Questions Unit 3 (28 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
WJEC Unit 3 Specimen Q14
14. (a) A cylindrical water tank has base area \(4 \mathrm {~m} ^ { 2 }\). The depth of the water at time \(t\) seconds is \(h\) metres. Water is poured in at the rate \(0.004 \mathrm {~m} ^ { 3 }\) per second. Water leaks from a hole in the bottom at a rate of \(0.0008 \mathrm { hm } ^ { 3 }\) per second. Show that $$5000 \frac { \mathrm {~d} h } { \mathrm {~d} t } \equiv 5 - h$$ [Hint: the volume, \(V\), of the cylindrical water tank is given by \(V = 4 h\).]
(b) Given that the tank is empty initially, find \(h\) in terms of \(t\).
(c) Find the depth of the water in the tank when \(t = 3600 \mathrm {~s}\), giving your answer correct to 2 decimal places.
WJEC Unit 3 Specimen Q15
15. Prove by contradiction the following proposition. When \(x\) is real and positive, $$4 x + \frac { 9 } { x } \geq 12$$ The first line of the proof is given below.
Assume that there is a positive and a real value of \(x\) such that $$4 x + \frac { 9 } { x } < 12$$
WJEC Unit 3 2019 June Q1
\(\mathbf { 1 }\) & \(\mathbf { 0 }\)
\hline \end{tabular} \end{center} a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\)
ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\)
iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\)
b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
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1
The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
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1
A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\).
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a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places.