Questions Unit 3 (82 questions)

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WJEC Unit 3 Specimen Q9
6 marks Standard +0.3
\includegraphics{figure_9} The diagram above shows a sketch of the curves \(y = x^2 + 4\) and \(y = 12 - x^2\). Find the area of the region bounded by the two curves. [6]
WJEC Unit 3 Specimen Q10
15 marks Standard +0.3
The equation $$1 + 5x - x^4 = 0$$ has a positive root \(\alpha\).
  1. Show that \(\alpha\) lies between 1 and 2. [2]
  2. Use the iterative sequence based on the arrangement $$x = \sqrt[4]{1+5x}$$ with starting value 1.5 to find \(\alpha\) correct to two decimal places. [3]
  3. Use the Newton-Raphson method to find \(\alpha\) correct to six decimal places. [6]
WJEC Unit 3 Specimen Q11
11 marks Standard +0.3
  1. The curve \(C\) is given by the equation $$x^4 + x^2 y + y^2 = 13.$$ Find the value of \(\frac{dy}{dx}\) at the point \((-1, 3)\). [4]
  2. Show that the equation of the normal to the curve \(y^2 = 4x\) at the point \(P(p^2, 2p)\) is $$y + px = 2p + p^3.$$ Given that \(p \neq 0\) and that the normal at \(P\) cuts the \(x\)-axis at \(B(b, 0)\), show that \(b > 2\). [7]
WJEC Unit 3 Specimen Q12
9 marks Standard +0.3
  1. Differentiate \(\cos x\) from first principles. [5]
  2. Differentiate the following with respect to \(x\), simplifying your answer as far as possible.
    1. \(\frac{3x^2}{x^3+1}\) [2]
    2. \(x^3 \tan 3x\) [2]
WJEC Unit 3 Specimen Q13
12 marks Standard +0.3
  1. Solve the equation $$\operatorname{cosec}^2 x + \cot^2 x = 5$$ for \(0^{\circ} \leq x \leq 360^{\circ}\). [5]
    1. Express \(4\sin \theta + 3\cos \theta\) in the form \(R\sin(\theta + \alpha)\), where \(R > 0\) and \(0^{\circ} \leq \alpha \leq 90^{\circ}\). [4]
    2. Solve the equation $$4\sin \theta + 3\cos \theta = 2$$ for \(0^{\circ} \leq \theta \leq 360^{\circ}\), giving your answer correct to the nearest degree. [3]
WJEC Unit 3 Specimen Q14
10 marks Standard +0.3
  1. A cylindrical water tank has base area 4 m\(^2\). The depth of the water at time \(t\) seconds is \(h\) metres. Water is poured in at the rate 0.004 m\(^3\) per second. Water leaks from a hole in the bottom at a rate of 0.0008\(h\) m\(^3\) per second. Show that $$5000 \frac{dh}{dt} = 5 - h.$$ [2] [Hint: the volume, \(V\), of the cylindrical water tank is given by \(V = 4h\).]
  2. Given that the tank is empty initially, find \(h\) in terms of \(t\). [7]
  3. Find the depth of the water in the tank when \(t = 3600\) s, giving your answer correct to 2 decimal places. [1]
WJEC Unit 3 Specimen Q15
3 marks Moderate -0.5
Prove by contradiction the following proposition. When \(x\) is real and positive, $$4x + \frac{9}{x} \geq 12.$$ The first line of the proof is given below. Assume that there is a positive and a real value of \(x\) such that $$4x + \frac{9}{x} < 12.$$ [3]