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Chapter 2: Polynomials

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Curated Multiple Choice Questions (MCQ) worksheet on Class 10 Polynomials, covering zeroes of linear and quadratic polynomials, relationship between zeroes and coefficients, and graphical parabolas.

MCQ Worksheet

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  1. Question 1: If one zero of the quadratic polynomial $x^2 + 3x + k$ is 2, then the value of $k$ is:
    (A) 10
    (B) -10
    (C) 5
    (D) -5
    Solution: (B) -10
    Since 2 is a zero, $p(2) = 0$.
    $(2)^2 + 3(2) + k = 0 \Rightarrow 4 + 6 + k = 0 \Rightarrow k = -10$.
  2. Question 2: A quadratic polynomial, whose zeroes are -3 and 4, is:
    (A) $x^2 - x + 12$
    (B) $x^2 + x + 12$
    (C) $x^2 - x - 12$
    (D) $2x^2 + 2x - 24$
    Solution: (C) $x^2 - x - 12$
    Sum of zeroes $= -3 + 4 = 1$. Product $= -3 \times 4 = -12$.
    Polynomial is $k[x^2 - (\text{Sum})x + \text{Product}] = k(x^2 - x - 12)$. For $k=1$, it is $x^2 - x - 12$.
  3. Question 3: The zeroes of the quadratic polynomial $x^2 + 99x + 127$ are:
    (A) both positive
    (B) both negative
    (C) one positive and one negative
    (D) both equal
    Solution: (B) both negative
    Since all coefficients are positive, both zeroes must be negative. (Sum is negative, Product is positive).
  4. Question 4: If the zeroes of the quadratic polynomial $x^2 + (a+1)x + b$ are 2 and -3, then:
    (A) $a = -7, b = -1$
    (B) $a = 5, b = -1$
    (C) $a = 2, b = -6$
    (D) $a = 0, b = -6$
    Solution: (D) $a = 0, b = -6$
    Sum $= 2 + (-3) = -1 = -(a+1) \Rightarrow a+1 = 1 \Rightarrow a=0$.
    Product $= 2(-3) = -6 = b$.
  5. Question 5: The number of polynomials having zeroes as -2 and 5 is:
    (A) 1
    (B) 2
    (C) 3
    (D) more than 3
    Solution: (D) more than 3
    There are infinite polynomials of the form $k(x+2)(x-5)$ where $k$ is any non-zero real number.
  6. Question 6: If one of the zeroes of the cubic polynomial $x^3 + ax^2 + bx + c$ is -1, then the product of the other two zeroes is:
    (A) $b - a + 1$
    (B) $b - a - 1$
    (C) $a - b + 1$
    (D) $a - b - 1$
    Solution: (A) $b - a + 1$
    Let zeroes be $\alpha, \beta, \gamma$ with $\alpha = -1$.
    $P(-1) = -1 + a - b + c = 0 \Rightarrow c = 1 - a + b$.
    Product $\alpha\beta\gamma = -c \Rightarrow (-1)\beta\gamma = -c \Rightarrow \beta\gamma = c = 1 - a + b$.
  7. Question 7: The zeroes of the quadratic polynomial $x^2 + kx + k, k \neq 0$,
    (A) cannot both be positive
    (B) cannot both be negative
    (C) are always unequal
    (D) are always equal
    Solution: (A) cannot both be positive
    For real roots, $D = k^2 - 4k \ge 0 \Rightarrow k(k-4) \ge 0$. So $k<0$ or $k \ge 4$.
    If $k \ge 4$, sum ($-k$) is negative and product ($k$) is positive $\Rightarrow$ both negative.
    If $k < 0$, sum is positive and product is negative $\Rightarrow$ one positive, one negative.
    In neither case can both be positive.
  8. Question 8: If the zeroes of the quadratic polynomial $ax^2 + bx + c, c \neq 0$ are equal, then:
    (A) c and a have opposite signs
    (B) c and b have opposite signs
    (C) c and a have the same sign
    (D) c and b have the same sign
    Solution: (C) c and a have the same sign
    For equal roots, $D = b^2 - 4ac = 0 \Rightarrow b^2 = 4ac$. Since $b^2 > 0$, $4ac$ must be positive, which implies $a$ and $c$ must have the same sign.
  9. Question 9: If one of the zeroes of a quadratic polynomial of the form $x^2 + ax + b$ 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.
    Solution: (A) has no linear term and the constant term is negative.
    Zeroes are $\alpha, -\alpha$. Sum $= 0 = -a \Rightarrow a=0$ (no linear term).
    Product $= -\alpha^2 = b$. Since $\alpha^2 > 0$, $b$ is negative.
  10. Question 10: Which of the following is not the graph of a quadratic polynomial?
    (A) Parabola opening up
    (B) Parabola opening down
    (C) Parabola cutting x-axis at 2 points
    (D) S-shaped curve
    Solution: (D) S-shaped curve
    The graph of a quadratic polynomial is always a parabola. An S-shaped curve typically represents a cubic polynomial.
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