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NCERT Exemplar · Class 10 Mathematics Polynomials

29 questions · 29 still being checked

EXERCISE 2.2 1–2 (part 2 of 4)

  1. Exercise 1

    Answer the following and justify:
    (i)
    Can x2−1\displaystyle x^2-1 be the quotient on division of x6+2x3+x−1\displaystyle x^6+2 x^3+x-1 by a polynomial in x\displaystyle x of degree 5\displaystyle 5?
    (ii)
    What will the quotient and remainder be on division of ax2+bx+c\displaystyle a x^2+b x+c by px3+qx2+rx+s,p≠0\displaystyle p x^3+q x^2+r x+s, p \neq 0?
    (iii)
    If on division of a polynomial p(x)\displaystyle p(x) by a polynomial g(x)\displaystyle g(x), the quotient is zero, what is the relation between the degrees of p(x)\displaystyle p(x) and g(x)\displaystyle g(x)?
    (iv)
    If on division of a non-zero polynomial p(x)\displaystyle p(x) by a polynomial g(x)\displaystyle g(x), the remainder is zero, what is the relation between the degrees of p(x)\displaystyle p(x) and g(x)\displaystyle g(x)?
    (v)
    Can the quadratic polynomial x2+kx+k\displaystyle x^2+k x+k have equal zeroes for some odd integer k>1\displaystyle k>1?

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    NCERT’s answer
    (i)
    No
    (ii)
    \(\displaystyle 0, a x^2+b x+c\)
    (iii)
    \(\displaystyle \operatorname{deg} p(x)<\operatorname{deg} g(x)\)
    (iv)
    \(\displaystyle \operatorname{deg} g(x) \leq \operatorname{deg} p(x)\)
    (v)
    No
    (i)
    No — \[\deg p=\deg g+\deg q \] forces \(\displaystyle \deg q=6-5=1\), but \(\displaystyle x^2-1\) has degree \(\displaystyle 2\).
    (ii)
    Quotient \(\displaystyle 0\), remainder \(\displaystyle ax^2+bx+c\) — \[\deg(ax^2+bx+c)<\deg(px^3+qx^2+rx+s) \] so division stops immediately.
    (iii)
    \[p(x)=g(x)\cdot 0+r(x)=r(x),\quad \deg r(x)<\deg g(x) \Rightarrow \deg p(x)<\deg g(x) \]
    (iv)
    \[p(x)=g(x)q(x),\ q\neq 0 \Rightarrow \deg p(x)=\deg g(x)+\deg q(x)\ge \deg g(x) \]
    (v)
    No — \[k^2-4k=0 \Rightarrow k=0,\,4 \] neither is an odd integer greater than \(\displaystyle 1\).
    Answer: (i) No (ii) quotient \(\displaystyle 0\), remainder \(\displaystyle ax^2+bx+c\) (iii) \(\displaystyle \deg p(x)<\deg g(x)\) (iv) \(\displaystyle \deg p(x)\ge \deg g(x)\) (v) No.

  2. Exercise 2

    Are the following statements 'True' or 'False'? Justify your answers.
    (i)
    If the zeroes of a quadratic polynomial ax2+bx+c\displaystyle a x^2+b x+c are both positive, then a,b\displaystyle a, b and c\displaystyle c all have the same sign.
    (ii)
    If the graph of a polynomial intersects the x\displaystyle x-axis at only one point, it cannot be a quadratic polynomial.
    (iii)
    If the graph of a polynomial intersects the x\displaystyle x-axis at exactly two points, it need not be a quadratic polynomial.
    (iv)
    If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
    (v)
    If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
    (vi)
    If all three zeroes of a cubic polynomial x3+ax2−bx+c\displaystyle x^3+a x^2-b x+c are positive, then at least one of a,b\displaystyle a, b and c\displaystyle c is non-negative.
    (vii)
    The only value of k\displaystyle k for which the quadratic polynomial kx2+x+k\displaystyle k x^2+x+k has equal zeros is 12\displaystyle \frac{1}{2}

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    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    False
    (ii)
    False
    (iii)
    True
    (iv)
    True
    (v)
    True
    (vi)
    False
    (vii)
    False
    (i)
    False — \[\alpha+\beta=-\frac{b}{a}>0,\quad \alpha\beta=\frac{c}{a}>0 \] so \(\displaystyle b\) is opposite in sign to \(\displaystyle a\); only \(\displaystyle a,c\) match.
    (ii)
    False — \[x^2-2x+1=(x-1)^2 \] touches the axis once yet is quadratic.
    (iii)
    True — \[x^3-x^2=x^2(x-1) \] meets the axis at \(\displaystyle 0,1\) but is cubic.
    (iv)
    True — zeroes \(\displaystyle 0,0,\gamma\) give \[f(x)=ax^2(x-\gamma)=ax^3-a\gamma x^2 \] no \(\displaystyle x\)-term or constant term.
    (v)
    True — zeroes \(\displaystyle -\alpha,-\beta,-\gamma\ (\alpha,\beta,\gamma>0)\) give \[f(x)=a\big[x^3+(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x+\alpha\beta\gamma\big] \] every coefficient shares the sign of \(\displaystyle a\).
    (vi)
    False — \[a=-(\alpha+\beta+\gamma),\quad b=-(\alpha\beta+\beta\gamma+\gamma\alpha),\quad c=-\alpha\beta\gamma \] all come out negative for positive \(\displaystyle \alpha,\beta,\gamma\).
    (vii)
    False — \[1-4k^2=0 \Rightarrow k=\pm\tfrac12 \] both values work, not only \(\displaystyle \tfrac12\).