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Can you solve it? Mathematics

Ten problems that get harder as you go. Each has a hint and a full worked solution, but try properly before you look. Your progress is saved on this device.

0 of 10 solved
  1. 1Very hard

    Zeros at the end

    100! means 100 × 99 × 98 × … × 2 × 1. How many zeros are at the end of this number?

  2. 2Hard

    The leaning ladder

    A 10 m ladder leans against a wall, making an angle of 60° with the ground. How high up the wall does it reach? Give your answer to 2 decimal places.

  3. 3Very hard

    No calculator needed

    Without a calculator, find the exact value of sin²15° + sin²75°.

  4. 4Hard

    The biggest field

    A farmer has 40 m of fencing to make a rectangular pen. What is the largest area, in m², that the pen can have?

  5. 5Fiendish

    The best box

    From a 12 cm × 12 cm square of card, you cut equal squares of side x from each corner and fold up the sides to make an open box. What is the largest possible volume, in cm³?

  6. 6Very hard

    Area under a parabola

    Find the exact area between the curve y = x², the x-axis, and the lines x = 0 and x = 3.

  7. 7Hard

    Seven or eleven

    Two fair dice are rolled. What is the probability that the total is 7 or 11? Give it as a percentage to 1 decimal place.

  8. 8Fiendish

    The egg seller

    An egg seller finds that when she counts her eggs in 2s, 3s, 4s, 5s or 6s, there is always exactly 1 egg left over, but when she counts them in 7s there are none left over. What is the smallest number of eggs she could have?

  9. 9FiendishLong problem

    Hills and valleys of a cubic

    For the curve y = x³ − 6x² + 9x + 1, work through all four parts, then type your answer to part (d).

    Equation sheet
    f(x) = x³ − 6x² + 9x + 1Power rule: d/dx (xⁿ) = n xⁿ⁻¹Stationary points: f′(x) = 0Second derivative: f″(x) > 0 ⟹ minimum (valley), f″(x) < 0 ⟹ maximum (hill)Integration: ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + CDefinite integral: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′(x) = f(x)Area under a curve that stays above the x-axis = its definite integral
    1. (a) Find f′(x) and factorise it.
    2. (b) Find the coordinates of both stationary points.
    3. (c) Use f″(x) to decide which is a maximum and which is a minimum.
    4. (d) Find the exact area between the curve, the x-axis and the lines x = 0 and x = 3. Type it as a decimal.
  10. 10FiendishLong problem

    The triangle and its circle

    Triangle ABC has sides a = BC = 7 cm and b = CA = 8 cm, with angle C = 60° between them. Work through all four parts, then type your answer to part (d).

    Equation sheet
    Cosine rule: c² = a² + b² − 2ab cos CSine rule: a / sin A = b / sin B = c / sin C = 2R (R is the radius of the circle through all three corners)Area: Area = ½ ab sin CExact values: sin 60° = √3 / 2, cos 60° = ½Angle sum: A + B + C = 180°
    1. (a) Find the third side c exactly, and as a decimal.
    2. (b) Find the exact area of the triangle.
    3. (c) Find angles A and B to 1 decimal place.
    4. (d) Find the radius R of the circle that passes through A, B and C. Type it to 2 decimal places.