Class 12 IMO — Full Syllabus Mock Test

Q1 Mathematical Reasoning

Evaluate: ∫ dx / (3 + 2x - x²)

Complete square: 3 + 2x - x² = 4 - (x-1)². Integral is ∫ dx / (2² - (x-1)²). Formula gives (1/(2*2)) log|(2+(x-1)) / (2-(x-1))| = (1/4) log|(x+1)/(3-x)| + C.

Q2 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.

Q3 Everyday Mathematics

A Delhi Transport Corporation (DTC) depot maintains a fleet of buses. Matrix A represents the number of buses on 3 routes, and Matrix B represents the number of trips per bus on each route. To find total trips on all routes combined, we must:

To calculate the total trips, we multiply the quantity matrix (buses per route) with the rate matrix (trips per route) via matrix multiplication.

Q4 Mathematical Reasoning

A line makes angle θ with the x-axis, and angle φ with the y-axis. What angle does it make with the z-axis if θ = φ = 60°?

cos²60° + cos²60° + cos²γ = 1. 1/4 + 1/4 + cos²γ = 1 → 1/2 + cos²γ = 1 → cos²γ = 1/2. cos γ = ±1/√2. So γ = 45° or 135°.

Q5 Logical Reasoning

For an odd function f(x), the integral ∫(−a to a) f(x) dx = 0. What does this imply about the geometric areas bounded by the curve and the x-axis on the intervals [−a, 0] and [0, a]?

Odd symmetry means the region below the axis perfectly mirrors the region above the axis, causing the signed areas to sum to zero.

Q6 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).

Q7 Logical Reasoning

The differential equation dy/dx = y is:

It can be written as dy/y = dx, hence variable separable.

Q8 Everyday Mathematics

Sachin cuts a wooden board bounded by the curve y = x³ and the line y = 8, along with the y-axis, to make an E-learning Hub sign. What is the area of the board?

Integrating with respect to y: Area = ∫(0 to 8) x dy = ∫(0 to 8) y^(1/3) dy = [3/4 y^(4/3)] from 0 to 8 = 3/4 × 16 = 12 sq units.

Q9 Mathematical Reasoning

Find the interval in which the function f(x) = x³ - 12x² + 36x + 17 is strictly increasing.

f'(x) = 3x² - 24x + 36 = 3(x - 2)(x - 6). For f'(x) > 0, x < 2 or x > 6. Interval is (-∞, 2) ∪ (6, ∞).

Q10 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.

Q11 Mathematical Reasoning

Evaluate ∫₋₂² |x| dx.

|x| is even, so ∫₋₂² |x| dx = 2 ∫₀² x dx = 2 · [x²/2]₀² = 2 · 2 = 4.

Q12 Mathematical Reasoning

The area bounded by y = sin 2x, x = 0, x = π/2 and the x-axis is:

Area = ∫₀^(π/2) |sin 2x| dx. sin 2x ≥ 0 in [0, π/2], so area = ∫₀^(π/2) sin 2x dx = [−1/2 cos 2x]₀^(π/2) = −1/2(−1 − 1) = 1.

Q13 Logical Reasoning

For the curve y = x² - 5x + 6, the tangent at point (2, 0) is parallel to the line:

dy/dx = 2x - 5. At x = 2, slope m = 4 - 5 = -1. The line x + y = 0 can be written as y = -x, which has slope -1. Thus, they are parallel.

Q14 Logical Reasoning

Consider f(x) = |x − 1| + |x − 2|. The number of points where f(x) is not differentiable is:

The sum of absolute value functions has sharp turns at the roots of the expressions inside the modulus. Here, the roots are x = 1 and x = 2. So, it is not differentiable at 2 points.

Q15 Mathematical Reasoning

If the determinant of the matrix with rows (x, 2) and (8, x) is zero, then the value of x is:

x² − 16 = 0, which gives x² = 16, so x = ±4.

Q16 Achievers Section

If A and B are non-singular 3×3 matrices such that |A| = 2 and |B| = 3, then |2AB⁻¹| equals:

|2AB⁻¹| = 2³ |A| |B⁻¹| = 8 × 2 × (1/3) = 16/3.

Q17 Mathematical Reasoning

If a matrix M = [[1, 2], [2, 1]] and N is a matrix such that M + N is a zero matrix, then N is:

M + N = 0 implies N = −M = [[−1, −2], [−2, −1]].

Q18 Mathematical Reasoning

The area bounded by y = log x, y = 0 and x = e is:

Area = ∫[1 to e] log x dx = [x log x − x]₁ᵉ = (e−e) − (0−1) = 1 sq unit.

Q19 Achievers Section

If matrix A = [[1, 1, 1], [1, 1, 1], [1, 1, 1]], then Aⁿ (where n is a positive integer) is equal to:

A² = A × A = [[3, 3, 3], [3, 3, 3], [3, 3, 3]] = 3A. A³ = A² × A = 3A × A = 3A² = 3(3A) = 9A = 3²A. By induction, Aⁿ = 3^(n−1) A.

Q20 Mathematical Reasoning

Evaluate: ∫ dx / ((x-1)(x-2))

Using partial fractions 1/(x-2) - 1/(x-1), integration yields log|x-2| - log|x-1| = log|(x-2)/(x-1)| + C.

Q21 Achievers Section

The solution of dy/dx = (y+1) is:

dy/(y+1)=dx. Hence ln(y+1)=x+C.

Q22 Mathematical Reasoning

The volume of a cube is increasing at the rate of 8 cm³/s. How fast is the surface area increasing when the length of an edge is 12 cm?

V = x³. dV/dt = 3x²(dx/dt) = 8, so dx/dt = 8/(3x²). Surface area S = 6x². dS/dt = 12x(dx/dt) = 12x(8/(3x²)) = 32/x. When x = 12, dS/dt = 32/12 = 8/3 cm²/s.

Q23 Everyday Mathematics

A steel plant in Bhilai has a production rate given by P'(t) = 50 + 4t tonnes/hour. What is the total steel produced in the first 5 hours?

Total production = ∫[0 to 5] (50 + 4t) dt = [50t + 2t²] from 0 to 5 = 250 + 2(25) = 250 + 50 = 300 tonnes.

Q24 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.

Q25 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.

Q26 Logical Reasoning

A triangle has vertices (a, b+c), (b, c+a) and (c, a+b). Its area is:

Area = (1/2) |a(c+a−a−b) + b(a+b−b−c) + c(b+c−c−a)| = (1/2) |a(c−b) + b(a−c) + c(b−a)| = (1/2)|ac−ab+ab−bc+bc−ac| = 0.

Q27 Logical Reasoning

The planes 2x + 3y + 4z = 5 and 4x + 6y + 8z = 12 are:

Ratio of normals: 2/4 = 3/6 = 4/8 = 1/2. Since the normals are proportional, the planes are parallel.

Q28 Logical Reasoning

The maximum slope of the curve y = -x³ + 3x² + 9x - 27 is:

Slope m = dy/dx = -3x² + 6x + 9. To maximize m, dm/dx = -6x + 6 = 0 → x = 1. Max slope = -3(1)² + 6(1) + 9 = 12.

Q29 Everyday Mathematics

Ravi's kite string forms a straight line given by (x−1)/2 = (y−2)/−1 = z/2. If the ground is the xy-plane, what is the sine of the angle the string makes with the ground?

Ground is xy-plane, normal is z-axis (0, 0, 1). Line DRs are 2, −1, 2. Angle with plane has sin θ = |a*l + b*m + c*n| / (mag1 * mag2). sin θ = |(2)(0) + (−1)(0) + (2)(1)| / (3 * 1) = 2/3.

Q30 Logical Reasoning

In the expression ∫ f(x) dx, what is the term 'f(x)' called?

The function f(x) which is to be integrated is called the integrand.

Q31 Everyday Mathematics

In a factory in Surat, Machine A applies function f(x) = 2x to weight x. Machine B applies g(x) = x + 15. If a material passes through Machine A and then B, the combined operation is:

The material goes through A first, yielding f(x) = 2x. Then it goes through B, which applies g to the result. So we need g(f(x)) = g(2x) = 2x + 15.

Q32 Everyday Mathematics

Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. The numbers are:

Let numbers be x and 15-x. Sum of squares S = x² + (15-x)². S' = 2x - 2(15-x) = 4x - 30. S' = 0 gives x = 7.5. The numbers are 7.5 and 7.5.

Q33 Logical Reasoning

Let L be the set of all lines in a plane and R be the relation on L defined as L₁ R L₂ if L₁ is perpendicular to L₂. Then R is:

A line cannot be perpendicular to itself (not reflexive). If L₁ ⊥ L₂, then L₂ ⊥ L₁ (symmetric). If L₁ ⊥ L₂ and L₂ ⊥ L₃, then L₁ is parallel to L₃, not perpendicular (not transitive).

Q34 Mathematical Reasoning

If x² + y² = 25, then y″ at (3, 4) is:

2x + 2y y′ = 0 → y′ = −x/y. At (3,4): y′ = −3/4. y″ = −(y − x y′)/y² = −(4 − 3(−3/4))/16 = −(4 + 9/4)/16 = −(25/4)/16 = −25/64.

Q35 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.

Q36 Mathematical Reasoning

Using partial fractions, ∫(dx)/(x² − 1) equals:

1/(x²−1) = 1/[(x−1)(x+1)] = A/(x−1) + B/(x+1). Solving: A=1/2, B=−1/2. Integral = (1/2)ln|x−1| − (1/2)ln|x+1| + C = (1/2)ln|(x−1)/(x+1)| + C.

Q37 Mathematical Reasoning

The normal to the curve y = x² + 2x + 3 at the point where x = 1 is:

y = 1 + 2 + 3 = 6. dy/dx = 2x + 2 = 4. Normal slope = −1/4. Eq: y − 6 = −1/4(x − 1) → 4y − 24 = −x + 1 → x + 4y = 25.

Q38 Mathematical Reasoning

The value of ∫₀² x² [x] dx, where [x] is greatest integer function, is:

∫₀² x² [x] dx = ∫₀¹ x²·0 dx + ∫₁² x²·1 dx = 0 + [x³/3]₁² = 8/3 − 1/3 = 7/3. We check: x in [0,1): [x]=0. x in [1,2): [x]=1. At x=2, [2]=2 but point measure zero. So ∫ = 0 + ∫₁² x² dx = [x³/3]₁² = 8/3 − 1/3 = 7/3. Option 3 is 7/3. So correct index 3.

Q39 Logical Reasoning

Find the derivative of cos⁻¹(sin x) with respect to x, for x in (0, π/2).

We can write sin x as cos(π/2 − x). So, cos⁻¹(cos(π/2 − x)) = π/2 − x. The derivative of π/2 − x is −1.

Q40 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.

Q41 Mathematical Reasoning

The operation a ∗ b = (a + b)/2 on R is:

Commutative: (a+b)/2 = (b+a)/2. Associative: (a∗b)∗c = ((a+b)/2 + c)/2 = (a+b+2c)/4. a∗(b∗c) = (a + (b+c)/2)/2 = (2a+b+c)/4. Not equal in general, so not associative.

Q42 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.

Q43 Mathematical Reasoning

If y = (log x)ˣ + x^(log x), then at x = e, dy/dx is:

From earlier calculation, u′ = (log x)ˣ [log(log x) + 1/log x], v′ = x^(log x) [2 log x / x]. At x=e, log x=1, u=1, u′ = 1[0+1]=1. v=e, v′ = e[2/e]=2. Total = 3.

Q44 Everyday Mathematics

For a physics demonstration at Yudgam Foundation, a student tracks a particle whose velocity curve forms a region bounded by y = |x − 3| and the x-axis from x = 0 to x = 6. What is the total distance (area)?

Graph forms two right triangles of base 3 and height 3. Area = 2 × (1/2 × 3 × 3) = 9 units.

Q45 Mathematical Reasoning

If f : R → R is defined by f(x) = 2ˣ, then f is:

2ˣ is strictly increasing → one-one. Range is (0, ∞), not all real numbers → not onto.

Q46 Mathematical Reasoning

The interval on which f(x) = x/log x is increasing is:

f'(x) = (log x − 1)/(log x)². f'(x) > 0 when log x > 1 → x > e. Domain is x > 0, x ≠ 1. Increasing on (e, ∞).

Q47 Achievers Section

Evaluate the integral involving the greatest integer function: ∫[0 to 1.5] [x] dx.

Split the integral where the integer value changes: ∫[0 to 1] 0 dx + ∫[1 to 1.5] 1 dx = 0 + [x] from 1 to 1.5 = 1.5 - 1 = 0.5.

Q48 Everyday Mathematics

A plot is in the shape of the region between y = √x, y = x. The cost of fencing at ₹50 per unit length of perimeter is approximately (use integration for arc length if needed, but area in sq units is asked here). Actually, if land price is ₹200 per sq unit, the total land cost is:

Area = 1/6 sq unit. Cost = (1/6) × 200 = ₹100/3.

Q49 Everyday Mathematics

A shop owner knows 60% customers buy notebooks and 40% buy pens. If 25% buy both, P(notebook|pen) is:

0.25/0.4=0.625.

Q50 Logical Reasoning

The ILATE rule for integration by parts prioritizes functions in which order?

ILATE stands for Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential — this is the priority order for choosing the first function u.