Chapter 2 · Polynomials

NCERT Exemplar Class 10 Maths Exercise 2.2

All 2 questions of NCERT Exemplar Class 10 Maths Exercise 2.2 (Polynomials), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   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 $a{x}^2+bx+c$ by $p{x}^3+q{x}^2+rx+s$, $p$ 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 $k>1$?
   Show answer

   (i) No, degrees do not match ($5+2=7$ not 6). (ii) Quotient $=0$, remainder $=a{x}^2+bx+c$. (iii) $\mathrm{deg}p(x)<\mathrm{deg}g(x)$. (iv) $\mathrm{deg}p(x)\ge \mathrm{deg}g(x)$. (v) No - equal zeroes need $k=0$ or $4$, neither an odd integer greater than 1.

   Step-by-step solution →
2. Q2
   Are the following statements 'True' or 'False'? Justify your answers.

   1. (i)If the zeroes ​of a quadratic polynomial $a{x}^2+bx+c$ are both positive, then $a$, $b$ and $c$ all have ​the same sign.
   2. (ii)If the graph ​of a polynomial ​intersects ​the $x$-axis at only one point, it cannot be a quadratic polynomial.
   3. (iii)If the graph of a polynomial intersects the $x$-axis at exactly two points, it need ​not be ​a quadratic polynomial.
   4. (iv)If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
   5. (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.
   6. (vi)If all three zeroes of a cubic polynomial $x^3+a{x}^2-bx+c$ are positive, then at least one of $a$, $b$ and $c$ is non-negative.
   7. (vii)The only value of $k$ for which the ​quadratic polynomial $k{x}^2+x+k$ has equal zeros is $\frac{1}{2}$.
   Show answer

   (i) False, (ii) False, (iii) True, (iv) True, (v) True, (vi) False, (vii) False

   Step-by-step solution →