Questions — Edexcel C3 (377 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 C3 Q8
8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
  2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
  3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
  4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.
Edexcel C3 Q1
  1. (a) Given that \(\cos x = \sqrt { 3 } - 1\), find the value of \(\cos 2 x\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
    (b) Given that
$$2 \cos ( y + 30 ) ^ { \circ } = \sqrt { 3 } \sin ( y - 30 ) ^ { \circ }$$ find the value of \(\tan y\) in the form \(k \sqrt { 3 }\) where \(k\) is a rational constant.
Edexcel C3 Q2
2. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv x ^ { 2 } - 3 x + 7 , \quad x \in \mathbb { R }
& \mathrm {~g} ( x ) \equiv 2 x - 1 , \quad x \in \mathbb { R } \end{aligned}$$
  1. Find the range of f .
  2. Evaluate \(\operatorname { gf } ( - 1 )\).
  3. Solve the equation $$\mathrm { fg } ( x ) = 17$$
Edexcel C3 Q3
  1. \(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }\).
    1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that
    $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$ The point \(P\) on the curve \(y = \mathrm { f } ( x )\) has \(x\)-coordinate 1.
  2. Show that the normal to the curve \(y = \mathrm { f } ( x )\) at \(P\) has the equation $$x + 5 y + 9 = 0$$
Edexcel C3 Q4
  1. (a) Given that
$$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$ (b) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).
Edexcel C3 Q5
5. $$\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , \quad x \in \mathbb { R } .$$
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 7\).
  4. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(y = 7\).
Edexcel C3 Q6
6. (a) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z } .$$ (b) Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7 ,$$ giving your answers to 2 decimal places.
Edexcel C3 Q7
7. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$
  1. State the range of f .
  2. Evaluate fg(-2).
  3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
  4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
  5. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
  6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures.
Edexcel C3 Q1
  1. (a) Find the exact value of \(x\) such that
$$3 \arctan ( x - 2 ) + \pi = 0$$ (b) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).
Edexcel C3 Q2
2. (a) Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
(b) Simplify $$\frac { x ^ { 3 } - 8 } { 3 x ^ { 2 } - 8 x + 4 } .$$
Edexcel C3 Q3
  1. Differentiate each of the following with respect to \(x\) and simplify your answers.
    1. \(\cot x ^ { 2 }\)
    2. \(x ^ { 2 } \mathrm { e } ^ { - x }\)
    3. \(\frac { \sin x } { 3 + 2 \cos x }\)
    4. (a) Find, as natural logarithms, the solutions of the equation
    $$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$
  2. Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.
Edexcel C3 Q5
5. The function f is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R } .$$
  1. State the range of f.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R } .$$ Find, in terms of e,
  3. the value of \(\mathrm { gf } ( \ln 2 )\),
  4. the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4 .$$
Edexcel C3 Q6
6. $$f ( x ) = 2 x ^ { 2 } + 3 \ln ( 2 - x ) , \quad x \in \mathbb { R } , \quad x < 2 .$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = 2 - \mathrm { e } ^ { k x ^ { 2 } } ,$$ where \(k\) is a constant to be found. The root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) is 1.9 correct to 1 decimal place.
  2. Use the iteration formula $$x _ { n + 1 } = 2 - \mathrm { e } ^ { k x _ { n } ^ { 2 } } ,$$ with \(x _ { 0 } = 1.9\) and your value of \(k\), to find \(\alpha\) to 3 decimal places and justify the accuracy of your answer.
  3. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\).
Edexcel C3 Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7c3dd501-0545-4166-aaf9-5e1ac1f369c5-4_552_771_248_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = \mathrm { f } ( | x | )\),
    2. \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
  2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ } ,$$ where \(R > 0\) and \(0 < \alpha < 90\),
  3. state the value of \(A\),
  4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
Edexcel C3 Q1
  1. \(f ( x ) \equiv \frac { 2 x - 3 } { x - 2 } , \quad x \in \mathbb { R } , \quad x > 2\).
    1. Find the range of f .
    2. Show that \(\operatorname { ff } ( x ) = x\) for all \(x > 2\).
    3. Hence, write down an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    4. Solve each equation, giving your answers in exact form.
    5. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
    6. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
    7. The curve \(C\) has the equation \(y = 2 \mathrm { e } ^ { x } - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate 1.
    8. Find an equation for the tangent to \(C\) at \(P\).
    The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
  2. Show that the area of triangle \(O Q R\), where \(O\) is the origin, is \(\frac { 9 } { 3 - \mathrm { e } }\).
Edexcel C3 Q4
4. (a) Express $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form.
(b) Hence, show that the equation $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) } = 1$$ has no real roots.
Edexcel C3 Q5
5. Find the values of \(x\) in the interval \(- 180 < x < 180\) for which $$\tan ( x + 45 ) ^ { \circ } - \tan x ^ { \circ } = 4$$ giving your answers to 1 decimal place.
Edexcel C3 Q6
6. (a) Sketch on the same diagram the graphs of \(y = | x | - a\) and \(y = | 3 x + 5 a |\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes.
(b) Solve the equation $$| x | - a = | 3 x + 5 a | .$$
Edexcel C3 Q7
  1. (a) Use the identity
$$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$ (b) Prove that, for \(\sin x \neq 0\), $$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$ (c) Find the values of \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) for which $$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$ giving your answers to 1 decimal place where appropriate.
Edexcel C3 Q8
8. A curve has the equation \(y = ( 2 x + 3 ) \mathrm { e } ^ { - x }\).
  1. Find the exact coordinates of the stationary point of the curve. The curve crosses the \(y\)-axis at the point \(P\).
  2. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) meets the curve again at \(Q\).
  3. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([ - 2 , - 1 ]\).
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 3 - 3 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { x _ { n } } - 2 }$$ with \(x _ { 0 } = - 1\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 2 decimal places.
  5. Show that your value for \(x _ { 4 }\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. END
Edexcel C3 Q1
  1. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta\).
    (b) Solve, for \(0 \leq \theta < 360 ^ { \circ }\), the equation
$$2 \tan ^ { 2 } \theta + \sec \theta = 1 ,$$ giving your answers to 1 decimal place.
Edexcel C3 Q2
2. (a) Differentiate with respect to \(x\)
  1. \(3 \sin ^ { 2 } x + \sec 2 x\),
  2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , x \neq 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).
Edexcel C3 Q3
3. The function \(f\) is defined by $$f : x \mapsto \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } + 5 , \quad x \in \mathbb { R } .$$
  3. Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 4 }\).
Edexcel C3 Q4
4. $$\mathrm { f } ( x ) = 3 \mathrm { e } ^ { x } - \frac { 1 } { 2 } \ln x - 2 , \quad x > 0 .$$
  1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) has a turning point at \(P\). The \(x\)-coordinate of \(P\) is \(\alpha\).
  2. Show that \(\alpha = \frac { 1 } { 6 } \mathrm { e } ^ { - \alpha }\). The iterative formula $$x _ { n + 1 } = \frac { 1 } { 6 } \mathrm { e } ^ { - x _ { n } } , \quad x _ { 0 } = 1$$ is used to find an approximate value for \(\alpha\).
  3. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  4. By considering the change of sign of \(\mathrm { f } ^ { \prime } ( x )\) in a suitable interval, prove that \(\alpha = 0.1443\) correct to 4 decimal places.
Edexcel C3 Q5
5. (a) Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$ (b) Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$ (c) Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(d) Hence, for \(0 \leq \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
Hence, for \(0 \leq \theta < \pi\), solve
\includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.