11 questions · 100 marks · ≈ 2 hours
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Z Diagram NOT accurately drawn h cm X Y Figure 1 In Figure 1, △XYZ has base XY and height h cm. The length of XY is 7.2 cm, to 2 significant figures.
Write down the lower bound for the length of XY. The area of △XYZ is 29 cm², to 2 significant figures.
Write down the upper bound for the area of △XYZ.
Calculate, to 3 decimal places, the upper bound of h.
Complete the table of values for
On the grid opposite, plot the points from your completed table and join them to form a smooth curve.
Use your graph to find an estimate for the minimum value of y.
Use your graph to find estimates, to one decimal place, for the solutions of the equation x² − 3x + 1 = 0
By drawing a suitable straight line on your grid, find estimates, to one decimal place, for the solutions of the equation x² − 4x + 2 = 0
Hugh and Pau are planning to sell ice creams at a charity event. They buy enough ingredients to make exactly 270 ice creams. Hugh and Pau make vanilla flavour, strawberry flavour and chocolate flavour ice creams. The numbers of vanilla flavour, strawberry flavour and chocolate flavour ice creams they make are in the ratios 2 : 3 : 4
Calculate the number of chocolate flavour ice creams they make. Hugh and Pau buy all the ingredients for the 270 ice creams at a total cost of £64.80 They calculate the selling price of each ice cream so that if they sell all 270 ice creams, they would make a profit of £1.20 on each ice cream.
Calculate the percentage profit that Hugh and Pau would make if they sell all 270 ice creams at their calculated selling price. At the charity event, Hugh and Pau sell of the 270 ice creams at their calculated selling price. 9 They then sell the remainder of the ice creams at half this selling price.
Calculate the total profit, in pounds, that they make by selling all 270 ice creams.
B 3a C Diagram NOT 6b accurately drawn M O 6a A Figure 2 → → → In Figure 2, OACB is a trapezium in which OA = 6a, OB = 6b and BC = 3a M is the point on AC such that AM : AC = 1 : 3
AB (ii) AC (iii) AM → → The point N is such that ON = μOM where μ > 1 and such that BCN is a straight line. →
Find and simplify an expression, in terms of a and b, for ON . Given that the area of △OAM is 12
…………… cm²
find the area, in cm², of △NMC.
On the grid opposite, triangle B is the image of triangle A under a single transformation.
Describe fully the single transformation. Triangle B is transformed to triangle C under the transformation with matrix M where ⎛ −1 0⎞ M= −3 −1
On the grid, draw and label triangle C. Triangle C is transformed to triangle D under the transformation with matrix N where ⎛ 3 −1⎞ N= −1 0
On the grid, draw and label triangle D. Triangle D is the image of triangle B under a single transformation.
Describe fully this transformation.
All 35 students in class 11T were asked which of rugby (R), cricket (C ) and football (F ) they like. Of these 35 students 14 like rugby 20 like cricket 12 like football 6 like both cricket and football 7 like both rugby and football 4 like all three sports 5 do not like any of rugby, cricket or football. Let x be the number of students in class 11T who like both rugby and cricket.
Show all this information on the Venn diagram, giving the number of elements in each appropriate subset, in terms of x where necessary. E R C F
Find the value of x.
n(R ∪ F )
n(R ∩ [C ∪ F ]) Some of the 14 students in class 11T, who like rugby, play for a rugby club team. The incomplete probability tree diagram gives information about these 14 students. plays for a club team boy does not play for a club team plays for a club team girl does not play for a club team
Complete the probability tree diagram. A student who likes rugby in class 11T is to be chosen at random.
Find the probability that this student plays rugby for a club team.
B Diagram NOT accurately drawn A O F 59° D C 27° 12 cm E Figure 3 In Figure 3, ABCD is a circle, centre O. EA is the tangent to the circle at A. ECF is the tangent to the circle at C. EDO is a straight line. ∠OEC = 27° ∠BCF = 59° EC = 12 cm
Explain why ∠OCE = 90°
Calculate the area, in cm² to 3 significant figures, of △OEC.
Giving reasons, calculate the size, in degrees, of ∠ABC.
Calculate the size, in degrees, of ∠ADC.
Calculate the size, in degrees, of ∠BAO.
Solve the simultaneous equations 2x2 + y² = 6 x − 2y = 3 Show clear algebraic working.
Figure 4 shows a histogram that gives information about the time, in minutes, each of 440 people spent in a supermarket. Two people are chosen at random from the 440 people. Calculate an estimate, to 3 significant figures, for the probability that both of these people spent more than 45 minutes in the supermarket.
Diagram NOT accurately drawn 2x cm y cm 2x cm Figure 5 Figure 5 shows a cuboid with width 2x cm, height 2x cm and length y cm. Given that the total surface area of the cuboid is 240 cm²
show that x² + xy − 30 = 0
Using calculus, calculate the exact value of x for which the volume of the cuboid is a maximum.
The three functions, f, g and h, are defined as f : x ↦ x² + 3x − 4 g : x ↦ 2x + 3 x h:x ↦ 3 x − 25
Write down the value of x that must be excluded from any domain of h.
Find f(6)
Solve the equation h(x) = 7
Find hg(4)
Express the inverse function h−1 in the form h−1 : x ↦ …
Solve the equation fg(x) = 0