt{2}x+\sqrt{3})^2+x^2=2x^2+2\sqrt{6}x+3+x^2=3x^2+2\sqrt{6}x+3 \] \[3x^2+2\sqrt{6}x+3=3x^2-5x \] \[(2\sqrt{6}+5)x+3=0 \]The \(\displaystyle x^2\) terms cancel, leaving a linear equation; (A), (B), (D) keep an \(\displaystyle x^2\) term.

  3. Exercise 3

    Which of the following equations has 2\displaystyle 2 as a root?
    (A)
    x2−4x+5=0\displaystyle x^2-4 x+5=0
    (B)
    x2+3x−12=0\displaystyle x^2+3 x-12=0
    (C)
    2x2−7x+6=0\displaystyle 2 x^2-7 x+6=0
    (D)
    3x2−6x−2=0\displaystyle 3 x^2-6 x-2=0

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    NCERT’s answer
    (C)
    (C)\[2x^2-7x+6=0 \] \[2(2)^2-7(2)+6=8-14+6=0 \](A), (B), (D) do not vanish at \(\displaystyle x=2\).

  4. Exercise 4

    If 12\displaystyle \frac{1}{2} is a root of the equation x2+kx−54=0\displaystyle x^2+k x-\frac{5}{4}=0, then the value of k\displaystyle k is
    (A)
    2\displaystyle 2 (B) −2\displaystyle -2 (C) 14\displaystyle \frac{1}{4}
    (D)
    12\displaystyle \frac{1}{2}

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    NCERT’s answer
    (A)
    (A) \(\displaystyle 2\)\[x^2+kx-\frac{5}{4}=0,\quad x=\frac{1}{2} \] \[\left(\frac{1}{2}\right)^2+k\left(\frac{1}{2}\right)-\frac{5}{4}=0 \] \[\frac{1}{4}+\frac{k}{2}-\frac{5}{4}=0 \] \[\frac{k}{2}=1 \] \[k=2 \]

  5. Exercise 5

    Which of the following equations has the sum of its roots as 3\displaystyle 3?
    (A)
    2x2−3x+6=0\displaystyle 2 x^2-3 x+6=0
    (B)
    −x2+3x−3=0\displaystyle -x^2+3 x-3=0
    (C)
    2x2−32x+1=0\displaystyle \sqrt{2} x^2-\frac{3}{\sqrt{2}} x+1=0
    (D)
    3x2−3x+3=0\displaystyle 3 x^2-3 x+3=0

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    NCERT’s answer
    (B)
    (B)\[ax^2+bx+c=0 \implies \text{sum of roots}=-\frac{b}{a} \] \[-x^2+3x-3=0:\quad -\frac{3}{-1}=3 \](A), (C), (D) give sum \(\displaystyle \frac{3}{2},\ \frac{3}{2},\ 1\) respectively.

  6. Exercise 6

    Values of k\displaystyle k for which the quadratic equation 2x2−kx+k=0\displaystyle 2 x^2-k x+k=0 has equal roots is
    (A)
    0\displaystyle 0 only
    (B)
    4\displaystyle 4 (C) 8\displaystyle 8 only
    (D)
    0\displaystyle 0, 8\displaystyle 8

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    NCERT’s answer
    (D)
    (D) \(\displaystyle 0,\ 8\)\[2x^2-kx+k=0,\quad b^2-4ac=0 \] \[(-k)^2-4(2)(k)=0 \] \[k^2-8k=0 \] \[k(k-8)=0 \] \[k=0 \ \text{or}\ k=8 \]

  7. Exercise 7

    Which constant must be added and subtracted to solve the quadratic equation 9x2+34x−2=0\displaystyle 9 x^2+\frac{3}{4} x-\sqrt{2}=0 by the method of completing the square?
    (A)
    18\displaystyle \frac{1}{8}
    (B)
    164\displaystyle \frac{1}{64}
    (C)
    14\displaystyle \frac{1}{4}
    (D)
    964\displaystyle \frac{9}{64}

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    NCERT’s answer
    (B)
    (B) \(\displaystyle \dfrac{1}{64}\)Write as part of a square: \[(3x)^2+2(3x)\left(\frac18\right)=9x^2+\frac34x \] \[\left(3x+\frac18\right)^2=9x^2+\frac34x+\frac{1}{64} \] So \(\displaystyle \frac{1}{64}\) is added and subtracted.

  8. Exercise 8

    The quadratic equation 2x2−5x+1=0\displaystyle 2 x^2-\sqrt{5} x+1=0 has
    (A)
    two distinct real roots
    (B)
    two equal real roots
    (C)
    no real roots
    (D)
    more than 2\displaystyle 2 real roots

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    NCERT’s answer
    (C)
    (C) no real roots\[D=b^2-4ac=(-\sqrt5)^2-4(2)(1)=5-8=-3 \] Since \(\displaystyle D<0\), the equation has no real roots.

  9. Exercise 9

    Which of the following equations has two distinct real roots?
    (A)
    2x2−32x+94=0\displaystyle 2 x^2-3 \sqrt{2} x+\frac{9}{4}=0
    (B)
    x2+x−5=0\displaystyle x^2+x-5=0
    (C)
    x2+3x+22=0\displaystyle x^2+3 x+2 \sqrt{2}=0
    (D)
    5x2−3x+1=0\displaystyle 5 x^2-3 x+1=0

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    NCERT’s answer
    (B)
    (B) \(\displaystyle x^2+x-5=0\)\[D_A=(3\sqrt2)^2-4(2)\left(\frac94\right)=18-18=0 \] \[D_B=1^2-4(1)(-5)=1+20=21 \] \[D_C=3^2-4(1)(2\sqrt2)=9-8\sqrt2<0 \] \[D_D=(-3)^2-4(5)(1)=9-20=-11 \] Only \(\displaystyle D_B>0\), so (B) has two distinct real roots.

  10. Exercise 10

    Which of the following equations has no real roots?
    (A)
    x2−4x+32=0\displaystyle x^2-4 x+3 \sqrt{2}=0
    (B)
    x2+4x−32=0\displaystyle x^2+4 x-3 \sqrt{2}=0
    (C)
    x2−4x−32=0\displaystyle x^2-4 x-3 \sqrt{2}=0
    (D)
    3x2+43x+4=0\displaystyle 3 x^2+4 \sqrt{3} x+4=0

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    NCERT’s answer
    (A)
    (A) \(\displaystyle x^2-4x+3\sqrt2=0\)\[D_A=(-4)^2-4(1)(3\sqrt2)=16-12\sqrt2 \] \[D_B=4^2-4(1)(-3\sqrt2)=16+12\sqrt2 \] \[D_C=(-4)^2-4(1)(-3\sqrt2)=16+12\sqrt2 \] \[D_D=(4\sqrt3)^2-4(3)(4)=48-48=0 \] \[12\sqrt2>16 \implies D_A<0 \] The rest are \(\displaystyle \ge0\), so only (A) has no real roots.

  11. Exercise 11

    (x2+1)2−x2=0\displaystyle \left(x^2+1\right)^2-x^2=0 has
    (A)
    four real roots
    (B)
    two real roots
    (C)
    no real roots
    (D)
    one real root.

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    NCERT’s answer
    (C)
    (C) no real roots\[(x^2+1)^2-x^2=(x^2-x+1)(x^2+x+1) \] \[D_1=(-1)^2-4(1)(1)=-3,\quad D_2=1^2-4(1)(1)=-3 \] Both factors have \(\displaystyle D<0\), so the equation has no real roots.