Questions Unit 3 (82 questions)

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WJEC Unit 3 2023 June Q13
7 marks Standard +0.3
The curve \(C_1\) has parametric equations \(x = 3p + 1\), \(y = 9p^2\). The curve \(C_2\) has parametric equations \(x = 4q\), \(y = 2q\). Find the Cartesian coordinates of the points of intersection of \(C_1\) and \(C_2\). [7]
WJEC Unit 3 2023 June Q14
8 marks Moderate -0.3
  1. Use integration by parts to evaluate \(\int_0^1 (3x-1)e^{2x}\,dx\). [4]
  2. Use the substitution \(u = 1 - 2\cos x\) to find \(\int \frac{\sin x}{1 - 2\cos x}\,dx\). [4]
WJEC Unit 3 2024 June Q1
11 marks Standard +0.3
The function \(f\) is given by $$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$
  1. Express \(f(x)\) in terms of partial fractions. [4]
  2. Show that \(\int_1^2 f(x) dx = -\ln P\), where \(P\) is an integer whose value is to be found. [5]
  3. Show that the sign of \(f(x)\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f(x) = 0\) in this case. [2]
WJEC Unit 3 2024 June Q2
11 marks Standard +0.3
  1. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$3\cot\theta + 4\cosec^2\theta = 5.$$ [5]
  2. By writing \(24\cos x - 7\sin x\) in the form \(R\cos(x + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\), solve the equation $$24\cos x - 7\sin x = 16$$ for values of \(x\) between \(0°\) and \(360°\). [6]
WJEC Unit 3 2024 June Q3
7 marks Standard +0.3
The diagram below shows a badge \(ODC\). The shape \(OAB\) is a sector of a circle centre \(O\) and radius \(r\) cm. The shape \(ODC\) is a sector of a circle with the same centre \(O\). The length \(AD\) is \(5\) cm and angle \(AOB\) is \(\frac{\pi}{5}\) radians. The area of the shaded region, \(ABCD\), is \(\frac{13\pi}{2}\) cm\(^2\). \includegraphics{figure_3}
  1. Determine the value of \(r\). [4]
  2. Calculate the perimeter of the shaded region. [3]
WJEC Unit 3 2024 June Q4
6 marks Moderate -0.8
A function \(f\) is given by \(f(x) = |3x + 4|\).
  1. Sketch the graph of \(y = f(x)\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis. [3]
  2. On a separate set of axes, sketch the graph of \(y = \frac{1}{2}f(x) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A'\) and \(B'\). Clearly label the coordinates of the points \(A'\) and \(B'\). [3]
WJEC Unit 3 2024 June Q5
4 marks Standard +0.3
Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac{81}{x} \geq 18\). [4]
WJEC Unit 3 2024 June Q6
13 marks Standard +0.8
  1. Differentiate \(\cos x\) from first principles. [5]
  2. Differentiate \(e^{3x}\sin 4x\) with respect to \(x\). [3]
  3. Find \(\int x^2\sin 2x dx\). [5]
WJEC Unit 3 2024 June Q7
7 marks Moderate -0.8
Showing all your working, evaluate
  1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
  2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [3]
WJEC Unit 3 2024 June Q8
7 marks Standard +0.3
The function \(f\) is defined by $$f(x) = x^3 + 4x^2 - 3x - 1.$$
  1. Show that the equation \(f(x) = 0\) has a root in the interval \([0, 1]\). [1]
  2. Using the Newton-Raphson method with \(x_0 = 0 \cdot 8\),
    1. write down in full the decimal value of \(x_1\) as given in your calculator,
    2. determine the value of this root correct to six decimal places. [4]
  3. Explain why the Newton-Raphson method does not work if \(x_0 = \frac{1}{3}\). [2]
WJEC Unit 3 2024 June Q9
9 marks Standard +0.3
The diagram below shows a sketch of the curve \(C_1\) with equation \(y = -x^2 + \pi x + 1\) and a sketch of the curve \(C_2\) with equation \(y = \cos 2x\). The curves intersect at the points where \(x = 0\) and \(x = \pi\). \includegraphics{figure_9} Calculate the area of the shaded region enclosed by \(C_1\), \(C_2\) and the \(x\)-axis. Give your answer in terms of \(\pi\). [9]
WJEC Unit 3 2024 June Q10
14 marks Standard +0.3
The function \(f\) has domain \([4, \infty)\) and is defined by $$f(x) = \frac{2(3x + 1)}{x^2 - 2x - 3} + \frac{x}{x + 1}.$$
  1. Show that \(f(x) = \frac{x + 2}{x - 3}\). [4]
  2. Determine the range of \(f(x)\). [2]
  3. Find an expression for \(f^{-1}(x)\) and write down the domain and range of \(f^{-1}\). [4]
  4. Find the value of \(x\) when \(f(x) = f^{-1}(x)\). [4]
WJEC Unit 3 2024 June Q11
10 marks Standard +0.3
A curve is defined parametrically by $$x = 2\theta + \sin 2\theta, \quad y = 1 + \cos 2\theta.$$
  1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(-\tan\theta\). [6]
  2. Find the equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{4}\). [4]
WJEC Unit 3 2024 June Q12
6 marks Standard +0.3
  1. Given that \(\theta\) is small, show that \(2\cos\theta + \sin\theta - 1 \approx 1 + \theta - \theta^2\). [2]
  2. Hence, when \(\theta\) is small, show that $$\frac{1}{2\cos\theta + \sin\theta - 1} \approx 1 + a\theta + b\theta^2,$$ where \(a\), \(b\) are constants to be found. [4]
WJEC Unit 3 2024 June Q13
3 marks Standard +0.8
The diagram below shows a sketch of the graph of \(y = f'(x)\) for the interval \([x_1, x_5]\). \includegraphics{figure_13}
  1. Find the interval on which \(f(x)\) is both decreasing and convex. Give reasons for your answer. [2]
  2. Write down the \(x\)-coordinate of a point of inflection of the graph of \(y = f(x)\). [1]
WJEC Unit 3 2024 June Q14
7 marks Standard +0.3
  1. Given that \(y = \frac{1 + \ln x}{x}\), show that \(\frac{dy}{dx} = \frac{-\ln x}{x^2}\). [2]
  2. Hence, solve the differential equation $$\frac{dx}{dt} = \frac{x^2 t}{\ln x},$$ given that \(t = 3\) when \(x = 1\). Give your answer in the form \(t^2 = g(x)\), where \(g\) is a function of \(x\). [5]
WJEC Unit 3 2024 June Q15
5 marks Standard +0.3
Robert wants to deposit \(£P\) into a savings account. He has a choice of two accounts. • Account \(A\) offers an annual compound interest rate of \(1\%\). • Account \(B\) offers an interest rate of \(5\%\) for the first year and an annual compound interest rate of \(0.6\%\) for each subsequent year. After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\). [5]
WJEC Unit 3 Specimen Q1
4 marks Standard +0.3
Find a small positive value of \(x\) which is an approximate solution of the equation. $$\cos x - 4\sin x = x^2.$$ [4]
WJEC Unit 3 Specimen Q2
3 marks Standard +0.3
Air is pumped into a spherical balloon at the rate of 250 cm\(^3\) per second. When the radius of the balloon is 15 cm, calculate the rate at which the radius is increasing, giving your answer to three decimal places [3]
WJEC Unit 3 Specimen Q3
8 marks Moderate -0.3
  1. Sketch the graph of \(y = x^2 + 6x + 13\), identifying the stationary point. [2]
  2. The function \(f\) is defined by \(f(x) = x^2 + 6x + 13\) with domain \((a,b)\).
    1. Explain why \(f^{-1}\) does not exist when \(a = -10\) and \(b = 10\). [1]
    2. Write down a value of \(a\) and a value of \(b\) for which the inverse of \(f\) does exist and derive an expression for \(f^{-1}(x)\). [5]
WJEC Unit 3 Specimen Q4
4 marks Moderate -0.8
  1. Expand \((1-x)^{-\frac{1}{2}}\) in ascending power of \(x\) as far as the term in \(x^2\). State the range of \(x\) for which the expansion is valid. [2]
  2. By taking \(x = \frac{1}{10}\), find an approximation for \(\sqrt{10}\) in the form \(\frac{a}{b}\), where \(a\) and \(b\) are to be determined. [2]
WJEC Unit 3 Specimen Q5
5 marks Standard +0.3
Aled decides to invest £1000 in a savings scheme on the first day of each year. The scheme pays 8% compound interest per annum, and interest is added on the last day of each year. The amount of savings, in pounds, at the end of the third year is given by $$1000 \times 1 \cdot 08 + 1000 \times 1 \cdot 08^2 + 1000 \times 1 \cdot 08^3$$ Calculate, to the nearest pound, the amount of savings at the end of thirty years. [5]
WJEC Unit 3 Specimen Q6
4 marks Standard +0.3
The lengths of the sides of a fifteen-sided plane figure form an arithmetic sequence. The perimeter of the figure is 270 cm and the length of the largest side is eight times that of the smallest side. Find the length of the smallest side. [4]
WJEC Unit 3 Specimen Q7
16 marks Standard +0.8
The curve \(y = ax^4 + bx^3 + 18x^2\) has a point of inflection at \((1, 11)\).
  1. Show that \(2a + b + 6 = 0\). [2]
  2. Find the values of the constants \(a\) and \(b\) and show that the curve has another point of inflection at \((3, 27)\). [8]
  3. Sketch the curve, identifying all the stationary points including their nature. [6]
WJEC Unit 3 Specimen Q8
14 marks Standard +0.3
  1. Integrate
    1. \(e^{-3x+5}\) [2]
    2. \(x^2 \ln x\) [4]
  2. Use an appropriate substitution to show that $$\int_0^{\frac{1}{2}} \frac{x^2}{\sqrt{1-x^2}} dx = \frac{\pi}{12} - \frac{\sqrt{3}}{8}.$$ [8]