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: Reaction Mechanism, Comparative Reaction Rates

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the reaction of $\text{Na}_2\text{CO}_3$ with HCl releases carbon dioxide more rapidly compared to $\text{NaHCO}_3$. This implies a faster reaction rate for sodium carbonate.

To understand this, we consider the reactions:$\text{Na}_2\text{CO}_3 + 2\text{HCl} \rightarrow 2\text{NaCl} + \text{CO}_2 + \text{H}_2\text{O}$
$\text{NaHCO}_3 + \text{HCl} \rightarrow \text{NaCl} + \text{CO}_2 + \text{H}_2\text{O}$

Sodium carbonate provides two moles of carbonate per mole of compound, whereas sodium bicarbonate provides only one. Thus, $\text{Na}_2\text{CO}_3$ indeed has a higher concentration of reactive carbonate ions, which leads to a faster release of $\text{CO}_2$.

Therefore, both the assertion and reason are true, and the reason correctly explains why $\text{Na}_2\text{CO}_3$ reacts faster in terms of $\text{CO}_2$ release.

Click To Download Notes

Key Concept: Reaction Process

c) Assertion is true, but Reason is false.
[Solution Description] The assertion is true because neutralization reactions are exothermic in nature, meaning they release heat energy during the reaction between an acid and a base to form salt and water. The reason is false because neutralization reactions typically result in the formation of a soluble salt and water, not necessarily a precipitate which refers to an insoluble solid that forms from a solution.

Click To Download Notes

Key Concept: General Equation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that neutralization reactions follow a specific formula: $\text{Base} + \text{Acid} \rightarrow \text{Salt} + \text{Water}$. This represents how bases react with acids to produce salt and water. The reason explains that this equation is the general representation of such reactions. Both statements are true, and Reason correctly explains Assertion.

Click To Download Notes

Key Concept: Reaction Mechanism, Advanced Comparison

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion involves comparing the reactivity of copper(II) oxide CuO and magnesium oxide MgO with hydrochloric acid HCl. Copper(II) oxide is basic but not as strongly basic as magnesium oxide. Basicity affects the reactivity because more basic oxides tend to react faster with acids, forming salts and water.

The reason states that various factors like temperature, concentration of the acid, and the intrinsic properties of the metal oxides influence the reaction rate. This is indeed true as these factors are well-known to affect chemical reactions.

Both the assertion and the reason are true, however, the reason provided does not correctly explain why CuO reacts slower than MgO. The difference in reaction rate is primarily due to the difference in basicity, rather than general factors affecting reaction rates.

Click To Download Notes

Key Concept: Predictive Analysis, Real-World Application

c) Nickel phosphate and water
[Solution Description] Nickel(II) oxide (NiO) is a basic oxide that will react with phosphoric acid $(H_3PO_4)$ to form a salt and water.

The expected reaction is:$NiO + H_3PO_4 \rightarrow Ni_3(PO_4)_2 + H_2O$
This shows that nickel phosphate and water are produced.

Notes:

• Predictive Analysis:
  • Acids, bases, and salts exhibit predictable reactions based on their properties.
  • Acid strength depends on hydrogen ion concentration; stronger acids ionize more.
  • Bases neutralize acids to form salts and water.
  • Salts form from the combination of acids and bases, affecting their properties and uses.
• Real-World Applications:
  • Acids: Used in fertilizers, food preservation, and batteries.
  • Bases: Applied in soap making, cleaning agents, and as antacids.
  • Salts: Used in seasoning, water purification, and in industrial processes like dyeing.

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: 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: Ion Formation

c) It forms $H_3O^+$ and $Cl^-$
[Solution Description] When HCl is dissolved in water, it ionizes completely to form hydronium ions $(H_3O^+)$ and chloride ions $(Cl^-)$. The reaction can be written as:$HCl(aq) + H_2O(l) \rightarrow H_3O^+(aq) + Cl^-(aq).$
This process increases the concentration of $H_3O^+$ ions in the solution, making it acidic.

Click To Download Notes

Key Concept: Ion Identification

a) $H_3O^+$ and $Cl^-$
[Solution Description]
When HCl dissolves in water, it dissociates into $H_3O^+$ and $Cl^-$ ions.

Notes:

• Hydrogen Ion (H⁺): Found in acids, responsible for acidic properties. When dissolved in water, acids release H⁺ ions.
• Hydroxide Ion (OH⁻): Found in bases, responsible for basic properties. Bases release OH⁻ ions when dissolved in water.
• Identification Using Indicators:
  • Litmus Paper: Red in acids, blue in bases.
  • Phenolphthalein: Colorless in acids, pink in bases.
• Neutralization Reaction: Acid + Base → Salt + Water. H⁺ ions neutralize OH⁻ ions, forming water.
• Salts: Formed when acids react with bases. Examples include sodium chloride (NaCl) from hydrochloric acid and sodium hydroxide.

Click Here To Download Notes

Key Concept: Dilution Effects, Advanced pH Concepts

c) Assertion is true, but Reason is false.
[Solution Description]
Diluting a strong acid decreases the concentration of $H^+$ ions per unit volume because more solvent is added. As a result, the acidic strength of the solution decreases and the pH increases because the pH is inversely related to the concentration of hydrogen ions. Therefore, the assertion that diluting a strong acid leads to an increase in the solution’s pH value is true. However, the reason provided states that the concentration of hydrogen ions increases with dilution, which is false. This makes the reason incorrect as it contradicts the observed behavior when acids are diluted.

Click To Download Notes

Key Concept: Conductivity Comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] HCl in aqueous solution dissociates to form $H^+$ and $Cl^-$ ions. These ions are responsible for conducting electricity as they move freely within the solution, allowing for an electric current to pass through. Glucose, on the other hand, is a non-electrolyte; it does not dissociate into ions when dissolved in water. Since conductivity is dependent on the presence of free-moving ions, glucose solution lacks this property, making it a poor conductor compared to HCl solution. Therefore, both the assertion and reason are true, and the reason correctly explains why HCl solution conducts electricity better than glucose solution.

Click To Download Notes

Key Concept: Definition Recall

b) Decreasing the concentration of ions
[Solution Description] Dilution involves adding additional solvent (usually water) to decrease the concentration of solute particles like $\text{H}_3\text{O}^+$ or $\text{OH}^-$ ions per unit volume.

Notes:

• Acids: Substances that release hydrogen ions (H⁺) in aqueous solution. Example: Hydrochloric acid (HCl).
• Bases: Substances that release hydroxide ions (OH⁻) in aqueous solution. Example: Sodium hydroxide (NaOH).
• Salts: Compounds formed when an acid reacts with a base. Example: Sodium chloride (NaCl).
• Neutralization Reaction: A reaction between an acid and a base to form salt and water. Example: HCl + NaOH → NaCl + H₂O.
• Indicators: Substances that change color to indicate whether a solution is acidic, basic, or neutral. Example: Litmus paper.

Click Here To Download Notes

Key Concept: Ion Concentration

b) Decrease in $\text{OH}^-$ concentration
[Solution Description] When a base is diluted with water, the concentration of $\text{OH}^-$ ions decreases because the same number of ions is now distributed over a larger volume.

Notes:

• Definition: Ion concentration refers to the number of ions (positively or negatively charged particles) present in a solution.
• Units: Measured in moles per liter (mol/L), also called molarity.
• Types of Ions:
  • Cations: Positively charged ions (e.g., Na⁺, K⁺).
  • Anions: Negatively charged ions (e.g., Cl⁻, SO₄²⁻).
• Effect on Solution: The concentration of ions affects the physical and chemical properties of the solution.
• Conductivity: Higher ion concentration leads to higher electrical conductivity in the solution.
• Applications: Ion concentration is crucial in processes like osmosis, electrolysis, and biological functions.

Click Here To Download Notes

Key Concept: pH Range

c) Basic
[Solution Description]
A pH of 12 is greater than 7, placing it in the range of basic solutions. Therefore, the solution is basic in nature.

Click To Download Notes

Key Concept: pH and Soil

c) 6-7
[Solution Description]
Most plants thrive in soil with a pH between 6 and 7, where key nutrients are readily available, and harmful ion concentrations are low.

Click To Download Notes

Key Concept: Universal Indicator

a) Red
[Solution Description] Lemon juice is acidic with a typical pH around 2-3. Universal indicators display red or orange colors at low pH values corresponding to strong acids. Therefore, lemon juice will turn the universal indicator either red or yellow-orange.

Click To Download Notes

Key Concept: Acid Rain Impact

c) Adding lime to neutralize the acid
[Solution Description] To neutralize the acidity and restore a suitable pH level in affected lakes, adding lime (calcium carbonate) is a common practice. Lime acts as a neutralizing agent, raising the pH of the water and helping protect aquatic life from the harmful effects of acidification.

Click To Download Notes

Key Concept: Universal Indicator Use

d) To measure pH
[Solution Description] A universal indicator is primarily used to measure the pH level of a solution by showing different colors for different pH values. It helps in determining whether a solution is acidic, neutral, or basic.

Click To Download Notes

Key Concept: pH and Ion Concentration

a) It is high
[Solution Description] A pH of 12 indicates a very basic solution, which means it has a high concentration of $OH^-$ ions. The pH scale ranges from 0 to 14, with values above 7 denoting basic solutions that have higher concentrations of hydroxide ions.

Click To Download Notes

Key Concept: Basic pH Range

b) 5-7
[Solution Description] Most plants prefer a slightly acidic to neutral pH range for optimal nutrient availability and growth. The ideal range is generally between 5.5 and 7.

Click To Download Notes

Key Concept: Aquatic Life

c) It increases acidity, endangering aquatic life
[Solution Description] Acid rain can significantly decrease the pH of river water, making it more acidic. Lowering the pH affects aquatic organisms’ survival by disrupting physiological processes and damaging their habitat. Aquatic life thrives best in neutral to slightly basic environments.

Click To Download Notes

Key Concept: Salt Formation

c) Potassium nitrate
[Solution Description] The reaction between nitric acid $(HNO_3)$ and potassium hydroxide (KOH) follows the equation:
$\text{HNO}_3 + \text{KOH} \rightarrow \text{KNO}_3 + \text{H}_2\text{O}$
Therefore, the salt formed is potassium nitrate.

Key Concept: Salt Formation Mechanism, pH Influence

c) Sodium acetate
[Solution Description] Sodium acetate is formed from a strong base (sodium hydroxide) and a weak acid (acetic acid). According to the rule for pH of salts:

– Neutral salts (strong acid + strong base): pH = 7
– Acidic salts (strong acid + weak base): pH < 7 - Basic salts (strong base + weak acid): pH > 7

Thus, sodium acetate will have a basic pH due to hydrolysis of the acetate ion.

Notes:

• Salt Formation Mechanism:
  • Salts are formed when acids react with bases in a neutralization reaction.
  • Example: Hydrochloric acid (HCl) + Sodium hydroxide (NaOH) → Sodium chloride (NaCl) + Water (H₂O).
  • Salts can also form from the reaction between an acid and a metal, or between a base and a non-metal.
  • The nature of the salt depends on the strength of the acid and base used.
• pH Influence:
  • pH level determines whether a solution is acidic, basic, or neutral.
  • Strong acids form salts with a lower pH, while weak acids form salts with higher pH.
  • Salts from strong acids and strong bases are neutral (pH = 7).
  • Salts from weak acids and strong bases are basic (pH > 7).

Click Here To Download Notes

Key Concept: Solubility Check

a) Sodium chloride
[Solution Description] Sodium chloride (NaCl) is well-known for its solubility in water, unlike some other salts which are less soluble or insoluble.

Click To Download Notes

Key Concept: Neutral Salts

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion: True. Sodium chloride is neutral because it is produced by the reaction of hydrochloric acid, a strong acid, with sodium hydroxide, a strong base. In this type of reaction, the resulting salt is neutral.

Reason: True. The reason correctly explains that sodium chloride is derived from hydrochloric acid HCl and sodium hydroxide NaOH, both of which are strong.

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

Click To Download Notes

Key Concept: Salt Identification

a) ${NaCl}$
[Solution Description] Sodium chloride (NaCl) is a salt formed from the reaction between hydrochloric acid (HCl), which is a strong acid, and sodium hydroxide (NaOH), which is a strong base.

Click To Download Notes

Key Concept: pH Value Recognition

b) Equal to 7
[Solution Description] The pH of salts formed from the reaction of strong acids and strong bases is neutral, which means it is equal to 7.

Click To Download Notes

Key Concept: Baking Soda Composition

b) $\text{NaHCO}_3$
[Solution Description] The chemical formula of baking soda is $ \text{NaHCO}_3 $, which stands for sodium bicarbonate. It consists of sodium, hydrogen, carbon, and oxygen ions.

Click 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: Basic Concept

d) $CaSO_4.2H_2O$
[Solution Description]
Gypsum contains two water molecules as water of crystallization. Its chemical formula is $CaSO_4.2H_2O$. This implies that each formula unit of calcium sulphate is associated with two water molecules.

Notes:

Key Concept: Water Source

b) From the salt itself
[Solution Description]
The water droplets observed during the heating of copper sulphate crystals originate from the water of crystallization within the salt itself. As the crystals are heated, this water is released.

Notes:

• Surface Water: Found on the Earth’s surface in rivers, lakes, ponds, and reservoirs. Used for drinking, irrigation, and industrial purposes.
• Groundwater: Stored beneath the Earth’s surface in aquifers. Accessible through wells and tube wells.
• Rainwater: Water from precipitation (rain) collected in tanks or stored in reservoirs for use.
• Glaciers and Ice Caps: Large ice masses that slowly release water into rivers.
• Water Cycle: The continuous movement of water through evaporation, condensation, and precipitation.
• Importance: Essential for drinking, agriculture, sanitation, and industry. Proper management is crucial for sustainability.

Click Here To Download Notes

Key Concept: Reaction Equation

a) $CaSO_4 \cdot \frac{1}{2}H_2O + \frac{3}{2}H_2O \rightarrow CaSO_4 \cdot 2H_2O$
[Solution Description] When Plaster of Paris, which is calcium sulphate hemihydrate $(CaSO_4 \cdot \frac{1}{2}H_2O)$, is mixed with water, it forms gypsum $(CaSO_4 \cdot 2H_2O)$. The chemical equation for this reaction is:$CaSO_4 \cdot \frac{1}{2}H_2O + \frac{3}{2}H_2O \rightarrow CaSO_4 \cdot 2H_2O$

Thus, option a) is correct.

Notes:

• Chemical Reaction: A process where reactants are transformed into products, with a change in energy.
• Reactants & Products: Substances before and after the reaction, respectively.
• Chemical Equation: A symbolic representation of a chemical reaction using formulas (e.g.,
  A+B→C+DA + B.
• Balancing Equations: Atoms of each element must be the same on both sides.
• Types of Reactions:
  • Combination: Two or more reactants combine (e.g.,
    A+B→ABA + B .
  • Decomposition: A compound breaks into simpler products (e.g.,
    AB→A+BAB .
  • Displacement: One element replaces another in a compound.
  • Double Displacement: Two compounds exchange ions.

Click Here To Download Notes

Key Concept: Industrial Application

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion that “Plaster of Paris is extensively used in making casts for setting broken bones” is true since its ability to form a hard, solid mass upon mixing with water makes it ideal for supporting and immobilizing broken bones. The reason given that “Plaster of Paris cannot be easily molded into desired shapes or forms” is false because one of the principal advantages of Plaster of Paris is its ease of molding when mixed with water before it hardens. This allows it to accurately conform to the shape needed, such as a limb needing support. Therefore, the correct response is (c).

Click To Download Notes

Key Concept: Natural Indicators

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that litmus is used as an indicator to test acids and bases by observing a color change. This statement is true because litmus paper is a common natural indicator for distinguishing between acidic and basic solutions.

The reason provides specific information on how litmus paper reacts with acids and bases: acids indeed turn blue litmus red, and bases turn red litmus blue. This correlation explains why litmus can be used to identify acids and bases effectively.

Since both the assertion and the reason are true, and the reason correctly explains the assertion, option (a) is correct.

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 Sensitivity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both phenolphthalein and methyl orange are synthetic indicators, but they have different pH ranges where they exhibit color changes. Phenolphthalein changes color from colorless to pink in the pH range of approximately 8.2 to 10, which is a narrow range compared to methyl orange, which changes color from red to yellow in the pH range of around 3.1 to 4.4. Due to its narrower pH transition range, phenolphthalein is considered more sensitive to slight pH changes compared to methyl orange. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: 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: Indicator Properties

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that phenolphthalein is colorless in acidic solutions. This is true because phenolphthalein only shows its pink color in basic solutions. The reason given is that phenolphthalein changes color within the pH range of 8.2 to 10, which is also true. However, while the reason correctly describes the behavior of phenolphthalein, it does not explain why phenolphthalein is colorless in acidic solutions, as this property is due to the structure of phenolphthalein in acidic conditions where it does not exhibit any color change until the solution becomes basic.

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: Simple Odour Test

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Vanilla essence is an olfactory indicator that loses its smell in basic solutions, but retains its smell in acidic solutions because it does not undergo any chemical reaction in an acidic environment to cause a change in odour.

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: 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: Real-world Application, Reaction Equations

a) $\text{Ca(OH)}_2 + \text{CO}_2 \rightarrow \text{CaCO}_3 + \text{H}_2\text{O}$

[Solution Description]

The objective is to identify the balanced chemical reaction occurring when calcium hydroxide reacts with carbon dioxide in water to produce calcium carbonate and water. We begin by writing the unbalanced equation:

$\text{Ca(OH)}_2 + \text{CO}_2 \rightarrow \text{CaCO}_3 + \text{H}_2\text{O}$

To balance, ensure equal numbers of each type of atom on both sides:

Start with calcium (Ca):

– 1 Ca on both sides.

Next, balance the oxygen atoms:

– On the left: 2 from $\text{Ca(OH)}_2$ + 2 from $\text{CO}_2$ = 4 oxygens.

– On the right: 3 in $\text{CaCO}_3$ + 1 in $\text{H}_2\text{O}$ = 4 oxygens.

Finally, check hydrogen:

– 2 H on both sides from $\text{H}_2\text{O}$.

Therefore, the balanced equation is:

$\text{Ca(OH)}_2 + \text{CO}_2 \rightarrow \text{CaCO}_3 + \text{H}_2\text{O}$

Click To Download Notes

Key Concept: Hydrogen Ion Production, Dilution Effects

b) 0.7

[Solution Description]

First calculate the initial concentration of $\text{H}^+$ ions before dilution using $c_1v_1 = c_2v_2$ where $c_1$ is the initial concentration and $v_1$ the initial volume:

$1 \times 100 = c_2 \times 500$

Solving for $c_2$:

$c_2 = \frac{100}{500} = 0.2 \, \text{M}$

The $\text{pH}$ is given by:

$\text{pH} = -\log_{10}(c_2)$

Calculate:

$\text{pH} = -\log_{10}(0.2) = 0.69897 \approx 0.7$

Click To Download Notes

Key Concept: pH Identification

c) Lemon juice

[Solution Description]

The pH scale measures how acidic or basic a solution is, with values below 7 indicating acidity. Among the options given:

– Lemon juice is known as an acidic substance.

– Detergent solution and baking soda are typically basic.

– Pure water is neutral.

Thus, lemon juice has a pH less than 7.

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: Observation Identification

c) Evolution of carbon dioxide gas

[Solution Description]

Sodium carbonate $(\text{Na}_2\text{CO}_3)$ reacts with dilute sulphuric acid $(\text{H}_2\text{SO}_4)$ to form sodium sulfate $(\text{Na}_2\text{SO}_4)$, carbon dioxide $(\text{CO}_2)$, and water $(\text{H}_2\text{O})$. The reaction is:

$\text{Na}_2\text{CO}_3 + \text{H}_2\text{SO}_4 \rightarrow \text{Na}_2\text{SO}_4 + \text{CO}_2 + \text{H}_2\text{O}$

You will observe the evolution of carbon dioxide gas bubbles.

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: 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: Zero Polynomial Degree

d) Assertion is false, but Reason is true.
[Solution Description]
The zero polynomial is unique in that it does not have a definitive degree, as it consists entirely of zeroes and does not fit standard polynomial classification. The statement about its degree being greater than any other is a theoretical convention used in advanced mathematics, but not definitive. Thus, the assertion is true, while the reason is true within certain contexts, but not universally accepted as an explanation for having no defined degree.

Key Concept: Degree of a Polynomial

c) 5
[Solution Description] The degree of a polynomial is defined as the highest power of the variable present in the expression. In the polynomial $4x^5 + 3x^3 – 2x + 7$, the term with the highest power of $x$ is $4x^5$. Therefore, the degree of this polynomial is 5.

Key Concept: Degree of a Polynomial, Algebraic Expressions

c) 7
[Solution Description]
To find the degree of the product of two polynomials, we add the degrees of the individual polynomials.
The degree of $P(x) = 4x^5 – x^4 + 7x^3 – x + 6$ is 5, as the highest power of $x$ in $P(x)$ is 5.
The degree of the polynomial $(2x^2 + 3)$ is 2, as the highest power of $x$ is 2.
Therefore, the degree of $Q(x) = P(x) \cdot (2x^2 + 3)$ will be:
$\text{Degree of } Q(x) = 5 + 2 = 7$
Hence, the degree of $Q(x)$ is 7.

Key Concept: Cubic Polynomial

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] In algebra, the Fundamental Theorem of Algebra states that a polynomial of degree $n$ can have at most $n$ roots. Therefore, a cubic polynomial, which is of degree 3, can have at most three real roots. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Nature of Roots

d) Assertion is false, but Reason is true.
[Solution Description] A cubic polynomial cannot have all non-real roots due to the fact that complex roots appear in conjugate pairs. Since there are only three roots, having all non-real roots would violate this rule. The assertion is false, but the reason is true because complex roots do indeed occur in conjugate pairs.

Key Concept: Graphical interpretation, axis intersection

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion correctly notes that the graph of a quadratic polynomial, a parabola, can intersect the x-axis at most twice, reflecting its maximum two real roots. The reason states that these intersections correspond to the roots of the polynomial equation, which is accurate. Here, the reason actually serves as an explanation for the assertion since the number of x-intercepts corresponds to the roots’ nature.

Key Concept: Identifying Coefficients

c) $a = 3$, $b = -6$, $c = 5$
[Solution Description] The general form of a quadratic equation is $ax^2 + bx + c$. For the expression $3x^2 – 6x + 5$, we have the coefficients:
$a = 3$, $b = -6$, and $c = 5$.

Key Concept: Linear Polynomials, Degree of Polynomial

Correct Answer option
[Solution Description] The assertion states that a linear polynomial has a degree of one, which is true by definition because a polynomial of degree one is called a linear polynomial. The reason provided states that the difference between successive values of a linear polynomial at integer points is constant. This is also true because when you substitute integers into a linear polynomial $ax + b$, the resulting values differ by $a$ for each unit increase in $x$. However, while both statements are true, the reason does not directly explain why a linear polynomial has a degree of one; rather, it describes a property of its values.

Key Concept: Definition of Linear Polynomials

b) $ax + b$
[Solution Description] A linear polynomial is defined by having a degree of one. The standard form of a linear polynomial is expressed as $ax + b$, where:
$a$ is a constant (coefficient of $x$),
$b$ is a constant (the y-intercept),
and $x$ is the variable.
The key feature of a linear polynomial is that the exponent of the variable $x$ is always 1.

Key Concept: Constant Polynomials

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion that a constant polynomial can never be zero is incorrect. A constant polynomial can indeed be zero if it simply equals zero everywhere, i.e., $f(x) = 0$.
On the other hand, non-zero constant polynomials have no roots, as they do not cross or touch the x-axis at any point. Therefore, reasoning is false.

Key Concept: Constant Polynomials

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
When differentiating a constant polynomial $f(x) = c$, we obtain $f'(x) = 0$, which is itself a constant polynomial. Hence, the assertion is true.
The reason given states that differentiating reduces the degree of a polynomial by one. While this generally applies to polynomials of degree greater than zero, a constant polynomial already has a degree of zero, and differentiation results in a zero polynomial. Thus, though the reasoning is true for non-constant polynomials, it does not specifically explain the assertion about constant polynomials.

Key Concept: Linear Polynomials, Perimeter Calculation

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the formula for calculating the perimeter of a rectangle forms a linear polynomial in terms of the variables $l$ and $w$. Indeed, if we expand $2(l + w)$, it simplifies to $2l + 2w$, where both terms are linear in either $l$ or $w$. Thus, this is correct.
The reason claims that any polynomial with two variables and degree one in each does not form a linear polynomial. This statement is false because a polynomial like $ax + by$, where $a$ and $b$ are constants, is also considered a linear polynomial as long as the degree of each variable term is one. Therefore, the reason is incorrect.
Hence, Assertion is true, but Reason is false.

Key Concept: Linear Polynomial Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a square with side $x$, the perimeter is $4x$. If the side length is doubled to $2x$, the new perimeter becomes $4(2x) = 8x$. Thus, doubling the side results in double the original perimeter, confirming that both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Linear Polynomials in Real-Life Contexts – Cost Calculation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In the assertion, the formula $200 + 50m$ represents the cost of playing $m$ matches, where Rs.200 is the fixed joining fee and Rs.50 is charged per match played. This representation is an example of a linear polynomial where one term is constant, and the other is dependent on the number of matches ($m$).
The reason states that a linear polynomial can include both fixed and variable components, as demonstrated in the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Linear Patterns, Linear Growth

c) 35
[Solution Description] The nth term of the sequence is given by $a_n = 4n – 5$. To find the 10th term, substitute $n = 10$ into the equation:
$a_{10} = 4(10) – 5$
$a_{10} = 40 – 5$
$a_{10} = 35$
Therefore, the 10th term is 35.

Key Concept: Linear Relationship: Slope Calculation

c) 3
[Solution Description] In the linear equation $y = ax + b$, the coefficient of $x$ represents the slope of the line. Hence, for the equation $y = 3x + 2$, the slope is 3.

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: Linear Equations, Intersection with Axes

c) -12
[Solution Description]
When the line intersects the x-axis, $y = 0$. Substitute the values into the equation $y = ax + b$:
$0 = 2 \times 6 + b$
Simplify to find $b$:
$0 = 12 + b$
Solving for $b$ gives:
$b = -12$
Thus, the y-intercept $b$ is $-12$.

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: Application of Linear Polynomials in Real-Life Contexts

b) $T(m) = 0.1m + 30$
[Solution Description] The total monthly cost is composed of a fixed monthly fee plus a variable cost depending on the number of text messages. This can be expressed as:
$T(m) = 0.1m + 30$
Where $0.1m$ is the charge for text messages and $30$ is the monthly subscription fee.

Key Concept: Predicting future stages

b) 15
[Solution Description] Using the formula $2n – 1$ for Stage 8, substitute $n = 8$:
Calculate:
$2 \times 8 – 1 = 16 – 1 = 15$
Thus, Stage 8 will have 15 square tiles.

Key Concept: Analyzing Stages with Specific Tile Counts

c) Assertion is true, but Reason is false.
[Solution Description]
Set up the equation using the number of tiles $21$:
$2n – 1 = 21$
Adding 1 to both sides gives:
$2n = 22$
Dividing both sides by 2 yields:
$n = 11$
Therefore, the stage containing 21 tiles is the 11th, not the 10th. The assertion is true, but the reason is false due to calculation error.

Key Concept: Generalising Linear Patterns

b) 29
[Solution Description]
To find the number of tiles in the 15th stage, substitute $n = 15$ into the expression $2n – 1$.
$2(15) – 1 = 30 – 1 = 29$
Therefore, there are 29 tiles in the 15th stage.

Key Concept: Determining Stage from Tile Count, Parallel Lines Concept

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To find the stage corresponding to 21 tiles, set $2n – 1 = 21$:
$2n – 1 = 21 \\ \\ 2n = 22 \\ \\ n = 11$
Thus, the assertion that 21 tiles exist at Stage 11 is true.
Parallel lines indeed share the same slope but differ by their y-intercepts. Hence, the reason holds true as well.
Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

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 Pattern, Linear Polynomial

c) Assertion is true, but Reason is false.
[Solution Description]
To find the number of tiles at Stage 10 using the correct expression $2n – 1$, we substitute $n = 10$:
$2 \times 10 – 1 = 20 – 1 = 19$
This confirms the Assertion as true.
However, the Reason states a different formula $3n – 1$. Using this incorrect formula for n = 10:
$3 \times 10 – 1 = 30 – 1 = 29$
Which is not equal to 19. Hence, the Reason is false.

Key Concept: Linear Growth, Finding Time for a Specific Value

c) 35 minutes
[Solution Description]
The problem involves finding the time taken for the water in the tank to reach a certain volume with a linear growth pattern. Initially, there are 200 liters.
Let $V(t)$ be the volume at time $t$, and since the rate of increase is 8 liters per minute, we have:
$V(t) = 200 + 8t$
Set $V(t) = 480$ to find $t$:
$480 = 200 + 8t$
Subtract 200 from both sides:
$280 = 8t$
Divide both sides by 8:
$t = \frac{280}{8} = 35$
So, it will take 35 minutes to reach 480 liters.

Key Concept: Applications of Linear Growth

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion accurately captures a linear growth where height increases steadily by 0.25 meters monthly. The reason incorrectly characterizes linear growth; instead, it should indicate constancy in increments, not progression in rate change.

Key Concept: Linear Decay Application, Amount Decreases Linearly

c) Assertion is true, but Reason is false.
[Solution Description]
For assertion: If debt reduces by equal amounts regularly, it is a case of linear decay, since the principal decreased uniformly over intervals.
For reason: Geometric progressions involve multiplicative factors, indicating exponential change, not linear.
While the assertion accurately captures linear decay, the reason is flawed, suggesting an exponential model instead. Therefore, Assertion is true, but Reason is false.

Key Concept: Linear Decay

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion is true because the phone’s value decreasing by `800 each year represents linear decay since it decreases by a fixed amount annually.
The reason describes exponential decay, not linear decay. Exponential decay involves a percentage decrease, not a constant amount.
Thus, Assertion is true, but Reason is false.

Key Concept: Y-intercept Understanding

c) 7
[Solution Description]
The y-intercept of a line in the equation $y = ax + b$ is given by the term $b$. In this case, $b = 7$.
Therefore, the y-intercept is 7.

Key Concept: Slopes of Linear Functions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that the slope of the line $y = ax + b$ is constant, which is true since $a$ represents the slope and does not change with different values of $x$. The reason correctly describes what a slope signifies; it measures how steep a line is. Therefore, both statements are true and the reason correctly explains the assertion.

Key Concept: Finding Values of $a$ and $b$

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the values of $a$ and $b$, we use the given points $(10, 400)$ and $(14, 500)$ in the linear equation $y = ax + b$.
1. First, calculate the slope $a$:
$a = \frac{500 – 400}{14 – 10} = \frac{100}{4} = 25$
2. Substitute $a = 25$ into one of the points to solve for $b$. Using $(10, 400)$:
$400 = 25(10) + b$
$b = 400 – 250 = 150$
Thus, the equation $y = 25x + 150$ accurately describes the relationship.

Key Concept: Temperature Conversion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion is true as there exists a linear correlation between °C and °F that can be derived from the given data points (freezing: $0^\circ C$, $32^\circ F$; boiling: $100^\circ C$, $212^\circ F$). For the reason, $a$ does represent the ratio of differences $(\frac{100 – 0}{212 – 32}) = \frac{5}{9}$ which confirms the correct explanation. So, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Slope of Line

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both lines are written in the form $y = ax + b$, where $a$ represents the slope. For $y = 3x + 2$, the slope $a = 3$, and for $y = x + 2$, the slope $a = 1$. Since $3 > 1$, the line $y = 3x + 2$ is indeed steeper than the line $y = x + 2$. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Effect of Slope Value

b) They are parallel with identical slopes.
[Solution Description]
Both equations have the same slope ($a = 1$), so they are equally inclined to the axes. However, they have different y-intercepts. The first equation cuts the y-axis at $-2$, while the second one cuts it at $2$. This means they are parallel and equidistant from each other along the x-axis.

Key Concept: Effect of Slope on Steepness

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
If $a < 1$, it implies the slope of the line $y = ax$ is less than the slope of the line $y = x$. Hence, such a line will be less steep compared to $y = x$ as stated in the assertion. The reason provided accurately describes this relationship between the value of $a$ and the steepness, thereby supporting the assertion. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Effect of Changing ‘a’ in $y = ax$

b) The line becomes more steep.
[Solution Description] Increasing the value of $a$ in the equation $y = ax$ results in a steeper line because $a$ represents the slope of the line, and a larger slope makes the line steeper.

Key Concept: Comparing Lines with Identical Slopes

c) $|c + 8|$
[Solution Description]
The slopes of both lines are $a = 5$, indicating that they are parallel. To find the distance between their y-intercepts, calculate the difference in their $b$ values. For $L_1$, the y-intercept is $c$, and for $L_2$, it is $-8$. The distance between the y-intercepts is $\lvert c + 8 \rvert$. Given no specific value for $c$, the answer would depend on $c$. However, without any additional information about $c$, we assume the simplest case where $c=0$, giving $\lvert -8 \rvert = 8$.

Key Concept: y-Intercept of a Line

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When $a = 0$, the equation becomes $y = b$, which means that the line is horizontal and parallel to the x-axis. Here, the value of $b$ indeed identifies the line’s vertical position on the y-axis. Thus, both statements are true, and the reason correctly explains the assertion.

Key Concept: Relationship Between Slope and Line Direction

b) The line turns from rising to falling
[Solution Description] When a line’s slope changes from positive to negative while the intercept $b$ remains zero, it implies a shift from an upward-sloping line ($a > 0$) to a downward-sloping line ($a < 0$). As such, the line initially goes up from left to right and later goes down from left to right once the slope turns negative.

Key Concept: Comparing Slopes

b) The first line is steeper than the second
[Solution Description] The slope of the first line is 4, and the slope of the second line is 1. Since 4 > 1, the first line is steeper than the second one.

Key Concept: Y-intercept change impact

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] For these parallel lines, both possessing the identical slope of 5, the vertical distance remains consistent over any shared x-value. The difference between the y-intercepts determines this y-axis separation, which is $|12 – 7| = 5$. The reason correctly asserts that such lines maintain uniform spacing due to their parallel nature, ensuring both statements hold true, with the reason effectively explaining the assertion. Correct answer: Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Role of ‘b’ as Y-Intercept

b) It shifts downwards by 6 units
[Solution Description] Both equations have the same slope of 5. Therefore, both lines are parallel. The change in the y-intercept from 2 to -4 shifts the second line down by 6 units on the y-axis.

Key Concept: Linear Growth Represented by Positive Slope

c) 3
[Solution Description] The general form of a linear equation is given by $y = ax + b$, where $a$ represents the slope. In this case, the equation is $y = 3x + 5$. By comparing this to the general form, we can see that $a = 3$. Therefore, the slope of the line is 3.

Key Concept: Slope Interpretation

b) $y = -3x – 1$
[Solution Description] To determine the steepest negative slope, we need to compare the slope coefficients (the values of $a$ in $y = ax + b$) from each option. The more negative the slope coefficient, the steeper the negative slope.
Comparing the slopes:
For $y = -2x + 3$, the slope is -2.
For $y = -3x – 1$, the slope is -3.
For $y = -1x + 2$, the slope is -1.
For $y = -0.5x + 4$, the slope is -0.5.
Since -3 is the smallest value among these, $y = -3x – 1$ has the steepest negative slope.