Questions C3 (1301 questions)

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Edexcel C3 Q1
6 marks Standard +0.2
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & f : x \rightarrow 2 - x ^ { 2 } , \quad x \in \mathbb { R } , \\ & g : x \rightarrow \frac { 3 x } { 2 x - 1 } , \quad x \in \mathbb { R } , \quad x \neq \frac { 1 } { 2 } \end{aligned}$$
  1. Evaluate fg(2).
  2. Solve the equation \(\operatorname { gf } ( x ) = \frac { 1 } { 2 }\).
Edexcel C3 Q2
7 marks Challenging +1.2
2. Giving your answers to 1 decimal place, solve the equation $$5 \tan ^ { 2 } 2 \theta - 13 \sec 2 \theta = 1 ,$$ for \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).
Edexcel C3 Q3
7 marks Standard +0.3
3.
  1. Simplify $$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$
  2. Solve the equation $$\ln \left( 2 x ^ { 2 } + 3 x - 9 \right) = 2 + \ln \left( 2 x ^ { 2 } - 7 x + 6 \right)$$ giving your answer in terms of e.
Edexcel C3 Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3bd9d8a3-a324-4649-9357-392a48a4a1de-3_508_771_255_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\). The graph has a minimum at \(\left( \frac { \pi } { 2 } , - 1 \right)\), a maximum at \(\left( \frac { 3 \pi } { 2 } , - 5 \right)\) and an asymptote with equation \(x = \pi\).
  1. Showing the coordinates of any stationary points, sketch the graph of \(y = | \mathrm { f } ( x ) |\). Given that $$f : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi ,$$
  2. find the values of the constants \(a\) and \(b\),
  3. find, to 2 decimal places, the \(x\)-coordinates of the points where the graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
Edexcel C3 Q5
10 marks Moderate -0.3
5. The number of bacteria present in a culture at time \(t\) hours is modelled by the continuous variable \(N\) and the relationship $$N = 2000 \mathrm { e } ^ { k t } ,$$ where \(k\) is a constant. Given that when \(t = 3 , N = 18000\), find
  1. the value of \(k\) to 3 significant figures,
  2. how long it takes for the number of bacteria present to double, giving your answer to the nearest minute,
  3. the rate at which the number of bacteria is increasing when \(t = 3\).
Edexcel C3 Q6
11 marks Standard +0.3
6.
  1. Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
  2. Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
  3. Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).
Edexcel C3 Q7
12 marks Standard +0.3
7. \(\quad f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1\).
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. State the range of f.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  4. Describe fully two transformations that would map the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) onto the graph of \(y = \sqrt { x } , x \geq 0\).
  5. Find an equation for the normal to the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point where \(x = 8\).
Edexcel C3 Q8
13 marks Standard +0.3
8. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    [0pt]
  2. Show that the curve has a stationary point in the interval [1.3,1.4]. The point \(A\) on the curve has \(x\)-coordinate 2 .
  3. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
    The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
  4. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.
Edexcel C3 Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8b85e00-4549-4219-a75d-85f67ccb8e16-2_638_675_644_445} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.
  1. Find the exact length \(A B\). The two curves intersect at the point \(C\).
  2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).
Edexcel C3 Q3
8 marks Standard +0.3
3. $$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$
  1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
Edexcel C3 Q4
8 marks Standard +0.2
4. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).
  1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
  2. Find the \(x\)-coordinates of the stationary points of \(C\).
Edexcel C3 Q5
9 marks Moderate -0.3
5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } \end{aligned}$$
  1. Evaluate \(\operatorname { gf } ( 1 )\).
  2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
  3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$
Edexcel C3 Q6
11 marks Standard +0.8
  1. Use the identities for \(\cos ( A + B )\) and \(\cos ( A - B )\) to prove that $$\cos P - \cos Q \equiv - 2 \sin \frac { P + Q } { 2 } \sin \frac { P - Q } { 2 }$$
  2. Hence find all solutions in the interval \(0 \leq x < 180\) to the equation $$\cos 5 x ^ { \circ } + \sin 3 x ^ { \circ } - \cos x ^ { \circ } = 0$$
Edexcel C3 Q7
12 marks Standard +0.3
7. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R } ,$$ where \(a\) is a positive constant.
  1. Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of
    1. \(\quad y = | \mathrm { f } ( x ) |\),
    2. \(y = \mathrm { f } ( | x | )\). The function g is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
  2. Find fg(a) in terms of \(a\).
  3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$
Edexcel C3 Q8
14 marks Standard +0.8
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
8 marks Standard +0.3
  1. 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.
  2. 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
9 marks Moderate -0.3
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 $$\mathrm { fg } ( x ) = 17$$
    1. \(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }\).
    2. 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.
  4. Show that the normal to the curve \(y = \mathrm { f } ( x )\) at \(P\) has the equation $$x + 5 y + 9 = 0$$
    1. (a) Given that
    $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$
  5. 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 Q3
10 marks Standard +0.3
\(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }\).
  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
11 marks Standard +0.3
  1. Given that $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$
  2. 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
11 marks Moderate -0.3
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
11 marks Standard +0.8
6.
  1. Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z } .$$
  2. 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
15 marks Standard +0.3
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. \end{document}
OCR MEI C3 Q3
4 marks Easy -1.2
  1. \(\quad y = 2 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C3 Q9
18 marks Standard +0.3
9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f8be5ab-d241-4027-af26-c49da9394adc-4_401_799_488_593} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. On the Insert sketch graphs of
    (A) \(y = 2 \mathrm { f } ( x )\),
    (B) \(y = \mathrm { f } ( - x )\),
    (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
  2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
  3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
  4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
    Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
  5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
  6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
  7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
  8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
  9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}
OCR MEI C3 Q9
18 marks Moderate -0.3
9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.
  1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
  2. The data collected are as follows.
    Time \(( t\) years \()\)01234
    Number of bats, \(P\)100170300340360
    Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
    (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
  3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
  4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
  1. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
  2. Sketch the graph of \(y = \mathrm { f } ( 2 x )\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.} (ii) Plot \(\ln P\) against \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
    (iii) Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph. \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}