Official 2023 Paper 1
20 questions · 75 min · Difficulty 42
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Questions in this paper
- Question 1Given that ∫_0^1(ax+b)d x=1 and ∫_0^1x(ax+b)d x=1 find the value of a+b.
- Question 2The graphs of y=x^2+5x+6 and y=mx-3, where m is a constant, are plotted on the same set of axes. Given that the graphs do not meet, what is…
- Question 3For any integer n≥0, ∫_n^(n+1)f(x)d x=n+1. Evaluate ∫_0^3f(x)d x+∫_1^3f(x)d x+∫_2^3f(x)d x+∫_4^3f(x)d x+∫_5^3f(x)d x.
- Question 4Evaluate Σ_(n=0)^∞(sin(nπ+fracπ3))/(2^n).
- Question 5The following shape has two lines of reflectional symmetry. MNOP is a square of perimeter 40 cm. The vertices of rectangle RSTU lie on the…
- Question 6In the simplified expansion of (2+3x)^(12), how many of the terms have a coefficient that is divisible by 12?
- Question 7P(x) and Q(x) are defined as follows: P(x)=2^x+4 Q(x)=2^((2x-2))-2^((x+2))+16. Find the largest value of x such that P(x) and Q(x) are in…
- Question 8A triangle XYZ is called *fun* if it has the following properties: angle YXZ&=30°XY&=√3 a YZ&=a where a is a constant. For a given value of…
- Question 9How many solutions are there to (1+3cos3θ)^2=4 in the interval 0°≤θ≤180°?
- Question 10The trapezium rule with 4 strips is used to estimate the integral: ∫_(-2)^2√(4-x^2)d x. What is the positive difference between the…
- Question 11It is given that f(x)=x^2-6x. The curves y=f(kx) and y=f(x-c) have the same minimum point, where k>0 and c>0. Which of the following is a…
- Question 12How many solutions are there to the equation frac2^(tan^2x)4^(sin^2x)=1 in the range 0≤ x≤2π?
- Question 13Point P lies on the circle with equation (x-2)^2+(y-1)^2=16. Point Q lies on the circle with equation (x-4)^2+(y+5)^2=16. What is the…
- Question 14The function f(x)=frac23x^3+2mx^2+n, m>0 has three distinct real roots. What is the complete range of possible values of n, in terms of m?
- Question 15The difference between the maximum and minimum values of the function f(x)=a^(cos x), where a>0 and x is real, is 3. Find the sum of the…
- Question 16A right-angled triangle has vertices at (2,3), (9,-1) and (5,k). Find the sum of all the possible values of k.
- Question 17A circle C_n is defined by x^2+y^2=2n(x+y) where n is a positive integer. C_1 and C_2 are drawn and the area between them is shaded. Next…
- Question 18You are given that S=4+frac8k7+(16k^2)/(49)+(32k^3)/(343)+…+4(frac2k7)^n+…. The value for k is chosen as an integer in the range -5≤ k≤5…
- Question 19The solution to the differential equation (d y)/(d x)=|-6x| all x is y=f(x)+c, where c is a constant. Which one of the following is a…
- Question 20The diagram shows the graph of y=f(x). The function f attains its maximum value of 2 at x=1, and its minimum value of -2 at x=-1. Find the…