Questions F3 (135 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
Edexcel F3 2018 Specimen Q2
2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).
  1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
  2. find the exact area of triangle \(O P M\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2 \theta\)
    VIIIV SIHI NI JAIIM ION OCVIIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM IONOO
Edexcel F3 2018 Specimen Q3
3. Without using a calculator, find
  1. \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\), giving your answer as a multiple of \(\pi\),
  2. \(\int _ { - 1 } ^ { 4 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 34 } } \mathrm {~d} x\), giving your answer in the form \(p \ln ( q + r \sqrt { 2 } )\),
    where \(p , q\) and \(r\) are rational numbers to be found.
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel F3 2018 Specimen Q5
  1. Given that \(y = \operatorname { artanh } ( \cos x )\)
    1. show that
    $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \cos x \operatorname { artanh } ( \cos x ) d x$$ giving your answer in the form \(a \ln ( b + c \sqrt { 3 } ) + d \pi\), where \(a , b , c\) and \(d\) are rational numbers to be found.
    VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIIIM ION OCVEXV SIHIL NI JIIIM ION OO
Edexcel F3 2018 Specimen Q6
  1. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are \(( 1,2,3 )\),
The point \(D\) has coordinates \(( k , 4,14 )\) where \(k\) is a positive constant.
Given that the volume of the tetrahedron \(A B C D\) is 6 cubic units,
(b) find the value of \(k\). \section*{\(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).
(a) Find a cartesian equation of the plane \(\Pi\).
6. \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane (a) Find a cartesian equation of the plane \(\Pi\).}
Edexcel F3 2018 Specimen Q7
  1. The curve \(C\) has parametric equations
$$x = 3 t ^ { 4 } , \quad y = 4 t ^ { 3 } , \quad 0 \leqslant t \leqslant 1$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).
  1. Show that $$S = k \pi \int _ { 0 } ^ { 1 } t ^ { 5 } \left( t ^ { 2 } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} t$$ where \(k\) is a constant to be found.
  2. Use the substitution \(u ^ { 2 } = t ^ { 2 } + 1\) to find the value of \(S\), giving your answer in the form \(p \pi ( 11 \sqrt { 2 } - 4 )\) where \(p\) is a rational number to be found.
Edexcel F3 Specimen Q4
4. \(I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a > 0 , \quad n \geqslant 0\)
  1. Show that, for \(n \geqslant 2\), $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
  2. Hence evaluate \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x d x\)
Edexcel F3 Specimen Q5
5. Given that \(y = ( \operatorname { arcosh } 3 x ) ^ { 2 }\), where \(3 x > 1\), show that
  1. \(\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y\),
  2. \(\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18\).
Edexcel F3 Specimen Q6
6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3
0 & - 2 & 1
k & 0 & 1 \end{array} \right)\), where \(k\) is a constant．
Given that \(\left( \begin{array} { l } 6
1
6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) ，
（a）find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6
1
6 \end{array} \right)\) ，
（b）show that \(k = 3\) ，
（c）show that \(\mathbf { M }\) has exactly two eigenvalues． A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\) ．
The transformation \(T\) maps the line \(l _ { 1 }\) ，with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\) ，onto the line \(l _ { 2 }\) ．
6. \(\mathbf { M } = \left( \begin{array} { c c c } 0 & - 2 & 1
k & 0 & 1 \end{array} \right)\), where \(k\) is a constant． Given that \(\left( \begin{array} { l } 6
1
6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) ，
（a）find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6
1
6 \end{array} \right)\)
（d）Taking \(k = 3\) ，find cartesian equations of \(l _ { 2 }\) ．
Edexcel F3 Specimen Q7
  1. The plane \(\Pi\) has vector equation
$$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
  1. Find an equation of \(\Pi\) in the form \(\mathbf { r } . \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant. The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
  2. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
  3. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.
Edexcel F3 Specimen Q8
  1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\).
The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).
  1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
  2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$