Free IB Maths AI SL Practice Exam (PDF)

This free IB Mathematics: Applications and Interpretation SL practice exam follows the Paper 1 format: ten short-response questions, every one solvable with a graphic display calculator, across all five syllabus topics. It covers the questions AI students most often drop marks on — TVM loan repayments, percentage error, the cosine rule, Voronoi vertices, the chi-square test of independence, normal-distribution quantiles and the trapezoidal rule — and every solution is written as a markscheme, showing the method and accuracy marks. View it online or download the print-ready PDF; no account needed.

1. Question 16 marks · Financial mathematics
   Maya invests 8000 USD in a savings account that pays a nominal annual interest rate of 4.2%, compounded monthly. She leaves the money in the account for 5 years. Maya also borrows 15 000 USD to buy a car. The loan charges a nominal annual interest rate of 6%, compounded monthly, and is repaid in equal monthly instalments over 4 years.
   (a)

   Calculate the value of Maya's investment at the end of the 5 years. Give your answer correct to two decimal places.

   [2]
   (b)

   Tom estimates the value of Maya's investment by assuming simple interest of 4.2% per year for 5 years. Calculate the percentage error in Tom's estimate.

   [2]
   (c)

   Find the monthly repayment on the car loan, and hence the total interest Maya pays.

   [2]
2. Question 27 marks · Arithmetic & geometric sequences
   Maya and Tom each start a savings plan on 1 January. Maya saves 40 USD in the first month and increases the amount she saves by 5 USD every month. Tom saves 40 USD in the first month and increases the amount he saves by 8% every month.
   (a)

   Find the amount Maya saves in the twelfth month.

   [2]
   (b)

   Find the total amount Maya saves in the first two years.

   [2]
   (c)

   Find the total amount Tom saves in the first two years.

   [2]
   (d)

   State who saves more over the two years, and by how much.

   [1]
3. Question 37 marks · Trigonometry
   A triangular field ABC has AB = 85 m, AC = 62 m and angle BAC = 48 degrees. A vertical flagpole of height 12 m stands at C.
   (a)

   Find the length of BC.

   [3]
   (b)

   Find the area of the field.

   [2]
   (c)

   Find the angle of elevation of the top of the flagpole from B.

   [2]
4. Question 46 marks · Voronoi diagrams
   A town has three mobile-phone masts at A(0, 0), B(8, 0) and C(4, 6), where the coordinates are in kilometres. Every phone connects to the nearest mast, so the town is divided into three regions by a Voronoi diagram with sites A, B and C.
   (a)

   Write down the equation of the perpendicular bisector of AB.

   [1]
   (b)

   Find the equation of the perpendicular bisector of AC. Give your answer in the form $ax + by = c$, where $a$, $b$ and $c$ are integers.

   [3]
   (c)

   Hence find the coordinates of the vertex of the Voronoi diagram, the point that is equidistant from all three masts.

   [1]
   (d)

   A new house is built at (5, 2). Determine which mast the phones in this house will connect to.

   [1]
5. Question 56 marks · Volume & surface area
   A grain silo consists of a cylinder of radius 3 m and height 8 m with a cone of height 4 m on top. The cone has the same radius as the cylinder. The silo stands on flat ground.
   (a)

   Calculate the volume of the silo.

   [3]
   (b)

   Show that the slant height of the cone is 5 m.

   [1]
   (c)

   The outside of the silo, excluding the base on the ground, is to be painted. Calculate the area to be painted.

   [2]
6. Question 66 marks · Descriptive statistics
   The times, in minutes, taken by 12 students to complete a puzzle were 14, 18, 22, 23, 25, 25, 27, 29, 31, 34, 38 and 52.
   (a)

   Write down the median time.

   [1]
   (b)

   Find the interquartile range.

   [2]
   (c)

   Show that the time of 52 minutes is an outlier.

   [2]
   (d)

   Find the mean time.

   [1]
7. Question 76 marks · Normal & binomial distributions
   The masses of the apples grown on a farm are normally distributed with mean 165 g and standard deviation 12 g.
   (a)

   Find the probability that a randomly chosen apple has a mass greater than 180 g.

   [2]
   (b)

   The heaviest 5% of the apples are sold as premium apples. Find the minimum mass of a premium apple.

   [2]
   (c)

   Six apples are chosen at random. Find the probability that at least one of them has a mass greater than 180 g.

   [2]
8. Question 86 marks · Conditional probability
   A factory makes light bulbs on two machines. Machine A makes 60% of the bulbs, and 3% of its bulbs are faulty. Machine B makes the remaining 40% of the bulbs, and 5% of its bulbs are faulty. A bulb is chosen at random from the factory's output.
   (a)

   Find the probability that the bulb is faulty.

   [2]
   (b)

   Given that the bulb is faulty, find the probability that it was made by machine B.

   [2]
   (c)

   Two bulbs are chosen at random and independently. Find the probability that exactly one of them is faulty.

   [2]
9. Question 910 marks · Correlation, regression & chi-square test
   A teacher records the number of hours, $x$, that each of eight students spent revising for a test, and their scores, $y$, out of 100. The pairs $(x, y)$ are (1, 46), (2, 50), (3, 58), (4, 57), (5, 66), (6, 70), (7, 71) and (8, 82). The teacher then surveys 150 students across the school about their main revision method and whether they passed the test. Of the 50 students who mainly re-read their notes, 28 passed; of the 40 who used flashcards, 26 passed; of the 60 who did practice tests, 48 passed.
   (a)

   Find Pearson's product-moment correlation coefficient, $r$, and describe the correlation.

   [2]
   (b)

   Write down the equation of the regression line of $y$ on $x$.

   [2]
   (c)

   Use your regression line to estimate the score of a student who revised for 5.5 hours.

   [1]
   (d)

   The teacher carries out a chi-square test of independence at the 5% significance level. Write down the null hypothesis.

   [1]
   (e)

   Show that the expected number of students who did practice tests and passed is 40.8.

   [1]
   (f)

   Find the chi-square statistic and the p-value for the test.

   [2]
   (g)

   State, with a reason, the conclusion of the test.

   [1]
10. Question 1010 marks · Calculus: turning points & trapezoidal rule
    A garden designer models the width of a flower bed. One edge of the bed is a straight path along the x-axis, and the other edge follows the curve $y = w(x)$, where $w(x) = 0.1x^3 - 0.9x^2 + 2.4x$ for $0 \leq x \leq 5$, with $x$ and $w(x)$ in metres. The bed ends with a straight edge at $x = 5$.
    $w(x) = 0.1x^3 - 0.9x^2 + 2.4x, \qquad 0 \leq x \leq 5$
    (a)

    Find $w'(x)$.

    [2]
    (b)

    Find the coordinates of the local maximum point and the local minimum point of the curve.

    [4]
    (c)

    Use the trapezoidal rule with five intervals of width 1 m to estimate the area of the flower bed.

    [3]
    (d)

    Use your GDC to find the actual area of the bed, and state whether the trapezoidal rule gives an underestimate or an overestimate.

    [1]

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