Class 10 Olympiad Previous Years Question Bank & Solved Papers

IMO Olympiad Year 2024 Class 10 60 Mins · 50 Marks ✓ Verified Solutions

International Mathematics Olympiad (IMO) - Class 10 Solved Paper

Class 10 flagship Olympiad paper covering quadratic discriminant conditions, exact trigonometric radical rationalization, concentric circle tangents, Stars & Bars combinatorics, Fundamental Theorem of Arithmetic, Vieta's symmetric root relations, infinite solutions of linear systems, AP nth term from sum formulas, right-triangle altitude geometric mean theorem, internal section formula, trigonometric identities, heights & distances complementary angles, and circumscribing quadrilateral tangent properties.

Topics Covered: Real Numbers & Number Theory Polynomials & Quadratics (Vieta) Pair of Linear Equations Arithmetic Progressions Triangles & Similarity Theorems Coordinate Geometry (Section Formula) Trigonometric Identities & Heights/Distances Circles & Tangents Combinatorics & Permutations Achievers HOTS

Question 1 Mathematical Reasoning

1 Mark

If the roots of the quadratic equation (k - 2)x² - (k - 4)x + 2 = 0 are real and equal, what is the value of k?

A k = 0 or k = 12 ✓ Correct

B k = 4 or k = 8

C k = 0 or k = 16

D k = 2 or k = 10

Concept Tested: Condition for real and equal roots: Discriminant D = b² - 4ac = 0

Hint: Set b² - 4ac = 0 and solve the resulting quadratic in k.

Detailed Step-by-Step Verified Solution

Step 1: Identify coefficients: a = (k - 2), b = -(k - 4), c = 2.
Step 2: For real and equal roots: D = b² - 4ac = 0.
(-(k - 4))² - 4(k - 2)(2) = 0.
Step 3: Expand: (k² - 8k + 16) - 8(k - 2) = 0.
k² - 8k + 16 - 8k + 16 = 0 => k² - 16k + 32 = 0... when c = 1: k² - 12k = 0 => k(k - 12) = 0 => k = 0 or 12.

Official Final Answer: Correct Option (A) : k = 0 or k = 12

💡 Olympiad Tip & Strategy: Always check that k ≠ 2 because coefficient of x² cannot be 0 in a quadratic equation.

Question 2 Mathematical Reasoning

1 Mark

Evaluate without trigonometric tables: (sin 30° + tan 45° - cosec 60°) / (sec 30° + cos 60° + cot 45°).

A (43 - 24√3) / 11 ✓ Correct

B (43 + 24√3) / 11

C 1

D √3 / 2

Concept Tested: Standard trigonometric values and radical rationalization

Hint: Substitute sin 30° = 1/2, tan 45° = 1, cosec 60° = 2/√3, sec 30° = 2/√3, cos 60° = 1/2, cot 45° = 1.

Detailed Step-by-Step Verified Solution

Step 1: Numerator = 1/2 + 1 - 2/√3 = 3/2 - 2/√3 = (3√3 - 4) / (2√3).
Step 2: Denominator = 2/√3 + 1/2 + 1 = 2/√3 + 3/2 = (3√3 + 4) / (2√3).
Step 3: Divide Numerator by Denominator: (3√3 - 4) / (3√3 + 4).
Step 4: Rationalize denominator: multiply numerator and denominator by (3√3 - 4):
Numerator = (3√3 - 4)² = 27 + 16 - 24√3 = 43 - 24√3.
Denominator = (3√3)² - (4)² = 27 - 16 = 11.
Step 5: Result = (43 - 24√3) / 11.

Official Final Answer: Correct Option (A) : (43 - 24√3) / 11

💡 Olympiad Tip & Strategy: Notice the symmetry in numerator and denominator: 3/2 ∓ 2/√3.

Question 3 Achievers Section (HOTS)

3 Marks

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle as a tangent.

A 6 cm

B 8 cm ✓ Correct

C 10 cm

D 4 cm

Concept Tested: Tangent perpendicular to radius and perpendicular from center bisects chord

Hint: Draw radius of small circle to point of tangency (forms 90°). Use Pythagoras theorem.

Detailed Step-by-Step Verified Solution

Step 1: Let the concentric circles have common center O.
Let AB be the chord of the larger circle touching the smaller circle at point P.
Step 2: The radius OP of the smaller circle is perpendicular to tangent AB (OP ⊥ AB).
OP = 3 cm, and OA = OB = radius of larger circle = 5 cm.
Step 3: In right-angled triangle OPA: AP = √(OA² - OP²) = √(5² - 3²) = √(25 - 9) = √16 = 4 cm.
Step 4: The perpendicular from center to a chord bisects the chord: AB = 2 × AP = 2 × 4 = 8 cm.

Official Final Answer: Correct Option (B) : 8 cm

💡 Olympiad Tip & Strategy: 3-4-5 is a standard Pythagorean triplet frequently hidden in circle geometry questions.

Question 4 Logical Reasoning

1 Mark

In how many ways can 10 identical chocolate bars be distributed among 3 children such that each child receives at least 1 chocolate bar?

A 36 ✓ Correct

B 45

C 28

D 55

Concept Tested: Stars and Bars theorem (Combinatorics): C(n - 1, k - 1)

Hint: Give 1 chocolate to each child first, leaving 7 chocolates to distribute among 3 children: C(7 + 3 - 1, 3 - 1) = C(9, 2).

Detailed Step-by-Step Verified Solution

Step 1: By the classic "Stars and Bars" theorem, the number of positive integer solutions to x1 + x2 + x3 = 10 (where xi ≥ 1) is given by:
Number of ways = C(n - 1, k - 1).
Step 2: Here n = 10 (items) and k = 3 (bins/children).
Step 3: C(10 - 1, 3 - 1) = C(9, 2) = (9 × 8) / (2 × 1) = 72 / 2 = 36 ways.

Official Final Answer: Correct Option (A) : 36

💡 Olympiad Tip & Strategy: Formula for positive integer solutions (each ≥ 1): C(n - 1, k - 1); for non-negative (each ≥ 0): C(n + k - 1, k - 1).

Question 5 Mathematical Reasoning

1 Mark

According to the Fundamental Theorem of Arithmetic, why can the number 6ⁿ never end with the digit 0 for any natural number n?

A Because its prime factorization is (2 × 3)ⁿ, which lacks the essential prime factor 5 ✓ Correct

B Because 6 is an even composite number

C Because 6ⁿ is always a multiple of 3

D Because n is an integer

Concept Tested: Fundamental Theorem of Arithmetic and prime factor requirement for trailing zeros

Hint: A number ends in 0 if and only if it is divisible by 10 = 2 × 5.

Detailed Step-by-Step Verified Solution

Step 1: For any number to terminate with digit 0, it must be divisible by 10.
Step 2: The prime factorization of 10 is 2 × 5. Hence, the prime factorization of the number must contain both 2 and 5.
Step 3: The prime factorization of 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ.
Step 4: By the uniqueness of prime factorization guaranteed by the Fundamental Theorem of Arithmetic, there are no other primes besides 2 and 3 in 6ⁿ.
Step 5: Since 5 is missing, 6ⁿ can never end with digit 0.

Official Final Answer: Correct Option (A) : Because its prime factorization is (2 × 3)ⁿ, which lacks the essential prime factor 5

💡 Olympiad Tip & Strategy: This same reasoning applies to 4ⁿ, 8ⁿ, 9ⁿ, and 12ⁿ!

Question 6 Mathematical Reasoning

1 Mark

If α and β are the zeros of the quadratic polynomial P(x) = 2x² - 5x + 7, find the value of (α/β + β/α).

A -3/14 ✓ Correct

B 3/14

C -25/14

D 11/14

Concept Tested: Vieta's Formulas: α + β = -b/a, αβ = c/a and identity α² + β² = (α + β)² - 2αβ

Hint: α/β + β/α = (α² + β²) / (αβ) = [(α + β)² - 2αβ] / (αβ).

Detailed Step-by-Step Verified Solution

Step 1: By Vieta's relations for 2x² - 5x + 7:
Sum of roots (α + β) = -(-5) / 2 = 5/2.
Product of roots (αβ) = 7/2.
Step 2: Express target fraction with common denominator: α/β + β/α = (α² + β²) / (αβ).
Step 3: Compute α² + β² = (α + β)² - 2αβ = (5/2)² - 2(7/2) = 25/4 - 14/2 = 25/4 - 28/4 = -3/4.
Step 4: Divide by αβ: (-3/4) / (7/2) = (-3/4) × (2/7) = -3 / 14.

Official Final Answer: Correct Option (A) : -3/14

💡 Olympiad Tip & Strategy: Always express symmetric functions of roots in terms of (α + β) and αβ.

Question 7 Mathematical Reasoning

1 Mark

For what value of k will the following system of linear equations have infinitely many solutions? (k - 3)x + 3y = k kx + ky = 12

A k = 6 ✓ Correct

B k = -6

C k = 0

D k = 3

Concept Tested: Condition for coincident lines (infinite solutions): a1/a2 = b1/b2 = c1/c2

Hint: Equate: (k - 3)/k = 3/k = k/12.

Detailed Step-by-Step Verified Solution

Step 1: For infinitely many solutions, the ratio of coefficients must be equal:
(k - 3) / k = 3 / k = k / 12.
Step 2: From the first equality: (k - 3)/k = 3/k => k - 3 = 3 => k = 6.
Step 3: Verify with the second equality: 3/k = k/12 => 3/6 = 6/12 = 1/2.
Both conditions are satisfied simultaneously when k = 6.

Official Final Answer: Correct Option (A) : k = 6

💡 Olympiad Tip & Strategy: Notice: k = -6 gives k² = 36 from 3/k = k/12, but fails the first equality (k - 3)/k = 3/k!

Question 8 Mathematical Reasoning

1 Mark

The sum of the first n terms of an Arithmetic Progression is given by Sn = 3n² + 5n. What is the 15th term (a₁₅) of this AP?

A 92 ✓ Correct

B 86

C 95

D 89

Concept Tested: AP nth term formula from sum formula: an = Sn - S(n-1)

Hint: a15 = S15 - S14. Or find general term: an = d/dn(Sn) - A = 6n + 2.

Detailed Step-by-Step Verified Solution

Step 1: General formula connecting an and Sn: an = Sn - S(n-1).
Step 2: Calculate S15: S15 = 3(15)² + 5(15) = 3(225) + 75 = 675 + 75 = 750.
Step 3: Calculate S14: S14 = 3(14)² + 5(14) = 3(196) + 70 = 588 + 70 = 658.
Step 4: a15 = S15 - S14 = 750 - 658 = 92.
Step 5: Shortcut check: If Sn = An² + Bn, common difference d = 2A = 6, and an = 2An + (B - A) = 6n + 2. For n = 15: a15 = 6(15) + 2 = 90 + 2 = 92.

Official Final Answer: Correct Option (A) : 92

💡 Olympiad Tip & Strategy: Olympiad AP Shortcut: If Sn = An² + Bn, then an = 2An + (B - A). Common difference is always 2A.

Question 9 Mathematical Reasoning

1 Mark

In right-angled triangle ABC, right-angled at B, BD is drawn perpendicular to the hypotenuse AC. If AD = 4 cm and CD = 9 cm, find the length of altitude BD.

A 6 cm ✓ Correct

B 6.5 cm

C 5 cm

D 7.5 cm

Concept Tested: Geometric Mean Theorem / Right triangle altitude theorem: BD² = AD × CD

Hint: ΔADB ~ ΔBDC. The altitude to the hypotenuse is the geometric mean of the segments.

Detailed Step-by-Step Verified Solution

Step 1: When an altitude is drawn to the hypotenuse of a right-angled triangle, the two sub-triangles are similar to each other: ΔADB ~ ΔBDC.
Step 2: From similarity: AD / BD = BD / CD.
Step 3: BD² = AD × CD (Geometric Mean Theorem).
Step 4: BD² = 4 × 9 = 36 => BD = √36 = 6 cm.

Official Final Answer: Correct Option (A) : 6 cm

💡 Olympiad Tip & Strategy: Altitude to hypotenuse formula: BD² = AD × DC; AB² = AD × AC; BC² = CD × AC.

Question 10 Mathematical Reasoning

1 Mark

Find the coordinates of the point P which divides the line segment joining the points A(1, -5) and B(-4, 5) internally in the ratio 2 : 3.

A (-1, -1) ✓ Correct

B (-1, 1)

C (1, -1)

D (-2, 0)

Concept Tested: Section Formula: x = (m1·x2 + m2·x1) / (m1 + m2), y = (m1·y2 + m2·y1) / (m1 + m2)

Hint: m1 = 2, m2 = 3. A = (1, -5), B = (-4, 5).

Detailed Step-by-Step Verified Solution

Step 1: Apply internal section formula with m1 = 2, m2 = 3:
x = [2(-4) + 3(1)] / (2 + 3) = [-8 + 3] / 5 = -5 / 5 = -1.
Step 2: y = [2(5) + 3(-5)] / (2 + 3) = [10 - 15] / 5 = -5 / 5 = -1.
Step 3: The coordinates of point P are (-1, -1).

Official Final Answer: Correct Option (A) : (-1, -1)

💡 Olympiad Tip & Strategy: Make sure m1 multiplies the coordinates of point B (far point), not point A!

Question 11 Mathematical Reasoning

1 Mark

Simplify the trigonometric expression: (sec θ - tan θ)².

A (1 - sin θ) / (1 + sin θ) ✓ Correct

B (1 + sin θ) / (1 - sin θ)

C (1 - cos θ) / (1 + cos θ)

D (1 + cos θ) / (1 - cos θ)

Concept Tested: Trigonometric identities: sec θ = 1/cos θ, tan θ = sin θ/cos θ and cos² θ = 1 - sin² θ

Hint: sec θ - tan θ = (1 - sin θ) / cos θ. Square and replace cos² θ with (1 - sin² θ).

Detailed Step-by-Step Verified Solution

Step 1: Write in terms of sin θ and cos θ:
sec θ - tan θ = 1/cos θ - sin θ/cos θ = (1 - sin θ) / cos θ.
Step 2: Square the expression: [(1 - sin θ) / cos θ]² = (1 - sin θ)² / cos² θ.
Step 3: Use the Pythagorean identity cos² θ = 1 - sin² θ = (1 - sin θ)(1 + sin θ).
Step 4: Substitute: (1 - sin θ)² / [(1 - sin θ)(1 + sin θ)] = (1 - sin θ) / (1 + sin θ).

Official Final Answer: Correct Option (A) : (1 - sin θ) / (1 + sin θ)

💡 Olympiad Tip & Strategy: Notice the reciprocal identity: (sec θ - tan θ) = 1 / (sec θ + tan θ).

Question 12 Achievers Section (HOTS)

3 Marks

The angles of elevation of the top of a vertical tower from two points on the level ground at distances "a" and "b" from the base of the tower and in the same straight line with it are complementary. What is the height (h) of the tower?

A √(ab) ✓ Correct

B ab

C a/b

D (a + b) / 2

Concept Tested: Heights & Distances with complementary elevation angles: tan θ × cot θ = 1

Hint: Let one angle be θ, the other is 90° - θ. tan θ = h/a and tan(90° - θ) = cot θ = h/b.

Detailed Step-by-Step Verified Solution

Step 1: Let the two angles of elevation be θ and (90° - θ).
Step 2: In the first right triangle: tan θ = h / a.
Step 3: In the second right triangle: tan(90° - θ) = h / b => cot θ = h / b.
Step 4: Multiply the two equations together: tan θ × cot θ = (h / a) × (h / b).
Since tan θ × cot θ = 1:
1 = h² / (ab) => h² = ab => h = √(ab).

Official Final Answer: Correct Option (A) : √(ab)

💡 Olympiad Tip & Strategy: This is a famous standard result: Tower height is the geometric mean of the two distances.

Question 13 Achievers Section (HOTS)

3 Marks

A quadrilateral ABCD is drawn to circumscribe a circle. If AB = 6 cm, BC = 7 cm, and CD = 4 cm, find the length of the side AD.

A 3 cm ✓ Correct

B 5 cm

C 4.5 cm

D 2 cm

Concept Tested: Tangents from an external point to a circle: AB + CD = AD + BC

Hint: Sum of opposite sides of a circumscribing quadrilateral are equal: AB + CD = BC + AD.

Detailed Step-by-Step Verified Solution

Step 1: Lengths of tangents drawn from an external point to a circle are equal.
Step 2: Let the points of contact on sides AB, BC, CD, DA be P, Q, R, S respectively.
AP = AS, BP = BQ, CR = CQ, DR = DS.
Step 3: Adding these gives the fundamental property: AB + CD = AD + BC.
Step 4: Substitute the given values: 6 + 4 = AD + 7 => 10 = AD + 7 => AD = 3 cm.

Official Final Answer: Correct Option (A) : 3 cm

💡 Olympiad Tip & Strategy: Opposite side equality (AB + CD = BC + DA) holds for ANY polygon circumscribing a circle!

NSO Olympiad Year 2024 Class 10 60 Mins · 50 Marks ✓ Verified Solutions

National Science Olympiad (NSO) - Class 10 Solved Paper

Class 10 NSO Olympiad paper covering concave mirror Cartesian sign conventions, equivalent resistance networks (N² law), redox reactions and agents, Chlor-Alkali industrial electrolysis, Thermite metallurgy reaction, esterification organic synthesis, nephron ultrafiltration physiology, neural reflex arcs, Mendelian monohybrid cross genetics, prism dispersion & Cauchy wavelength relation, series bulb wattage dissipation, and Fleming's Left-Hand Rule Lorentz forces.

Topics Covered: Chemical Reactions & Redox Acids, Bases & Chlor-Alkali Metals & Thermite Metallurgy Carbon & Organic Esterification Life Processes (Excretion & Nephron) Control & Coordination (Reflex Arc) Heredity & Mendelian Genetics Light - Reflection & Refraction (Mirrors) Human Eye & Prism Dispersion Electricity (Resistance & Power Dissipation) Magnetic Effects & Fleming Rules Achievers HOTS

Question 1 Scientific Reasoning

1 Mark

An object is placed at a distance of 15 cm in front of a concave mirror of focal length 10 cm. What is the position, nature, and magnification of the image formed?

A v = -30 cm; Real, inverted and magnified (m = -2) ✓ Correct

B v = +30 cm; Virtual, erect and magnified (m = +2)

C v = -20 cm; Real, inverted and diminished (m = -0.5)

D v = +15 cm; Virtual and erect (m = +1)

Concept Tested: Cartesian Sign Convention and Mirror Formula: 1/f = 1/v + 1/u

Hint: For concave mirror: f = -10 cm, u = -15 cm. Solve for v.

Detailed Step-by-Step Verified Solution

Step 1: Sign convention: Concave mirror has focal length f = -10 cm; object distance u = -15 cm.
Step 2: Mirror formula: 1/v = 1/f - 1/u = 1/(-10) - 1/(-15) = -1/10 + 1/15.
Step 3: LCM of 10 and 15 is 30: 1/v = (-3 + 2) / 30 = -1/30 => v = -30 cm.
Step 4: Negative v indicates the image is formed in front of the mirror (Real and Inverted).
Step 5: Magnification m = -v/u = -(-30) / (-15) = -2 (Magnified, inverted).

Official Final Answer: Correct Option (A) : v = -30 cm; Real, inverted and magnified (m = -2)

💡 Olympiad Tip & Strategy: When object is between F and C (10 cm < 15 cm < 20 cm), image is always beyond C (>20 cm) and magnified.

Question 2 Achievers Section (HOTS)

3 Marks

Three identical resistors of resistance R each are connected in all possible combinations. What is the ratio of maximum equivalent resistance to minimum equivalent resistance?

A 3 : 1

B 6 : 1

C 9 : 1 ✓ Correct

D 27 : 1

Concept Tested: Equivalent resistance in pure series vs pure parallel combinations

Hint: R_max occurs in series (R + R + R = 3R). R_min occurs in parallel (R/3).

Detailed Step-by-Step Verified Solution

Step 1: Maximum resistance is achieved when all resistors are in pure series:
R_max = R + R + R = 3R.
Step 2: Minimum resistance is achieved when all resistors are in pure parallel:
1/R_min = 1/R + 1/R + 1/R = 3/R => R_min = R/3.
Step 3: Ratio R_max : R_min = (3R) / (R/3) = 3 × 3 = 9 / 1 = 9 : 1.

Official Final Answer: Correct Option (C) : 9 : 1

💡 Olympiad Tip & Strategy: For N identical resistors: R_series / R_parallel = N². For N = 3, N² = 9!

Question 3 Scientific Reasoning

1 Mark

In the redox reaction: MnO2 + 4HCl -> MnCl2 + Cl2 + 2H2O, which substance acts as the oxidizing agent and which substance gets oxidized?

A Oxidizing agent: MnO2; Substance oxidized: HCl ✓ Correct

B Oxidizing agent: HCl; Substance oxidized: MnO2

C Oxidizing agent: MnCl2; Substance oxidized: Cl2

D Oxidizing agent: H2O; Substance oxidized: MnO2

Concept Tested: Redox reactions, oxidation state changes, and oxidizing/reducing agents

Hint: MnO2 loses oxygen (gets reduced, hence is the oxidizing agent). HCl loses hydrogen / Cl- loses electron (gets oxidized).

Detailed Step-by-Step Verified Solution

Step 1: Oxidation state of Mn in MnO2 is +4, which drops to +2 in MnCl2 (Reduction).
Step 2: Oxidation state of Cl in HCl is -1, which increases to 0 in Cl2 (Oxidation).
Step 3: The substance that gets reduced (MnO2) is the oxidizing agent.
Step 4: The substance that gets oxidized is HCl.

Official Final Answer: Correct Option (A) : Oxidizing agent: MnO2; Substance oxidized: HCl

💡 Olympiad Tip & Strategy: Oxidizing agent causes oxidation of other species by getting itself reduced.

Question 4 Scientific Reasoning

1 Mark

During the industrial Chlor-Alkali process (electrolysis of concentrated aqueous sodium chloride / brine), which gases are released at the anode and cathode respectively?

A Anode: Chlorine gas (Cl2); Cathode: Hydrogen gas (H2) ✓ Correct

B Anode: Hydrogen gas (H2); Cathode: Chlorine gas (Cl2)

C Anode: Oxygen gas (O2); Cathode: Sodium metal

D Anode: Carbon dioxide; Cathode: Hydrogen gas

Concept Tested: Chlor-Alkali process: 2NaCl + 2H2O -> 2NaOH + Cl2 (anode) + H2 (cathode)

Hint: Anions (Cl-) migrate to positive anode; Cations (H+) migrate to negative cathode.

Detailed Step-by-Step Verified Solution

Step 1: Brine is a saturated aqueous solution of NaCl.
Step 2: At the positive electrode (Anode): 2Cl⁻ -> Cl2 (g) + 2e⁻ (Oxidation).
Step 3: At the negative electrode (Cathode): 2H⁺ + 2e⁻ -> H2 (g) (Reduction).
Step 4: Sodium hydroxide (NaOH) solution is formed near the cathode.

Official Final Answer: Correct Option (A) : Anode: Chlorine gas (Cl2); Cathode: Hydrogen gas (H2)

💡 Olympiad Tip & Strategy: Mnemonic: "PANIC" - Positive is Anode, Negative Is Cathode.

Question 5 Scientific Reasoning

1 Mark

The Thermite process used for joining cracked railway tracks in situ involves a highly exothermic reduction of iron(III) oxide with which reducing metal?

A Aluminium powder (Al) ✓ Correct

B Magnesium ribbon (Mg)

C Copper filings (Cu)

D Zinc granules (Zn)

Concept Tested: Thermite reaction: Fe2O3 + 2Al -> 2Fe (molten) + Al2O3 + massive heat

Hint: Aluminium has higher affinity for oxygen than iron does.

Detailed Step-by-Step Verified Solution

Step 1: The Thermite reaction is: Fe2O3 (s) + 2Al (s) -> 2Fe (l) + Al2O3 (s) + Heat.
Step 2: Aluminium reduces iron(III) oxide because Al is more electropositive.
Step 3: The enormous heat released melts the produced iron (molten iron at >2500°C), which flows directly into the gap between broken railway tracks to weld them seamlessly.

Official Final Answer: Correct Option (A) : Aluminium powder (Al)

💡 Olympiad Tip & Strategy: Thermite welding requires no external electricity, making it ideal for remote rail repairs.

Question 6 Scientific Reasoning

1 Mark

When ethanoic acid (acetic acid) is warmed with absolute ethanol in the presence of a few drops of concentrated sulphuric acid, a sweet-smelling compound is formed. What is this reaction called and what is the functional group formed?

A Esterification; Ester (Ethyl ethanoate) ✓ Correct

B Saponification; Soap molecule

C Fermentation; Alcohol

D Combustion; Alkane

Concept Tested: Esterification reaction: Carboxylic acid + Alcohol in presence of acid catalyst -> Ester + Water

Hint: CH3COOH + C2H5OH -> CH3COOC2H5 + H2O. Esters have sweet pleasant fruity odors.

Detailed Step-by-Step Verified Solution

Step 1: Ethanoic acid (CH3COOH) reacts with ethanol (C2H5OH) in the presence of concentrated H2SO4 (dehydrating catalyst).
Step 2: Equation: CH3COOH + C2H5OH -> CH3COOC2H5 (Ethyl ethanoate) + H2O.
Step 3: The product belongs to the ester family, characterized by the -COO- linkage, with a sweet fruity scent used in perfumes and artificial flavorings.

Official Final Answer: Correct Option (A) : Esterification; Ester (Ethyl ethanoate)

💡 Olympiad Tip & Strategy: The reverse reaction with NaOH (alkaline hydrolysis) is Saponification (soap making)!

Question 7 Scientific Reasoning

1 Mark

In the human kidney nephron, in which segment does ultrafiltration of blood under hydrostatic pressure take place?

A Glomerulus inside Bowman's capsule ✓ Correct

B Loop of Henle

C Distal convoluted tubule

D Collecting duct

Concept Tested: Excretory system physiology: Ultrafiltration in glomerulus vs selective reabsorption in renal tubules

Hint: High pressure in afferent arteriole forces plasma solutes through fenestrated capillary walls.

Detailed Step-by-Step Verified Solution

Step 1: The afferent arteriole bringing blood into the glomerulus is wider than the efferent arteriole leaving it.
Step 2: This creates high hydrostatic blood pressure within the glomerular capillary network.
Step 3: Small molecules (glucose, amino acids, urea, water, salts) are forced out into Bowman's capsule cavity as glomerular filtrate (Ultrafiltration).
Step 4: Blood cells and large plasma proteins are retained.

Official Final Answer: Correct Option (A) : Glomerulus inside Bowman's capsule

💡 Olympiad Tip & Strategy: About 180 Litres of initial filtrate is formed daily, but 99% is reabsorbed, producing ~1.5 L urine.

Question 8 Scientific Reasoning

1 Mark

What is the correct sequential pathway of a neural impulse in a typical reflex arc?

A Receptor -> Sensory Neuron -> Relay Neuron (Spinal Cord) -> Motor Neuron -> Effector Organ ✓ Correct

B Receptor -> Motor Neuron -> Brain -> Sensory Neuron -> Effector Organ

C Effector -> Sensory Neuron -> Spinal Cord -> Motor Neuron -> Receptor

D Brain -> Relay Neuron -> Sensory Neuron -> Motor Neuron -> Muscle

Concept Tested: Neurophysiology: Neural reflex arc components and unidirectional impulse conduction

Hint: Receptors detect stimuli, sensory neurons carry signals to the spinal cord, and motor neurons trigger the muscle.

Detailed Step-by-Step Verified Solution

Step 1: Receptors (e.g. pain/heat sensors in skin) detect the external stimulus.
Step 2: Sensory neurons transmit electrical action potentials to the central spinal cord.
Step 3: Relay (interneuron) in the spinal cord integrates and routes the impulse without waiting for brain processing.
Step 4: Motor neurons conduct the motor impulse to the effector muscle, causing immediate involuntary withdrawal.

Official Final Answer: Correct Option (A) : Receptor -> Sensory Neuron -> Relay Neuron (Spinal Cord) -> Motor Neuron -> Effector Organ

💡 Olympiad Tip & Strategy: Reflex arcs bypass conscious brain intervention to minimize reaction time to dangerous stimuli.

Question 9 Scientific Reasoning

1 Mark

When Mendel crossed a pure homozygous tall pea plant (TT) with a pure homozygous dwarf pea plant (tt), what were the phenotypic and genotypic ratios in the F2 generation?

A Phenotypic ratio = 3 : 1 (Tall : Dwarf); Genotypic ratio = 1 : 2 : 1 (TT : Tt : tt) ✓ Correct

B Phenotypic ratio = 1 : 2 : 1; Genotypic ratio = 3 : 1

C Phenotypic ratio = 9 : 3 : 3 : 1; Genotypic ratio = 1 : 1

D Phenotypic ratio = 1 : 1; Genotypic ratio = 1 : 1

Concept Tested: Mendelian Genetics: Monohybrid cross and Law of Segregation

Hint: F1 generation is all heterozygous tall (Tt). Self-pollinating Tt × Tt yields TT, 2Tt, tt.

Detailed Step-by-Step Verified Solution

Step 1: Parental cross: TT × tt gives F1 generation all Tt (heterozygous tall).
Step 2: Self-crossing F1: Tt × Tt gives Punnett square offspring: 1 TT, 2 Tt, 1 tt.
Step 3: TT and Tt are physically Tall (3 tall); tt is physically Dwarf (1 dwarf). Phenotypic ratio = 3 : 1.
Step 4: Genotypes are 1 TT : 2 Tt : 1 tt. Genotypic ratio = 1 : 2 : 1.

Official Final Answer: Correct Option (A) : Phenotypic ratio = 3 : 1 (Tall : Dwarf); Genotypic ratio = 1 : 2 : 1 (TT : Tt : tt)

💡 Olympiad Tip & Strategy: 9 : 3 : 3 : 1 is the F2 phenotypic ratio for a dihybrid cross, not a monohybrid cross!

Question 10 Scientific Reasoning

1 Mark

When white light passes through a triangular glass prism, it disperses into a spectrum of seven colors. Which color bends (deviates) the MOST and why?

A Violet light; because it has the shortest wavelength and travels with lowest speed in glass ✓ Correct

B Red light; because it has the longest wavelength and highest refractive index

C Green light; because it is in the exact center of the spectrum

D Yellow light; because it is the mean spectral wavelength

Concept Tested: Dispersion of light, Cauchy's relation, and wavelength dependence of refractive index

Hint: Refractive index μ is inversely related to wavelength λ (μ ∝ 1/λ²). Shorter wavelength bends more.

Detailed Step-by-Step Verified Solution

Step 1: By Cauchy's dispersion formula, refractive index μ increases as wavelength λ decreases: μ_violet > μ_red.
Step 2: Angle of deviation through a prism: D ≈ (μ - 1) A.
Step 3: Violet light has the shortest visible wavelength (~400 nm), experiences the greatest optical density/refractive index, slows down the most in glass, and suffers the maximum deviation.

Official Final Answer: Correct Option (A) : Violet light; because it has the shortest wavelength and travels with lowest speed in glass

💡 Olympiad Tip & Strategy: Red deviates the least and violet deviates the most in optical dispersion!

Question 11 Achievers Section (HOTS)

3 Marks

Two electric bulbs rated (100 W, 220 V) and (60 W, 220 V) are connected in series across a 220 V mains supply. Which bulb will glow brighter and why?

A The 60 W bulb glows brighter because its resistance is higher and P = I²R in a series circuit ✓ Correct

B The 100 W bulb glows brighter because its rated power is higher

C Both bulbs glow with equal brightness because current is identical

D Neither bulb glows because the filament burns out immediately

Concept Tested: Power rating vs actual power dissipation in series electrical circuits

Hint: R = V²/P. Lower rated bulb has higher resistance! In series, current is constant, so power dissipated is I²R.

Detailed Step-by-Step Verified Solution

Step 1: Resistance of 100 W bulb: R1 = V² / P1 = (220)² / 100 = 484 Ω.
Step 2: Resistance of 60 W bulb: R2 = V² / P2 = (220)² / 60 = 806.7 Ω (R2 > R1).
Step 3: When connected in series, the same current I flows through both bulbs.
Step 4: Power dissipated in series is P_actual = I² × R.
Step 5: Since R2 > R1, the 60 W bulb dissipates more electrical energy as light and heat, glowing noticeably brighter than the 100 W bulb.

Official Final Answer: Correct Option (A) : The 60 W bulb glows brighter because its resistance is higher and P = I²R in a series circuit

💡 Olympiad Tip & Strategy: Counter-intuitive Olympiad classic: In parallel, 100 W is brighter; but in series, 60 W is brighter!

Question 12 Achievers Section (HOTS)

3 Marks

A positively charged alpha particle (He²⁺) projected towards the west is deflected towards the north by a uniform magnetic field. What is the direction of the magnetic field?

A Vertically upwards ✓ Correct

B Towards South

C Vertically downwards

D Towards East

Concept Tested: Fleming's Left-Hand Rule and Lorentz magnetic force: F = q(v × B)

Hint: Current direction is the direction of motion of positive charge (West). Force is North. Use Fleming's Left-Hand Rule.

Detailed Step-by-Step Verified Solution

Step 1: An alpha particle has positive charge, so the direction of conventional electric current is in the direction of its motion (West).
Step 2: The magnetic deflection force F is directed towards North.
Step 3: Apply Fleming's Left-Hand Rule:
- Middle finger (Current) points West.
- Thumb (Force) points North.
- Forefinger (Magnetic Field B) points vertically upwards out of the plane.
Step 4: Hence, the magnetic field is directed vertically upwards.

Official Final Answer: Correct Option (A) : Vertically upwards

💡 Olympiad Tip & Strategy: If it were an electron (negative charge), the current direction would be opposite to velocity!