Questions Unit 3 (82 questions)

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WJEC Unit 3 2019 June Q1

a) Express \(\frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } }\) in terms of partial fractions. b) Find \(\int \frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } } \mathrm {~d} x\).

WJEC Unit 3 2019 June Q2

Expand \(\frac { 4 - x } { \sqrt { 1 + 2 x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). State the range of values of \(x\) for which the expansion is valid.

WJEC Unit 3 2019 June Q3

The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.

WJEC Unit 3 2019 June Q6

A curve \(C\) has parametric equations \(x = \sin \theta , y = \cos 2 \theta\). a) The equation of the tangent to the curve \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 4 }\) is \(y = m x + c\). Find the exact values of \(m\) and \(c\).
b) Find the coordinates of the points of intersection of the curve \(C\) and the straight line \(x + y = 1\).

\(\mathbf { 0 }\)

The diagram below shows a sketch of the graph of \(y = f ( x )\). The graph crosses the \(y\)-axis at the point \(( 0 , - 2 )\), and the \(x\)-axis at the point \(( 8,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-3_784_1080_1407_513}
a) Sketch the graph of \(y = - 4 f ( x + 3 )\). Indicate the coordinates of the point where the graph crosses the \(x\)-axis and the \(y\)-coordinate of the point where \(x = - 3\).
b) Sketch the graph of \(y = 3 + f ( 2 x )\). Indicate the \(y\)-coordinate of the point where \(x = 4\).

WJEC Unit 3 2019 June Q8

a) The \(3 ^ { \text {rd } } , 19 ^ { \text {th } }\) and \(67 ^ { \text {th } }\) terms of an arithmetic sequence form a geometric sequence. Given that the arithmetic sequence is increasing and that the first term is 3 , find the common difference of the arithmetic sequence. b) A firm has 100 employees on a particular Monday. The next day it adds 12 employees onto its staff and continues to do so on every successive working day, from Monday to Friday.
i) Find the number of employees at the end of the \(8 { } ^ { \text {th } }\) week.
ii) Each employee is paid \(\pounds 55\) per working day. Determine the total wage bill for the 8 week period.

WJEC Unit 3 2019 June Q9

a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$

WJEC Unit 3 2019 June Q10

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 \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
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. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.

WJEC Unit 3 2019 June Q14

a) Find \(\int \left( \mathrm { e } ^ { 2 x } + 6 \sin 3 x \right) \mathrm { d } x\). b) Find \(\int 7 \left( x ^ { 2 } + \sin x \right) ^ { 6 } ( 2 x + \cos x ) \mathrm { d } x\).
c) Find \(\int \frac { 1 } { x ^ { 2 } } \ln x \mathrm {~d} x\).
d) Use the substitution \(u = 2 \cos x + 1\) to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$

WJEC Unit 3 2019 June Q15

Use proof by contradiction to show that \(\sqrt { 6 }\) is irrational.

WJEC Unit 3 2022 June Q1

Solve the equation $$6 \sec ^ { 2 } x - 8 = \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

WJEC Unit 3 2022 June Q2

Differentiate the following functions with respect to \(x\). a) \(x ^ { 3 } \ln ( 5 x )\) b) \(( x + \cos 3 x ) ^ { 4 }\)

WJEC Unit 3 2022 June Q3

The diagram below shows a plan of the patio Eric wants to build.

\includegraphics[max width=\textwidth, alt={}]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-2_517_746_1505_632}

The walls \(O A\) and \(O C\) are perpendicular. The straight line \(A B\) is of length 4 m and is perpendicular to \(O A\). The shape \(O B C\) is a sector of a circle with centre \(O\) and radius OC.
The angle \(B O C\) is \(\frac { \pi } { 3 }\) radians. Calculate the area of the patio \(O A B C\). Give your answer correct to 2 decimal places. The sum to infinity of a geometric series with first term \(a\) and common ratio \(r\) is 120 . The sum to infinity of another geometric series with first term \(a\) and common ratio \(4 r ^ { 2 }\) is \(112 \frac { 1 } { 2 }\). Find the possible values of \(r\) and the corresponding values of \(a\).

The function \(f ( x )\) is defined by $$f ( x ) = \frac { 6 x + 4 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) }$$ a) Express \(f ( x )\) in terms of partial fractions.
b) Find \(\int \frac { 3 x + 2 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) } \mathrm { d } x\), giving your answer in the form \(a \ln | g ( x ) |\), where \(a\) is a real number and \(g ( x )\) is a function of \(x\).

Geraint opens a savings account. He deposits \(\pounds 10\) in the first month. In each subsequent month, the amount he deposits is 20 pence greater than the amount he deposited in the previous month.
a) Find the amount that Geraint deposits into the savings account in the 12th month.
b) Determine the number of months it takes for the total amount in the savings account to reach \(\pounds 954\).
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0
The diagram below shows a sketch of the curves \(y = x ^ { 2 }\) and \(y = 8 \sqrt { x }\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-3_508_869_2094_623} Find the area of the region bounded by the two curves.

WJEC Unit 3 2022 June Q8

Find the first three terms in the binomial expansion of \(\frac { 2 - x } { \sqrt { 1 + 3 x } }\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac { 1 } { 22 }\) in your expansion, find an approximate value for \(\sqrt { 22 }\) in the form \(\frac { a } { b }\), where \(a , b\) are integers whose values are to be found.

WJEC Unit 3 2022 June Q9

For each of the following sequences, find the first 5 terms, \(u _ { 1 }\) to \(u _ { 5 }\). Describe the behaviour of each sequence. a) \(\quad u _ { n } = \sin \left( \frac { n \pi } { 2 } \right)\) b) \(u _ { 6 } = 33 , u _ { n } = 2 u _ { n - 1 } - 1\)

WJEC Unit 3 2022 June Q10

Solve the equation $$\frac { 6 x ^ { 5 } - 17 x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } } { ( 3 x + 2 ) } = 0$$

WJEC Unit 3 2022 June Q11

a) Express \(9 \cos x + 40 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). b) Find the maximum possible value of \(\frac { 12 } { 9 \cos x + 40 \sin x + 47 }\).

WJEC Unit 3 2022 June Q12

The diagram below shows a sketch of the graph of \(y = f ( x )\), where $$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$ The graph intersects the \(x\)-axis at the points \(( p , 0 ) , ( q , 0 )\) and the \(y\)-axis at the point \(( 0,10 )\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of \(f f ( p )\).
b) Determine the values of \(p\) and \(q\).
c) Express \(f ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why \(f ^ { - 1 } ( x )\) does not exist.
e) The function \(g ( x )\) is defined as $$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$ i) Find an expression for \(g ^ { - 1 } ( x )\).
ii) Sketch the graph of \(y = g ^ { - 1 } ( x )\), indicating the coordinates of the points where the graph intersects the \(x\)-axis and the \(y\)-axis.

WJEC Unit 3 2022 June Q13

A function is defined by \(f ( x ) = 2 x ^ { 3 } + 3 x - 5\). a) Prove that the graph of \(f ( x )\) does not have a stationary point.
b) Show that the graph of \(f ( x )\) does have a point of inflection and find the coordinates of the point of inflection.
c) Sketch the graph of \(f ( x )\).

Evaluate the integral \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x\).

WJEC Unit 3 2022 June Q15

A rectangle is inscribed in a semicircle with centre \(O\) and radius 4. The point \(P ( x , y )\) is the vertex of the rectangle in the first quadrant as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area \(A\) of the rectangle as a function of \(x\).
b) Show that the maximum value of \(A\) occurs when \(y = x\).

WJEC Unit 3 2022 June Q16

The parametric equations of the curve \(C\) are $$x = 3 - 4 t + t ^ { 2 } , \quad y = ( 4 - t ) ^ { 2 }$$ a) Find the coordinates of the points where \(C\) meets the \(y\)-axis.
b) Show that the \(x\)-axis is a tangent to the curve \(C\).

a) Prove that $$\cos ( \alpha - \beta ) + \sin ( \alpha + \beta ) \equiv ( \cos \alpha + \sin \alpha ) ( \cos \beta + \sin \beta )$$ b) i) Hence show that \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta }\) can be expressed as \(\cos \theta + \sin \theta\).
ii) Explain why \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta } \neq \cos \theta + \sin \theta\) when \(\theta = \frac { 3 \pi } { 16 }\).

WJEC Unit 3 2022 June Q18

a) Use a suitable substitution to find $$\int \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x$$ b) Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x\). END OF PAPER \end{document}

WJEC Unit 3 2018 June Q1

4 marks Standard +0.3

Solve the equation $$|2x + 1| = 3|x - 2|.$$ [4]

WJEC Unit 3 2018 June Q2

4 marks Moderate -0.8

The diagram below shows a circle centre O, radius 4 cm. Points A and B lie on the circumference such that arc AB is 5 cm. \includegraphics{figure_2}

Calculate the angle subtended at O by the arc AB. [2]
Determine the area of the sector OAB. [2]

WJEC Unit 3 2018 June Q3

6 marks Moderate -0.8

The diagram below shows a sketch of the graph of \(y = f(x)\). The graph passes through the points \((-2, 0)\), \((0, 8)\), \((4, 0)\) and has a maximum point at \((1, 9)\). \includegraphics{figure_3}

Sketch the graph of \(y = 2f(x + 3)\). Indicate the coordinates of the stationary point and the points where the graph crosses the \(x\)-axis. [3]
Sketch the graph of \(y = 5 - f(x)\). Indicate the coordinates of the stationary point and the point where the graph crosses the \(y\)-axis. [3]

WJEC Unit 3 2018 June Q4

5 marks Standard +0.8

Solve the equation $$2\tan^2\theta + 2\tan\theta - \sec^2\theta = 2,$$ for values of \(\theta\) between \(0°\) and \(360°\). [5]