SECTION F — Assertion and Reasoning Type Questions [1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
- (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c) Assertion (A) is true, but Reason (R) is false.
- (d) Assertion (A) is false, but Reason (R) is true.
-
Q1. Assertion (A): The polynomial $p(x) = x^2 + 3x + 3$ has no real zeroes.
Reason (R): A quadratic polynomial $ax^2 + bx + c$ has no real zeroes if its discriminant $D = b^2 - 4ac < 0$. -
Q2. Assertion (A): If the product of the zeroes of the quadratic polynomial $p(x) = kx^2 - 4x - 6$ is $3$, then the value of $k$ is $-2$.
Reason (R): The product of the zeroes of a quadratic polynomial $ax^2 + bx + c$ is given by $\frac{c}{a}$. -
Q3. Assertion (A): A quadratic polynomial whose zeroes are $-3$ and $4$ is $x^2 - x - 12$.
Reason (R): If $\alpha$ and $\beta$ are the zeroes of a quadratic polynomial, then the polynomial is given by $p(x) = k[x^2 - (\alpha + \beta)x + \alpha\beta]$, where $k$ is any non-zero real number. -
Q4. Assertion (A): The degree of a non-zero constant polynomial is $0$.
Reason (R): The degree of a zero polynomial is not defined. -
Q5. Assertion (A): If $1$ is a zero of the polynomial $p(x) = ax^2 - 3(a - 1)x - 1$, then $a = \frac{1}{2}$.
Reason (R): If $k$ is a zero of $p(x)$, then $p(k) = 0$. -
Q6. Assertion (A): The quadratic polynomial $p(x) = x^2 - 6x + 9$ has only one distinct real zero, which is $3$.
Reason (R): When the discriminant $b^2 - 4ac = 0$, the quadratic equation has coincident (equal) roots, resulting in a single repeated zero. -
Q7. Assertion (A): $(x - 2)$ is a factor of the polynomial $p(x) = x^3 - 3x^2 + 4$.
Reason (R): According to the Factor Theorem, if $p(a) = 0$, then $(x - a)$ is a factor of $p(x)$. -
Q8. Assertion (A): The zero of the linear polynomial $p(x) = 2x + 5$ is $-\frac{5}{2}$.
Reason (R): A linear polynomial $ax + b$ ($a \neq 0$) has exactly one zero, given by $x = -\frac{b}{a}$. -
Q9. Assertion (A): The degree of the polynomial $p(x) = 4x^3 - 3x^5 + 7x - 2$ is $3$.
Reason (R): The degree of a polynomial is the highest power of the variable $x$ in the polynomial. -
Q10. Assertion (A): A quadratic polynomial whose sum and product of zeroes are $-3$ and $2$ respectively is $x^2 + 3x + 2$.
Reason (R): A quadratic polynomial with zeroes $\alpha$ and $\beta$ is given by $x^2 - (\alpha + \beta)x + \alpha\beta$. -
Q11. Assertion (A): The graph of the quadratic polynomial $y = x^2 - 4x + 4$ touches the $x$-axis at exactly one point.
Reason (R): A quadratic polynomial can have at most two distinct zeroes; if the discriminant $D = 0$, it has two equal real zeroes and the corresponding parabola is tangent to the $x$-axis. -
Q12. Assertion (A): The value of the polynomial $p(x) = x^3 - 3x^2 + 2x - 5$ at $x = 2$ is $-5$.
Reason (R): The value of a polynomial $p(x)$ at $x = k$ is obtained by substituting $x = k$ in $p(x)$. -
Q13. Assertion (A): If the graph of a polynomial intersects the $x$-axis at 3 points, then the number of zeroes of the polynomial is 3.
Reason (R): The zeroes of a polynomial $p(x)$ are precisely the $x$-coordinates of the points where the graph of $y = p(x)$ intersects the $x$-axis. -
Q14. Assertion (A): The product of the zeroes of the quadratic polynomial $p(x) = 3x^2 - 5x + 7$ is $\frac{7}{3}$.
Reason (R): For a quadratic polynomial $ax^2 + bx + c$, the product of zeroes is equal to $-\frac{b}{a}$. -
Q15. Assertion (A): If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = 2x^2 - 8x + 6$, then $\alpha + \beta = 4$.
Reason (R): The sum of the zeroes of a quadratic polynomial $ax^2 + bx + c$ is given by $-\frac{b}{a}$. -
Q16. Assertion (A): If $p(x)$ and $g(x)$ are two polynomials with $g(x) \neq 0$, then we can find polynomials $q(x)$ and $r(x)$ such that $p(x) = g(x)q(x) + r(x)$, where $r(x) = 0$ or degree of $r(x) <$ degree of $g(x)$.
Reason (R): This statement is known as the Division Algorithm for polynomials. -
Q17. Assertion (A): The expression $p(x) = \sqrt{x} + 3x - 5$ is a polynomial of degree $\frac{1}{2}$.
Reason (R): An algebraic expression in which the powers of the variable are non-negative integers is called a polynomial. -
Q18. Assertion (A): The quadratic polynomial $p(x) = x^2 + 1$ has two distinct real zeroes.
Reason (R): A quadratic polynomial can have real zeroes only if its discriminant $D = b^2 - 4ac \geq 0$. -
Q19. Assertion (A): A cubic polynomial can have at most three real zeroes.
Reason (R): The maximum number of zeroes of a polynomial is equal to its degree. -
Q20. Assertion (A): If both zeroes of the quadratic polynomial $ax^2 + bx + c$ are negative, then $a$, $b$, and $c$ all have the same sign.
Reason (R): If $\alpha < 0$ and $\beta < 0$, then their sum $(\alpha + \beta) < 0$ (implying $\frac{b}{a} > 0$) and their product $\alpha\beta > 0$ (implying $\frac{c}{a} > 0$).
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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