class 9 maths chapter 2

Key Concept: Reaction Analysis, Reaction Mechanism

b) Sulfuric acid
[Solution Description]
The balanced chemical equation is:$MgCO_3 + H_2SO_4 \rightarrow MgSO_4 + H_2O + CO_2$
Moles of $MgCO_3$: $\frac{50}{84.31} \approx 0.593$ $\text{Molar mass} = 24.31 + 12 + 16 \times 3 = 84.31 \, \text{g/mol}$

For sulfuric acid, $100\text{ ml}$ of $2M$ solution contains:$\left(\frac{100}{1000}\right) \times 2 = 0.2 \text{ mol}$

Since the ratio from the balanced equation is 1:1, $H_2SO_4$ is the limiting reagent as its moles $(0.2)$ are lesser than those of $MgCO_3$.

Notes:

Key Concept: Complex Equation, Reaction Prediction

a) 1
[Solution Description]
Balance the equation by equating the number of atoms for each element on both sides:

$K_2CO_3 + 2HNO_3 \rightarrow 2KNO_3 + H_2O + CO_2$
Here, two H atoms from $$2HNO_3$$ form one molecule of $H_2O$. Hence, the coefficient for water is 1.

Notes:

• Reaction of Acids with Bases (Neutralization): Acid + Base → Salt + Water. Example:
  HCl+NaOH→NaCl+H2O
• Metal Carbonates/Bicarbonates with Acids: Produces salt, water, and
  CO2 Example: NaHCO3+HCl→NaCl+H2O+CO2 
• Predicting Reactions:
  • Use reactivity series to predict displacement reactions.
  • Double displacement occurs when ions exchange, forming precipitates or gas.
• Acids React with Metals: Produces hydrogen gas. Example:
  Zn+H2SO4→ZnSO4+H2
• Complex Equations: Identify the nature of reactants (acid/base/salt) and predict outcomes based on solubility and product stability.

Click Here To Download Notes

Key Concepts: Complex Reaction Analysis, Real-World Application

c) 98 g
[Solution Description] The balanced chemical equation for the reaction of calcium hydroxide with sulfuric acid is:
$\text{Ca(OH)}_2(s) + \text{H}_2\text{SO}_4(aq) \rightarrow \text{CaSO}_4(s) + 2\text{H}_2\text{O(l)}$
From stoichiometry, 1 mole of $\text{Ca(OH)}_2 $ reacts with 1 mole of $ \text{H}_2\text{SO}_4 $. Given that 74 g of $ \text{Ca(OH)}_2 $ equals 1 mole, then 1 mole of $ \text{H}_2\text{SO}_4 $ was neutralized, which weighs:$1 \, \text{mol} \times 98 \, \frac{\text{g}}{\text{mol}} = 98 \, \text{g}$

Notes:

• Complex Reactions: Acids and bases often react with each other in neutralization reactions to form salts and water.
• Real-World Reactions:
  • Titration: Used in laboratories to determine the concentration of unknown solutions.
  • Antacid Tablets: Neutralize stomach acid, providing relief from acidity.
  • Cleaning Agents: Bases like sodium hydroxide are used in cleaning agents, while acids like vinegar remove stains.
• Buffer Solutions: Maintain pH balance in biological systems like blood.
• Environmental Applications: Acid rain affects soil and water bodies; limestone neutralizes acidity in lakes.

Click Here To Download Notes

Key Concept: Complex Reaction Analysis, Reaction Heat

b) 2.865 kJ
[Solution Description] The balanced neutralization reaction between HCl and NaOH can be written as:
$\text{HCl(aq)} + \text{NaOH(aq)} \rightarrow \text{NaCl(aq)} + \text{H}_2\text{O(l)}$
Since both solutions are in equal concentrations and volumes, they completely neutralize each other. The number of moles of HCl or NaOH is given by:$n = M \times V = 1\, \text{mol/L} \times 0.05\, \text{L} = 0.05\, \text{mol}$
Hence, the heat released is calculated using the formula:$q = n \times \Delta H = 0.05\, \text{mol} \times (-57.3\, \text{kJ/mol}) = -2.865\, \text{kJ}$
Therefore, the heat released is 2.865 kJ.

Notes:

Complex Reaction Analysis & Heat of Reactions

• Complex Reactions: Involves multiple steps where reactants undergo intermediate stages before reaching final products.
• Exothermic Reactions: Release energy in the form of heat. Example: Neutralization of an acid with a base.
• Endothermic Reactions: Absorb energy from the surroundings. Example: Dissolving ammonium nitrate in water.
• Heat of Reaction: The heat energy released or absorbed during a chemical reaction.
  • Enthalpy Change (ΔH): Measures the heat change at constant pressure.
  • Reaction Types: Exothermic (-ΔH), Endothermic (+ΔH).

Click Here To Download Notes

Key Concept: Observational Skills

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] When copper oxide (CuO) is added to hydrochloric acid (HCl), it reacts to form copper(II) chloride $\text{CuCl}_2$
and water $H_2O$ The chemical equation for the reaction is $CuO + 2HCl \rightarrow CuCl_2 + H_2O$. The formation of copper(II) chloride, which has a distinct blue-green color, is responsible for the change observed in the color of the solution. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Click To Download Notes

Key Concept: Basic Reaction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is stating the general outcome of the reaction between metal oxides and acids, which indeed results in the formation of salt and water. On the other hand, the reason explains that this process resembles a neutralization reaction, where a base reacts with an acid resulting in the formation of salt and water—a characteristic feature of neutralization reactions. Therefore, both statements are true, and the Reason correctly explains the Assertion.

Click To Download Notes

Key Concept: Real-world Application, Multi-step Solutions

b) Both gases react with $Ca(OH)_2$ forming solids
[Solution Description]
Limewater $(Ca(OH)_2)$ can be used to scrub both $CO_2$ and $SO_2$ from emissions by forming insoluble salts.

For $CO_2$:$Ca(OH)_2 + CO_2 \rightarrow CaCO_3 + H_2O$

For $SO_2$:$Ca(OH)_2 + SO_2 \rightarrow CaSO_3 + H_2O$

Both reactions produce precipitates $CaCO_3$ and $CaSO_3$, removing gaseous pollutants from the emissions effectively. Implementing large-scale spray towers or packed bed reactors ensures intimate contact between the gas stream and limewater, facilitating pollution control in industrial setups.

Notes:

• Antacid Tablets (Bases): Used to neutralize excess stomach acid (HCl), providing relief from acidity.
• Soaps and Detergents (Salts): Soaps are made from fatty acids and sodium hydroxide, which help clean by removing oils and dirt.
• Preservation of Food (Salts): Sodium chloride (table salt) is used to preserve food by inhibiting bacterial growth.
• Agriculture (Fertilizers): Fertilizers like ammonium sulfate (acidic) and calcium phosphate (basic) adjust soil pH for optimal plant growth.
• Cleaning Products (Acids and Bases): Vinegar (acetic acid) and baking soda (sodium bicarbonate) are used in cleaning due to their neutralizing properties.

Click Here To Download Notes

Key Concept: Reaction Mechanism, Conceptual Understanding

d) It acts as an acid forming $Na_2SO_3$ and $H_2O$
[Solution Description]
To understand this reaction mechanism, we start by recognizing that $SO_2$ is a non-metallic oxide and behaves as an acid when dissolved in water, forming sulfurous acid $(H_2SO_3)$. The balanced chemical reaction with sodium hydroxide is:$SO_2 + 2NaOH \rightarrow Na_2SO_3 + H_2O$

The $SO_2$ dissolves in water to form sulfurous acid:$SO_2 + H_2O \rightarrow H_2SO_3$

This sulfurous acid reacts with the base (sodium hydroxide) to form the salt (sodium sulfite) and water:$H_2SO_3 + 2NaOH \rightarrow Na_2SO_3 + 2H_2O$

Overall, $SO_2$ acts as an acid and neutralizes the base NaOH, creating $Na_2SO_3$ and $H_2O$.

Notes:

• Reaction Mechanism involves the step-by-step process by which reactants are converted into products.
• Acid-Base Reactions: Acids donate protons (H⁺), and bases accept protons (OH⁻).
• Neutralization Reaction: An acid reacts with a base to form salt and water.
• Indicator: Substances like litmus paper, phenolphthalein, and methyl orange show the pH of a solution.
• Salts: Formed when an acid reacts with a base. Example: NaCl from HCl and NaOH.
• Strong & Weak Acids/Bases: Strong acids/bases ionize completely, while weak ones only partially ionize.
• Buffer Solutions: Maintain a stable pH despite adding an acid or base.

Click Here To Download Notes

Key Concept: Comparative Analysis, Advanced Reasoning

b) Solution Y is weaker as a base than X is as an acid
[Solution Description] A pH of 1 indicates a very strong acid since its $H^+$ concentration is quite high, while a pH of 11 indicates a strong base due to a low $H^+$ concentration and high $OH^-$ concentration. From this information, solution X is a strong acid, and solution Y is a strong base. Comparatively, solution X is stronger as an acid than solution Y is as a base.

Click To Download Notes

Key Concept: Basic Properties

c) $OH^-$
[Solution Description]
The dissolution of NaOH in water produces $Na^+(aq)$ and $OH^-(aq)$ ions. The $OH^-$ ion is responsible for the basic properties.

Notes:

• Acids:
  • Sour taste (e.g., lemon, vinegar).
  • Turn blue litmus paper red.
  • Release hydrogen ions (H⁺) in water.
  • Conduct electricity (electrolytes).
  • React with metals (e.g., zinc) to produce hydrogen gas.
• Bases:
  • Bitter taste, slippery feel.
  • Turn red litmus paper blue.
  • Release hydroxide ions (OH⁻) in water.
  • Conduct electricity.
  • React with acids to form salts and water (neutralization).
• Salts:
  • Formed by the reaction between acids and bases.
  • Generally neutral in nature.
  • Conduct electricity in molten or aqueous form.

Click Here To Download Notes

Key Concept: Base Conductivity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that sodium hydroxide NaOH) solution conducts electricity, which is true because NaOH dissociates in water to produce $Na^+$ and $OH^-$ ions. These ions are charge carriers, allowing the solution to conduct electricity. The reason given also correctly explains why this conductivity occurs by highlighting the ion formation due to dissociation. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Click To Download Notes

Key Concept: Advanced Reaction Analysis, Conceptual Understanding

b) Formation of $NH_4NO_3$, increasing conductivity
[Solution Description] Mixing $HNO_3$ (nitric acid) with $NH_3$ (ammonia) results in the formation of $NH_4NO_3$, a soluble salt. This reaction increases the number of ions $(NH_4^+)$ and $(NO_3^-)$ in the solution, thereby increasing conductivity.

Click To Download Notes

Key Concept: Real-world Application, Process Implications

b) Molarity decreases, volume increases
[Solution Description] Using the formula $M_1V_1 = M_2V_2$, where $M_1$ is 6 M and $M_2$ is 1 M, the initial volume $V_1$ is altered by adding sufficient water to increase the total volume $(V_2)$ such that the concentration meets the desired 1 M standard. This significantly increases the volume while lowering the molarity.

Click To Download Notes

Key Concept: Safety Precaution

b) Add acid to water
[Solution Description] To safely dilute sulphuric acid, it should always be added to water slowly while stirring continuously. This helps to prevent splashing and reduces the risk of burns from the exothermic reaction.

Notes:

• Wear Safety Gear: Always wear lab coat, safety goggles, and gloves to protect from chemicals and physical hazards.
• Handle Chemicals Carefully: Read labels before use; avoid contact with skin and eyes.
• Proper Ventilation: Ensure good air circulation, especially when working with toxic substances.
• Use Equipment Properly: Familiarize yourself with equipment before use; handle with care.
• No Food or Drink: Never consume food or drink in the lab to avoid contamination.
• Fire Safety: Keep fire extinguishers nearby and know how to use them.
• Report Accidents: Immediately report any spills, accidents, or injuries to the teacher.
• Dispose Waste Properly: Dispose of chemicals and materials as per instructions.

Click Here To Download Notes

Key Concept: pH and Biological Systems, Industrial Applications

b) Reduced microbial activity, slowing down waste degradation.
[Solution Description] A pH drop below 5.0 creates an overly acidic environment, hindering the microorganisms responsible for breaking down organic matter. Their enzymatic activities slow down significantly, reducing the efficiency of the treatment process and leading to potential buildup of untreated waste.

Click To Download Notes

Key Concept: Acid Rain Impact

b) It lowers the pH, creating adverse conditions for aquatic survival.
[Solution Description]
Acid rain, with a pH less than 5.6, can lower the pH of river water, leading to an acidic environment. This alteration in pH makes it difficult for many aquatic species to survive as it disrupts biological functions and habitat conditions.

Click To Download Notes

Key Concept: pH and Everyday Life

c) Baking Soda Solution
[Solution Description] Basic substances have a pH greater than 7. Among common household items, baking soda solution typically has a pH around 8-9 making it slightly basic compared to other options such as vinegar and lemon juice, which are acidic.

Click To Download Notes

Key Concept: pH and Buffer Systems, pH and Soil Health

c) Assertion is true, but Reason is false.
[Solution Description]
In this question, we need to evaluate the truthfulness of both the assertion and reason and determine if the reason correctly explains the assertion.

Firstly, analyzing the Assertion: Buffer solutions are indeed used to maintain stable pH levels in various environments, including soil, to ensure that plants have an optimal growing environment. This is true because certain pH levels are essential for nutrient availability and microbial activity critical for plant health.

Now, evaluating the Reason: While it is true that the pH of a buffer depends on the concentration ratio of its conjugate acid-base pair, the statement that it is not affected by changes in temperature is false. Temperature can affect the dissociation constants $K_a$ and $K_b$ of acids and bases in the buffer, thus altering the pH balance indirectly.

Therefore, while the assertion is true, the reason provided is incorrect as it fails to acknowledge the effect of temperature changes.

Hence, the correct option is c) Assertion is true, but Reason is false.

Click To Download Notes

Key Concept: Universal Indicator Use

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a universal indicator can display various colors depending on the pH level of a solution, which is true. This is because a universal indicator is designed to provide an approximate pH value visually by exhibiting specific color changes across the pH scale.

The reason provided is that a universal indicator consists of a mixture of several indicators, each sensitive to certain pH ranges and contributing to the overall range of color changes observed in the universal indicator. This explanation is correct and directly related to why the assertion is true.

Thus, both the Assertion and the Reason are true, and the Reason correctly explains the Assertion.

Click To Download Notes

Key Concept: pH Sensitivity in Organisms, pH and Chemical Reactions

c) By reducing acidity, they inhibit the effectiveness of digestive enzymes.
[Solution Description] Antacids neutralize excess stomach acid, raising the pH level closer to neutrality. This can impact the stomach’s digestive efficiency, as certain enzymes require highly acidic conditions to function optimally. The reduction in acidity can hinder protein breakdown and absorption of nutrients like iron and calcium, thereby affecting overall nutrient uptake from food.

Click To Download Notes

Key Concept: Antacid Function

b) Neutralize excess acid
[Solution Description] Antacids are substances that neutralize excess stomach acid, relieving symptoms of heartburn and indigestion.

Click To Download Notes

Key Concept: Acid Rain Impact, pH in Soil

c) Reduces biodiversity and harms aquatic organisms
[Solution Description] Acid rain with a lower pH than normal rain can lead to several detrimental effects on the aquatic ecosystem. It results in increased water acidity, affecting aquatic life by leaching toxic metals such as aluminum from soils into rivers and lakes, reducing biodiversity, and harming organisms that require specific pH levels to survive.

Click To Download Notes

Key Concept: Reaction Analysis

a) Calcium sulfate, carbon dioxide, and water
[Solution Description] The reaction between calcium carbonate $(CaCO_3)$ and sulfuric acid $H_2$ $SO_4$ follows the equation:
$\text{CaCO}_3 + \text{H}_2\text{SO}_4 \rightarrow \text{CaSO}_4 + \text{CO}_2 + \text{H}_2\text{O}$
Hence, the products are calcium sulfate, carbon dioxide, and water.

Click To Download Notes

Key Concept: pH and Composition

b) Strong acid and weak base
[Solution Description] Ammonium chloride is formed from the neutralization of a strong acid (hydrochloric acid, HCl) and a weak base ammonia, $(NH_3)$ This results in an acidic salt solution with pH < 7. Click To Download Notes

Key Concept: Salt Formation

c) Assertion is true, but Reason is false.
[Solution Description] Sodium acetate $(CH_3COONa)$ is indeed a basic salt because it is derived from the reaction of acetic acid $(CH_3COOH)$, which is a weak acid, and sodium hydroxide NaOH, which is a strong base. The pH of a solution containing sodium acetate is greater than 7 due to hydrolysis of the acetate ion. However, the reason provided incorrectly states that basic salts are produced from strong acids and weak bases, when in fact, they result from the combination of strong bases and weak acids. Therefore, the assertion is true, but the reason is false.

Click To Download Notes

Key Concept: Salt and pH Correlation

a) Less than 7
[Solution Description] Sodium acetate is a salt formed from acetic acid (a weak acid) and sodium hydroxide (a strong base). Such salts generally have a basic nature with $pH > 7$. However, if this solution turns blue litmus paper red, it indicates an acidic medium suggesting contamination or incorrect labeling since sodium acetate normally doesn’t exhibit such behavior.

Notes:

• Salts and pH: Salts are formed when acids and bases react together. The nature of the salt depends on the strength of the acid and base involved.
• Types of Salts:
  • Neutral salts: Formed from strong acids and strong bases (e.g., NaCl). They do not affect pH.
  • Acidic salts: Formed from a strong acid and a weak base (e.g., NH₄Cl). They lower pH.
  • Basic salts: Formed from a weak acid and a strong base (e.g., Na₂CO₃). They raise pH.
• pH Change: The pH of a salt solution can be determined by its acidic or basic nature, influencing its effect on the environment.

Click Here To Download Notes

Key Concept: Neutralization Reaction Analysis, Salt pH Analysis

c) Ammonium nitrate; it makes the solution slightly acidic.
[Solution Description] Nitric acid $(\text{HNO}_3)$ is a strong acid, while ammonium hydroxide $(\text{NH}_4\text{OH})$ is a weak base. The reaction forms ammonium nitrate $(\text{NH}_4\text{NO}_3)$:$\text{HNO}_3 + \text{NH}_4\text{OH} \rightarrow \text{NH}_4\text{NO}_3 + \text{H}_2\text{O}$
Due to the strong acid and weak base, the salt will be slightly acidic and will lower the pH below 7.

Notes:

Neutralization Reaction

• A neutralization reaction occurs when an acid reacts with a base to form water and a salt.
• Example:
  HCl+NaOH→NaCl+H2OHCl + NaOH .
• The acidic and basic properties cancel each other out, resulting in a neutral solution (pH = 7).
• This reaction is exothermic, releasing heat.

Salt pH Analysis

• Salts formed in neutralization reactions can have different pH values based on the strength of the acid and base used.
  • Neutral Salts: Formed by strong acid + strong base (e.g., NaCl, KNO₃), pH = 7.
  • Acidic Salts: Formed by strong acid + weak base (e.g., NH₄Cl), pH < 7.
  • Basic Salts: Formed by weak acid + strong base (e.g., Na₂CO₃), pH > 7.

Click Here To Download Notes

Key Concept: Neutralization Reaction

d) Sodium chloride $(NaCl)$
[Solution Description] The neutralization reaction involves a strong acid (HCl) reacting with a strong base (NaOH). The general reaction for neutralization is:
$H^+(aq) + OH^-(aq) \rightarrow H_2O(l)$
For this specific case, we have:$HCl(aq) + NaOH(aq) \rightarrow NaCl(aq) + H_2O(l)$
Hence, sodium chloride (NaCl) is the salt formed.

Notes:

• Definition: A neutralization reaction occurs when an acid reacts with a base to form water and a salt.
• General Equation:
  Acid + Base → Salt + Water
• Example:
  HCl (acid) + NaOH (base) → NaCl (salt) + H₂O (water)
• Characteristics:
  • Heat is usually released in the reaction (exothermic).
  • pH of the solution becomes neutral (pH = 7).
• Applications:
  • Used in antacids to neutralize excess stomach acid.
  • Lime (CaO) is used to neutralize acidic soil.
• Important Note: Neutralization can be complete or partial depending on the strength of acid/base.

Click Here To Download Notes

Key Concept: Complex Reaction Pathways, Reaction Analysis

b) Sodium carbonate, water, and carbon dioxide
[Solution Description] Heating $(2NaHCO_3)$ leads to the formation of sodium carbonate $(Na_2CO_3)$, water $(H_2O)$, and carbon dioxide $(CO_2)$. This decomposition is significant in producing sodium carbonate used in glass manufacturing and as a detergent component.

Notes:

• Complex Reaction Pathways: Reactions that involve multiple steps or stages, where reactants go through intermediate states before forming the final products.
• Reaction Mechanism: The detailed step-by-step process by which reactants are converted into products.
• Elementary Reactions: Simple reactions occurring in a single step with a single transition state.
• Intermediate Compounds: Short-lived substances formed during the reaction that are not present in the final products.
• Rate Determining Step: The slowest step in a reaction mechanism that limits the overall reaction rate.
• Chain Reactions: Reactions where products from one step catalyze further steps, creating a cycle.
• Example: Photosynthesis involves multiple steps with intermediates like ATP and NADPH.

Click Here To Download Notes

Key Concept: Industrial Process Understanding, Complex Reaction Pathways

c) $NaCl \rightarrow NaHCO_3 \rightarrow Na_2CO_3 \rightarrow Na_2CO_3.10H_2O$
[Solution Description] The production of washing soda from common salt follows: $NaCl \rightarrow NaHCO_3 \rightarrow Na_2CO_3 \rightarrow Na_2CO_3.10H_2O$. This includes initial conversion to sodium bicarbonate and then to sodium carbonate before recrystallization into washing soda.

Notes:

Key Concept: Critical Thinking, Advanced Application

d) They help stabilize active ingredients in formulations.
[Solution Description]
Compounds with water of crystallization often stabilize active ingredients, which helps maintain drug efficacy and shelf life.

Notes:

• Critical Thinking involves analyzing, evaluating, and synthesizing information to form reasoned judgments.
• It is essential in solving complex problems, making decisions, and drawing conclusions based on evidence.
• Key skills:
  • Questioning assumptions
  • Identifying biases
  • Evaluating evidence
  • Recognizing patterns and inconsistencies
• In scientific research, critical thinking helps in formulating hypotheses, designing experiments, and interpreting results.
• Advanced Applications of critical thinking are seen in:
  • Innovative technology development
  • Medical advancements like new treatments
  • Environmental solutions to global challenges.
• It aids in adapting to new information and evolving understanding in various fields.

Click Here To Download Notes

Key Concept: Problem Solving, Critical Thinking

c) 36%
[Solution Description]
Determine the mass of the water lost:$\text{Water lost} = 25 – 16 = 9 \text{ grams}$
Then calculate the percentage of water:$\text{Percentage of Water} = \left(\frac{9}{25}\right) \times 100\% = 36\%$

Notes:

• Problem Solving: Involves identifying a problem, analyzing it, and finding a solution using logical steps.
• Critical Thinking: The ability to think clearly, logically, and objectively. It helps in evaluating information and arguments.
• Steps in Problem Solving:
  • Understanding the problem.
  • Breaking it into smaller parts.
  • Analyzing the components.
  • Testing possible solutions.
  • Reaching a conclusion.
• Importance of Critical Thinking: Helps in making informed decisions, avoiding biases, and solving complex problems effectively.
• Applications: Used in scientific experiments, real-life situations, and daily decision-making processes.

Click Here To Download Notes

Key Concept: Chemical Transformation, Detailed Mechanism

b) Absorption of water molecules
[Solution Description] The setting of Plaster of Paris involves the hydration of calcium sulfate hemihydrate to form calcium sulfate dihydrate. During this process, water molecules are absorbed into the crystalline structure of the gypsum, resulting in the formation of strong crystalline bonds that provide rigidity and hardness to the material.

Notes:

Chemical Transformation: Chemical transformation refers to the process in which one or more substances are converted into different substances. This occurs through chemical reactions, where bonds between atoms are broken and new bonds are formed.

Detailed Mechanism of Chemical Reactions:

1. Reactants and Products: A chemical reaction involves reactants transforming into products.
2. Breaking of Bonds: Reactants undergo bond breaking, which requires energy.
3. Formation of New Bonds: New bonds form in the product molecules, releasing energy.
4. Energy Changes: Chemical reactions involve exothermic or endothermic energy changes.
5. Conservation of Mass: The mass of reactants equals the mass of products (Law of Conservation of Mass).
6. Reaction Conditions: Reactions may require catalysts, temperature, or pressure changes.

Click Here To Download Notes

Key Concept: Comparative Analysis, Chemical Process Insight

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that Plaster of Paris has less water of crystallization than gypsum, which is correct. Gypsum $(CaSO_4 \cdot 2H_2O)$ has two water molecules per formula unit, while Plaster of Paris $(CaSO_4 \cdot \frac{1}{2}H_2O)$ has half a water molecule shared between two formula units, effectively having one water per formula unit. This reduction occurs during heating gypsum at 373 K, confirming that both statements are true, and the reason explains why the assertion is true.

Starting with gypsum:

$\text{CaSO}_4 \cdot 2\text{H}_2\text{O} \xrightarrow{373 \, \text{K}} \text{CaSO}_4 \cdot \frac{1}{2}\text{H}_2\text{O} + \text{H}_2\text{O}$

Thus, this chemical reaction supports the explanation provided in Reason for the Assertion’s truth.

Click To Download Notes

Key Concept: Olfactory Indicator Mechanism, Reaction Mechanism

c) Assertion is true, but Reason is false.
[Solution Description] In the given assertion, it is noted that the smell of vanilla persists in acidic solutions but changes in basic ones. This aligns with the behavior of many olfactory indicators, which alter their odor in response to the pH level of the solution they are in. Therefore, the assertion is true.

The reason states that olfactory indicators change odor only during neutralization processes. However, this statement is incorrect because olfactory indicators can display different odors simply by being exposed to either acidic or basic conditions, independent of a complete neutralization reaction. Thus, the reason does not correctly explain why the assertion holds true.

Hence, while both statements are separately accurate regarding the behavior of olfactory indicators, the reason provided is not a valid explanation for the given assertion.

Click To Download Notes

Key Concept: Indicator Comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both the assertion and reason are correct. Phenolphthalein is a synthetic indicator that shows a distinct color change from colorless to pink in basic solutions, generally above pH 8.5, which aligns well with the pH values of many common bases. Turmeric, meanwhile, does not provide such a clear indication across this range as it changes color only between specific acidic and basic conditions. Therefore, phenolphthalein can be considered more effective in identifying bases due to its clear transition within the typical pH range of bases, supporting the assertion that phenolphthalein is more effective, with the reason being the correct explanation.

Click To Download Notes

Key Concept: Indicator Chemistry, Complex Scenarios, Advanced Applications

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To evaluate this assertion and reason, we must consider both the chemical properties of phenolphthalein and its effectiveness in various scenarios:

– Assertion: Phenolphthalein indeed changes color from colorless to pink when transitioning from an acidic to a basic environment because its hydroxyl group loses a proton at higher pH levels. This makes it suitable for detecting strong bases as well as weak bases that are above the pH transition range. Thus, the assertion is a true statement.

– Reason: The reason correctly identifies that phenolphthalein has a color transition interval from pH 8.3 to 10.0. This narrow range enables it to precisely signal the endpoint during a titration of strong acids with strong bases; however, it does not explain why phenolphthalein is only useful for strong bases or how it changes color chemically.

Therefore, while both the assertion and the reason are true individually, the reason does not serve as the correct explanation of the assertion.

Click To Download Notes

Key Concept: Basic Indicators

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that red litmus paper turns blue when exposed to a base, which is true. The reason given is that bases change the color of red litmus to blue, which correctly explains why the assertion occurs.

Click To Download Notes

Key Concept: Indicator Effectiveness, Indicator Limitations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether both the assertion and reason are true and if the reason correctly explains the assertion, we must understand the effectiveness of these indicators in colored solutions. Litmus changes from red to blue when exposed to a base, which is a distinct change and generally noticeable even in slightly colored solutions. On the other hand, turmeric turns reddish-brown in basic conditions, but this change might not be as apparent in colored solutions due to the overlap with the solution’s inherent color.

The assertion that litmus is more effective than turmeric for detecting bases in colored solutions is accurate because litmus provides a clearer transition from red to blue compared to turmeric’s change in hue, which can be obscured by the solution’s color. The reason given supports this assertion as it highlights the distinctness of the color change provided by litmus.

Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Click To Download Notes

Key Concept: Litmus Source

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion is true because litmus is indeed extracted from lichen, as it is mentioned in the syllabus that litmus is a natural indicator coming from this source. However, the reason provided is false. Lichens are a symbiotic association between a fungus and an alga or cyanobacterium, but they are not classified solely within the fungi kingdom; instead, they are considered a unique entity due to their dual nature. Therefore, the correct option is that the assertion is true, and the reason is false.

Click To Download Notes

Key Concept: Complex Indicator Interaction, Advanced Application

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that phenolphthalein is used over methyl orange for a specific titration because it provides a clearer endpoint in basic solutions. This is true because phenolphthalein turns pink in basic environments, making it suitable for titrations that end at a higher pH. The reason mentions that methyl orange changes color at a lower pH range than phenolphthalein, which is also true; methyl orange transitions from red to yellow around a pH of 3.1 to 4.4, while phenolphthalein transitions above pH 8. However, while both statements are true, the reason does not directly explain why phenolphthalein is preferred over methyl orange in this context; instead, it simply states their different pH ranges.

Click To Download Notes

Key Concept: Basic Identification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Methyl orange is known to change colors depending on the pH of the solution it is placed in. Specifically, it turns red if the solution is acidic and transitions to yellow when the solution becomes basic. This behavior is due to its effective range between pH 3.1 and 4.4. Hence, both statements are true, and the reason correctly explains the assertion.

Click To Download Notes

Key Concept: Reaction Specificity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Clove oil is known to change its odour specifically in basic solutions, as mentioned in the syllabus. This implies that it does not alter its odour in acidic environments, supporting the assertion that clove oil retains its characteristic smell when exposed to acids. Hence, both the assertion and reason are true, and the reason correctly explains why the assertion holds.

Click To Download Notes

Key Concept: Complex Scenario Analysis, Advanced Application

d) Assertion is false, but Reason is true.
[Solution Description]
The problem requires determining the behavior of clove oil and vanilla essence at a given pH level. According to the syllabus, clove oil’s odour changes in basic solutions. Basic solutions are those with a pH greater than 7. Hence, for $pH = 9$, which is basic, clove oil will change its odour.

Vanilla essence loses its smell in acidic solutions according to the syllabus. Acidic solutions have a pH less than 7, so when the pH is 9, vanilla essence would not lose its odour since it remains unchanged in basic solutions as per the provided information. Therefore, the Assertion is false because only clove oil changes its odour in this scenario; vanilla essence does not lose its odour. The Reason states that clove oil changes its odour in basic solutions, which is true, but it incorrectly claims that vanilla essence retains its smell, which contradicts the assertion about losing odour in acidic conditions.

Thus, the Assertion is false and the Reason is true.

Click To Download Notes

Key Concept: Complex Reaction Analysis, Reaction Mechanism

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion suggests that copper can react with sulfuric acid to yield copper sulfate and hydrogen gas. In such reactions involving metals and acids, hydrogen gas is typically produced if the metal is more reactive than hydrogen according to the reactivity series. However, copper is below hydrogen in the reactivity series, indicating that copper does not have enough reactivity to displace hydrogen from an acid. Therefore, no reaction occurs between copper and sulfuric acid under standard conditions.

Moreover, the reason correctly states that copper’s lower position relative to hydrogen in the reactivity series implies its lesser reactivity compared to hydrogen. Hence, both the Assertion and Reason are false because copper cannot displace hydrogen from sulfuric acid due to its lower reactivity.

According to our analysis:
– Assertion: False (Copper does not react with sulfuric acid under standard conditions)
– Reason: True (Copper is indeed less reactive than hydrogen)

Thus, option (d) is correct where the Assertion is false, but the Reason is true.

Click To Download Notes

Key Concept: Basic Reaction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The reaction between magnesium (Mg) and hydrochloric acid (HCl) can be represented by the following equation:
$\text{Mg(s)} + 2\text{HCl(aq)} \rightarrow \text{MgCl}_2\text{(aq)} + \text{H}_2\text{(g)}$
In this reaction, magnesium displaces hydrogen ions from hydrochloric acid, resulting in the formation of magnesium chloride $(MgCl_2)$ and the release of hydrogen gas $(H_2)$. Hence, both the assertion that hydrogen gas is evolved and the reason that magnesium displaces hydrogen are true. Additionally, the reason provided is the correct explanation for the assertion as it describes the chemical process taking place.

Click To Download Notes

Key Concept: pH Scale Understanding

c) It is weakly acidic.

[Solution Description] The pH scale ranges from 0 to 14, where values below 7 indicate acidity, and values above 7 indicate alkalinity. A pH of 7 is neutral. Since the given pH value is 5, it indicates that the solution is acidic. The closer the pH value is to 0, the stronger the acid; thus, a pH of 5 represents a weak acid.

Click To Download Notes

Key Concept: Reaction with Metals

d) Hydrogen

[Solution Description] When hydrochloric acid $(\text{HCl})$ reacts with zinc $(\text{Zn}),$ hydrogen gas $(\text{H}_2)$ is evolved. The balanced chemical equation for this reaction is:

$\text{Zn} + 2\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2 \uparrow$

In this reaction, zinc displaces the hydrogen ions from hydrochloric acid, resulting in the formation of zinc chloride and the release of hydrogen gas.

Click To Download Notes

Key Concept: Natural Indicators

c) Red cabbage leaves

[Solution Description] Red cabbage leaves are known to change color when exposed to different pH levels, making them useful as natural indicators for testing acidity and alkalinity.

Click To Download Notes

Key Concept: Neutralization Reaction

c) Salt, water, and carbon dioxide

[Solution Description] When an acid reacts with a metal carbonate, it produces a salt, water, and carbon dioxide gas. This reaction is a typical characteristic of acids reacting with carbonates.

Click To Download Notes

Key Concept: eaction Products

a) Calcium sulfate, carbon dioxide, and water

[Solution Description]

Calcium carbonate $(\text{CaCO}_3)$ reacts with sulfuric acid $(\text{H}_2\text{SO}_4)$ to form calcium sulfate $(\text{CaSO}_4),$ carbon dioxide $(\text{CO}_2),$ and water $(\text{H}_2\text{O}).$ The balanced chemical equation is:

$\text{CaCO}_3\text{(s)} + \text{H}_2\text{SO}_4\text{(aq)} \rightarrow \text{CaSO}_4\text{(s)} + \text{CO}_2\text{(g)} + \text{H}_2\text{O(l)}$

This reaction produces a salt, gas, and water as expected.

Click To Download Notes

Key Concept: Basic Reaction

b) Salt and hydrogen gas

[Solution Description]

When hydrochloric acid $(\text{HCl})$ reacts with magnesium $(\text{Mg})$, it forms magnesium chloride $(\text{MgCl}_2)$ and hydrogen gas $(\text{H}_2)$. The reaction is represented as:

$\text{Mg} + 2\text{HCl} \rightarrow \text{MgCl}_2 + \text{H}_2$

So, the products are salt (magnesium chloride) and hydrogen gas.

Click To Download Notes

Key Concept: Degree and Coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Analyzing the assertion: The polynomial given is $3x^4 – 5x^2 + x – 7$, where the term with the highest power of $x$ is $x^4$, thus, making the degree 4. This confirms the assertion is true.
Analyzing the reason: According to the definition of the degree of a polynomial, it is determined by the highest power of the variable present in it. Hence, this reason correctly explains why the degree of the polynomial is 4.
Both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Degree of Univariate Polynomials

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
A polynomial is classified based on its degree, which is the highest power of the variable present in the expression. The given polynomial $x^3 – 4x + 5$ is indeed a cubic polynomial because the highest power of $x$ is 3. However, the number of terms does not determine if a polynomial is cubic; it is solely determined by the degree. Therefore, while both the assertion and reason are true, the reason does not correctly explain why the polynomial is cubic.

Key Concept: Types of Polynomials

c) Assertion is true, but Reason is false.
[Solution Description]
A linear polynomial is defined as a polynomial with the highest power of the variable equal to 1. Therefore, the assertion that a linear polynomial has a degree of 1 is true. However, the polynomial $7y + 9$ has the highest power of $y$ as 1, which makes it a linear polynomial, not a cubic one. Thus, the reason is false.

Key Concept: Product of Polynomials, Highest Term Analysis

d) 8
[Solution Description]
To determine the degree of the product, consider the highest degree term from each polynomial:
Degree of $z^5 – z^2 + 1$ is 5.
Degree of $3z^3 + 4$ is 3.
Therefore, the degree of $U(z)$ is:
$\text{Degree of } U(z) = 5 + 3 = 8$
Thus, the degree of $U(z)$ is 8.

Key Concept: Standard form of cubic polynomial

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both the assertion and reason are true, and the reason explained correctly justifies the assertion. The definition of a cubic polynomial requires an $x^3$ term with a non-zero coefficient to ensure its highest degree is three.

Key Concept: Derivative Significance, Critical Points

d) Assertion is false, but Reason is true.
[Solution Description] For a cubic polynomial $f(x) = ax^3 + bx^2 + cx + d$, the first derivative $f'(x) = 3ax^2 + 2bx + c$ is a quadratic equation. It has two solutions (critical points) unless the discriminant of $f'(x)$ is zero, resulting in coincident critical points, or negative, indicating no real critical points. Consequently, the assertion is incorrect; however, the reason is valid because critical points are defined via the derivative equaling zero.

Key Concept: Quadratic Polynomials

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion is true as the graph of a quadratic polynomial $ax^2 + bx + c$ forms a parabola, determined by the sign of $a$. The reason discussing cubic polynomials is unrelated to the nature of the quadratic polynomial graph. Hence, the assertion is true, but the reason is not relevant.

Key Concept: Zeroes of a Quadratic Polynomial

b) Two
[Solution Description] A quadratic polynomial, which is of degree 2, typically has two zeroes. These can be real or complex numbers.

Key Concept: Y-intercept in Linear Equations

a) -5
[Solution Description] The y-intercept in a linear equation of the form $y = ax + b$ is the constant term ‘b’. Here, the equation is $y = 4x – 5$, so the y-intercept is -5.

Key Concept: Linear Polynomials, Coefficients

Correct Answer option
[Solution Description] In the polynomial $3x + 2$, the number 3 is indeed the coefficient of the variable $x$. The reason explains what coefficients are: they are the numbers multiplying the variables in a polynomial. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Applications of Constant Polynomials

d) $3x + 2$
[Solution Description] A constant polynomial is an expression that contains only a constant term without any variable component. Among the given options, the expression $3x + 2$ includes a term with a variable $x$ and thus is not a constant polynomial.

Key Concept: Constant Polynomials, Polynomial Degree, Algebraic Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Analyzing both statements:
– Assertion is true because a constant polynomial like $f(x) = c$, where $c \neq 0$, has no variable component, resulting in a degree of zero.
– Reason is also correct because the degree is defined by the highest power of the variable term present in the polynomial.
– Here, the Reason correctly explains why a constant polynomial with a non-zero value has a degree of zero.

Key Concept: Field Boundary

a) $4b + 2a$
[Solution Description]
Let the shorter side be $b$ meters. The longer side is $(b + a)$ meters. The perimeter $P$ is given by:
$P = 2(b + (b + a))$
Simplifying,
$P = 2(2b + a) = 4b + 2a$
Hence, the linear polynomial is $4b + 2a$.

Key Concept: Chess Club Matches Calculation

c) 15
[Solution Description]
Use the total cost equation:
$\text{Total Cost} = 200 + 50m$
Plug in Rs.950 for the total cost:
$200 + 50m = 950$
Subtract 200 from both sides:
$50m = 750$
Solve for $m$ by dividing both sides by 50:
$m = \frac{750}{50} = 15$
Therefore, the player played 15 matches.

Key Concept: Linear Polynomials in Payments

c) Rs.1300
[Solution Description]
The joining fee is Rs.100 and the monthly fee is Rs.200. For 6 months, the calculation becomes:
$200 \times 6 = 1200$
Adding the joining fee gives:
$100 + 1200 = 1300$
Thus, the total cost for 6 months is Rs.1300.

Key Concept: Cost Calculation using Linear Polynomials

d) Rs.400
[Solution Description] The cost to rent the car is given by the equation $C = 300 + 20h$.
Substitute $h = 5$ into the equation:
$C = 300 + 20 \times 5$
$C = 300 + 100 = 400$
Therefore, the total cost for renting the car for 5 hours is Rs.400.

Key Concept: Linear Growth: Constant Difference

d) 32
[Solution Description] The $n$-th term ($a_n$) of an arithmetic sequence with the first term $a_1$ and common difference $d$ is given by $a_n = a_1 + (n-1)d$. Here, $a_1 = 5$, $d = 3$, and $n = 10$. Thus, $a_{10} = 5 + (10-1)\times 3 = 5 + 27 = 32$.

Key Concept: Linear Decay: Constant Difference

c) 26
[Solution Description] For a linear decay sequence, the $n$-th term ($a_n$) can be calculated using $a_n = a_1 – (n-1)d$, where $a_1 = 50$, $d = 4$, and $n = 7$. So, $a_7 = 50 – (7-1)\times 4 = 50 – 24 = 26$.

Key Concept: Graph intersection

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion claims that the line $y = x + 4$ intersects the $y$-axis at $(0, 4)$. This is correct since when $x = 0$, $y = 4$.
The reason explains that for an equation of the form $y = ax + b$, the intersect point on the $y$-axis is at $(0, b)$. This is also correct as it matches with the standard definition of the $y$-intercept.
Since both statements are true and the reason correctly explains the assertion, option a) is correct.

Key Concept: Equation Rearrangement and Y-intercept Calculation

c) 4
[Solution Description]
Rearrange the equation $3y – 2x = 12$ to solve for $y$:
$3y = 2x + 12$
Divide each term by 3 to isolate $y$:
$y = \frac{2}{3}x + 4$
From this expression, the y-intercept $c$ is $4$.

Key Concept: Finding an Unknown in a Linear Equation

d) 3
[Solution Description]
Start with the linear equation $y = 2x + 1$ and set $y = 7$:
Substitute $7$ for $y$:
$7 = 2x + 1$
Subtract $1$ from both sides:
$7 – 1 = 2x$
$6 = 2x$
Divide both sides by $2$:
$x = \frac{6}{2} = 3$
Thus, the value of $x$ is $3$.

Key Concept: Determining Input Given Output

b) 4
[Solution Description]
The equation given is $4x – 7 = 9$. We need to solve this equation for $x$.
First, add $7$ to both sides to isolate the term with $x$:
$4x – 7 + 7 = 9 + 7$
$4x = 16$
Next, divide both sides by $4$ to solve for $x$:
$x = \frac{16}{4} = 4$
Therefore, the input value $x$ is $4$.

Key Concept: Pattern Extension and Real-World Application

c) 225
[Solution Description]
For a linear pattern described by $2n – 1$, sum the sequence from Stage 1 to Stage 15.
The formula for the sum of the first $n$ terms of this arithmetic sequence is:
$S_n = \frac{n}{2} \times (\text{{first term}} + \text{{last term}})$
The first term ($a_1$) is 1, and the last term for Stage 15 ($a_{15}$) is $2(15) – 1 = 29$.
Substituting these values gives:
$S_{15} = \frac{15}{2} \times (1 + 29)$
Simplifying inside the parenthesis:
$S_{15} = \frac{15}{2} \times 30$
Multiply:
$S_{15} = 15 \times 15 = 225$
Therefore, the total number of tiles used is 225.

Key Concept: Generalization of sequence

b) 2
[Solution Description] The common difference in a linear pattern can be found by calculating the difference between consecutive terms.
From the sequence 1, 3, 5, 7, 9, the difference is:
$3 – 1 = 2 \\ \\ 5 – 3 = 2 \\ \\ 7 – 5 = 2 \\ \\ 9 – 7 = 2$
Hence, the common difference is 2.

Key Concept: Generalising Linear Patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To find the expression for the number of square tiles at Stage $n$, observe that the equation $2n – 1$ suggests doubling the stage number and subtracting one to get the total tiles in that stage. For example, at Stage 3: $2(3) – 1 = 5$. This matches the given sequence. The reason explains why this formula works, as each term increases linearly by 2 units with every subsequent stage, confirming the assertion as true with the correct reason.

Key Concept: Application of Linear Expression for Larger Stages

c) 29 tiles
[Solution Description]
The formula for the number of tiles in Stage $n$ is given by $2n – 1$.
Substitute $n = 15$ into the formula:
$2(15) – 1 = 30 – 1$
Simplifying the expression gives:
$29$
Therefore, the 15th stage will have 29 tiles.

Key Concept: Generalisation of Pattern Formula

c) $2n – 1$
[Solution Description] The relationship between stage number $n$ and the number of tiles follows a linear pattern given by the formula:
$2n – 1$
This formula states that each stage has two less than twice the stage number in terms of tiles.

Key Concept: Linear Relationship Calculation

b) 51
[Solution Description]
Using the formula $T = 2n – 1$ for the number of tiles at stage $n$, where $n=26$:
\begin{align*}
T &= 2(26) – 1
&= 52 – 1
&= 51
\end{align*}
Therefore, there are 51 tiles in the 26th stage.

Key Concept: Linear Growth: Finding Future Value

c) \$125
[Solution Description]
The total cost $C$ for renting a car is given by the function $C = C_0 + rd$, where $C_0$ is the base cost and $r$ is the daily rate.
Given $C_0 = 50$ and $r = 15$, and $d = 5$ days:
$C = 50 + 15 \times 5$
Calculating further,
$C = 50 + 75 = 125$
Thus, the cost for renting the car for 5 days is \$125.

Key Concept: Linear Growth: Savings

c) \$2900
[Solution Description]
The savings grow by \$150 every month.
Let $B$ represent the balance and $m$ represent time in months.
Initial balance, $B_0 = 2000$
Monthly deposit = \$150
Balance after 6 months, $B(m)$, is given by:
$B(m) = B_0 + 150 \times m$
Substitute $B_0 = 2000$ and $m = 6$:
$B(6) = 2000 + 150 \times 6$
$B(6) = 2000 + 900 = 2900$
Thus, the total balance after 6 months will be \$2900.

Key Concept: Linear Decay

c) 2600
[Solution Description] This scenario illustrates linear decay as calories decrease steadily over time.
The initial calorie count is $c_0 = 5,000$. The burn rate is 200 calories per hour. Thus, the remaining calories after $x$ hours is computed as:
$c(x) = 5000 – 200x$
Plug in $x=12$:
$c(12) = 5000 – 200 \times 12 = 5000 – 2400 = 2600$
Therefore, the calories remaining after 12 hours are 2600.

Key Concept: Linear Decay, Real-World Application

c) 68 years
[Solution Description]
Model the depreciation with the function $V(t) = 30000 – 300t$. We need to solve for $t$ when $V(t) 30000 – 300t$
Subtracting 30000 from both sides:
$-20000 > -300t$
Dividing through by -300 and reversing the inequality:
$t > \frac{20000}{300} = \frac{200}{3} \approx 66.67$
Thus, the car will be more than 67 years old when its value falls below \$10,000.

Key Concept: Y-intercepts in Linear Equations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] If $b = 0$, substituting this into the equation gives $y = ax$. At $x = 0$, $y = 0$, indicating the line passes through the origin $(0, 0)$. The reason statement correctly identifies that the y-intercept $b$ determines the intersection point on the y-axis. Hence, both the assertion and reason are true and related to each other but the reason does not explain why the line passes through the origin when $b = 0$.

Key Concept: Line through Origin

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] An equation in the form $y = ax$ does not contain a constant term $b$, meaning there is no vertical shift. Setting $x = 0$ gives $y = 0$, confirming the line passes through the origin $(0, 0)$. This logical consequence supports both the assertion and its explanation.

Key Concept: Understanding Linear Relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A linear equation in the form $y = ax + b$ has a slope $a$ which represents the rate of change of $y$ with respect to $x$. The y-intercept $b$ tells us the value at which the line crosses the y-axis, meaning it indicates the value of $y$ when $x$ is zero.

Key Concept: Understanding Linear Relationships, Deriving Constants

b) $a = \frac{2}{3}, \, b = 16.67$
[Solution Description]
From the equation $y = ax + b$:
For 500 miles:
$350 = 500a + b$
For 800 miles:
$550 = 800a + b$
Subtracting these:
$550 – 350 = 800a + b – 500a – b$
$200 = 300a$
Solving for $a$:
$a = \frac{200}{300} = \frac{2}{3}$
Substitute $a$ back into the first equation:
$350 = 500\left(\frac{2}{3}\right) + b$
$350 = \frac{1000}{3} + b$
$b = 350 – \frac{1000}{3}$
$b = \frac{1050}{3} – \frac{1000}{3} = \frac{50}{3} \approx 16.67$
Therefore, $a = \frac{2}{3}$ and $b \approx 16.67$.

Key Concept: Influence of Slope

c) Assertion is true, but Reason is false.
[Solution Description]
In the assertion, it’s claimed that increasing the absolute value of $a$ makes lines $y = ax$ steeper. The slope $a$ quantifies the steepness; thus, this statement holds true.
Conversely, the reason erroneously states that higher slopes mean less steep lines, which contradicts the actual definition of slope. Positive or negative, larger magnitudes indicate steeper lines.
Thus, the assertion is true while the reason is false.

Key Concept: Intercepts and Slopes

d) Assertion is false, but Reason is true.
[Solution Description]
The line $y = 5x – 7$ has a slope of $5$, which is the coefficient of $x$. However, the line intersects the y-axis at $(0, -7)$; thus, the y-intercept is $(-7)$, not the point $(-7, 0)$. The statement about the slope is correct. Although the reason regarding the slope is true, it does not justify the erroneous claim about where the line intersects the axes. Therefore, the assertion is false, though the reason itself is true.

Key Concept: Graphical Analysis of Slope

c) Assertion is true, but Reason is false.
[Solution Description]
When $a = 0$ in $y = ax$, the equation becomes $y = 0$, resulting in a horizontal line along the x-axis. This means the assertion is true. However, a horizontal line actually has a slope of 0, not an undefined slope. Vertical lines have an undefined slope. Therefore, the reason is false.

Key Concept: Y-Intercept and Line Position

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion is false, as changing the value of $b$ affects the position of the line but not its slope. The slope is determined by $a$. The reason is true since $b$ indeed represents the y-intercept where the line crosses the y-axis. Thus, the reason is true but does not prove the assertion.

Key Concept: Vertical Shifts and Intercepts

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion observes that when $b$ increases from 1 to 3, the line indeed shifts upwards by 2 units on the graph. This is due to the nature of $b$ being the y-intercept.
The reason simply restates that an increment in $b$ corresponds to a direct vertical shift by the same magnitude on the y-axis. It affirms the observation described in the assertion.
Consequently, both the assertion and the reason are true, and the reason provides a suitable explanation for the assertion.
Thus, the correct choice is: Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Slope and y-Intercept Applications

a) $(0, 5)$
[Solution Description]
First, calculate the slope using the formula $a = \frac{y_2 – y_1}{x_2 – x_1}$. Substituting gives $a = \frac{17 – 11}{4 – 2} = \frac{6}{2} = 3$. Now use one of the points, say $(2, 11)$, and the slope to find the y-intercept using $y = ax + b$: $11 = 3(2) + b \Rightarrow 11 = 6 + b \Rightarrow b = 5$. So, the y-intercept is 5 or coordinate $(0, 5)$.

Key Concept: Understanding Slope Change

d) The slope increases
[Solution Description] The original slope of the line is 3, as represented by the coefficient of $x$. In the new equation $y = 5x + 4$, the slope has changed to 5. Therefore, the slope becomes steeper since 5 is greater than 3.

Key Concept: Effect of Changing ‘a’ with ‘b’ Fixed

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] When $a 1$) results in a steeper line. Hence, both statements are true and the reason provides the correct explanation for the assertion.

Key Concept: Effect of Changing ‘b’

c) They are parallel
[Solution Description]
Both lines have the same slope $7$, hence they are parallel according to the property that lines with equal slopes remain parallel irrespective of their difference in $b$.

Key Concept: Visualizing Line Translation, Fixed Slope

b) $y = -5x – 2$
[Solution Description]
The translation of a line vertically alters its y-intercept without affecting its slope. The original line has the equation $y = -5x + 6$. Moving it downward by 8 units involves reducing the y-intercept by 8:
$b_{new} = 6 – 8 = -2$
Therefore, the equation of the new line becomes:
$y = -5x – 2$

Key Concept: Y-intercept Interpretation

c) It represents the initial value of $y$
[Solution Description] In the equation $y = -4x + 12$, the y-intercept is the constant term when $x = 0$, which is 12. This means that when $x = 0$, the value of $y$ is 12. The y-intercept represents the point where the line crosses the y-axis, indicating the initial value of $y$ before any changes in $x$.
Therefore, the y-intercept represents the starting value of $y$ when $x = 0$.

Key Concept: Graphical Representation

d) Assertion is false, but Reason is true.
[Solution Description] The assertion is incorrect because a straight line with a negative slope, such as $y = -3x + 4$, actually indicates linear decay or decrease. The reason correctly explains that the negative sign indicates a downward slope. Thus, the assertion is false and the reason is true.