Class 10 Mathematics Chapter 5 Quadratic Equations

Key Concept: Equations reducible to quadratic equations, Substitution method

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the given equation can be transformed into a quadratic form by substituting $y = \frac{x + 1}{x - 1}$. Let's verify this:
Step 1: Substitute $y = \frac{x + 1}{x - 1}$ into the equation:
$y^2 + y - 2 = 0$
Step 2: Solve the quadratic equation:
$y = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm 3}{2} \Rightarrow y = 1 \text{ or } y = -2$
Step 3: Resubstitute $y = \frac{x + 1}{x - 1}$:
For $y = 1$:
$1 = \frac{x + 1}{x - 1} \Rightarrow x + 1 = x - 1 \Rightarrow 1 = -1 \text{ (No solution)}$
For $y = -2$:
$-2 = \frac{x + 1}{x - 1} \Rightarrow -2x + 2 = x + 1 \Rightarrow -3x = -1 \Rightarrow x = \frac{1}{3}$
Thus, the assertion is true because the substitution successfully reduced the equation to a quadratic form and yielded a valid solution ($x = \frac{1}{3}$). However, the reason claims that any rational equation of the form $f\left(\frac{P(x)}{Q(x)}\right) = 0$ can always be reduced to a quadratic equation, which is false because not all such equations result in a quadratic form after substitution (e.g., $\left(\frac{x + 1}{x - 1}\right)^3 - 1 = 0$ becomes cubic after substitution). Therefore, the assertion is true but the reason is false.

Key Concept: Equations Reducible to Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Substitute $y = \frac{x - 1}{x + 1}$. Then, $\frac{x + 1}{x - 1} = \frac{1}{y}$.
Step 2: The given equation becomes $y + \frac{1}{y} = \frac{10}{3}$, as stated in Reason (R).
Step 3: Solve for $y$ by multiplying through by $3y$ to eliminate denominators:
$3y^2 + 3 = 10y$.
Step 4: Rearrange to standard quadratic form:
$3y^2 - 10y + 3 = 0$.
Step 5: Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains how the substitution reduces the equation to quadratic form.

Key Concept: Substitution Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The given equation is $2x^4 - 5x^2 + 3 = 0$. By substituting $y = x^2$, the equation becomes $2y^2 - 5y + 3 = 0$, which is a quadratic equation. Thus, the assertion is true. The reason correctly explains the transformation by substitution, so it is also true and is the correct explanation.

Key Concept: Solving Quadratic Equations by Factorisation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, let's analyze the quadratic equation $x^2 - 6x + 9 = 0$.
Step 1: Identify coefficients: $a = 1$, $b = -6$, $c = 9$.
Step 2: Calculate the discriminant ($D$):
$D = b^2 - 4ac = (-6)^2 - 4(1)(9) = 36 - 36 = 0.$
Step 3: Interpretation: Since $D = 0$, the equation has exactly one real root (a repeated root).
For the Reason: The condition for a quadratic equation to have exactly one real root is indeed that its discriminant must be zero, making the Reason true.
Conclusion: Both Assertion and Reason are true, and Reason correctly explains Assertion because the zero discriminant directly implies a single real root.

Key Concept: Solving Quadratic Equations by Factorisation

c) Assertion is true, but Reason is false.
[Solution Description]
To solve $(x + 4)(x - 5) = 0$, we set each factor equal to zero:
- $x + 4 = 0$ gives $x = -4$ (correct as per Assertion).
- $x - 5 = 0$ gives $x = 5$ (correct as per Assertion).
However, the Reason states that for $(x + a)(x + b) = 0$, the solutions are $x = -a$ and $x = -b$. This is incorrect because the second term is $(x - 5)$, not $(x + b)$. The correct general rule is that if $(x + p)(x + q) = 0$, then $x = -p$ or $x = -q$. Here, since one term is $(x - 5)$, the solution is $x = 5$, not $x = -(-5)$. Thus, Assertion is true, but Reason is false.

Key Concept: Zero Product Rule

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The equation $x^2 - 9 = 0$ can be rewritten as $(x + 3)(x - 3) = 0$. According to the Zero Product Rule, if $(x + 3)(x - 3) = 0$, then either $x + 3 = 0$ or $x - 3 = 0$. Solving these gives $x = -3$ or $x = 3$. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Nature of Roots, Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, calculate the discriminant $D$ for the given equation $3x^2 - \sqrt{3}x + 1 = 0$.
Here, $a = 3$, $b = -\sqrt{3}$, $c = 1$.
The discriminant is calculated as:
$D = b^2 - 4ac = (-\sqrt{3})^2 - 4(3)(1) = 3 - 12 = -9$
Since $D = -9 < 0$, the roots are complex and not real.
Now, check the reason: It states that when $D < 0$, the roots are complex and conjugate. This is true because for any quadratic equation with real coefficients, if $D < 0$, the roots are always complex conjugates.
Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine the nature of the roots, calculate the discriminant ($D$) of the given quadratic equation $x^2 - 4x + 4 = 0$. Here, $a = 1$, $b = -4$, and $c = 4$.
The discriminant formula is:
$D = b^2 - 4ac$
Substituting the values:
$D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$
Since $D = 0$, the roots are real and equal. Thus, both Assertion (A) and Reason (R) are true, and Reason correctly explains Assertion because a zero discriminant implies real and equal roots.

Key Concept: Nature of roots using discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To determine if the assertion and reason are correct, first calculate the discriminant of the quadratic equation $x^2 - 4x + 4 = 0$. Here, $a = 1$, $b = -4$, and $c = 4$. The discriminant $D$ is given by $D = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$. Since $D = 0$, the roots are real and equal. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Verification of Roots, Multiple Roots and Simultaneous Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Substituting $x = 2$ into the equation $3x^2 - 2mx + 2n = 0$ gives:
$3(2)^2 - 2m(2) + 2n = 0$
$12 - 4m + 2n = 0$
Substituting $x = 3$ into the same equation gives:
$3(3)^2 - 2m(3) + 2n = 0$
$27 - 6m + 2n = 0$
Solving these two equations:
Subtract the first equation from the second to eliminate $2n$:
$(27 - 6m + 2n) - (12 - 4m + 2n) = 0$
$15 - 2m = 0 \Rightarrow m = \frac{15}{2}$
Substituting $m = \frac{15}{2}$ back into the first equation:
$12 - 4(\frac{15}{2}) + 2n = 0$
$12 - 30 + 2n = 0$
$-18 + 2n = 0 \Rightarrow n = 9$
Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Substituting Values to Check for Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's verify the assertion and reason step by step:
Step 1: Substitute $x = 2$ into the equation $x^2 - 5x + 6$.
Calculation: $(2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$.
Step 2: The result equals zero, confirming that $x = 2$ is indeed a root of the equation.
Step 3: The reason correctly demonstrates why the assertion is true.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Discriminant and Number of Solutions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine the number of solutions, we first calculate the discriminant ($D$) of the quadratic equation $x^2 - 4x + 4 = 0$. The discriminant formula is $D = b^2 - 4ac$, where $a = 1$, $b = -4$, and $c = 4$. Substituting the values, we get:
$D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$
Since $D = 0$, the equation has exactly one real root (a repeated root). Therefore, both Assertion (A) and Reason (R) are true, and Reason correctly explains the Assertion.

Key Concept: Application of the Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we analyze the Assertion: The quadratic equation is $x^2 - 4x + 4 = 0$. Here, $a = 1$, $b = -4$, and $c = 4$. We calculate the discriminant $D$ as follows:
$D = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0.$
Since $D = 0$, the Assertion is true as the roots are real and equal.
Next, the Reason states that for any quadratic equation $ax^2 + bx + c = 0$, if $D = 0$, the roots are real and equal. This is a correct statement based on the definition of the discriminant. Moreover, the Reason correctly explains why the Assertion is true because it directly applies to the given quadratic equation.
Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Application of the Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion and reason are correct, let's analyze the given quadratic equation:
For the equation $x^2 - 4x + 4 = 0$, the discriminant is calculated as:
$D = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$
Since $D = 0$, the roots are real and equal, which confirms that the Assertion is true. The Reason states that the discriminant is zero, which is also true. Moreover, the Reason correctly explains why the roots are equal because when $D = 0$, the quadratic equation has equal roots. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Application of the Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a quadratic equation $ax^2 + bx + c = 0$, equal roots occur when $D = b^2 - 4ac = 0$. Here, $a = 5$, $b = 4$, and $c = k$. Substituting into the discriminant formula:
$D = (4)^2 - 4(5)(k) = 16 - 20k.$
Setting $D = 0$ for equal roots:
$16 - 20k = 0 \implies 20k = 16 \implies k = \frac{16}{20} = \frac{4}{5}.$
Thus, the Assertion is true. The Reason correctly explains why the Assertion holds since $D = 0$ ensures equal roots.

Key Concept: Quadratic Equations, Solving Word Problems

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the equation $x^2 - 5x + 6 = 0$ corresponds to finding two numbers with a product of 6 and sum of 5. This is correct because the standard form $x^2 - (sum)x + product = 0$ aligns with this interpretation.
The reason states that for any quadratic equation $x^2 + bx + c = 0$, the sum of roots is $-b$ and the product is $c$. This is also true as it follows from Vieta's formulas.
Furthermore, the reason directly explains why the assertion is valid since the sum and product of the roots correspond to the coefficients in the quadratic equation. Hence, the reason is the correct explanation for the assertion.

Key Concept: Quadratic Equations - Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine the correctness of the Assertion (A) and Reason (R), we analyze the given quadratic equation $x^2 - 4x + 4 = 0$.
Step 1: Find the discriminant ($D$) of the equation.
The standard form is $ax^2 + bx + c = 0$, where $a = 1$, $b = -4$, and $c = 4$.
The discriminant is calculated as:
$D = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$
Step 2: Interpret the discriminant.
Since $D = 0$, the equation has real and equal roots, which confirms Assertion (A).
Step 3: Verify Reason (R).
The reason states that the discriminant is zero, which matches our calculation in Step 1.
Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Formulating Quadratic Equations from Word Problems

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let the first even integer be $x$. Since even numbers are multiples of 2, the next consecutive even integer will indeed be $x + 2$. Therefore, their product is correctly represented by the quadratic equation $x(x + 2)$. Both the Assertion and Reason are true, and the Reason logically explains why the Assertion is correct.

Key Concept: Shape of the Graph, Discriminant, Vertex

d) Assertion is false, but Reason is true.
[Solution Description]
To determine if the parabola intersects the x-axis, we check the discriminant $D = b^2 - 4ac$. For the given equation $y = -2x^2 + 8x - 7$, $a = -2$, $b = 8$, and $c = -7$.
Step 1: Calculate the discriminant:
$D = b^2 - 4ac = (8)^2 - 4(-2)(-7) = 64 - 56 = 8$
Since $D > 0$, the parabola intersects the x-axis at two distinct points. Therefore, the Assertion is false.
Step 2: Check the Reason:
The Reason states that $D 0$. Thus, the Reason is also false.
But in the options provided, the condition where both Assertion and Reason are false is not listed. Hence, based on the given options, the closest correct interpretation is that the Assertion is false, but the Reason is true (which is incorrect as per the calculation).
Given the discrepancy, the correct answer according to the standard form of Assertion-Reason questions would be to select the option where only one of them is false. However, in this case, both are false, but since that option is not provided, the best available choice is the closest match.

Key Concept: Shape of the Graph

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the graph of the quadratic equation $y = -x^2 + 4x - 3$ opens upwards. However, this is false because for a quadratic equation $y = ax^2 + bx + c$, if the coefficient of $x^2$ ($a$) is negative, the parabola opens downwards. The reason given is true since the coefficient of $x^2$ is $-1$ (negative). Thus, Assertion is false, but Reason is true.

Key Concept: Graph Basics

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the graph of $y = x^2$ is a parabola, which is true because quadratic functions of this form always produce parabolic graphs. The reason explains what a parabola is and its general shape, which is also correct. Moreover, the reason correctly explains why the assertion is true since $y = x^2$ specifically produces an upward-opening parabola.

Key Concept: Vertex Form, Axis of Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.
[Solution Description]
The assertion states the vertex of $y = 2(x-3)^2 + 5$ is $(3, 5)$, which is correct as the vertex form $y = a(x-h)^2 + k$ reveals the vertex at $(h, k)$. The reason explains this property of the vertex form, which justifies the assertion. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Graphical Representation of Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To analyze the assertion and reason:
1. **Assertion**: The quadratic equation $x^2 - 6x + 9 = 0$ has exactly one real root.
- Let us find the discriminant ($D$) of the equation. For a quadratic equation $ax^2 + bx + c = 0$, $D = b^2 - 4ac$.
- Here, $a = 1$, $b = -6$, $c = 9$.
- Calculating $D = (-6)^2 - 4(1)(9) = 36 - 36 = 0$.
- Since $D = 0$, the equation has exactly one real root (a repeated root). Hence, the assertion is true.
2. **Reason**: The discriminant of the equation $x^2 - 6x + 9 = 0$ is zero.
- As calculated above, $D = 0$ for this equation. Thus, the reason is also true.
3. **Explanation**:
- The reason correctly explains why there is only one real root for the equation, as a discriminant of zero implies a single repeated root. Therefore, the reason is the correct explanation for the assertion.

Key Concept: Graphical Representation of Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For the given quadratic equation $x^2 + 2x + 1 = 0$, we can calculate the discriminant $D$ using the formula $D = b^2 - 4ac$. Here, $a = 1$, $b = 2$, and $c = 1$. Substituting these values into the formula, we get:
$D = (2)^2 - 4(1)(1) = 4 - 4 = 0$
When the discriminant is zero ($D = 0$), the quadratic equation has exactly one real root, meaning the parabola touches the x-axis at exactly one point (the vertex). Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Pythagorean theorem

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion correctly states the Pythagorean theorem, which is true. The reason provides an application of the theorem, which is also true. However, the reason does not explain why the assertion is true; it merely describes an application. Therefore, both statements are true, but the reason does not correctly explain the assertion.

Key Concept: Properties of Triangles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states the mathematical relationship in a right triangle which is correct as per properties of triangles. The reason correctly identifies this relationship as Pythagoras theorem which indeed explains why this relationship holds true. Therefore both are true and reason correctly explains the assertion.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

The assertion (A) is correct because "Application in Motion" refers to understanding how forces influence the motion of objects—this is a core aspect of classical mechanics. The reason (R) is also correct, as Newton’s laws of motion form the foundational principles for analyzing such situations. Newton's First Law explains the need for a net force to change an object’s state of motion; the Second Law quantifies how force affects acceleration; and the Third Law describes action-reaction pairs. Since Newton’s laws directly apply to studying forces on moving objects, the reason correctly explains the assertion. Therefore, both (A) and (R) are true, and (R) is the correct explanation of (A).

Key Concept: Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether the assertion and reason are true:
1. First, find the discriminant ($D$) of the given quadratic equation $x^2 - 5x + 6 = 0$.
Formula for discriminant: $D = b^2 - 4ac$
Here, $a = 1$, $b = -5$, $c = 6$
Substituting values: $D = (-5)^2 - 4(1)(6) = 25 - 24 = 1$
Since $D > 0$, the roots are real and distinct.
2. The assertion states that the equation has real and distinct roots, which is true because $D = 1 > 0$.
3. The reason correctly explains why the assertion is true—because the discriminant is positive.
Therefore, both the assertion and reason are true, and the reason is the correct explanation.

Key Concept: Basic Motion Concepts

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

The given question includes placeholder statements for both the Assertion (A) and Reason (R), meaning the actual content is missing or yet to be provided. Without the specific details of the assertion and reason, it is not possible to evaluate their correctness or the logical connection between them. To determine the correct relationship—whether the reason correctly explains the assertion—we need the complete and meaningful content for both (A) and (R). Until then, this question cannot be answered accurately and remains incomplete.

Key Concept: Quadratic Equations, Word Problems

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the assertion, we first need to check if the quadratic equation formed using the given sum and product of roots matches the assertion.
1. Given the sum of roots ($S$) is $5$ and the product of roots ($P$) is $-6$.
2. A standard quadratic equation based on the sum and product of roots is:
$x^2 - Sx + P = 0$
3. Substituting the values:
$x^2 - 5x + (-6) = 0 \implies x^2 - 5x - 6 = 0$
4. However, the assertion claims the equation to be $x^2 - x - 30 = 0$, which does not match our calculation.
The reason states the correct method for forming a quadratic equation from its roots, but the assertion misapplies this method by incorrectly stating the equation. Therefore, the Reason is true, but the Assertion is false due to incorrect substitution or reasoning.

Key Concept: Word Problems Based on Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us solve the assertion step by step.
Step 1: Given the quadratic equation is $x^2 - 5x + 6 = 0$.
Step 2: Factorize the equation to find its roots. We look for two numbers whose sum is 5 and product is 6. These numbers are 2 and 3. So, the equation can be written as $(x-2)(x-3)=0$.
Step 3: The roots are $x=2$ and $x=3$, which are integers. Thus, Assertion (A) is true.
Step 4: Now analyze Reason (R). For a quadratic equation of the form $x^2 - (sum)x + (product) = 0$, if both the sum and product of the roots are integers, then the roots will also be integers. This is because the solutions will be factors of the product term that add up to the sum term.
Step 5: In the given equation, the sum of roots is 5 and the product is 6, both of which are integers, leading to integer roots. Thus, Reason (R) is true and correctly explains Assertion (A).
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Word Problems Based on Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the problem, we first factorize the quadratic equation $x^2 - 5x + 6 = 0$.
Step 1: Factorize the equation. We look for two numbers whose product is 6 and sum is -5. The numbers are -2 and -3. So, the equation becomes $(x-2)(x-3)=0$.
Step 2: Find the roots. Setting each factor to zero gives $x=2$ and $x=3$.
Step 3: Interpret the roots. Since length and width must be positive, the possible dimensions are 2 units and 3 units.
Therefore, both Assertion and Reason are true, and Reason correctly explains how to find the dimensions using the quadratic equation.

Key Concept: Nature of Roots Based on Discriminant, Rationality Condition

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For the quadratic equation $3x^2 + 5x + 2 = 0$, the coefficients are $a = 3$, $b = 5$, $c = 2$, all of which are rational. The discriminant is calculated as:
$D = (5)^2 - 4(3)(2) = 25 - 24 = 1$
Since $D = 1$ is a perfect square and $a$, $b$, $c$ are rational, the roots must be rational. Hence, both Assertion and Reason are true, and Reason correctly explains why the roots are rational.

Key Concept: Nature of Roots Based on Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Identify the coefficients of the quadratic equation $x^2 - 4x + 4 = 0$. Here, $a = 1$, $b = -4$, and $c = 4$.
Step 2: Calculate the discriminant using the formula $D = b^2 - 4ac$. Substituting the values, we get:
$D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$
Step 3: Interpret the discriminant value. Since $D = 0$, the roots are real and equal. Therefore, the Assertion (A) is true.
Step 4: Evaluate the Reason (R). The discriminant being zero ($D = 0$) directly explains why the roots are real and equal. Hence, the Reason (R) is true and correctly explains the Assertion (A).

Key Concept: Nature of Roots Based on Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given the quadratic equation $x^2 - 4x + 4 = 0$, we calculate the discriminant as follows:
Here, $a = 1$, $b = -4$, and $c = 4$.
The discriminant $D = b^2 - 4ac = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$.
Since $D = 0$, the roots are real and equal. Hence, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Sum of the Roots, Quadratic Equations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Let the roots be $\alpha$ and $2\alpha$ (since one root is twice the other). The sum of the roots is $\alpha + 2\alpha = 3\alpha$. According to the formula for the sum of the roots, we have $3\alpha = 2k + 1$. Since $3\alpha$ is clearly divisible by 3, this implies that $(2k + 1)$ must also be divisible by 3. Thus, the Assertion is true. The Reason states the correct formula for the sum of the roots of any quadratic equation, which is universally true. However, the Reason does not explain why the sum must be a multiple of 3 in this specific case; it only provides the general formula. Therefore, both statements are true, but the Reason does not correctly explain the Assertion.

Key Concept: Sum of the Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the sum of the roots of the quadratic equation $3x^2 - 5x + 2 = 0$, we use the formula for the sum of the roots of a general quadratic equation $ax^2 + bx + c = 0$, which is $\alpha + \beta = -\frac{b}{a}$. Here, $a = 3$, $b = -5$, and $c = 2$. Plugging these values into the formula, we get:
$\alpha + \beta = -\left(\frac{-5}{3}\right) = \frac{5}{3}.$
Therefore, the Assertion (A) is true as it correctly states the sum of the roots. The Reason (R) is also true as it provides the correct formula for the sum of the roots. Moreover, Reason (R) is indeed the correct explanation for Assertion (A) because the calculation in the assertion directly uses the formula stated in the reason.

Key Concept: Sum of the Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the sum of the roots of the quadratic equation $x^2 - 6x + 8 = 0$, we use the formula:
$\text{Sum of roots} = -\frac{b}{a}$
Here, $a = 1$, $b = -6$, and $c = 8$. Plugging in the values:
$\text{Sum of roots} = -\frac{-6}{1} = 6$
Therefore, the assertion is true.
The reason states the correct formula for the sum of the roots of a quadratic equation, which explains why the assertion is true. Hence, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Quadratic Equations, Relationship Between Roots and Coefficients

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if the sum of the roots is 5 and the product is 6, the quadratic equation can be written as $x^2 - 5x + 6 = 0$. This is true because, given the sum and product of the roots, we can reconstruct the quadratic equation using the formula $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.
The reason states the general relationship between the coefficients and the roots of a quadratic equation, which is also true. Moreover, the reason directly explains why the assertion is valid because it provides the mathematical foundation for reconstructing the quadratic equation from the sum and product of its roots.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Relationship Between Roots and Coefficients

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's verify the assertion and reason step by step.
1) The given quadratic equation is $x^2 - 5x + 6 = 0$. Comparing with the standard form $ax^2 + bx + c = 0$, we have $a = 1$, $b = -5$, and $c = 6$.
2) Using the relationship between roots and coefficients, the sum of the roots ($\alpha + \beta$) is given by:
$\alpha + \beta = -\frac{b}{a} = -\left(\frac{-5}{1}\right) = 5$
Hence, Assertion (A) is true.
3) Reason (R) states that for any quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$. This is a standard result in algebra and is correct.
4) Since Assertion (A) is directly derived from Reason (R), Reason correctly explains the Assertion.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Relationship Between Roots and Coefficients

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for the equation $x^2 - 5x + 6 = 0$, the sum of roots $\alpha + \beta = 5$. This is true because according to the relationship between roots and coefficients, $\alpha + \beta = -\frac{b}{a}$. Here, $a = 1$ and $b = -5$, so $\alpha + \beta = -\frac{-5}{1} = 5$. The reason correctly explains the assertion because it provides the formula used to derive the sum of roots. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Quadratic Formula, Discriminant

d) Assertion is false, but Reason is true.
[Solution Description]
For the given quadratic equation $x^2 - 4x + 5 = 0$, let's compute the discriminant:
$D = (-4)^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4$
Since $D < 0$, the roots are not real (they are complex conjugates). Therefore, Assertion (A) is false. However, Reason (R) is a true statement about the discriminant condition for real and distinct roots. Hence, the correct relationship is that Assertion is false, but Reason is true.

Key Concept: Quadratic Formula and Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For the quadratic equation $2x^2 - 5x + 3 = 0$, let's calculate the discriminant ($D$) to determine the nature of the roots:
Given: $a = 2$, $b = -5$, $c = 3$
Discriminant formula: $D = b^2 - 4ac$
Substituting values: $D = (-5)^2 - 4(2)(3) = 25 - 24 = 1$
Since $D > 0$, the roots are real and distinct. Thus, the Assertion (A) is true. The Reason (R) states that the discriminant is positive, which is also true as calculated ($D = 1$). Further, the Reason correctly explains why the roots are real and distinct because a positive discriminant implies two distinct real roots. Therefore, both (A) and (R) are true, and (R) is the correct explanation of (A).

Key Concept: Quadratic Formula

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the quadratic formula can solve any quadratic equation, which is true as it applies universally to equations of the form $ax^2 + bx + c = 0$ where $a \neq 0$. The Reason explains the derivation method (completing the square), which is also true and directly justifies why the formula works.

Key Concept: Factorization Method, Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that substituting the roots $\alpha$ or $\beta$ into the quadratic equation $(x - \alpha)(x - \beta) = 0$ satisfies the equation. This is true because when we substitute $x = \alpha$, the equation becomes $(\alpha - \alpha)(\alpha - \beta) = 0 \times (\alpha - \beta) = 0$, which holds true. Similarly for $x = \beta$. The reason correctly explains the assertion since the Zero Product Property justifies why substituting either root results in a valid equality (the product being zero implies at least one factor is zero).

Key Concept: Factorization Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's factorize the given equation $x^2 - 5x + 6 = 0$.
Step 1: Find two numbers whose product is $6$ (which is $\frac{c}{a} = \frac{6}{1}$) and sum is $-(-5)/1 = 5$.
Step 2: The numbers are $2$ and $3$ because $2 \times 3 = 6$ and $2 + 3 = 5$.
Step 3: So, the factors are $(x - 2)(x - 3) = 0$.
Step 4: Solving gives $x = 2$ or $x = 3$, which matches Assertion (A).
The Reason (R) correctly states the conditions for factorization of a quadratic equation, and it explains why the factorization is possible in this case. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Factorization Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we factorize the quadratic equation $x^2 - 5x + 6 = 0$. The factors are $(x - 2)(x - 3) = 0$, which gives the roots $x = 2$ and $x = 3$. Since both the assertion and reason are true, and the reason correctly explains why the roots are 2 and 3, the correct option is a).

Key Concept: Nature of Roots, Quadratic Formula

d) Assertion is false, but Reason is true.
[Solution Description]
To determine the correctness of the Assertion and Reason, let us analyze them step by step.
1. **Assertion Analysis**:
The given quadratic equation is $x^2 - 4x + 5 = 0$.
Here, $a = 1$, $b = -4$, $c = 5$.
Calculate the discriminant:
$D = b^2 - 4ac = (-4)^2 - 4(1)(5) = 16 - 20 = -4$.
Since $D 0$, the roots are real and distinct. This is correct as per the nature of roots. However, the assertion was false, making the Reason irrelevant in explaining the Assertion.
Conclusion: Assertion is false, but Reason is true.

Key Concept: Solving quadratic equations by factorisation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the equation $x^2 - 5x + 6 = 0$ has roots $x = 2$ and $x = 3$.
Step 1: Factorise the quadratic expression $x^2 - 5x + 6$.
We look for two numbers that multiply to 6 and add to $-5$. These numbers are $-2$ and $-3$.
Step 2: Rewrite the equation as $(x-2)(x-3) = 0$.
Step 3: Set each factor equal to zero and solve:
$x - 2 = 0 \Rightarrow x = 2$,
$x - 3 = 0 \Rightarrow x = 3$.
Therefore, the roots are indeed $x = 2$ and $x = 3$, confirming the assertion is true.
The reason states that the factors of the quadratic expression $x^2 - 5x + 6$ are $(x-2)(x-3)$, which is correct as shown in Step 2. Moreover, the reason correctly explains the assertion because the roots are derived from these factors.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Solving quadratic equations by factorisation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the given quadratic equation $x^2 - 5x + 6 = 0$, we can factorise it as $(x - 2)(x - 3) = 0$. Setting each factor equal to zero gives the solutions $x = 2$ and $x = 3$. Thus, the Assertion is true. The Reason states that a quadratic equation can be solved by factorising, which is also true and correctly explains why the Assertion holds.

Key Concept: Discriminant, Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion and reason are true, let's verify the condition for real and distinct roots.
Given the quadratic equation: $4x^2 - 12x + k = 0$, we identify:
- $a = 4$
- $b = -12$
- $c = k$
The discriminant $D$ is calculated as:
$D = b^2 - 4ac = (-12)^2 - 4 \times 4 \times k = 144 - 16k$
For real and distinct roots, $D > 0$:
$144 - 16k > 0$
Subtract 144 from both sides:
$-16k > -144$
Divide both sides by $-16$ (remembering to reverse the inequality sign):
$k < 9$
Therefore, the Assertion (A) is true since $k$ must indeed be less than 9 for real and distinct roots. The Reason (R) correctly states the condition for real and distinct roots, and it explains why the Assertion is true. Hence, both are true, and Reason is the correct explanation of Assertion.

Key Concept: Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine the nature of the roots, we first calculate the discriminant ($D$) of the given quadratic equation $3x^2 - 4x + 5 = 0$.
Step 1: Identify coefficients:
$a = 3$, $b = -4$, $c = 5$.
Step 2: Calculate discriminant:
$D = b^2 - 4ac = (-4)^2 - 4(3)(5) = 16 - 60 = -44$.
Step 3: Interpret discriminant:
Since $D < 0$, the roots are imaginary.
Conclusion: Both Assertion and Reason are true, and the Reason correctly explains why the roots are imaginary (as $D < 0$).

Key Concept: Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For the equation $x^2 - 6x + 9 = 0$, the discriminant is calculated as:
$D = (-6)^2 - 4 \times 1 \times 9 = 36 - 36 = 0$
Since $D = 0$, the roots are real and equal. Thus, both Assertion (A) and Reason (R) are true, and R correctly explains A.

Key Concept: Nature of Roots, Discriminant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether the assertion and reason are true, let's analyze the given quadratic equation $x^2 + 2\sqrt{2}x + 2 = 0$. Here, $a = 1$, $b = 2\sqrt{2}$, and $c = 2$. The discriminant is calculated as:
$D = b^2 - 4ac = (2\sqrt{2})^2 - 4(1)(2) = 8 - 8 = 0$
Since $D = 0$, the roots are indeed real and equal. The reason correctly states the condition for real and equal roots when $D = 0$. Therefore, both Assertion and Reason are true, and Reason explains the Assertion correctly.

Key Concept: Roots of Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine the nature of the roots, we need to calculate the discriminant ($D$) of the given quadratic equation $x^2 - 4x + 4 = 0$. Here, $a = 1$, $b = -4$, and $c = 4$.
The discriminant is calculated as:
$D = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0.$
Since $D = 0$, the quadratic equation has real and equal roots. Therefore, both Assertion (A) and Reason (R) are true, and Reason correctly explains Assertion.

Key Concept: Roots of Quadratic Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine the nature of the roots, we calculate the discriminant $D$ of the quadratic equation $x^2 - 5x + 6 = 0$.
Here, $a = 1$, $b = -5$, and $c = 6$.
The discriminant $D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1$.
Since $D > 0$, the roots are real and distinct.
Therefore, the Assertion (A) is true.
The Reason (R) correctly explains why the roots are real and distinct as it states that $D > 0$.

Key Concept: Quadratic Equations, Discriminant, Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For the given quadratic equation $$ x^2 - 4x + 4 = 0$$, let us calculate the discriminant $D$. Here, $a = 1$, $b = -4$, and $c = 4$. The discriminant is calculated as $D = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$. Since $D = 0$, the roots are real and equal, making the Assertion true. The Reason states that the discriminant is zero, which matches our calculation. Since the zero discriminant explains the nature of the roots being real and equal, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Degree of the Equation

d) Assertion is false, but Reason is true.
[Solution Description]
To determine if the assertion and reason are correct, we analyze the given equation:
1. The equation is $5x^4 - 3x^2 + 7 = 0$. The highest power of the variable $x$ is 4 ($x^4$). A quadratic equation must have the highest power as 2. Hence, the assertion that the given equation is quadratic is false.
2. The reason states that the highest power of the variable is 4, which is true based on the equation.
Thus, Assertion (A) is false, but Reason (R) is true.

Key Concept: Degree of the Equation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The equation $x^2 - 4 = 0$ has only the square term ($x^2$) and no linear term ($x$). According to the definition, a pure quadratic equation contains only the square term of the unknown variable. Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Quadratic Equations, Discriminant, Nature of Roots

d) Assertion is false, but Reason is true.
[Solution Description]
First, let's analyze the Assertion: The given quadratic equation is $(3x^2 - 2\sqrt{5}x + \sqrt{5} = 0)$. To determine if it has real and equal roots, we calculate its discriminant $(D = b^2 - 4ac)$.
Here, $a = 3$, $b = -2\sqrt{5}$, and $c = \sqrt{5}$. Calculating discriminant:
$D = (-2\sqrt{5})^2 - 4(3)(\sqrt{5}) = 4 \times 5 - 12\sqrt{5} = 20 - 12\sqrt{5}$
Now, evaluate $(20 - 12\sqrt{5})$:
Approximate $\sqrt{5} \approx 2.236$
$12\sqrt{5} \approx 12 \times 2.236 = 26.832$
So,
$20 - 26.832 = -6.832 < 0$
Since $(D < 0)$, the roots are imaginary (not real). Thus, Assertion (A) is false.
Now, Reason (R): It correctly states that for a quadratic equation to have real and equal roots, its discriminant must be zero. This is always true by definition.
Since Assertion is false but Reason is true, the correct answer is d).

Key Concept: Nature of Roots

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For the quadratic equation $3x^2 - 6x + 3 = 0$, we calculate the discriminant $(D)$.
Here, $a = 3$, $b = -6$, and $c = 3$.
The discriminant is calculated as:
$D = b^2 - 4ac = (-6)^2 - 4(3)(3) = 36 - 36 = 0$
Since $D = 0$, the roots are real and equal.
Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Quadratic Equations - Standard Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $3x^2 + 5x - 2 = 0$ is a quadratic equation in standard form, which is correct because it follows the general form $ax^2 + bx + c = 0$ with $a = 3$, $b = 5$, and $c = -2$. The reason correctly defines the standard form of a quadratic equation, explaining why the assertion is true. Thus, both are true, and the reason explains the assertion.

Key Concept: Fractional equations reducible to quadratic

a) $x = \frac{17}{6}$
[Solution Description]
Cross-multiplying gives:
$(2x - 1)(x - 2) = (2x - 5)(x + 3)$.
Expanding both sides:
$2x^2 - 5x + 2 = 2x^2 + x - 15$.
Simplifying:
$-5x + 2 = x - 15 \Rightarrow -6x = -17 \Rightarrow x = \frac{17}{6}$.
Verify that $x = \frac{17}{6}$ does not make any denominator zero.
Thus, the solution is $x = \frac{17}{6}$.

Key Concept: Substitution method

a) $x = \frac{-5 \pm \sqrt{57}}{2}, \frac{-5 \pm \sqrt{17}}{2}$
[Solution Description]
Let $y = x^2 + 5x$. The equation becomes $y^2 - 6y - 16 = 0$. Solving this quadratic equation:
$y = \frac{6 \pm \sqrt{36 + 64}}{2} = \frac{6 \pm \sqrt{100}}{2} = \frac{6 \pm 10}{2}$
So, $y = 8$ or $y = -2$.
Back-substituting:
For $y = 8$, $x^2 + 5x - 8 = 0 \Rightarrow x = \frac{-5 \pm \sqrt{25 + 32}}{2} = \frac{-5 \pm \sqrt{57}}{2}$.
For $y = -2$, $x^2 + 5x + 2 = 0 \Rightarrow x = \frac{-5 \pm \sqrt{25 - 8}}{2} = \frac{-5 \pm \sqrt{17}}{2}$.
Thus, the solutions are $x = \frac{-5 \pm \sqrt{57}}{2}, \frac{-5 \pm \sqrt{17}}{2}$.

Key Concept: Equations reducible to quadratic form

a) $x = -3, -1, 1, 3$
[Solution Description]
Let $y = x^2$. The equation becomes $y^2 - 10y + 9 = 0$. Solving this quadratic equation:
$y = \frac{10 \pm \sqrt{100 - 36}}{2} = \frac{10 \pm \sqrt{64}}{2} = \frac{10 \pm 8}{2}$
So, $y = 1$ or $y = 9$.
Back-substituting:
For $y = 1$, $x^2 = 1 \Rightarrow x = \pm 1$.
For $y = 9$, $x^2 = 9 \Rightarrow x = \pm 3$.
Thus, the solutions are $x = -3, -1, 1, 3$.

Key Concept: Substitution to Quadratic Form

b) No real solutions
[Solution Description]
Cross-multiply to eliminate denominators:
$(x - 2)(2x - 1) = (x - 6)(x + 3)$. Expand both sides:
Left side: $2x^2 - x - 4x + 2 = 2x^2 - 5x + 2$.
Right side: $x^2 + 3x - 6x - 18 = x^2 - 3x - 18$.
Bring all terms to one side:
$2x^2 - 5x + 2 - x^2 + 3x + 18 = 0 \Rightarrow x^2 - 2x + 20 = 0$.
Now solve the quadratic equation using the discriminant:
Discriminant: $D = (-2)^2 - 4(1)(20) = 4 - 80 = -76 < 0$.
Since the discriminant is negative, there are no real solutions.

Key Concept: Substitution to Quadratic Form

c) $x = \frac{3}{2}, \frac{5}{2}$
[Solution Description]
Let $y = 2x - 1$. Then the equation becomes:
$y^2 - 6y + 8 = 0$. Solve this quadratic equation by factoring:
$(y - 2)(y - 4) = 0$, so $y = 2$ or $y = 4$.
Substitute back $y = 2x - 1$:
For $y = 2$, $2x - 1 = 2 \Rightarrow 2x = 3 \Rightarrow x = \frac{3}{2}$.
For $y = 4$, $2x - 1 = 4 \Rightarrow 2x = 5 \Rightarrow x = \frac{5}{2}$.
The solutions are $x = \frac{3}{2}, \frac{5}{2}$.

Key Concept: Substitution to Quadratic Form

b) $\pm \sqrt{2}, \pm \frac{\sqrt{3}}{3}$
[Solution Description]
Let $y = x^2$, then the equation becomes:
$3y^2 - 7y + 2 = 0$. Solve this quadratic equation using the quadratic formula:
$y = \frac{7 \pm \sqrt{49 - 24}}{6} = \frac{7 \pm 5}{6}$. So, $y = 2$ or $y = \frac{1}{3}$.
Substitute back $y = x^2$:
For $y = 2$, $x^2 = 2 \Rightarrow x = \pm \sqrt{2}$.
For $y = \frac{1}{3}$, $x^2 = \frac{1}{3} \Rightarrow x = \pm \frac{\sqrt{3}}{3}$.
The solutions are $x = \sqrt{2}, -\sqrt{2}, \frac{\sqrt{3}}{3}, -\frac{\sqrt{3}}{3}$.

Key Concept: Fractional equation reducible to quadratic

c) $x = -2$, $x = 1$
[Solution Description]
Multiply through by $2x(x + 1)$ to eliminate denominators:
$2x^2 + 2(x + 1)^2 = 5x(x + 1)$
Simplify:
$2x^2 + 2(x^2 + 2x + 1) = 5x^2 + 5x$
Expand:
$2x^2 + 2x^2 + 4x + 2 = 5x^2 + 5x$
Combine terms:
$4x^2 + 4x + 2 = 5x^2 + 5x$
Rearrange to standard quadratic form:
$-x^2 - x + 2 = 0$
Multiply by $-1$:
$x^2 + x - 2 = 0$
Factorize:
$(x + 2)(x - 1) = 0$
Solutions: $x = -2$ and $x = 1$.
Verify for extraneous roots (both valid).

Key Concept: Biquadratic equation

b) $x = \pm 1$, $x = \pm 2$
[Solution Description]
Let $y = x^2$. The equation becomes:
$2y^2 - 10y + 8 = 0$
Divide by 2:
$y^2 - 5y + 4 = 0$
Factorize:
$(y - 1)(y - 4) = 0$
So, $y = 1$ or $y = 4$.
Back-substitute $y = x^2$:
For $y = 1$, $x^2 = 1 \implies x = \pm 1$
For $y = 4$, $x^2 = 4 \implies x = \pm 2$
The solutions are $x = 1$, $x = -1$, $x = 2$, $x = -2$.

Key Concept: Equation reducible to quadratic form

b) $x = 1$, $x = 2$
[Solution Description]
Let $y = x + 1$. Substituting, the equation becomes:
$y^2 - 5y + 6 = 0$
Factorizing:
$(y - 2)(y - 3) = 0$
Therefore, $y = 2$ or $y = 3$.
Back-substitute $y = x + 1$:
For $y = 2$, $x + 1 = 2 \implies x = 1$
For $y = 3$, $x + 1 = 3 \implies x = 2$
The solutions are $x = 1$ and $x = 2$.

Key Concept: Forming Quadratic Equations from Roots

a) $x^2 + 5x - 14 = 0$
[Solution Description]
To form a quadratic equation with roots $2$ and $-7$, we use the fact that if $\alpha$ and $\beta$ are roots, the equation can be written as $(x - \alpha)(x - \beta) = 0$. Substituting $\alpha = 2$ and $\beta = -7$, we get $(x - 2)(x + 7) = 0$. Expanding this gives $x^2 + 5x - 14 = 0$.

Key Concept: Finding Unknown Coefficients, Quadratic Roots

b) $k = -1$
[Solution Description]
Since $x = 3$ is a root of the equation $2x^2 + kx - 15 = 0$, substituting $x = 3$ into the equation must satisfy it: $2(3)^2 + k(3) - 15 = 0$. Simplifying, $18 + 3k - 15 = 0$ leads to $3k + 3 = 0$. Solving for $k$, we get $k = -1$.

Key Concept: Solving quadratic equations, Zero Product Rule

b) $x = 4, x = 5$
[Solution Description]
To solve $x^2 - 9x + 20 = 0$, we factorise the quadratic expression. We look for two numbers that multiply to $20$ and add up to $-9$. These numbers are $-5$ and $-4$. Thus, the equation can be written as $(x - 5)(x - 4) = 0$. Applying the Zero Product Rule, we set each factor to zero: $x - 5 = 0$ or $x - 4 = 0$. Solving these gives $x = 5$ or $x = 4$.

Key Concept: Forming Quadratic Equations from Solution Sets

b) $x^2 - 3x - 10 = 0$
[Solution Description]
A quadratic equation with roots $-2$ and $5$ can be formed as:
$(x + 2)(x - 5) = 0.$
Expanding this gives:
$x^2 - 5x + 2x - 10 = x^2 - 3x - 10 = 0.$
Hence, the required equation is $x^2 - 3x - 10 = 0$.

Key Concept: Zero Product Rule

b) $x = -\frac{1}{2}$
[Solution Description]
Using the Zero Product Rule, set each factor equal to zero:
$2x + 1 = 0 \implies x = -\frac{1}{2},$
$x - 4 = 0 \implies x = 4.$
Thus, the solutions are $x = -\frac{1}{2}$ and $x = 4$.

Key Concept: Solving quadratic equations by factorisation

c) 2 and 3
[Solution Description]
To find the roots of the equation $x^2 - 5x + 6 = 0$, we first factorise it:
$x^2 - 5x + 6 = (x - 2)(x - 3) = 0.$
Setting each factor equal to zero gives:
$x - 2 = 0 \implies x = 2,$
$x - 3 = 0 \implies x = 3.$
Therefore, the roots are $x = 2$ and $x = 3$.

Key Concept: Finding roots

b) $x = 3$
[Solution Description]
To find the roots of $x^2 - 8x + 15 = 0$, we factorise it as:
$(x - 3)(x - 5) = 0$
Applying the zero product rule:
$x - 3 = 0 \quad \text{or} \quad x - 5 = 0$
Thus, the roots are:
$x = 3 \quad \text{or} \quad x = 5$
So, one of the roots is $3$ or $5$.

Key Concept: Zero Product Rule

d) $x = -4$ or $x = 3$
[Solution Description]
Given the equation $(x + 4)(x - 3) = 0$, we apply the zero product rule:
$x + 4 = 0 \quad \text{or} \quad x - 3 = 0$
Solving these gives:
$x = -4 \quad \text{or} \quad x = 3$
The possible values of $x$ are $-4$ and $3$.

Key Concept: Solving quadratic equations by factorisation

a) $x = 0$
[Solution Description]
To solve the equation $x^2 - 5x = 0$, we first factorise the left-hand side:
$x(x - 5) = 0$
Using the zero product rule, set each factor to zero:
$x = 0 \quad \text{or} \quad x - 5 = 0$
Solving these gives:
$x = 0 \quad \text{or} \quad x = 5$
The solutions are $x = 0$ and $x = 5$.

Key Concept: Imaginary roots, Condition analysis

c) $k > -\frac{2}{3}, k \neq -1$
[Solution Description]
For imaginary roots, discriminant $D < 0$. Here $a = k+1$, $b = 2k$, $c = k+2$. Discriminant:
$D = (2k)^2 - 4(k+1)(k+2) = 4k^2 - 4(k^2 + 3k + 2) = -12k - 8$
Setting $D < 0$:
$-12k - 8 < 0 \Rightarrow -12k -\frac{2}{3}$
Additionally, coefficient of $x^2$ must not be zero: $k+1 \neq 0 \Rightarrow k \neq -1$.
Combining these, $k > -\frac{2}{3}$ but $k \neq -1$.

Key Concept: Equal roots, Parameter finding

c) $\frac{1}{8}$
[Solution Description]
When two quadratics have both roots equal, their coefficients are proportional. Therefore:
$\frac{1}{1} = \frac{p}{1} = \frac{8}{q}$
From $\frac{1}{1} = \frac{p}{1}$ we get $p = 1$.
From $\frac{1}{1} = \frac{8}{q}$ we get $q = 8$.
Thus $\frac{p}{q} = \frac{1}{8}$.

Key Concept: Discriminant analysis, Nature of roots

d) $p 3$
[Solution Description]
For real and distinct roots, discriminant $D > 0$. Here $a = p-1$, $b = 2(p-1)$, $c = 2$. Calculating discriminant:
$D = b^2 - 4ac = [2(p-1)]^2 - 4(p-1)(2) = 4(p-1)^2 - 8(p-1)$
Setting $D > 0$:
$4(p-1)^2 - 8(p-1) > 0$
Factor out $4(p-1)$:
$4(p-1)[(p-1) - 2] > 0 \Rightarrow 4(p-1)(p-3) > 0$
The critical points are $p=1$ and $p=3$. Testing intervals:
For $p 0$ (valid)
For $1<p<3$, say $p=2$: $4(1)(-1)=-43$, say $p=4$: $4(3)(1)=12>0$ (valid)
Also, $p \neq 1$ to keep it quadratic. Thus solution is $p 3$.

Key Concept: Finding Unknown Coefficient

b) $p = \frac{2}{3}$
[Solution Description]
Since $x = 2$ is a root of the equation, substitute it into the equation:
$2^2 + (p - 3)(2) + p = 0 \implies 4 + 2p - 6 + p = 0 \implies 3p - 2 = 0 \implies 3p = 2 \implies p = \frac{2}{3}$
Therefore, the value of $p$ is $\frac{2}{3}$.

Key Concept: Value of Parameter for Equal Roots

b) $k = 8$
[Solution Description]
For a quadratic equation $ax^2 + bx + c = 0$ to have equal roots, the discriminant must be zero ($D = 0$). Here, $a = 2$, $b = -k$, and $c = 8$. Setting the discriminant to zero:
$D = (-k)^2 - 4(2)(8) = 0 \implies k^2 - 64 = 0 \implies k^2 = 64 \implies k = \pm 8$
Thus, the possible values of $k$ are $8$ or $-8$.

Key Concept: Discriminant and Nature of Roots

d) Roots are real and equal
[Solution Description]
The discriminant $D$ for the quadratic equation $ax^2 + bx + c = 0$ is given by $D = b^2 - 4ac$. For the given equation $x^2 - 6x + 9 = 0$, we have $a = 1$, $b = -6$, and $c = 9$. Calculating the discriminant:
$D = (-6)^2 - 4(1)(9) = 36 - 36 = 0$
Since $D = 0$, the roots are real and equal.

Key Concept: Identifying Quadratic Equations

b) $3x^2 - 7x + 2 = 0$
[Solution Description]
A quadratic equation in standard form is written as $ax^2 + bx + c = 0$, where $a \neq 0$. Let us verify each option:
Option A: $x^3 + 2x + 1 = 0$ → Not quadratic (highest power is 3).
Option B: $3x^2 - 7x + 2 = 0$ → Quadratic in standard form.
Option C: $4x + 5 = 0$ → Linear equation.
Option D: $2x^2 = 5x$ → Can be rewritten as $2x^2 - 5x = 0$, which is quadratic.
However, only Option B is already in standard form.

Key Concept: Nature of Roots Based on Discriminant

c) Imaginary
[Solution Description]
The discriminant $D = b^2 - 4ac$ determines the nature of the roots:
1. If $D > 0$: Real and unequal roots.
2. If $D = 0$: Real and equal roots.
3. If $D < 0$: Imaginary (not real) roots.
Here, $D < 0$, so the roots are not real.

Key Concept: Discriminant and Nature of Roots

b) 1
[Solution Description]
To find the discriminant $D$ of the quadratic equation $ax^2 + bx + c = 0$, use the formula $D = b^2 - 4ac$. For the equation $2x^2 - 5x + 3 = 0$, we identify:
$a = 2, \, b = -5, \, c = 3$
Now, calculate $D$:
$D = (-5)^2 - 4 \times 2 \times 3 = 25 - 24 = 1$
The discriminant $D$ is $1$.

Key Concept: Trigonometric Equations, Periodicity

c) 4
[Solution Description]
To find solutions for $\sin(2x) = \frac{1}{2}$ in $[0, 2\pi]$:
1. Find all angles θ where $\sin(\theta) = \frac{1}{2}$. These are $\theta = \frac{\pi}{6} + 2\pi n$ and $\theta = \frac{5\pi}{6} + 2\pi n$ for any integer n.
2. Since $2x = \theta$, we get $x = \frac{\pi}{12} + \pi n$ and $x = \frac{5\pi}{12} + \pi n$.
3. Within $[0, 2\pi]$, the solutions are:
- For n=0: $\frac{\pi}{12}, \frac{5\pi}{12}$
- For n=1: $\frac{\pi}{12}+\pi = \frac{13\pi}{12}, \frac{5\pi}{12}+\pi = \frac{17\pi}{12}$
- For n=2: would exceed $2\pi$
4. Total distinct solutions: 4.

Key Concept: Quadratic Equations, Discriminant

d) $b^2 - 4ac = 0$
[Solution Description]
A quadratic equation has exactly one real solution when its discriminant is zero. The discriminant D is given by $D = b^2 - 4ac$. For one real solution: $b^2 - 4ac = 0$.

Key Concept: Linear Systems, Determinants

c) 6
[Solution Description]
For a system of linear equations to have infinitely many solutions, the two equations must be proportional. This means the ratio of coefficients must be equal. For the first equation $2x + 3y = 7$ and second equation $4x + ky = 14$, we need:
1. Ratio of x-coefficients: $\frac{2}{4} = \frac{1}{2}$
2. Ratio of y-coefficients: $\frac{3}{k}$ must also equal $\frac{1}{2}$
3. Ratio of constants: $\frac{7}{14} = \frac{1}{2}$
Solving $\frac{3}{k} = \frac{1}{2}$ gives $k = 6$.

Key Concept: Verification of Solution

c) $x = 3$
[Solution Description]
Step 1: Solve $x^2 - 4x + 3 = 0$ → Factorize as $(x - 1)(x - 3) = 0$ → $x = 1$ or $x = 3$.
Step 2: Substitute $y = 1$ into $x + y = 4$ → $x + 1 = 4$ → $x = 3$.
Only $x = 3$ satisfies both equations simultaneously.

Key Concept: Substitution Method

b) $x^2 + x - 12 = 0$
[Solution Description]
Substitute $y = x + 1$ into $x^2 + y^2 = 25$:
Step 1: Replace $y$ → $x^2 + (x + 1)^2 = 25$.
Step 2: Expand $(x + 1)^2$ → $x^2 + x^2 + 2x + 1 = 25$.
Step 3: Combine like terms → $2x^2 + 2x + 1 = 25$.
Step 4: Subtract 25 from both sides → $2x^2 + 2x - 24 = 0$.
Step 5: Divide by 2 → $x^2 + x - 12 = 0$. This is the simplified quadratic equation.

Key Concept: Root Verification

b) (2, 3)
[Solution Description]
To verify if a pair $(a, b)$ satisfies the quadratic equation $x^2 - 5x + 6 = 0$, substitute $x = a$ and $x = b$ into the equation and check if it holds true.
Step 1: Substitute $x = 2$ → $(2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$.
Step 2: Substitute $x = 3$ → $(3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0$.
Thus, $(2, 3)$ is a valid pair. Other options must be checked similarly but do not satisfy the equation.

Key Concept: Quadratic Formula

b) $1$
[Solution Description]
The discriminant $D$ determines the number of real roots.
For $x^2 - 4x + 4 = 0$, $D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$.
Since $D = 0$, there is exactly one real root.

Key Concept: Finding Unknown Coefficients

a) $1$
[Solution Description]
Substitute $x = 1$ into the equation:
$k(1)^2 - 3(1) + 2 = 0$.
Simplify: $k - 3 + 2 = 0$.
Solve for $k$: $k - 1 = 0$, so $k = 1$.

Key Concept: Verification of Roots

b) $2$
[Solution Description]
To determine if a value is a root, substitute it into the equation and check for equality.
Substitute $x = 2$: $(2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$.
Hence, $x = 2$ is a root.

Key Concept: Combined concepts: Discriminant and parameter conditions

b) $-4 - 2\sqrt{2} < k < -4 + 2\sqrt{2}$
[Solution Description]
For the quadratic equation to have no real roots, the discriminant must be negative ($D < 0$). Here, $a = k + 1$, $b = 3k + 2$, and $c = 2k - 1$. The discriminant is:
$D = b^2 - 4ac = (3k + 2)^2 - 4(k + 1)(2k - 1)$
Expanding and simplifying:
$D = 9k^2 + 12k + 4 - 4(2k^2 - k + 2k - 1) = 9k^2 + 12k + 4 - 8k^2 - 4k + 4 = k^2 + 8k + 8$
For $D < 0$:
$k^2 + 8k + 8 < 0$
Solving the inequality, we find the roots of the equation $k^2 + 8k + 8 = 0$ using the quadratic formula:
$k = \frac{-8 \pm \sqrt{64 - 32}}{2} = \frac{-8 \pm \sqrt{32}}{2} = \frac{-8 \pm 4\sqrt{2}}{2} = -4 \pm 2\sqrt{2}$
The parabola opens upwards, so $D < 0$ between the roots:
$-4 - 2\sqrt{2} < k < -4 + 2\sqrt{2}$
This is the range of $k$ for which the original equation has no real roots.

Key Concept: Discriminant analysis, Nature of roots

b) $p 0$). Here, $a = 2$, $b = -5$, and $c = p$. The discriminant is:
$D = b^2 - 4ac = (-5)^2 - 4(2)(p) = 25 - 8p$
For distinct real roots:
$25 - 8p > 0 \implies 8p < 25 \implies p < \frac{25}{8}$
Thus, the range of $p$ for distinct real roots is all real numbers less than $\frac{25}{8}$.

Key Concept: Discriminant analysis, Equal roots

c) Both $k = 12$ and $k = -12$
[Solution Description]
For the quadratic equation $ax^2 + bx + c = 0$ to have equal roots, the discriminant must be zero ($D = 0$). Here, $a = 4$, $b = k$, and $c = 9$. The discriminant is calculated as:
$D = b^2 - 4ac = k^2 - 4(4)(9) = k^2 - 144$
Setting $D = 0$ gives:
$k^2 - 144 = 0 \implies k^2 = 144 \implies k = \pm 12$
Therefore, the values of $k$ are $\pm 12$.

Key Concept: Distinct and Real Roots

c) $-20 < p < 20$
[Solution Description]
For no real roots, the discriminant must be less than zero. The discriminant $D$ for the given equation $5x^2 + px + 20 = 0$ is calculated as:
$D = b^2 - 4ac = p^2 - 4(5)(20) = p^2 - 400$
For no real roots, $D < 0$:
$p^2 - 400 < 0 \implies p^2 < 400 \implies -20 < p < 20$
Therefore, $p$ must lie between $-20$ and $20$.

Key Concept: Finding the Value for Equal Roots

c) $m = 4.5$
[Solution Description]
For the roots to be equal, the discriminant must be zero. The discriminant $D$ for the given equation $2x^2 - 6x + m = 0$ is calculated as:
$D = b^2 - 4ac = (-6)^2 - 4(2)(m) = 36 - 8m$
For equal roots, $D = 0$:
$36 - 8m = 0 \implies 8m = 36 \implies m = \frac{36}{8} = 4.5$
Therefore, $m$ must be $4.5$.

Key Concept: Nature of Roots Based on Discriminant

c) k < 4
$16 - 4k > 0 \implies 4k < 16 \implies k < 4$
Therefore, $k$ must be less than 4.

Key Concept: Application in Problems

c) 2
[Solution Description]
The discriminant $D$ for the equation $x^2 + 6x + 9 = 0$ is:
$D = (6)^2 - 4 \times 1 \times 9 = 36 - 36 = 0$
Since $D = 0$, the equation has two equal real roots.

Key Concept: Finding Conditions for Equal Roots

c) 4.5
[Solution Description]
For the equation to have equal roots, the discriminant must be zero:
$D = b^2 - 4ac = 0$
Substituting the values from the equation $2x^2 + 6x + k = 0$:
$6^2 - 4 \times 2 \times k = 0 \implies 36 - 8k = 0 \implies k = \frac{36}{8} = 4.5$
Therefore, $k = 4.5$.

Key Concept: Nature of Roots Based on Discriminant

d) Two equal real roots
[Solution Description]
The discriminant $D$ for the quadratic equation $x^2 - 4x + 4 = 0$ is calculated as:
$D = b^2 - 4ac = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$
Since $D = 0$, the equation has two equal real roots.

Key Concept: Quadratic Equations, Substitution Method

b) 6
[Solution Description]
Square both sides to eliminate the square root:
$x + 3 = (x - 3)^2$
Expand the right side:
$x + 3 = x^2 - 6x + 9$
Rearrange to standard form:
$x^2 - 7x + 6 = 0$
Factorize:
$(x - 1)(x - 6) = 0$
So, $x = 1$ or $x = 6$. Check for extraneous solutions:
For $x = 1$: $\sqrt{1 + 3} = 1 - 3$ → $2 \neq -2$ (invalid).
For $x = 6$: $\sqrt{6 + 3} = 6 - 3$ → $3 = 3$ (valid).
Only $x = 6$ is valid.

Key Concept: Quadratic Equations, Quadratic Formula

d) Both a) and c)
[Solution Description]
Set $h = 15$ in the equation:
$20t - 5t^2 = 15$
Rearrange to standard form:
$-5t^2 + 20t - 15 = 0$
Divide by $-5$:
$t^2 - 4t + 3 = 0$
Solve using the quadratic formula:
$t = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2}$
Thus, $t = 3$ or $t = 1$ seconds.

Key Concept: Quadratic Equations, Nature of Roots

a) 3 m and 8 m
[Solution Description]
Let the width be $w$ meters. Then, the length is $(2w + 2)$ meters. The area is given by:
$w(2w + 2) = 24$
Simplify the equation:
$2w^2 + 2w - 24 = 0$
Divide by 2:
$w^2 + w - 12 = 0$
Factorize:
$(w + 4)(w - 3) = 0$
So, $w = -4$ or $w = 3$. Since width cannot be negative, $w = 3$ meters. The length is $2(3) + 2 = 8$ meters.

Key Concept: Word Problem - Sum of Roots

b) 5
[Solution Description]
For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by $\frac{-b}{a}$. In the given equation, $a = 1$ and $b = -5$. Therefore:
$\text{Sum of roots} = \frac{-(-5)}{1} = 5$
This matches the given condition. However, to find $k$, we use another property of quadratic equations: the product of the roots is $\frac{c}{a} = k$. But since we do not have additional information, we observe that the problem asks for $k$. Given the options, the correct value can be inferred from the standard form, where $k$ represents the constant term $c$.

Key Concept: Word Problem - Quadratic Formula

b) 4.5 meters
[Solution Description]
Let the width be $w$ meters. Then, the length is $2w + 3$ meters. The area of the rectangle is given by length times width:
$w(2w + 3) = 54$
Expand and rewrite the equation in standard quadratic form:
$2w^2 + 3w - 54 = 0$
Use the quadratic formula to solve for $w$:
$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Here, $a = 2$, $b = 3$, and $c = -54$. Substitute these values into the formula:
$w = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-54)}}{2(2)}$
$w = \frac{-3 \pm \sqrt{9 + 432}}{4}$
$w = \frac{-3 \pm \sqrt{441}}{4}$
$w = \frac{-3 \pm 21}{4}$
We discard the negative root because width cannot be negative:
$w = \frac{18}{4} = 4.5 \text{ meters}$

Key Concept: Word Problem - Finding Roots

d) 11
[Solution Description]
Let the smaller integer be $x$. Then, the next consecutive odd integer is $x + 2$. According to the problem, their product is 99:
$x(x + 2) = 99$
Expand and rewrite the equation in standard quadratic form:
$x^2 + 2x - 99 = 0$
Factorize the quadratic equation:
$(x + 11)(x - 9) = 0$
Solve for $x$:
$x = -11 \text{ or } x = 9$
Since the integers are positive, we take $x = 9$. The greater integer is $x + 2 = 11$.

Key Concept: Nature of Roots

b) 1
[Solution Description]
For a quadratic equation $ax^2 + bx + c = 0$, the discriminant $D = b^2 - 4ac$. Here, $a = 2$, $b = -5$, and $c = 3$. Substituting these values gives:
$D = (-5)^2 - 4(2)(3)$
$D = 25 - 24$
$D = 1$.

Key Concept: Steps to Solve Word Problems

c) 12
[Solution Description]
Substitute $x = 3$ into the equation to find $k$:
$(3)^2 - 7(3) + k = 0$
$9 - 21 + k = 0$
$-12 + k = 0$
Thus, $k = 12$.

Key Concept: Introduction to Quadratic Equations in Word Problems

c) 6 and 4
[Solution Description]
Let the numbers be $x$ and $10 - x$. According to the problem, $x(10 - x) = 24$. Simplifying, we get:
$10x - x^2 = 24$
Rearranging terms gives:
$-x^2 + 10x - 24 = 0$
Multiply by $-1$:
$x^2 - 10x + 24 = 0$
Factorize:
$(x - 6)(x - 4) = 0$
So, the numbers are $x = 6$ and $x = 4$.

Key Concept: Shape of the Graph, Axis of Symmetry, Vertex

c) $y = (x + 3)^2$
[Solution Description]
Step 1: Axis of symmetry is $x = -3$, so vertex form is $y = a(x + 3)^2 + k$.
Step 2: Since it opens upwards, $a > 0$.
Step 3: Substitute point $(-1, 4)$ into the equation: $4 = a(-1 + 3)^2 + k = 4a + k$.
Step 4: Possible solutions are infinite, but we can test given options. Let's assume $a = 1$, then $k = 0$.
Step 5: Equation becomes $y = (x + 3)^2$, which satisfies all conditions.

Key Concept: Shape of the Graph, Discriminant

c) 8
[Solution Description]
Step 1: For the graph to touch x-axis at one point, discriminant $D = 0$.
Step 2: Given $a = 2$, $b = -8$, $c = k$, compute discriminant: $D = b^2 - 4ac = (-8)^2 - 4(2)(k) = 64 - 8k$.
Step 3: Set $D = 0$: $64 - 8k = 0$.
Step 4: Solve for $k$: $k = \frac{64}{8} = 8$.

Key Concept: Shape of the Graph, Vertex, Axis of Symmetry

b) Opens downwards, vertex at (2, 4)
[Solution Description]
Step 1: Identify coefficients $a = -3$, $b = 12$, $c = -8$.
Step 2: Since $a < 0$, the parabola opens downwards.
Step 3: Calculate vertex using formula $\left( -\frac{b}{2a}, c - \frac{b^2}{4a} \right)$.
Step 4: Compute x-coordinate of vertex: $-\frac{12}{2 \times (-3)} = 2$.
Step 5: Compute y-coordinate of vertex: $-8 - \frac{(12)^2}{4 \times (-3)} = -8 + \frac{144}{12} = -8 + 12 = 4$.
Step 6: Vertex is at $(2, 4)$.

Key Concept: Graph Shape and Direction

d) Narrow and opens downwards
[Solution Description]
For the quadratic equation $y = ax^2 + bx + c$, if $a < 0$, the parabola opens downwards. A larger absolute value of $a$ makes the parabola narrower. Thus, the correct description is a narrow parabola opening downwards.

Key Concept: Nature of Roots

c) Never
[Solution Description]
To determine the number of x-intercepts, calculate the discriminant:
$D = b^2 - 4ac = (4)^2 - 4(-2)(-3) = 16 - 24 = -8$
Since $D < 0$, there are no real roots. Therefore, the graph does not intersect the x-axis.

Key Concept: Vertex and Axis of Symmetry

a) Vertex at $(1, -1)$, axis of symmetry is $x = 1$
[Solution Description]
To find the vertex, use the formula for the x-coordinate of the vertex:
$x = -\frac{b}{2a} = -\frac{-6}{2 \times 3} = \frac{6}{6} = 1$
Now, substitute $x = 1$ into the equation to find the y-coordinate:
$y = 3(1)^2 - 6(1) + 2 = 3 - 6 + 2 = -1$
So, the vertex is $(1, -1)$. The axis of symmetry is the vertical line passing through the vertex:
$x = 1$

Key Concept: Graph Shape Identification

c) The graph shows an S-shaped curve that represents the growth pattern.
[Solution Description]
A cubic function like $y = x^3$ has an S-shaped curve, known as a cubic curve. It starts from negative infinity, passes through the origin, and extends to positive infinity, forming a smooth S-shape. Thus, the correct answer is "S-shaped curve."

Key Concept: Graph Shape Identification

d) Parabola
[Solution Description]
A quadratic function has the standard form $y = ax^2 + bx + c$. Its graph is called a parabola, which is a U-shaped or inverted U-shaped curve depending on the sign of $a$. Therefore, the correct answer is "Parabola."

Key Concept: Graph Shape Identification

d) Straight line
[Solution Description]
A linear function has the general form $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. The graph of a linear function is always a straight line. Therefore, the correct answer is "Straight line."

Key Concept: Direction of Parabola

b) Downwards
[Solution Description]
The direction in which the parabola opens depends on the coefficient of $x^2$ (i.e., $a$). Here, $a = -3$. Since $a < 0$, the parabola opens downwards.

Key Concept: Roots of Quadratic Equation

b) One real root
[Solution Description]
The number of real roots can be determined using the discriminant $\Delta = b^2 - 4ac$. Here, $a = 1$, $b = -6$, and $c = 9$. Calculating the discriminant:
$\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0.$
Since $\Delta = 0$, the equation has exactly one real root (a repeated root).

Key Concept: Vertex of Quadratic Equation

b) $(2, -3)$
[Solution Description]
To find the vertex of the parabola, we use the formula for the x-coordinate of the vertex: $x = -\frac{b}{2a}$. Here, $a = 2$ and $b = -8$. Substituting, we get:
$x = -\frac{-8}{2 \times 2} = \frac{8}{4} = 2.$
To find the y-coordinate, substitute $x = 2$ back into the equation:
$y = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3.$
Thus, the vertex is at $(2, -3)$.

Key Concept: Roots of the Quadratic Equation

c) Real and equal roots
[Solution Description]
The discriminant $D$ is calculated as $b^2 - 4ac = (-6)^2 - 4(1)(9) = 36 - 36 = 0$. Since $D = 0$, the equation has real and equal roots.

Key Concept: Y-intercept

b) $(0, 5)$
[Solution Description]
The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation gives $y = (0)^2 - 3(0) + 5 = 5$. Thus, the y-intercept is $(0, 5)$.

Key Concept: Shape of the Parabola

b) Downwards
[Solution Description]
The coefficient of $x^2$ is $-2$, which is less than zero ($a < 0$). Therefore, the parabola opens downwards.

Key Concept: Area Problems, Quadratic Equations

d) No valid solution
[Solution Description]
First, factorize $x^2 + 9x + 20 = 0$ to get $(x+4)(x+5)=0$, so $x=-4$ or $x=-5$. Discard negative values. Let width be $y$, then length is $2y+2$. Area: $y(2y+2) = x^2 + 9x + 20$. However, since we already have roots, actual dimensions must correspond to valid positive solutions. This indicates inconsistency; thus, re-evaluate the problem setup. Correct interpretation: if $x^2 + 9x + 20$ represents area, then solving it gives non-realistic dimensions, suggesting no valid solution exists under given conditions.

Key Concept: Right-Angled Triangles, Pythagorean Theorem

c) 12 cm
[Solution Description]
Let the legs be $x$ and $x-7$. Using the Pythagorean theorem: $x^2 + (x-7)^2 = 13^2$. Simplify to $2x^2 - 14x + 49 = 169$. Then $2x^2 - 14x - 120 = 0$. Divide by 2: $x^2 - 7x - 60 = 0$. Solve using quadratic formula: $x = \frac{7 \pm \sqrt{289}}{2}$. Only positive root: $x = \frac{7+17}{2} = 12$ cm.