OCR MEI C3 — Question 2

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
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Mark schemeDownload PDF ↗
TopicTrig Graphs & Exact Values
TypeSolve inverse trig equation
DifficultyModerate -0.8 This is a straightforward question requiring only direct evaluation of arcsin (finding x = sin(π/6) = 1/2) and then applying the standard identity arcsin x + arccos x = π/2. Both parts are routine recall and basic manipulation with no problem-solving required, making it easier than average.
Spec1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs
Given that \(\arcsin x = \frac{1}{6}\pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).
Question 2:
2
Section A (36 marks)
1 Fig.1 shows the graphs of y (cid:1) (cid:1)x(cid:1) and y (cid:1) (cid:1)x (cid:3) 2(cid:1)(cid:2)1. The point P is the minimum point of
y (cid:1) (cid:1)x (cid:3) 2(cid:1)(cid:2)1, and Q is the point of intersection of the two graphs.
y
y = x-2 +1
Q
y = x
P
O x
Fig. 1
(i) Write down the coordinates of P. [1]
1
(ii) Verify that the y-coordinate of Q is1 . [4]
2
2
2 Evaluate Û Ù x2 lnxdx, giving your answer in an exact form. [5]
ı
1
3 The value £V of a car is modelled by the equation V (cid:1) Ae (cid:3)kt, where t is the age of the car in years
and A and k are constants. Its value when new is £10000, and after 3 years its value is £6000.
(i) Find the values of A and k. [5]
(ii) Find the age of the car when its value is £2000. [2]
4 Use the method of exhaustion to prove the following result.
No 1- or 2-digit perfect square ends in 2, 3, 7 or 8
State a generalisation of this result. [3]
x2
5 The equation of a curve is y (cid:1) .
2x(cid:2)1
dy 2x(x(cid:2)1)
(i) Show that (cid:1) . [4]
dx (2x(cid:2)1)2
(ii) Find the coordinates of the stationary points of the curve. You need not determine their nature.
[4]
2
Section A(36 marks)
1 (i) Differentiate 1+2x. [3]
1
(ii) Show that the derivative of ln (1 (cid:3) e (cid:3)x) is .
ex (cid:3) 1 [4]
2 Given that f(x) (cid:2) 1 (cid:3) x and g(x) (cid:2) (cid:1) x (cid:1), write down the composite function gf(x).
On separate diagrams, sketch the graphs of y (cid:2) f(x) and y (cid:2) gf(x). [3]
3 Acurve has equation 2y2(cid:1)y (cid:2) 9x2(cid:1)1.
dy
(i) Find in terms of x and y. Hence find the gradient of the curve at the point A(1, 2). [4]
dx
dy
(ii) Find the coordinates of the points on the curve at which (cid:2) 0. [4]
dx
4 Acup of water is cooling. Its initial temperature is 100°C. After 3 minutes, its temperature is 80°C.
(i) Given that T (cid:2) 25(cid:1)ae (cid:3)kt, where T is the temperature in °C, t is the time in minutes and
a and k are constants, find the values of a and k. [5]
(ii) What is the temperature of the water
(A) after 5 minutes,
(B) in the long term? [3]
5 Prove that the following statement is false.
For all integers n greater than or equal to 1, n2(cid:1)3n(cid:1)1is a prime number. [2]
Question 2:
2
Section A (36 marks)
1 Fig.1 shows the graphs of y (cid:1) (cid:1)x(cid:1) and y (cid:1) (cid:1)x (cid:3) 2(cid:1)(cid:2)1. The point P is the minimum point of
y (cid:1) (cid:1)x (cid:3) 2(cid:1)(cid:2)1, and Q is the point of intersection of the two graphs.
y
y = x-2 +1
Q
y = x
P
O x
Fig. 1
(i) Write down the coordinates of P. [1]
1
(ii) Verify that the y-coordinate of Q is1 . [4]
2
2
2 Evaluate Û Ù x2 lnxdx, giving your answer in an exact form. [5]
ı
1
3 The value £V of a car is modelled by the equation V (cid:1) Ae (cid:3)kt, where t is the age of the car in years
and A and k are constants. Its value when new is £10000, and after 3 years its value is £6000.
(i) Find the values of A and k. [5]
(ii) Find the age of the car when its value is £2000. [2]
4 Use the method of exhaustion to prove the following result.
No 1- or 2-digit perfect square ends in 2, 3, 7 or 8
State a generalisation of this result. [3]
x2
5 The equation of a curve is y (cid:1) .
2x(cid:2)1
dy 2x(x(cid:2)1)
(i) Show that (cid:1) . [4]
dx (2x(cid:2)1)2
(ii) Find the coordinates of the stationary points of the curve. You need not determine their nature.
[4]
2
Section A(36 marks)
1 (i) Differentiate 1+2x. [3]
1
(ii) Show that the derivative of ln (1 (cid:3) e (cid:3)x) is .
ex (cid:3) 1 [4]
2 Given that f(x) (cid:2) 1 (cid:3) x and g(x) (cid:2) (cid:1) x (cid:1), write down the composite function gf(x).
On separate diagrams, sketch the graphs of y (cid:2) f(x) and y (cid:2) gf(x). [3]
3 Acurve has equation 2y2(cid:1)y (cid:2) 9x2(cid:1)1.
dy
(i) Find in terms of x and y. Hence find the gradient of the curve at the point A(1, 2). [4]
dx
dy
(ii) Find the coordinates of the points on the curve at which (cid:2) 0. [4]
dx
4 Acup of water is cooling. Its initial temperature is 100°C. After 3 minutes, its temperature is 80°C.
(i) Given that T (cid:2) 25(cid:1)ae (cid:3)kt, where T is the temperature in °C, t is the time in minutes and
a and k are constants, find the values of a and k. [5]
(ii) What is the temperature of the water
(A) after 5 minutes,
(B) in the long term? [3]
5 Prove that the following statement is false.
For all integers n greater than or equal to 1, n2(cid:1)3n(cid:1)1is a prime number. [2]
Given that $\arcsin x = \frac{1}{6}\pi$, find $x$. Find $\arccos x$ in terms of $\pi$.

\hfill \mbox{\textit{OCR MEI C3  Q2}}
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