Questions — WJEC Unit 1 (36 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 1 Specimen Q9
9. The quadratic equation \(4 x ^ { 2 } - 12 x + m = 0\), where \(m\) is a positive constant, has two distinct real roots.
Show that the quadratic equation \(3 x ^ { 2 } + m x + 7 = 0\) has no real roots.
WJEC Unit 1 Specimen Q10
10. (a) Use the binomial theorem to express \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 5 }\) in the form \(a \sqrt { 3 } + b \sqrt { 2 }\), where \(a , b\) are integers whose values are to be found.
(b) Given that \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 5 } \approx 0\), use your answer to part (a) to find an approximate value for \(\sqrt { 6 }\) in the form \(\frac { c } { d }\), where \(c\) and \(d\) are positive integers whose values are to be found.
WJEC Unit 1 Specimen Q11
11.
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-4_609_951_1541_605} The diagram shows a sketch of the curve \(y = 6 + 4 x - x ^ { 2 }\) and the line \(y = x + 2\). The point \(P\) has coordinates ( \(a , b\) ). Write down the three inequalities involving \(a\) and \(b\) which are such that the point \(P\) will be strictly contained within the shaded area above, if and only if, all three inequalities are satisfied.
WJEC Unit 1 Specimen Q12
12. Prove that $$\log _ { 7 } a \times \log _ { a } 19 = \log _ { 7 } 19$$ whatever the value of the positive constant \(a\).
WJEC Unit 1 Specimen Q13
13. In triangle \(A B C , B C = 12 \mathrm {~cm}\) and \(\cos A \hat { B } C = \frac { 2 } { 3 }\). The length of \(A C\) is 2 cm greater than the length of \(A B\).
  1. Find the lengths of \(A B\) and \(A C\).
  2. Find the exact value of \(\sin B \hat { A } C\). Give your answer in its simplest form.
WJEC Unit 1 Specimen Q14
14. The diagram below shows a closed box in the form of a cuboid, which is such that the length of its base is twice the width of its base. The volume of the box is \(9000 \mathrm {~cm} ^ { 3 }\). The total surface area of the box is denoted by \(S \mathrm {~cm} ^ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-5_357_915_1190_543}
  1. Show that \(S = 4 x ^ { 2 } + \frac { 27000 } { x }\), where \(x \mathrm {~cm}\) denotes the width of the base.
  2. Find the minimum value of \(S\), showing that the value you have found is a minimum value.
WJEC Unit 1 Specimen Q15
15. The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).
  1. Interpret the constant \(A\) in the context of the question.
  2. Show that \(k = 0 \cdot 047\), correct to three decimal places.
  3. Find the size of the population when \(t = 20\).
WJEC Unit 1 Specimen Q16
16. Find the range of values of \(x\) for which the function $$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 13$$ is an increasing function.
WJEC Unit 1 Specimen Q17
17.
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-6_705_1130_623_438} The diagram above shows a sketch of the curve \(y = 3 x - x ^ { 2 }\). The curve intersects the \(x\)-axis at the origin and at the point \(A\). The tangent to the curve at the point \(B ( 2,2 )\) intersects the \(x\)-axis at the point C .
  1. Find the equation of the tangent to the curve at \(B\).
  2. Find the area of the shaded region.
WJEC Unit 1 Specimen Q18
18. (a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 2 \mathbf { i } - 3 \mathbf { j } , \mathbf { v } = - 4 \mathbf { i } + 5 \mathbf { j }\).
  1. Find the vector \(4 \mathbf { u } - 3 \mathbf { v }\).
  2. The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are the position vectors of the points \(U\) and \(V\), respectively. Find the length of the line UV.
    (b) Two villages \(A\) and \(B\) are 40 km apart on a long straight road passing through a desert. The position vectors of \(A\) and \(B\) are denoted by \(\mathbf { a }\) and \(\mathbf { b }\), respectively.
  3. Village \(C\) lies on the road between \(A\) and \(B\) at a distance 4 km from \(B\). Find the position vector of \(C\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\).
  4. Village \(D\) has position vector \(\frac { 2 } { 9 } \mathbf { a } + \frac { 5 } { 9 } \mathbf { b }\). Explain why village \(D\) cannot possibly be on the straight road passing through \(A\) and \(B\).
WJEC Unit 1 Specimen Q4
  1. Identify the statement which is false. Find a counter example to show that this statement is in fact false.
  2. Identify the statement which is true. Give a proof to show that this statement is in fact true. \item Figure 1 shows a sketch of the graph of \(y = f ( x )\). The graph has a minimum point at \(( - 3 , - 4 )\) and intersects the \(x\)-axis at the points \(( - 8,0 )\) and \(( 2,0 )\). \end{enumerate} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-3_540_992_422_518} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  3. Sketch the graph of \(y = f ( x + 3 )\), indicating the coordinates of the stationary point and the coordinates of the points of intersection of the graph with the \(x\)-axis.
  4. Figure 2 shows a sketch of the graph having one of the following equations with an appropriate value of either \(p , q\) or \(r\).
    \(y = f ( p x )\), where \(p\) is a constant
    \(y = f ( x ) + q\), where \(q\) is a constant
    \(y = r f ( x )\), where \(r\) is a constant \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-3_513_1072_1683_587} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Write down the equation of the graph sketched in Figure 2, together with the value of the corresponding constant.