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): $(x - 2)^2 + 1 = 2x - 3$ is a quadratic equation.
Reason (R): Any equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$ and $a, b, c$ are real numbers, is called a quadratic equation. -
Q2. Assertion (A): The polynomial $p(x) = 2x^3 - 3x^2 + x - 5$ is a quadratic equation.
Reason (R): A quadratic equation is a polynomial of degree 2. -
Q3. Assertion (A): $x = 3$ is a solution of the quadratic equation $x^2 - 5x + 6 = 0$.
Reason (R): If $x = \alpha$ satisfies the quadratic equation $p(x) = 0$, then $\alpha$ is called a root of the equation. -
Q4. Assertion (A): The quadratic equation $2x^2 - 4x + 3 = 0$ has no real roots.
Reason (R): The discriminant $D = b^2 - 4ac$ for this equation is negative ($D < 0$). -
Q5. Assertion (A): The quadratic equation $3x^2 - 5x + 2 = 0$ has two distinct real roots.
Reason (R): If the discriminant $D = b^2 - 4ac > 0$, the quadratic equation has two distinct real roots. -
Q6. Assertion (A): If the equation $2x^2 + kx + 3 = 0$ has equal roots, then the value of $k$ is $\pm 2\sqrt{6}$.
Reason (R): For a quadratic equation to have equal and real roots, its discriminant must be equal to zero ($D = 0$). -
Q7. Assertion (A): The roots of $x^2 - 3x - 10 = 0$ are $5$ and $-2$.
Reason (R): The roots of any quadratic equation $ax^2 + bx + c = 0$ can be found using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. -
Q8. Assertion (A): For the quadratic equation $2x^2 - 7x + 3 = 0$, the sum of its roots is $\frac{7}{2}$.
Reason (R): The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$. -
Q9. Assertion (A): The product of the roots of the equation $4x^2 - 12x + 9 = 0$ is $\frac{9}{4}$.
Reason (R): The product of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $\frac{c}{a}$. -
Q10. Assertion (A): The equation $x^2 + 1 = 0$ does not have any real roots.
Reason (R): The square of any real number is always non-negative, hence $x^2 = -1$ has no solution in the set of real numbers. -
Q11. Assertion (A): If $x^2 - 4px + p = 0$ has equal roots, then $p = \frac{1}{4}$ (for $p \neq 0$).
Reason (R): Setting $D = (-4p)^2 - 4(1)(p) = 0$ gives $16p^2 - 4p = 0$, which yields $p = \frac{1}{4}$. -
Q12. Assertion (A): The quadratic equation $x^2 - 5x + 6 = 0$ can be solved by splitting the middle term into $-2x$ and $-3x$.
Reason (R): Splitting the middle term involves finding two numbers whose sum is $b$ and whose product is $ac$. -
Q13. Assertion (A): The quadratic equation $x^2 + x + 1 = 0$ has real roots.
Reason (R): A quadratic equation has real roots if and only if $D \geq 0$. -
Q14. Assertion (A): The expression $4x^2 - 12x + 9 = 0$ represents a perfect square quadratic equation.
Reason (R): It can be factored as $(2x - 3)^2 = 0$, yielding two coincident equal roots. -
Q15. Assertion (A): If $\alpha$ is a root of $ax^2 + bx + c = 0$, then $\frac{1}{\alpha}$ is always a root of $cx^2 + bx + a = 0$.
Reason (R): Replacing $x$ with $\frac{1}{x}$ in a reciprocal quadratic equation inverts the coefficients $a$ and $c$. -
Q16. Assertion (A): If a train's speed is increased by $10\text{ km/h}$, the time taken to cover a fixed distance decreases, which is modeled by a rational equation reducible to a quadratic equation.
Reason (R): Speed is inversely proportional to time when distance is constant. -
Q17. Assertion (A): If $b^2 - 4ac < 0$, the quadratic equation $ax^2 + bx + c = 0$ has imaginary / non-real complex roots.
Reason (R): The square root of a negative number does not exist within the real number system. -
Q18. Assertion (A): Two quadratic equations can share the same set of real roots if their coefficients are proportional.
Reason (R): Proportional coefficients mean both equations represent the exact same curve scaled by a constant factor. -
Q19. Assertion (A): The length of a rectangular plot is $2$ more than its breadth, and its area is $120\text{ m}^2$. The quadratic equation representing this is $x^2 + 2x - 120 = 0$.
Reason (R): Area of a rectangle is equal to the product of its length and breadth. -
Q20. Assertion (A): The sum of the squares of two consecutive positive integers is $25$. This statement leads to a quadratic equation with integer coefficients.
Reason (R): Consecutive integers can be represented algebraically as $x$ and $x + 1$.
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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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