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Paper pattern

A · all 20 · 1 mark each · 18 multiple choice and 2 assertion-reason20

B · all 5 · 2 marks each · very short answer, internal choice in 210

C · all 6 · 3 marks each · short answer, internal choice in 318

D · all 4 · 5 marks each · long answer, internal choice in 220

E · all 3 · 4 marks each · case study, sub-parts of 1 + 1 + 2 or 2 + 212

Board paper total80

CBSE Mathematics (041) Class XII question paper design and sample question paper, 2026-27: 38 questions, 80 marks, 3 hours, internal choice in about a third of the questions, no calculators.

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2 marks · short answer

Let A = {1, 2, 3}. Write a relation on A that is symmetric and transitive but not reflexive. Justify your answer.Unit I On the set of all people in a town, R = {(x, y): x and y have the same blood group}. Show that R is reflexive and symmetric.Unit I Check whether the function f: R → R, f(x) = |x| + x is one-one. Give a reason.Unit I Show that the function f: N → N, f(x) = 3x is one-one but not onto.Unit I Find the domain of cos⁻¹(2x² − 1).Unit II Find the value of cosec⁻¹(−2) + cot⁻¹(−√3).Unit II Write the principal value branch of sec⁻¹ x, and explain why π/2 is left out of it.Unit II Find the value of tan⁻¹(tan 5π/6).Unit II Construct a 2 × 3 matrix A = [aᵢⱼ] whose elements are aᵢⱼ = |i − 2j|.Unit III If A = [3, −1; 2, 0] and B = [1, 4; −2, 5], verify that (A + B)′ = A′ + B′.Unit III If the matrix [2, x − 1, 4; 5, 0, y; 4, 7, 3] is symmetric, find x and y.Unit III Show that B = [1, −3; −1, 4] is the inverse of A = [4, 3; 1, 1].Unit III Find the values of x for which the matrix [x + 1, 3; 2, x] is singular.Unit IV If A is a square matrix of order 3 with |A| = 4, find |adj A| and |2A|.Unit IV Using determinants, show that the points (−1, 3), (2, 0) and (4, −2) are collinear.Unit IV Examine the consistency of the system 2x + 3y = 5, 6x + 9y = 12.Unit IV Show that f(x) = |x − 3| is continuous but not differentiable at x = 3.Unit V Find dy/dx if y = tan⁻¹ √((1 − cos 2x)/(1 + cos 2x)), 0 < x < π/2.Unit V If x² + 2xy + y³ = 29, find dy/dx at the point (3, 2).Unit V Differentiate e^(sin²x) with respect to x.Unit V If x = t + 1/t and y = t − 1/t, find dy/dx.Unit V The radius of a circular metal plate increases at 0.2 cm/s when it is heated. Find the rate at which its area increases when the radius is 5 cm.Unit VI Show that f(x) = 3x + cos 2x is strictly increasing on R.Unit VI Find the intervals in which f(x) = x² − 8x + 5 is strictly increasing or strictly decreasing.Unit VI Using the second derivative test, find the local maximum value of f(x) = 2x³ − 6x + 1.Unit VI Find ∫ (1 − cos 2x)/(1 + cos 2x) dx.Unit VII Find ∫ cos x/(3 + sin x)² dx.Unit VII Find ∫ eˣ (1 + x)/(2 + x)² dx.Unit VII Evaluate ∫ cos x/(1 + sin² x) dx from 0 to π/2.Unit VII Evaluate ∫ |x − 1| dx from 0 to 2.Unit VII Using integration, find the area of the region bounded by the line y = x + 2, the x-axis and the lines x = 0 and x = 2.Unit VIII Find the area of the region bounded by the curve y = √x, the x-axis and the line x = 9.Unit VIII Using integration, find the area of the region in the first quadrant enclosed by the circle x² + y² = 64 and the coordinate axes.Unit VIII Using integration, find the area enclosed by the ellipse x²/4 + y²/16 = 1.Unit VIII Find the order and degree (if defined) of: (i) (y″)² + (y′)³ = sin x, (ii) y‴ + e^(y′) = 0.Unit IX Verify that y = x² + C/x is a solution of x(dy/dx) + y = 3x² (x ≠ 0).Unit IX Find the general solution of dy/dx = eˣ cos² y.Unit IX Find the integrating factor of x(dy/dx) + (1 + x)y = x² (x > 0).Unit IX Find the direction cosines of the vector 2î − 6ĵ + 3k̂ and the angle it makes with the x-axis.Unit X Find a vector of magnitude 10 units in the direction of 3î − 4k̂.Unit X If |a⃗| = 3, |b⃗| = 4 and |a⃗ + b⃗| = 5, find |a⃗ − b⃗|.Unit X Find the area of the parallelogram whose adjacent sides are a⃗ = î + 2ĵ and b⃗ = 3î − ĵ + 2k̂.Unit X P and Q have position vectors 2î − ĵ + 3k̂ and î + ĵ − k̂. Find the position vector of the point that divides PQ externally in the ratio 3 : 1.Unit X Find the direction cosines of the line passing through the points (3, −1, 2) and (5, 2, 8).Unit XI A line makes angles α, β, γ with the coordinate axes. Show that cos 2α + cos 2β + cos 2γ = −1.Unit XI Find k so that the lines (x − 2)/1 = (y + 1)/k = (z − 3)/2 and (3 − x)/2 = (y − 1)/4 = z/(−1) are perpendicular.Unit XI Write the Cartesian equation of the line through (−1, 0, 2) and (3, 4, 6), and find its direction cosines.Unit XI Draw the feasible region for 2x + y ≤ 8, x ≤ 3, x ≥ 0, y ≥ 0 and write its corner points.Unit XII The corner points of a bounded feasible region are (0, 0), (4, 0), (3, 2) and (0, 3). Find the maximum and minimum values of Z = 5x + 7y.Unit XII Define (i) a feasible solution and (ii) an optimal solution of a linear programming problem.Unit XII A trader buys x pressure cookers and y mixer grinders. He can store at most 60 items. A cooker costs ₹ 1500 and a mixer grinder ₹ 2500 (example prices), and he can spend at most ₹ 1,20,000. Write the constraints.Unit XII If P(A) = 0.6, P(B) = 0.5 and P(A|B) = 0.4, find P(A ∪ B) and P(B|A).Unit XIII A die is thrown. Given that the number shown is greater than 3, use P(E′|F) = 1 − P(E|F) to find the probability that it is not even.Unit XIII Two cards are drawn one after another without replacement from a pack of 52 cards. Find the probability that the first is a king and the second is a queen.Unit XIII A and B are independent events with P(A) = 0.25 and P(B) = 0.6. Find P(A′ ∩ B′) and P(A ∪ B).Unit XIII

3 marks · short answer

Check whether the relation R = {(a, b): a ≤ b + 1} on the set R of real numbers is reflexive, symmetric or transitive.Unit I Show that the relation R = {(a, b): a − b + √3 is irrational} on the set R of real numbers is reflexive, but neither symmetric nor transitive.Unit I Show that the function f: R → R, f(x) = 2x³ − 7 is one-one and onto.Unit I Find the domain and the range of f(x) = 2 sin⁻¹(x − 1).Unit II Evaluate cos⁻¹(cos 5π/3) + sin⁻¹(sin 7π/6) + tan⁻¹(tan 5π/4).Unit II Draw the graph of y = cos⁻¹ x. Mark the points for x = −1, −1/2, 0, 1/2 and 1, and write its domain and range.Unit II Find the matrices X and Y if X + Y = [4, 0; 2, 2] and X − Y = [2, 2; −2, 2].Unit III Express the matrix A = [6, −2; 4, 3] as the sum of a symmetric and a skew symmetric matrix.Unit III If A = [5, 2; 2, 1], show that A² − 6A + I = O. Hence find A⁻¹.Unit III Using the cofactors of the elements of the second column, evaluate Δ = |2, −1, 3; 1, 0, 4; −2, 5, 1|.Unit IV If A = [2, −1; −1, 2], show that A² − 4A + 3I = O. Hence find A⁻¹.Unit IV Using determinants, find the equation of the line joining (−2, 3) and (4, 6). Also find k if the point (k, 5) lies on this line.Unit IV Examine the consistency of the system x + y + z = 1, x + 2y + 3z = 2, 2x + 3y + 4z = 5.Unit IV Find the values of a and b so that f(x) = {3, if x ≤ 1; ax + b, if 1 < x < 4; 9, if x ≥ 4} is continuous.Unit V If y = (sin x)ˣ, 0 < x < π, find dy/dx.Unit V If y = e^(2 tan⁻¹x), prove that (1 + x²)y₂ + 2(x − 1)y₁ = 0.Unit V If x = 2(θ + sin θ) and y = 2(1 − cos θ), find dy/dx and d²y/dx² at θ = π/2.Unit V Sand pours from a pipe at 18 cm³/s and forms a cone on the ground whose height is always one-third of its base radius. How fast is the height of the cone increasing when the height is 3 cm?Unit VI Find the intervals in which f(x) = x³ − 12x² + 36x + 17 is strictly increasing or strictly decreasing.Unit VI Find the absolute maximum and absolute minimum values of f(x) = 2 sin x + sin 2x on [0, 3π/2].Unit VI Find ∫ (x² + 1)/((x − 1)(x − 2)) dx.Unit VII Find ∫ (2x + 5)/√(x² + 2x + 5) dx.Unit VII Find ∫ x² tan⁻¹ x dx.Unit VII Evaluate ∫ dx/(1 + cot³ x) from π/6 to π/3.Unit VII Using integration, find the area of the region bounded by the parabola y² = 12x and its latus rectum.Unit VIII Using integration, find the area of the region bounded by y = |x − 1|, the x-axis and the lines x = −1 and x = 3.Unit VIII Find the area of the region bounded by the curve y = √(25 − x²), the x-axis and the lines x = 0 and x = 5/2.Unit VIII Find the particular solution of x(1 + y²) dx − y(1 + x²) dy = 0, given that y = 1 when x = 0.Unit IX Solve the differential equation x(dy/dx) = y + 2x tan(y/x).Unit IX Find the particular solution of x(dy/dx) − y = x²eˣ, given that y = 0 when x = 1.Unit IX Solve y dx + (x − y²eʸ) dy = 0, y > 0, by writing it as a linear equation in x.Unit IX Show that the differential equation y dx = (x + ye^(x/y)) dy is homogeneous. Find its particular solution, given that x = 0 when y = 1.Unit IX Find the particular solution of x(dy/dx) + 2y = x log x (x > 0), given that y = 0 when x = 1.Unit IX Find a vector of magnitude 6 that is perpendicular to both a⃗ = î + ĵ − k̂ and b⃗ = 2î − ĵ + k̂.Unit X Using vectors, show that A(1, 2, 3), B(2, 3, 5) and C(2, 3, 2) are the vertices of a right-angled triangle. Find its area.Unit X Let a⃗ = î + 2ĵ − 2k̂, b⃗ = 3î − ĵ + 4k̂ and c⃗ = −î + 3ĵ. Find the projection of b⃗ + c⃗ on a⃗. Is the angle between b⃗ + c⃗ and a⃗ acute or obtuse?Unit X Find the position vector of the point R that divides the join of P(2î + ĵ − k̂) and Q(î − ĵ + 2k̂) externally in the ratio 2 : 1. Show that Q is the midpoint of PR.Unit X For a⃗ = î + 2ĵ + 2k̂ and b⃗ = 2î − ĵ + 2k̂, find the angle between a⃗ and b⃗, and the area of the parallelogram with adjacent sides a⃗ and b⃗.Unit X Show that the lines (x − 1)/1 = (y − 2)/1 = (z + 2)/1 and x/2 = (y − 4)/(−1) = (z + 4)/3 intersect. Find their point of intersection.Unit XI Find the vector equation of the line passing through A(3, 4, −7) and B(1, −1, 6). Does it pass through the point (5, 9, −20)?Unit XI Find the distance between the lines r⃗ = (î − ĵ) + λ(2î + 2ĵ + k̂) and r⃗ = (4î − ĵ) + μ(4î + 4ĵ + 2k̂).Unit XI Solve graphically: Maximise Z = 2x + 3y subject to x + y ≤ 6, x + 3y ≤ 12, x ≤ 4, x ≥ 0, y ≥ 0.Unit XII Solve graphically: Minimise Z = 3x + 2y subject to x + y ≥ 4, x + 3y ≥ 6, x ≥ 0, y ≥ 0.Unit XII Show graphically that the constraints x + 2y ≤ 4, 3x + 4y ≥ 24, x ≥ 0, y ≥ 0 have no feasible region.Unit XII Two dice are thrown. Find the probability that the first die shows 6, given that the sum of the numbers is at least 9.Unit XIII The probabilities that A, B and C solve a problem on their own are 1/2, 1/4 and 1/6. Find the probability that the problem is solved.Unit XIII A dairy gets 60% of its milk from farm P and 40% from farm Q. 3% of P's cans and 5% of Q's cans fail a quality test. Find the probability that a can picked at random fails the test.Unit XIII

5 marks · long answer

Let A = {x ∈ Z: 0 ≤ x ≤ 15} and R = {(a, b): |a − b| is a multiple of 5}. Show that R is an equivalence relation on A, and write all its equivalence classes.Unit I Show that the function f: R − {−2} → R − {3}, f(x) = (3x − 1)/(x + 2) is one-one and onto.Unit I Sketch the graphs of y = sin x on [−π/2, π/2] and y = sin⁻¹ x on the same axes. Explain why each is the mirror image of the other in the line y = x. Use your graph to read off sin⁻¹(1/2) and sin⁻¹(−1).Unit II Explain why the domain of a trigonometric function must be restricted before its inverse can be defined, and write the principal value branches of sin⁻¹, cos⁻¹ and tan⁻¹. Hence find the domain of f(x) = sin⁻¹(2x − 1) + cos⁻¹(1 − x), and evaluate f(0) and f(1).Unit II Prove that for any square matrix A, A + A′ is symmetric and A − A′ is skew symmetric. Hence express A = [3, −1, 2; 5, 0, −4; 6, 2, 1] as the sum of a symmetric and a skew symmetric matrix, and check your answer.Unit III If A = [1, 2; 0, 1] and B = [1, 0; 3, 1], find AB and BA and show that AB ≠ BA. Find (A + B)² and A² + 2AB + B², and explain why they are different. Also verify that (AB)′ = B′A′.Unit III Find A⁻¹ for A = [1, 1, 1; 2, −1, 3; 1, 2, −1]. Hence solve the system x + y + z = 2, 2x − y + 3z = 8, x + 2y − z = −1.Unit IV If A = [1, 2, 0; 0, 1, 1; 1, 0, 2] and B = [2, −4, 2; 1, 2, −1; −1, 2, 1], find AB. Use it to solve x + 2y = 5, y + z = 5, x + 2z = 7.Unit IV Find dy/dx if y = x^(cos x) + (cos x)^(sin x), 0 < x < π/2.Unit V Find the values of a and b so that f(x) = {ax² + b, if x ≤ 1; 4x − 1, if x > 1} is differentiable at x = 1. Hence find f′(x) for all x, and check whether f′ is continuous at x = 1.Unit V A rectangle is drawn inside a semicircle of radius 10 cm, with one side along the diameter. Find the dimensions of the rectangle of greatest area, and that area.Unit VI A gram panchayat wants an open-top storage tank with a square base, made from 108 m² of sheet metal. Find the dimensions of the tank that give the greatest volume, and that volume.Unit VI Find ∫ (x² + 4)/(x⁴ + 16) dx. (Hint: divide the numerator and the denominator by x².)Unit VII Evaluate ∫ x sin³ x dx from 0 to π, stating each property of definite integrals that you use.Unit VII Find ∫ √(5 − 4x − x²) dx. Hence evaluate ∫ √(5 − 4x − x²) dx from −2 to 1.Unit VII Sketch the line y = 4 − 2x. Using integration, find the area of the region bounded by this line, the x-axis and the lines x = −1 and x = 4. Show that ∫ (4 − 2x) dx from −1 to 4 does not give this area, and explain why.Unit VIII Using integration, find the area of the region of the ellipse x²/25 + y²/9 = 1 that lies between the lines x = −5/2 and x = 5/2.Unit VIII Find the shortest distance between the lines r⃗ = (î − k̂) + λ(î − 2ĵ + 2k̂) and r⃗ = (2î + 3ĵ + 3k̂) + μ(2î + 2ĵ + k̂). Are the lines skew? Also find the angle between them.Unit XI The lines (x − 1)/2 = (y + 1)/k = (z − 2)/3 and (x + 2)/1 = (y − 3)/2 = (z − 1)/(−2) are perpendicular. (a) Find k. (b) Find the vector and Cartesian equations of the line through (3, 0, −2) that is perpendicular to both these lines.Unit XI A unit makes school bags and lunch bags. A school bag needs 3 hours on machine I and 1 hour on machine II; a lunch bag needs 1 hour on machine I and 2 hours on machine II. Machine I runs at most 15 hours a day and machine II at most 10 hours. The profit is ₹ 80 on a school bag and ₹ 100 on a lunch bag (example rates). Formulate the LPP and solve it graphically to find the maximum profit.Unit XII Find the minimum and maximum values of Z = 2x + y subject to x + y ≥ 3, x + 2y ≤ 10, x ≤ 4, x ≥ 0, y ≥ 0, by the graphical method.Unit XII Urn A has 2 white and 3 black balls, urn B has 4 white and 1 black ball, and urn C has 3 white and 2 black balls. An urn is chosen at random and a ball drawn from it is white. Find the probability that it came from urn B.Unit XIII A disease affects 0.5% of a town. A test is positive for 98% of people who have it, but also for 2% of healthy people (example values). State Bayes' theorem, and find the probability that a person who tests positive has the disease.Unit XIII

4 marks · case-based

In a mathematics club of 12 students, the teacher defines a relation R on the set S of members: (x, y) ∈ R if x and y were born in the same month. Asha and Bina were born in March, and Charu was born in July. No other member was born in March. (i) Is R reflexive? Give a reason. (1 mark) (ii) Is (Asha, Charu) ∈ R? Is R symmetric? (1 mark) (iii) Show that R is an equivalence relation, and write the equivalence class of Asha. (2 marks)Unit I A concert hall in Madurai has 60 numbered seats. For one show, 50 tickets are sold. Each ticket gets a seat, and no seat is given to two tickets. Let f be the function from the set T of 50 tickets to the set S of 60 seats. (i) Is f one-one? Give a reason. (1 mark) (ii) Is f onto? Give a reason. (1 mark) (iii) How many more tickets, each with a new seat, must be sold for the ticket-to-seat function to become a bijection? Explain using the meaning of a bijection. (2 marks)Unit I A metro station in Kochi is fitting wheelchair ramps. Ramp P rises 1 m along a sloping surface 2 m long. Ramp Q rises 1 m over a horizontal distance of √3 m. An engineer writes the angle of a third ramp as cos⁻¹(−1/2). (i) Write the slope angle of ramp P as an inverse sine and find its principal value. (1 mark) (ii) Write the slope angle of ramp Q as an inverse tangent and find its principal value. (1 mark) (iii) Find the principal value of cos⁻¹(−1/2), and explain why it cannot be the slope angle of a ramp. (2 marks)Unit II Meena stands on level ground d metres from the foot of a temple gopuram 30 m tall. The angle of elevation of its top is θ = tan⁻¹(30/d). (i) Find θ when d = 30√3 m. (1 mark) (ii) Write the domain and range of tan⁻¹ x. (1 mark) (iii) As Meena walks towards the gopuram, d decreases. How does θ change? Using the range of tan⁻¹ x, explain why θ can never reach π/2. (2 marks)Unit II A bakery in Chennai has two outlets and sells bread, buns and cakes. On Monday the numbers sold are given by Q = [30, 40, 10; 25, 50, 8], with one row for each outlet and one column for each item. The prices in ₹ per item (example rates) form the column matrix P = [40; 10; 300]. (i) Write the orders of Q and P. (1 mark) (ii) Is the product PQ defined? Give a reason. (1 mark) (iii) Find QP and say what each of its entries tells the owner. (2 marks)Unit III The road distances in km between three towns P, Q and R form the matrix D = [0, 120, 200; 120, 0, 90; 200, 90, 0], with rows and columns in the order P, Q, R. A transport company charges ₹ 2 per km (an example rate). (i) Show that D is a symmetric matrix. (1 mark) (ii) Why is every diagonal entry of D equal to 0? (1 mark) (iii) Write the fare matrix F = 2D, and use a property of transpose to show that F is also symmetric. (2 marks)Unit III At a weekly market, Priya buys 2 kg of onions, 1 kg of tomatoes and 1 kg of potatoes for ₹ 130. Ravi buys 1 kg of onions, 2 kg of tomatoes and 1 kg of potatoes for ₹ 120. Sana buys 1 kg of onions, 1 kg of tomatoes and 2 kg of potatoes for ₹ 110. All three pay the same price per kg (example prices). (i) Write these purchases as a matrix equation AX = B. (1 mark) (ii) Find |A| and say why the prices can be found uniquely. (1 mark) (iii) Using A⁻¹, find the price per kg of each vegetable. (2 marks)Unit IV A school is planning a triangular lawn. On the plan, its corners are P(1, 2), Q(5, 4) and R(3, 8), where 1 unit stands for 10 m. A lamp post L(k, 6) is to stand on the straight line through P and Q. (i) Find the area of the lawn in square units. (1 mark) (ii) Write this area in m². (1 mark) (iii) Using determinants, find k. (2 marks)Unit IV A power company bills electricity by the rule C(x) = 5x for 0 ≤ x ≤ 100 and C(x) = 7x − k for x > 100, where x is the number of units used and C(x) is in rupees (example rates). The company wants no sudden jump in the bill at 100 units. One family used 150 units last month. (i) Find k so that C is continuous at x = 100. (1 mark) (ii) Using this k, find this family's bill. (1 mark) (iii) Is C differentiable at x = 100? Give a reason. (2 marks)Unit V On sports day, a ball is thrown so that after t seconds it is x = 20t metres along the ground and y = 15t − 5t² metres above the ground. Both x and y are given in terms of the time t, which acts as the parameter. The coach wants to know the direction in which the ball moves and how high it goes. (i) Find dx/dt and dy/dt. (1 mark) (ii) Find dy/dx in terms of t. (1 mark) (iii) Find the time at which the ball is highest, and its height then. (2 marks)Unit V A handicraft unit in Moradabad makes open brass trays from square sheets of side 30 cm. It cuts equal squares of side x cm from each corner and folds up the sides. The volume of a tray depends on x. The unit wants the tray that holds the most. (i) Write the volume V(x) of a tray and the values that x can take. (1 mark) (ii) Find dV/dx. (1 mark) (iii) Find the value of x that gives the greatest volume, show that it is a maximum, and find that volume. (2 marks)Unit VI A mela opens at 10 a.m. and closes at 5 p.m. The number of visitors present, in hundreds, t hours after it opens is modelled by N(t) = −t³ + 9t² − 15t + 30, for 0 ≤ t ≤ 7. The organisers use this to plan extra staff. (i) Find N′(t). (1 mark) (ii) Between which times is the crowd growing? (1 mark) (iii) Using the second derivative test, find when the crowd is largest and how many visitors are present then. (2 marks)Unit VI A dairy co-operative makes flavoured milk. The marginal cost, the cost of making one more litre, is C′(x) = 0.04x + 20 rupees per litre when x litres are made in a day (example figures). The fixed cost, the cost when nothing is made, is ₹ 500 a day. The manager wants the total cost C(x) and the extra cost of raising output. (i) Explain why C(x) is found by integrating C′(x). (1 mark) (ii) Find C(x). (1 mark) (iii) Using a definite integral, find the extra cost of raising output from 100 to 200 litres a day. (2 marks)Unit VII In a maths club, Meera tells her friends that ∫ f(x) dx from 0 to a = ∫ f(a − x) dx from 0 to a for any continuous function f. Arjun wants to use this to find I = ∫ sin x/(sin x + cos x) dx from 0 to π/2. He notices that sin(π/2 − x) = cos x and cos(π/2 − x) = sin x. (i) Write the integrand after x is replaced by π/2 − x. (1 mark) (ii) Add the two forms of I and find 2I. (1 mark) (iii) Hence find I, and use the same idea to show that ∫ (sin³ x − cos³ x) dx from 0 to π/2 = 0. (2 marks)Unit VII A landscape designer plans an elliptical lawn in a park. Taking the centre of the lawn as the origin, with lengths in metres, its boundary is x²/400 + y²/100 = 1. Grass seed is bought by the square metre, so she needs the area. (i) Write the lengths of the major and minor axes of the lawn. (1 mark) (ii) Write y in terms of x for the part of the boundary above the x-axis. (1 mark) (iii) Using integration, find the area of the lawn. (2 marks)Unit VIII A food-safety lab studies bacteria in milk kept at room temperature. The rate of growth of the number N of bacteria (per mL) is proportional to N. At 8 a.m. there are 500 bacteria per mL; at 10 a.m. there are 2000 per mL. Let t be the time in hours after 8 a.m. (i) Write the differential equation for N, using a constant k. (1 mark) (ii) Write its general solution. (1 mark) (iii) Find the number of bacteria per mL at 11 a.m. (2 marks)Unit IX A cup of tea at 90 °C is placed in a room at 30 °C. Its temperature T after t minutes satisfies dT/dt = −k(T − 30), where k is a positive constant. Tea stalls use such results to know how long tea stays hot. (i) Write the equation in the form dT/dt + PT = Q and say why it is linear. (1 mark) (ii) Find the integrating factor. (1 mark) (iii) Find T in terms of t and k. (2 marks)Unit IX A drone delivery service in Bengaluru starts each flight at a depot O, taken as the origin, with distances in metres. A drone first rises to the point A with OA⃗ = 2î + 3ĵ + 6k̂. It then flies in a straight line to B with OB⃗ = 17î + 23ĵ + 6k̂. (i) How far is A from the depot? (1 mark) (ii) Find the unit vector in the direction of AB⃗. (1 mark) (iii) A charging pad C is placed on AB with AC : CB = 2 : 3. Find OC⃗. (2 marks)Unit X A farm worker pulls a loaded cart with a rope. The force on the cart is F⃗ = 6î + 2ĵ + 3k̂ newtons, and the cart moves through d⃗ = 4î + 3k̂ metres. The work done is W = F⃗ · d⃗. (i) Find the work done. (1 mark) (ii) Find the cosine of the angle between the force and the displacement. (1 mark) (iii) Find the projection of F⃗ on d⃗, the part of the force that acts along the motion. (2 marks)Unit X At a Republic Day drone show, two drones fly along straight paths. Drone A follows r⃗ = λ(2î + ĵ − 2k̂) and drone B follows r⃗ = 3k̂ + μ(î + 2ĵ + 2k̂), where 1 unit is 10 m. The organisers must know how close the two paths come, so that the drones never collide. (i) Write the direction cosines of path A. (1 mark) (ii) Find the angle between the two paths. (1 mark) (iii) Find the shortest distance between the paths, in metres. (2 marks)Unit XI Meena runs a home bakery in Madurai. A tray of cookies needs 1 hour of oven time and 2 kg of flour. A cake needs 2 hours of oven time and 1 kg of flour. Each day she has at most 12 hours of oven time and 12 kg of flour. She earns ₹ 150 on a tray of cookies and ₹ 200 on a cake (example rates). Let x be the number of trays of cookies and y the number of cakes. (i) Write the objective function. (1 mark) (ii) Write the constraints. (1 mark) (iii) Find the corner points of the feasible region and her maximum earnings. (2 marks)Unit XII A school plans a snack from two foods F₁ and F₂. One serving of F₁ gives 3 units of iron and 1 unit of protein; one serving of F₂ gives 1 unit of iron and 2 units of protein. The snack must give at least 9 units of iron and 8 units of protein. F₁ costs ₹ 20 and F₂ costs ₹ 30 per serving (example prices). (i) Write the LPP to minimise cost. (1 mark) (ii) Find the corner points of the feasible region. (1 mark) (iii) Find the minimum cost and explain why it is truly the minimum. (2 marks)Unit XII An online grocery service in Chennai sends orders from three warehouses W₁, W₂ and W₃, which handle 50%, 30% and 20% of the orders. Late deliveries happen for 4%, 6% and 5% of their orders. The manager wants to know where late orders come from, to decide which warehouse needs more delivery staff. (i) Find the probability that an order comes from W₂ and is late. (1 mark) (ii) Find the probability that an order is late. (1 mark) (iii) An order was late. Find the probability that it came from W₁. (2 marks)Unit XIII In a class of 40 students, 24 play cricket, 16 play chess and 10 play both. A student is picked at random. The sports teacher wants to know whether playing one game is linked to playing the other. (i) Find the probability that the student plays chess, given that the student plays cricket. (1 mark) (ii) Find the probability that the student plays cricket, given that the student plays chess. (1 mark) (iii) Are the events 'plays cricket' and 'plays chess' independent? Justify. (2 marks)Unit XIII

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How to lay out a long answer, with a write-up check. 92 frames so far.