Class 12 IMO — Full Syllabus Mock Test

Q1 Everyday Mathematics

A bag contains 5 red and 3 blue balls. If a ball drawn is known to be red or blue, the probability it is red is:

Total balls=8. Probability of red=5/8.

Q2 Mathematical Reasoning

If ∫₀ᵏ (2x + 3) dx = 10, then k equals:

∫₀ᵏ (2x+3) dx = [x² + 3x]₀ᵏ = k² + 3k = 10 → k² + 3k − 10 = 0 → (k+5)(k−2) = 0 → k = 2 (positive).

Q3 Mathematical Reasoning

Equation of the plane passing through a point with position vector a and normal to the vector n is:

For any point r on the plane, the vector (r − a) lies in the plane, so it is perpendicular to the normal vector n. Thus, their dot product is zero: (r − a) · n = 0.

Q4 Logical Reasoning

What is the area of the circle x² + y² = a² using definite integration in the first quadrant?

The area in the first quadrant is ∫[0 to a] √(a² - x²) dx. Using the standard formula, it evaluates to (πa²)/4.

Q5 Logical Reasoning

The solution of dy/dx=(x+y)/x can be obtained using:

RHS depends only on y/x.

Q6 Everyday Mathematics

A train from New Delhi to Agra follows a distance-time curve s(t) = t³ − 6t² + 15t + 10 (s in km, t in hours). What is its acceleration at t = 2 hours?

Velocity v(t) = s'(t) = 3t² − 12t + 15. Acceleration a(t) = v'(t) = 6t − 12. At t = 2, a(2) = 12 − 12 = 0.

Q7 Logical Reasoning

In a culture, the bacteria count is 100,000. The number is increasing at a rate proportional to the number present. In 2 hours, it increases by 10%. The DE is dN/dt = kN. What is N at time t=0?

The initial count at t=0 is explicitly given in the problem as 100,000.

Q8 Everyday Mathematics

Ravi from E-learning Hub Yudgam looks at a server tower. The angle of elevation is tan⁻¹(3/4). If he is 40m away, the tower's height is:

tan(θ) = height / base. 3/4 = height / 40. Height = 40 × (3/4) = 30m.

Q9 Logical Reasoning

If A is an involuntary matrix, which condition holds true?

An involutory matrix is a matrix that is its own inverse, meaning that when multiplied by itself, it yields the identity matrix, A² = I.

Q10 Achievers Section

Evaluate ∫₀¹ x (1 − x)⁹ dx.

Beta function: ∫₀¹ x^(m−1) (1−x)^(n−1) dx = B(m,n) = Γ(m)Γ(n)/Γ(m+n). Here x = x¹ so m−1=1 → m=2. (1−x)⁹ so n−1=9 → n=10. B(2,10) = Γ(2)Γ(10)/Γ(12) = 1! 9! / 11! = 1/(10×11) = 1/110.

Q11 Mathematical Reasoning

If the vectors a = 2î + λĵ + k̂ and b = î − 2ĵ + 3k̂ are perpendicular to each other, find the value of λ.

For perpendicular vectors, a.b = 0. So, (2)(1) + (λ)(−2) + (1)(3) = 0 → 2 − 2λ + 3 = 0 → 5 = 2λ → λ = 5/2.

Q12 Mathematical Reasoning

For the binary operation ∗ on Z⁺ defined by a ∗ b = LCM(a, b), the identity element is:

LCM(a, 1) = a for all a. LCM(1, a) = a. So 1 is the identity. 0 is not in Z⁺. No other number works for all a.

Q13 Logical Reasoning

Let f(x) = x³ − x² + x + 1 and g(x) = max{f(t): 0 ≤ t ≤ x} for 0 ≤ x ≤ 1, then g(x) is not differentiable at x = 1/3. True or False: g is discontinuous at x = 1/3.

g(x) is always continuous as the maximum function of a continuous function. It may not be differentiable where f stops being the maximum, but it remains continuous.

Q14 Everyday Mathematics

A manufacturing plant in Noida finds its marginal cost of producing x units is given by MC = 3x² - 2x + 5. If the fixed cost (cost at x=0) is ₹ 5000, find the total cost function C(x).

C(x) = ∫ (3x² - 2x + 5) dx = x³ - x² + 5x + K. Since C(0) = 5000, K = 5000.

Q15 Mathematical Reasoning

For any two vectors a and b, what is the geometric interpretation of |a × b|?

The magnitude of the cross product of two vectors represents the area of a parallelogram whose adjacent sides are given by those vectors.

Q16 Mathematical Reasoning

The value of the determinant |2 3; 4 5| is:

For a 2×2 matrix |a b; c d|, determinant = ad − bc. So (2)(5) − (3)(4) = 10 − 12 = −2.

Q17 Achievers Section

The function f(x) = xˣ is strictly increasing for:

Let y = xˣ. Taking log, log y = x log x. Differentiating, (1/y)(dy/dx) = 1 + log x. So f'(x) = xˣ(1 + log x). f'(x) > 0 implies log x > -1, so x > 1/e.

Q18 Mathematical Reasoning

What is the area of the region bounded by the curves y = √x and y = x²?

Intersections at x=0, x=1. Area = ∫(0 to 1) (√x − x²) dx = [2/3 x^(3/2) − x³/3] = 2/3 − 1/3 = 1/3 sq units.

Q19 Everyday Mathematics

A shop board is hung using two strings. The tension in the strings are T₁ = 2î + 3ĵ and T₂ = −2î + 4ĵ. The total upward pull supporting the weight of the board is:

Total upward force = T₁ + T₂ = (2î + 3ĵ) + (−2î + 4ĵ) = 0î + 7ĵ = 7ĵ.

Q20 Achievers Section

The number of real solutions of the equation √(1+cos 2x) = √2 cos⁻¹(cos x) in [π/2, π] is:

LHS = √(2cos²x) = √2|cos x| = −√2 cos x (since cos x ≤ 0 in [π/2, π]). RHS = √2 cos⁻¹(cos x). For x ∈ [π/2, π], cos⁻¹(cos x) = 2π−x? No, in [0,π], cos⁻¹(cos x)=x. So RHS = √2 x. Equation: −√2 cos x = √2 x → cos x = −x. Drawing graphs of y=cos x and y=−x in [π/2, π] shows 1 intersection.

Q21 Everyday Mathematics

The area of a circular oil spill in Mumbai harbour is expanding. If A = πr², the rate of change of area with respect to its radius r when r = 10 m is:

dA/dr = 2πr. When r = 10 m, the rate is 2π(10) = 20π square meters per meter.

Q22 Mathematical Reasoning

The value of k×i is:

k×i = j.

Q23 Logical Reasoning

Let * be defined on the set of non-zero rational numbers Q by a * b = a/b. Then * is:

a * b = a/b, but b * a = b/a, so not commutative. Also, (a * b) * c = (a/b)/c = a/(bc), whereas a * (b * c) = a/(b/c) = (ac)/b. Hence, not associative.

Q24 Everyday Mathematics

A sweet shop sells boxes of laddoos. The demand function is p(x) = 100 - x², where x is the quantity. If the equilibrium quantity is 6 boxes, what is the Consumer Surplus? (Assume equilibrium price is p(6))

Equilibrium price = 100 - 36 = 64. Consumer Surplus = ∫[0 to 6] (100 - x² - 64) dx = ∫[0 to 6] (36 - x²) dx = [36x - x³/3] from 0 to 6 = 216 - 72 = 144.

Q25 Mathematical Reasoning

∫dx/√(x² + 9) is equal to:

∫dx/√(x² + a²) = ln|x + √(x² + a²)| + C or sinh⁻¹(x/a) + C. Both forms are equivalent.

Q26 Mathematical Reasoning

If y = e^(a sin⁻¹ x), then (1 − x²)y₂ − xy₁ equals:

y₁ = e^(a sin⁻¹ x) × a/√(1−x²) = a y/√(1−x²). So √(1−x²) y₁ = a y. Squaring: (1−x²)y₁² = a² y². Differentiating: −2x y₁² + 2(1−x²)y₁ y₂ = 2a² y y₁. Divide by 2y₁: −x y₁ + (1−x²)y₂ = a² y.

Q27 Logical Reasoning

If A and B are matrices of same order, under what condition is (A + B)² = A² + 2AB + B² true?

Expanding (A + B)² gives (A + B)(A + B) = A² + AB + BA + B². For this to equal A² + 2AB + B², we must have AB = BA (the matrices commute).

Q28 Logical Reasoning

In a school, 40% of students study Maths, 25% study Physics, and 15% study both. If a student selected at random studies Physics, the probability they also study Maths is:

P(Maths | Physics) = P(Maths ∩ Physics) / P(Physics) = 15% / 25% = 3/5.

Q29 Mathematical Reasoning

The normal vector to the plane 2x−3y+6z−5=0 is:

Coefficients of x, y, z give the normal vector.

Q30 Everyday Mathematics

A drone at a Diwali festival flies along the vector r = 2i + j + k + μ(i + j). What is its shortest distance from a static firework rocket launched vertically along the z-axis (r = λk)?

Line 1: a₁ = 2i + j + k, b₁ = i + j. Line 2: a₂ = 0, b₂ = k. Shortest distance = |(a₂ − a₁) · (b₁ × b₂)| / |b₁ × b₂|. a₂ − a₁ = −2i − j − k. b₁ × b₂ = (i+j) × k = −j + i = i − j. (a₂ − a₁) · (b₁ × b₂) = (−2)(1) + (−1)(−1) + 0 = −2 + 1 = −1. |b₁ × b₂| = √2. SD = |−1| / √2 = 1/√2.

Q31 Mathematical Reasoning

The appropriate substitution to solve the homogeneous differential equation dx/dy = f(x/y) is:

When the equation is expressed as dx/dy = f(x/y), it is easier to substitute x = vy, which gives dx/dy = v + y(dv/dy).

Q32 Mathematical Reasoning

The point on the curve y² = 4x at which the tangent is perpendicular to the line 2x + y = 3 is:

Line slope = −2. Tangent slope = 1/2 (perpendicular). dy/dx for y² = 4x is 2y dy/dx = 4 → dy/dx = 2/y. Set 2/y = 1/2 → y = 4. Then 4² = 4x → 16 = 4x → x = 4. Point (4, 4) has tangent slope 2/4 = 1/2. Correct. But option (4, 4) is there. So point (4, 4). Tangent slope = 2/4 = 1/2, which is perpendicular to slope −2. That's correct. But option (1/4, −1) has y = −1, slope = 2/(−1) = −2, not 1/2. So (4, 4) is correct. We'll set correct index to 1.

Q33 Mathematical Reasoning

Let A and B be sets having 3 and 4 elements respectively. The number of one-one functions from A to B is:

The number of one-one functions from a set of size m to a set of size n (where m ≤ n) is ⁿPₘ. Here it is ⁴P₃ = 4 × 3 × 2 = 24.

Q34 Mathematical Reasoning

The area of a parallelogram with adjacent sides given by vectors (2,1) and (1,3) using determinant is:

Area = |det[2 1; 1 3]| = |6 − 1| = 5.

Q35 Logical Reasoning

Which of the following is equal to the scalar triple product [b c a]?

Cyclic permutations of the vectors in a scalar triple product leave its value unchanged. Hence, [b c a] = [a b c].

Q36 Mathematical Reasoning

The absolute maximum value of f(x) = x³ - 3x on [-2, 2] is:

f'(x) = 3x² - 3 = 0 → x = ±1. Evaluate f(x) at boundaries and critical points: f(-2) = -2, f(-1) = 2, f(1) = -2, f(2) = 2. Absolute maximum is 2.

Q37 Logical Reasoning

The minimum value of (sin⁻¹(x))² + (cos⁻¹(x))² is:

From the quadratic 2a² − πa + π²/4, the minimum occurs at the vertex a = -(-π)/4 = π/4. The value is 2(π/4)² − π(π/4) + π²/4 = π²/8.

Q38 Logical Reasoning

A scalar matrix is a diagonal matrix in which all the principal diagonal elements are:

A scalar matrix is a special type of diagonal matrix where all diagonal entries are equal to a constant scalar.

Q39 Mathematical Reasoning

The particular solution of dy/dx = 2x, given that y(1) = 2, is:

dy = 2x dx. Integrating gives y = x² + C. At x=1, y=2 → 2 = (1)² + C → C = 1. So, y = x² + 1.

Q40 Everyday Mathematics

Water flows into an overhead tank in Mumbai at a rate of R(t) = 10e^(0.2t) liters/minute. How much water is added to the tank in the first 5 minutes?

Volume = ∫[0 to 5] 10e^(0.2t) dt = [10/0.2 e^(0.2t)] from 0 to 5 = 50[e¹ - e⁰] = 50(e - 1) liters.

Q41 Mathematical Reasoning

The normal to the curve x² = 4y at the point (2, 1) is:

Differentiating x² = 4y yields 2x = 4(dy/dx), so dy/dx = x/2. At (2,1), tangent slope = 1. Normal slope = -1. Equation: y - 1 = -1(x - 2) → x + y = 3.

Q42 Logical Reasoning

For a homogeneous system of equations AX = O, a non-trivial solution exists only if:

If |A| ≠ 0, only the trivial solution (X=O) exists. For non-trivial (non-zero) solutions, |A| must be 0.

Q43 Logical Reasoning

To compute the area between two curves f(x) and g(x) where f(x) ≥ g(x) for all x in [a, b], which integral setup is correct?

The correct formulation for the area between curves is the integral of the upper curve minus the lower curve: ∫ (upper − lower) dx.

Q44 Achievers Section

What is the area of the region completely bounded by the downward opening parabola y = 4x − x² and the x-axis?

Roots are x=0, 4. Area = ∫(0 to 4) (4x − x²) dx = [2x² − x³/3] = 32 − 64/3 = 32/3 sq units.

Q45 Logical Reasoning

Evaluate lim n→∞ (1/n) [ (1/n)² + (2/n)² + ... + (n/n)² ].

This converts to ∫[0 to 1] x² dx. Evaluating this gives [x³/3] from 0 to 1 = 1/3.

Q46 Logical Reasoning

The transpose of a matrix A, denoted by A', is obtained by:

The transpose of a matrix is formed by turning all its rows into columns and all its columns into rows.

Q47 Mathematical Reasoning

Principal value of sin⁻¹[cos(sin⁻¹(1/2))] is:

sin⁻¹(1/2) = π/6. cos(π/6) = √3/2. sin⁻¹(√3/2) = π/3.

Q48 Mathematical Reasoning

A fair die is thrown 3 times. Probability of getting a six exactly once is:

³C₁(1/6)(5/6)²=75/216=25/72.

Q49 Achievers Section

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, only two consecutive letters 'TA' are visible. The probability that the letter came from TATANAGAR is:

TATANAGAR has 8 pairs, 'TA' appears twice (prob 2/8 = 1/4). CALCUTTA has 7 pairs, 'TA' appears once (prob 1/7). P(TATA|TA) = (1/2 × 1/4) / [(1/2 × 1/4) + (1/2 × 1/7)] = (1/4) / (1/4 + 1/7) = 7/11.

Q50 Everyday Mathematics

A bank account grows continuously according to dA/dt = 0.08A. If A(0)=₹5000, then after time t the amount is:

Continuous growth follows A=A₀e^(kt).