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OCR MEI Further Pure with Technology 2024 June Q1
17 marks Standard +0.8
1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter \(a\) is a real number.
    1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
      • \(a = - 0.5\)
  2. \(a = - 0.1\)
  3. \(a = 0.5\) (ii) State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
    (iii) By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
    (iv) Write down the equation of the vertical asymptote of the curve (}).
    4. The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
    (v) Show that \(A = 0\).
    (vi) Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$ (vii) Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
  4. In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
    1. Determine, in terms of \(a\), the area of the triangle XYZ.
    2. Find the maximum area of the triangle XYZ.
OCR MEI Further Pure with Technology 2024 June Q2
23 marks Challenging +1.8
2 Wilson's theorem states that a positive integer \(n > 1\) is prime if and only if \(( n - 1 ) ! \equiv n - 1 ( \bmod n )\).
    1. Calculate 136! (mod 137).
    2. Hence, determine if the integer 137 is prime.
    1. Create a program that uses Wilson's theorem to find all prime numbers less than or equal to \(n\).
      Write down your program in the Printed Answer Booklet.
    2. Using part (b)(i), write down all prime numbers \(x\), where \(260 \leqslant x \leqslant 300\).
    1. Explain why there is exactly one prime number congruent to \(2 ( \bmod 4 )\).
    2. Explain why no prime number is congruent to \(0 ( \bmod 4 )\).
    3. Using part (b)(ii), write down the three prime numbers \(y\), where \(260 \leqslant y \leqslant 300\), that are congruent to \(3 ( \bmod 4 )\). Label the three prime numbers in part (c)(iii) \(c _ { 1 } , c _ { 2 }\) and \(c _ { 3 }\). Define the integer \(N\) by \(N = 4 c _ { 1 } c _ { 2 } c _ { 3 } + 3\).
    4. Explain why \(N\) is not divisible by \(c _ { 1 } , c _ { 2 }\) or \(c _ { 3 }\).
    5. Write down the value of \(N ( \bmod 4 )\).
  4. The fundamental theorem of arithmetic states that every positive integer can be written uniquely as the product of powers of prime numbers. Suppose there are finitely many prime numbers congruent to \(3 ( \bmod 4 )\). Label these prime numbers \(p _ { 1 } , \ldots , \mathrm { p } _ { \mathrm { n } }\), where \(p _ { 1 } = 3\). Using the \(n - 1\) integers \(p _ { 2 } , \ldots , \mathrm { p } _ { \mathrm { n } }\), define the integer \(M\) by \(\mathrm { M } = 4 \mathrm { p } _ { 2 } \mathrm { p } _ { 3 } \ldots \mathrm { p } _ { \mathrm { n } } + 3\).
    1. Write down the value of \(M ( \bmod 4 )\).
    2. Explain why \(\mathrm { M } = \mathrm { q } _ { 1 } \mathrm { q } _ { 2 } \ldots \mathrm { q } _ { \mathrm { r } }\), where the integers \(q _ { 1 } , \ldots , \mathrm { q } _ { \mathrm { r } }\) are all prime.
    3. Prove that there is at least one integer \(i\), where \(1 \leqslant i \leqslant n\), such that \(q _ { i } \equiv 3 ( \bmod 4 )\).
    4. Hence, deduce that there are infinitely many prime numbers congruent to \(3 ( \bmod 4 )\).
OCR MEI Further Pure with Technology 2024 June Q3
20 marks Standard +0.8
3 This question concerns the family of differential equations $$\frac { d y } { d x } = x ^ { 2 } - y + \operatorname { acos } ( x ) \cos ( y ) \quad ( * * )$$ where \(a\) is a constant, \(x \geqslant 0\) and \(y > 0\).
  1. In this part of the question \(a = 0\).
    1. Find the solution to (\textbf{) in which \(y = 1\) when \(x = 0\).
    2. In this part of the question \(m\) is a real number. Show that the equation of the isocline \(\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { m }\) is a parabola.
    3. Using the result given in part (a)(ii), or otherwise, sketch the tangent field for (}) on the axes in the Printed Answer Booklet.
  2. Fig. 3.1 and Fig. 3.2 show the tangent fields for two distinct and unspecified values of \(a\). In each case, a sketch of the solution curve \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) which passes through the point \(( 0,2 )\) is shown for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 3.1} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_399_666_1324_317}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 3.2} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_397_661_1324_1192}
    \end{figure}
    1. In each case, continue the sketch of the solution curve for \(\frac { 1 } { 2 } \leqslant x \leqslant 3\) on the axes in the Printed Answer Booklet.
    2. State one feature which is present in the continued solution curve for Fig. 3.1 that is not a feature of the continued solution curve for Fig. 3.2.
    3. Using a slider for \(a\), or otherwise, estimate the value of \(a\) for the solution curve shown in Fig. 3.2.
  3. The Euler method for the solution of the differential equation \(\frac { d y } { d x } = f ( x , y )\) is as follows. $$\begin{aligned} & y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right) \\ & x _ { n + 1 } = x _ { n } + h \end{aligned}$$
    1. Construct a spreadsheet to solve (), so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 = 0\) and \(y _ { 0 } = 1\). State the formulae you have used in your spreadsheet.
    2. In this part of the question \(a = 0\). Use your spreadsheet with \(h = 0.1\) to approximate the value of \(y\) when \(x = 0.5\) for the solution to (}) in which \(y = 1\) when \(x = 0\).
    3. Using part (a)(i), state the accuracy of the approximate value of \(y\) given in part (c)(ii).
    4. State one change to your spreadsheet that could improve the accuracy of the approximate value of \(y\) found in part (c)(ii).
  4. The modified Euler method for the solution of the differential equation \(\frac { d y } { d x } = f ( x , y )\) is as follows. \(k _ { 1 } = h f \left( x _ { n } , y _ { n } \right)\) \(k _ { 2 } = h f \left( x _ { n } + h , y _ { n } + k _ { 1 } \right)\) \(y _ { n + 1 } = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)\) \(\mathrm { x } _ { \mathrm { n } + 1 } = \mathrm { x } _ { \mathrm { n } } + \mathrm { h }\)
    1. Adapt your spreadsheet from part (c)(i) to a spreadsheet to solve (**), so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 } = 0\) and \(y _ { 0 } = 1\). State the formulae you have used in your spreadsheet.
    2. In this part of the question \(a = - 0.5\). Use the spreadsheet from part (d)(i) with \(h = 0.1\) to approximate the value of \(y\) when \(x = 0.5\) for the solution to \(( * * )\) in which \(y = 1\) when \(x = 0\). In this part of the question \(a = - 0.5\). The solution to (**) in which \(y = 1\) when \(x = 0\) has a turning point with coordinates \(( c , d )\) where \(0 < c < 1\).
    3. Use the spreadsheet in part (d)(i) to determine the value of \(c\) correct to \(\mathbf { 1 }\) decimal place.
    4. Use the spreadsheet in part (d)(i) to determine the value of \(d\) correct to \(\mathbf { 3 }\) decimal places.
OCR MEI Further Pure with Technology Specimen Q1
19 marks Challenging +1.8
1 A family of curves has polar equation \(r = \cos n \left( \frac { \theta } { n } \right) , 0 \leq \theta < n \pi\), where \(n\) is a positive even integer.
  1. (A) Sketch the curve for the cases \(n = 2\) and \(n = 4\).
    (B) State two points which lie on every curve in the family.
    (C) State one other feature common to all the curves.
  2. (A) Write down an integral for the length of the curve for the case \(n = 4\).
    (B) Evaluate the integral.
  3. (A) Using \(t = \theta\) as the parameter, find a parametric form of the equation of the family of curves.
    (B) Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sin t \sin \left( \frac { t } { n } \right) - \cos t \cos \left( \frac { t } { n } \right) } { \sin t \cos \left( \frac { t } { n } \right) + \cos t \sin \left( \frac { t } { n } \right) }\).
  4. Hence show that there are \(n + 1\) points where the tangent to the curve is parallel to the \(y\)-axis.
  5. By referring to appropriate sketches, show that the result in part (iv) is true in the case \(n = 4\).
  6. (A) Create a program to find all the solutions to \(x ^ { 2 } \equiv - 1 ( \bmod p )\) where \(0 \leq x < p\). Write out your program in full in the Printed Answer Booklet.
    (B) Use the program to find the solutions to \(x ^ { 2 } \equiv - 1 ( \bmod p )\) for the primes
    • \(p = 809\),
    • \(p = 811\) and
    • \(p = 444001\).
    • State Wilson's Theorem.
    • The following argument shows that \(( 4 k ) ! \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p )\) for the case \(p = 4 k + 1\).
    $$\begin{aligned} ( 4 k ) ! & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( 2 k + 1 ) \times ( 2 k + 2 ) \times \ldots \times ( 4 k - 1 ) \times 4 k ( \bmod p ) \\ & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( - 2 k ) \times ( - ( 2 k - 1 ) ) \times \ldots \times ( - 2 ) \times ( - 1 ) ( \bmod p ) \\ & \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p ) \end{aligned}$$ (A) Explain why ( \(2 k + 2\) ) can be written as ( \(- ( 2 k - 1 )\) ) in line ( 2 ).
    (B) Explain how line (3) has been obtained.
    (C) Explain why, if \(p\) is a prime of the form \(p = 4 k + 1\), then \(x ^ { 2 } \equiv - 1 ( \bmod p )\) will have at least one solution.
    (D) Hence find a solution of \(x ^ { 2 } \equiv - 1 ( \bmod 29 )\).
  7. (A) Create a program that will find all the positive integers \(n\), where \(n < 1000\), such that \(( n - 1 ) ! \equiv - 1 \left( \bmod n ^ { 2 } \right)\). Write out your program in full.
    (B) State the values of \(n\) obtained.
    (C) A Wilson prime is a prime \(p\) such that \(( p - 1 ) ! \equiv - 1 \left( \bmod p ^ { 2 } \right)\). Write down all the Wilson primes \(p\) where \(p < 1000\).
OCR MEI Further Pure with Technology Specimen Q3
20 marks Challenging +1.2
3 This question explores the family of differential equations \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + a x + 2 y }\) for various values of the parameter \(a\). Fig. 3 shows the tangent field in the case \(a = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{141c85ec-5749-4f24-9f6d-fe7a01567511-4_691_696_452_696} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. (A) Sketch the tangent field in the case \(a = - 2\).
    (B) Explain why the tangent field is not defined for the whole coordinate plane.
    (C) Give an inequality which describes the region in which the tangent field is defined.
    (D) Find a value of \(a\) such that the region for which the tangent field is defined includes the entire \(x\)-axis.
  2. (A) For the case \(a = 1\), with \(y = 1\) when \(x = 0\), construct a spreadsheet for the Runge-Kutta method of order 2 with formulae as follows, where \(\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }\). $$\begin{aligned} k _ { 1 } & = h \mathrm { f } \left( x _ { n } , y _ { n } \right) \\ k _ { 2 } & = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 1 } \right) \\ y _ { n + 1 } & = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right) \end{aligned}$$ State the formulae you have used in your spreadsheet.
    (B) Use your spreadsheet to obtain the value of \(y\) correct to 4 decimal places when \(x = 1\) for
    • \(h = 0.1\) and
    • \(h = 0.05\).
  3. (A) For the case \(a = 0\) find the analytical solution that passes through the point ( 0,1 ).
    (B) Verify that the solution in part (iii) (A) is a solution to the differential equation.
    (C) Use the solution in part (iii) (A) to find the value of \(y\) correct to 4 decimal places when \(x = 1\).
  4. (A) Verify that \(y = - \frac { a } { 2 } x + \frac { a ^ { 2 } } { 8 } - \frac { 1 } { 2 }\) is a solution for all cases when \(a \leq 0\).
    (B) Show that this is the only straight line solution in these cases. \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
WJEC Unit 1 2018 June Q1
7 marks Moderate -0.8
\(\mathbf { 1 }\) & \(\mathbf { 3 }\) \hline \end{tabular} \end{center} A curve \(C\) has equation \(y = x ^ { 3 } - 3 x ^ { 2 }\). a) Find the stationary points of \(C\) and determine their nature.
b) Draw a sketch of \(C\), clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\) is positive or
negative, giving a reason for your answer.
14
In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d . \\ & \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true.
15
The value of a car, \(\pounds V\), may be modelled as a continuous variable. At time \(t\) years, the value of the car is given by \(V = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. When the car is new, it is worth \(\pounds 30000\). When the car is two years old, it is worth \(\pounds 20000\). Determine the value of the car when it is six years old, giving your answer correct to the nearest \(\pounds 100\).
16
The curve \(C\) has equation \(y = 7 + 13 x - 2 x ^ { 2 }\). The point \(P\) lies on \(C\) and is such that the tangent to \(C\) at \(P\) has equation \(y = x + c\), where \(c\) is a constant. Find the coordinates of \(P\) and the value of \(c\).
17
a) Solve \(2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2\).
b) Solve \(3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }\).
c) Express \(4 ^ { x } - 10 \times 2 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\). Hence solve the equation \(4 ^ { x } - 10 \times 2 ^ { x } = - 16\).
\(\mathbf { 1 }\)\(\mathbf { 8 }\)
The coordinates of three points \(A , B , C\) are \(( 4,6 ) , ( - 3,5 )\) and \(( 5 , - 1 )\) respectively. a) Show that \(B \widehat { A C }\) is a right angle.
b) A circle passes through all three points \(A , B , C\). Determine the equation of the circle.
WJEC Unit 1 2019 June Q1
Moderate -0.8
\(\mathbf { 1 }\) & \(\mathbf { 1 }\) \hline \end{tabular} \end{center} Two quantities are related by the equation \(Q = 1 \cdot 25 P ^ { 3 }\). Explain why the graph of \(\log _ { 10 } Q\) against \(\log _ { 10 } P\) is a straight line. State the gradient of the straight line and the intercept on the \(\log _ { 10 } Q\) axis of the graph.
\(\mathbf { 1 }\)\(\mathbf { 2 }\)
In the binomial expansion of \(( 2 - 5 x ) ^ { 8 }\), find a) the number of terms,
b) the \(4 ^ { \text {th } }\) term, when the expansion is in ascending powers of \(x\),
c) the greatest positive coefficient.
\(\mathbf { 1 }\)\(\mathbf { 3 }\)
A curve \(C\) has equation \(y = \frac { 1 } { 9 } x ^ { 3 } - k x + 5\). A point \(Q\) lies on \(C\) and is such that the tangent to \(C\) at \(Q\) has gradient - 9 . The \(x\)-coordinate of \(Q\) is 3 . a) Show that \(k = 12\).
b) Find the coordinates of each of the stationary points of \(C\) and determine their nature.
c) Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis.
\(\mathbf { 1 }\)\(\mathbf { 4 }\)
The diagram below shows a triangle \(A B C\) with \(A C = 5 \mathrm {~cm} , A B = x \mathrm {~cm} , B C = y \mathrm {~cm}\) and angle \(B A C = 120 ^ { \circ }\). The area of the triangle \(A B C\) is \(14 \mathrm {~cm} ^ { 2 }\). Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places.
\(\mathbf { 1 }\)\(\mathbf { 5 }\)
Prove that \(f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 13 x - 7\) is an increasing function.
\(\mathbf { 1 }\)\(\mathbf { 6 }\)
The diagram below shows a curve with equation \(y = ( x + 2 ) ( x - 2 ) ( x + 1 )\).
\includegraphics[max width=\textwidth, alt={}]{2c33cbe4-b65e-4eae-aa2f-9d1d0f5cb9bd-7_754_743_724_678}
Calculate the total area of the two shaded regions. \section*{END OF PAPER}
WJEC Unit 1 2022 June Q1
Moderate -0.3
\(\mathbf { 1 }\) & \(\mathbf { 2 }\) \hline \end{tabular} \end{center} a) Solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 5 x - 2 = 0\). b) Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 180 ^ { \circ }\) satisfying $$\cos \left( 2 \theta - 51 ^ { \circ } \right) = 0 \cdot 891$$
\(\mathbf { 1 }\)\(\mathbf { 3 }\)
Find the term which is independent of \(x\) in the expansion of \(\frac { ( 2 - 3 x ) ^ { 5 } } { x ^ { 3 } }\).
\(\mathbf { 1 }\)\(\mathbf { 4 }\)
A curve \(C\) has equation \(f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } + x - 6\). a) Find the coordinates of the stationary points of \(C\) and determine their nature.
b) Without solving the equations, determine the number of distinct real roots for each of the following:
i) \(3 x ^ { 3 } - 5 x ^ { 2 } + x + 1 = 0\),
ii) \(\quad 6 x ^ { 3 } - 10 x ^ { 2 } + 2 x + 1 = 0\).
\(\mathbf { 1 }\)\(\mathbf { 5 }\)
Solve the simultaneous equations $$\begin{aligned} & 3 \log _ { a } \left( x ^ { 2 } y \right) - \log _ { a } \left( x ^ { 2 } y ^ { 2 } \right) + \log _ { a } \left( \frac { 9 } { x ^ { 2 } y ^ { 2 } } \right) = \log _ { a } 36 \\ & \log _ { a } y - \log _ { a } ( x + 3 ) = 0 \end{aligned}$$
\(\mathbf { 1 }\)\(\mathbf { 6 }\)
The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are defined by \(\mathbf { a } = 2 \mathbf { i } - \mathbf { j }\) and \(\mathbf { b } = \mathbf { i } - 3 \mathbf { j }\). a) Find a unit vector in the direction of \(\mathbf { a }\).
b) Determine the angle \(\mathbf { b }\) makes with the \(x\)-axis.
c) The vector \(\mu \mathbf { a } + \mathbf { b }\) is parallel to \(4 \mathbf { i } - 5 \mathbf { j }\).
i) Find the vector \(\mu \mathbf { a } + \mathbf { b }\) in terms of \(\mu , \mathbf { i }\) and j.
ii) Determine the value of \(\mu\).
WJEC Unit 1 2023 June Q1
6 marks Moderate -0.3
\(\mathbf { 1 }\) & \(\mathbf { 0 }\) \hline \end{tabular} \end{center} Solve the following equations for values of \(x\). a) \(\quad \ln ( 2 x + 5 ) = 3\) b) \(\quad 5 ^ { 2 x + 1 } = 14\) c) \(\quad 3 \log _ { 7 } ( 2 x ) - \log _ { 7 } \left( 8 x ^ { 2 } \right) + \log _ { 7 } x = \log _ { 3 } 81\)
\(\mathbf { 1 }\)\(\mathbf { 1 }\)
The function \(f\) is defined by \(f ( x ) = \frac { 8 } { x ^ { 2 } }\). a) Sketch the graph of \(y = f ( x )\).
b) On a separate set of axes, sketch the graph of \(y = f ( x - 2 )\). Indicate the vertical asymptote and the point where the curve crosses the \(y\)-axis.
c) Sketch the graphs of \(y = \frac { 8 } { x }\) and \(y = \frac { 8 } { ( x - 2 ) ^ { 2 } }\) on the same set of axes. Hence state the number of roots of the equation \(\frac { 8 } { ( x - 2 ) ^ { 2 } } = \frac { 8 } { x }\).
\(\mathbf { 1 }\)\(\mathbf { 2 }\)
The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf { a } = - 3 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { b } = 5 \mathbf { i } + 8 \mathbf { j }$$ respectively.
a) Find the vector \(\mathbf { A B }\).
b) i) Find a unit vector in the direction of \(\mathbf { a }\).
ii) The point \(C\) is such that the vector \(\mathbf { O C }\) is in the direction of \(\mathbf { a }\). Given that the length of \(\mathbf { O C }\) is 7 units, write down the position vector of \(C\).
c) Calculate the angle \(A O B\). \section*{
\(\mathbf { 1 }\)\(\mathbf { 3 }\)
a) Find \(\int \left( 4 x ^ { - \frac { 2 } { 3 } } + 5 x ^ { 3 } + 7 \right) \mathrm { d } x\).} b) The diagram below shows the graph of \(y = x ( x + 6 ) ( x - 3 )\). \includegraphics[max width=\textwidth, alt={}, center]{631084a7-d827-401a-af0b-bbe1860dc027-7_614_1107_641_470} Calculate the total area of the regions enclosed by the graph and the \(x\)-axis. 1 4 a) Two variables, \(x\) and \(y\), are such that the rate of change of \(y\) with respect to \(x\) is proportional to \(y\). State a model which may be appropriate for \(y\) in terms of \(x\).
b) The concentration, \(Y\) units, of a certain drug in a patient's body decreases exponentially with respect to time. At time \(t\) hours the concentration can be modelled by \(Y = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are constants. A patient was given a dose of the drug that resulted in an initial concentration of 5 units.
i) After 4 hours, the concentration had dropped to 1.25 units. Show that \(k = 0 \cdot 3466\), correct to four decimal places.
ii) The minimum effective concentration of the drug is 0.6 units. How much longer would it take for the drug concentration to drop to the minimum effective level?
WJEC Unit 1 2024 June Q1
4 marks Easy -1.2
  1. Given that \(y = 12 \sqrt { x } - \frac { 27 } { x } + 4\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 9\).
  2. Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 180 ^ { \circ }\) that satisfy the equation
$$2 \sin 2 \theta = 1$$
WJEC Unit 1 2024 June Q3
3 marks Easy -1.3
3. Find \(\int \left( 5 x ^ { \frac { 1 } { 4 } } + 3 x ^ { - 2 } - 2 \right) \mathrm { d } x\).
WJEC Unit 1 2024 June Q4
3 marks Easy -1.2
4. Given that \(n\) is an integer such that \(1 \leqslant n \leqslant 6\), use proof by exhaustion to show that \(n ^ { 2 } - 2\) is not divisible by 3 .
WJEC Unit 1 2024 June Q5
4 marks Moderate -0.8
5. A triangle \(A B C\) has sides \(A B = 6 \mathrm {~cm} , B C = 11 \mathrm {~cm}\) and \(A C = 13 \mathrm {~cm}\). Calculate the area of the triangle.
WJEC Unit 1 2024 June Q6
7 marks Moderate -0.8
6. (a) Find the exact value of \(x\) that satisfies the equation $$\frac { 7 x ^ { \frac { 5 } { 4 } } } { x ^ { \frac { 1 } { 2 } } } = \sqrt { 147 }$$ (b) Show that \(\frac { ( 8 x - 18 ) } { ( 2 \sqrt { x } - 3 ) }\), where \(x \neq \frac { 9 } { 4 }\), may be written as \(2 ( 2 \sqrt { x } + 3 )\).
WJEC Unit 1 2024 June Q7
11 marks Easy -1.3
7. (a) The line \(L _ { 1 }\) passes through the points \(A ( - 3,0 )\) and \(B ( 1,4 )\). Determine the equation of \(L _ { 1 }\).
(b) The line \(L _ { 2 }\) has equation \(y = 3 x - 3\).
  1. Given that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(C\), find the coordinates of \(C\).
  2. The line \(L _ { 2 }\) crosses the \(x\)-axis at the point \(D\). Show that the coordinates of \(D\) are \(( 1,0 )\).
    (c) Calculate the area of triangle \(A C D\).
    (d) Determine the angle \(A C D\).
WJEC Unit 1 2024 June Q8
4 marks Moderate -0.5
8. Prove that \(x - 10 < x ^ { 2 } - 5 x\) for all real values of \(x\).
WJEC Unit 1 2024 June Q9
9 marks Moderate -0.3
9. (a) Write down the binomial expansion of \(( 2 - x ) ^ { 6 }\) up to and including the term in \(x ^ { 2 }\).
(b) Given that $$( 1 + a x ) ( 2 - x ) ^ { 6 } \equiv 64 + b x + 336 x ^ { 2 } + \ldots$$ find the values of the constants \(a , b\).
WJEC Unit 1 2024 June Q10
6 marks Easy -1.2
10. Water is being emptied out of a sink. The depth of water, \(y \mathrm {~cm}\), at time \(t\) seconds, may be modelled by $$y = t ^ { 2 } - 14 t + 49 \quad 0 \leqslant t \leqslant 7$$
  1. Find the value of \(t\) when the depth of water is 25 cm .
  2. Find the rate of decrease of the depth of water when \(t = 3\).
WJEC Unit 1 2024 June Q11
4 marks Easy -1.2
11. (a) Sketch the graph of \(y = 3 ^ { x }\). Clearly label the coordinates of the point where the graph crosses the \(y\)-axis.
(b) On the same set of axes, sketch the graph of \(y = 3 ^ { ( x + 1 ) }\), clearly labelling the coordinates of the point where the graph crosses the \(y\)-axis.
WJEC Unit 1 2024 June Q12
10 marks Moderate -0.3
12. A curve \(C\) has equation \(y = - x ^ { 3 } + 12 x - 20\).
  1. Find the coordinates of the stationary points of \(C\) and determine their nature.
  2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation.
WJEC Unit 1 2024 June Q13
8 marks Easy -1.3
13. The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf { a } = 4 \mathbf { i } + 7 \mathbf { j } , \quad \mathbf { b } = \mathbf { i } + 3 \mathbf { j }$$ respectively.
  1. Find the vector \(\mathbf { A B }\).
  2. Determine the distance between the points \(A\) and \(B\).
  3. The position vector of the point \(C\) is given by \(\mathbf { c } = - 2 \mathbf { i } + 5 \mathbf { j }\). The point \(D\) is such that the distance between \(C\) and \(D\) is equal to the distance between \(A\) and \(B\), and \(C D\) is parallel to \(A B\). Find the possible position vectors of the point \(D\).
WJEC Unit 1 2024 June Q14
8 marks Moderate -0.8
14. The diagram below shows a sketch of the curve \(C\) with equation \(y = 2 - 3 x - 2 x ^ { 2 }\) and the line \(L\) with equation \(y = x + 2\). The curve and the line intersect the coordinate axes at the points \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-18_775_970_589_543}
  1. Write down the coordinates of \(A\) and \(B\).
    (b) Calculate the area enclosed by \(C\) and \(L\).
    [6]
    		Examiner only
WJEC Unit 1 2024 June Q15
7 marks Standard +0.3
  1. The diagram shows a sketch of part of the curve with equation \(y = 2 \sin x + 3 \cos ^ { 2 } x - 3\). The curve crosses the \(x\)-axis at the points \(O , A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-20_620_1009_516_520}
Find the value of \(x\) at each of the points \(A , B\) and \(C\).
WJEC Unit 1 2024 June Q16
10 marks Moderate -0.8
16. (a) Find the range of values of \(k\) for which the quadratic equation \(x ^ { 2 } - k x + 4 = 0\) has no real roots.
(b) Determine the coordinates of the points of intersection of the graphs of \(y = x ^ { 2 } - 3 x + 4\) and \(y = x + 16\).
(c) Using the information obtained in parts (a) and (b), sketch the graphs of \(y = x ^ { 2 } - 3 x + 4\) and \(y = x + 16\) on the same set of axes.
WJEC Unit 1 2024 June Q17
7 marks Standard +0.3
17. A function \(f\) is defined by \(f ( x ) = \log _ { 10 } ( 2 - x )\). Another function \(g\) is defined by \(g ( x ) = \log _ { 10 } ( 5 - x )\). The diagram below shows a sketch of the graphs of \(y = f ( x )\) and \(y = g ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-24_782_1072_559_486}
  1. The point \(( c , 1 )\) lies on \(y = f ( x )\). Find the value of \(c\).
  2. A point \(P\) lies on \(y = f ( x )\) and has \(x\)-coordinate \(\alpha\). Another point \(Q\) lies on \(y = g ( x )\) and also has \(x\)-coordinate \(\alpha\). The distance between \(P\) and \(Q\) is 1.2 units. Find the value of \(\alpha\), giving your answer correct to three decimal places.