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NCERT Exemplar · Class 10 Mathematics Polynomials

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EXERCISE 2.1 1–11 (part 1 of 4)

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

    Exercise 1

    If one of the zeroes of the quadratic polynomial (k−1)x2+kx+1\displaystyle (k-1) x^2+k x+1 is -3\displaystyle 3, then the value of k\displaystyle k is
    (A)
    43\displaystyle \frac{4}{3}
    (B)
    −43\displaystyle \frac{-4}{3}
    (C)
    23\displaystyle \frac{2}{3}
    (D)
    −23\displaystyle \frac{-2}{3}

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    NCERT’s answer
    (A)
    (A) \(\displaystyle \frac{4}{3}\)\[(k-1)(-3)^2+k(-3)+1=0 \] \[9(k-1)-3k+1=0 \] \[6k-8=0 \] \[k=\frac{4}{3} \]

  2. Exercise 2

    A quadratic polynomial, whose zeroes are -3\displaystyle 3 and 4\displaystyle 4, is
    (A)
    x2−x+12\displaystyle x^2-x+12
    (B)
    x2+x+12\displaystyle x^2+x+12
    (C)
    x22−x2−6\displaystyle \frac{x^2}{2}-\frac{x}{2}-6
    (D)
    2x2+2x−24\displaystyle 2 x^2+2 x-24

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    NCERT’s answer
    (C)
    (C) \(\displaystyle \frac{x^2}{2}-\frac{x}{2}-6\)\[\alpha+\beta=-3+4=1,\quad \alpha\beta=(-3)(4)=-12 \] \[x^2-(\alpha+\beta)x+\alpha\beta=x^2-x-12 \] \[\tfrac{1}{2}\left(x^2-x-12\right)=\frac{x^2}{2}-\frac{x}{2}-6 \]

  3. Exercise 3

    If the zeroes of the quadratic polynomial x2+(a+1)x+b\displaystyle x^2+(a+1) x+b are 2\displaystyle 2 and -3\displaystyle 3, then
    (A)
    a=−7,b=−1\displaystyle a=-7, b=-1
    (B)
    a=5,b=−1\displaystyle a=5, b=-1
    (C)
    a=2,b=−6\displaystyle a=2, b=-6
    (D)
    a=0,b=−6\displaystyle a=0, b=-6

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    NCERT’s answer
    (D)
    (D) \(\displaystyle a=0,\ b=-6\)\[\alpha+\beta=2+(-3)=-1=-(a+1) \] \[\implies a=0 \] \[\alpha\beta=2(-3)=-6=b \]

  4. Exercise 4

    The number of polynomials having zeroes as -2\displaystyle 2 and 5\displaystyle 5 is
    (A)
    1\displaystyle 1 (B) 2\displaystyle 2 (C) 3\displaystyle 3 (D) more than 3\displaystyle 3

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    NCERT’s answer
    (D)
    (D) more than $\displaystyle 3$\[p(x)=k(x+2)(x-5),\quad k\neq 0 \] Every nonzero \(\displaystyle k\) gives a different polynomial with the same two zeroes.

  5. Exercise 5

    Given that one of the zeroes of the cubic polynomial ax3+bx2+cx+d\displaystyle a x^3+b x^2+c x+d is zero, the product of the other two zeroes is
    (A)
    −ca\displaystyle -\frac{c}{a}
    (B)
    ca\displaystyle \frac{c}{a}
    (C)
    0\displaystyle 0 (D) −ba\displaystyle -\frac{b}{a}

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    NCERT’s answer
    (B)
    (B) \(\displaystyle \frac{c}{a}\)\[\text{zeroes: } 0,\ \alpha,\ \beta \] \[(0)\alpha+(0)\beta+\alpha\beta=\frac{c}{a} \] \[\alpha\beta=\frac{c}{a} \]

  6. Exercise 6

    If one of the zeroes of the cubic polynomial x3+ax2+bx+c\displaystyle x^3+a x^2+b x+c is -1\displaystyle 1, then the product of the other two zeroes is
    (A)
    b−a+1\displaystyle b-a+1
    (B)
    b−a−1\displaystyle b-a-1
    (C)
    a−b+1\displaystyle a-b+1
    (D)
    a−b−1\displaystyle a-b-1

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    NCERT’s answer
    (A)
    (A) \(\displaystyle b-a+1\)\[\text{zeroes: } -1,\ \alpha,\ \beta \] \[-1+\alpha+\beta=-a \implies \alpha+\beta=1-a \] \[(-1)\alpha+(-1)\beta+\alpha\beta=b \] \[\alpha\beta=b+(\alpha+\beta)=b+(1-a)=b-a+1 \]

  7. Exercise 7

    The zeroes of the quadratic polynomial x2+99x+127\displaystyle x^2+99 x+127 are
    (A)
    both positive
    (B)
    both negative
    (C)
    one positive and one negative
    (D)
    both equal

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    NCERT’s answer
    (B)
    (B) both negative. \[\alpha+\beta=\frac{-99}{1}=-99, \qquad \alpha\beta=\frac{127}{1}=127 \] Product positive: same sign. Sum negative: both negative.

  8. Exercise 8

    The zeroes of the quadratic polynomial x2+kx+k,k≠0\displaystyle x^2+k x+k, k \neq 0,
    (A)
    cannot both be positive
    (B)
    cannot both be negative
    (C)
    are always unequal
    (D)
    are always equal

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    NCERT’s answer
    (A)
    (A) cannot both be positive. \[\alpha+\beta=-k, \qquad \alpha\beta=k \] If both positive: \(\displaystyle -k>0\) and \(\displaystyle k>0\) — contradiction.

  9. Exercise 9

    If the zeroes of the quadratic polynomial ax2+bx+c,c≠0\displaystyle a x^2+b x+c, c \neq 0 are equal, then
    (A)
    c\displaystyle c and a\displaystyle a have opposite signs
    (B)
    c\displaystyle c and b\displaystyle b have opposite signs
    (C)
    c\displaystyle c and a\displaystyle a have the same sign
    (D)
    c\displaystyle c and b\displaystyle b have the same sign

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    NCERT’s answer
    (C)
    (C) \(\displaystyle c\) and \(\displaystyle a\) have the same sign. \[\Delta=b^2-4ac=0 \Rightarrow b^2=4ac \] \(\displaystyle b^2\ge 0\) and \(\displaystyle a,c\neq0\), so \(\displaystyle ac>0\): same sign.

  10. Exercise 10

    If one of the zeroes of a quadratic polynomial of the form x2+ax+b\displaystyle x^2+a x+b is the negative of the other, then it
    (A)
    has no linear term and the constant term is negative.
    (B)
    has no linear term and the constant term is positive.
    (C)
    can have a linear term but the constant term is negative.
    (D)
    can have a linear term but the constant term is positive.

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    NCERT’s answer
    (A)
    (A) has no linear term and the constant term is negative. Let the zeroes be \(\displaystyle r,-r\). \[\text{Sum}=r+(-r)=0=-a \Rightarrow a=0 \] \[\text{Product}=r(-r)=-r^2=b \Rightarrow b<0 \]

  11. Exercise 11

    Which of the following is not the graph of a quadratic polynomial?
    (A)
    (B)
    (C)
    (D)
    NCERT_Question_Class10_Maths_Exemplar_Ch2_Ex2-1_Q11
    NCERT_Question_Class10_Maths_Exemplar_Ch2_Ex2-1_Q11_2
    NCERT_Question_Class10_Maths_Exemplar_Ch2_Ex2-1_Q11_3
    NCERT_Question_Class10_Maths_Exemplar_Ch2_Ex2-1_Q11_4

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    NCERT’s answer
    (D)
    (D) NCERT_Solution_Class10_Maths_Exemplar_Ch2_Ex2-1_Q11 A quadratic's graph is a parabola with exactly one turning point. This curve turns twice — a maximum then a minimum — the shape of a cubic.