Questions C3 (1301 questions)

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OCR C3 Q6
11 marks Standard +0.8
6. (i) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z }$$ (ii) 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.
OCR C3 Q7
11 marks Moderate -0.3
7. 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 $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4$$
OCR C3 Q8
13 marks Standard +0.3
  1. A curve has the equation \(y = x ^ { 2 } - \sqrt { 4 + \ln x }\).
    1. Show that the tangent to the curve at the point where \(x = 1\) has the equation
    $$7 x - 4 y = 11$$ The curve has a stationary point with \(x\)-coordinate \(\alpha\).
  2. Show that \(0.3 < \alpha < 0.4\)
  3. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with \(x _ { 0 } = 0.35\), to find \(\alpha\) correct to 5 decimal places.
    You should show the result of each iteration.
OCR C3 Q1
5 marks Standard +0.3
  1. (i) Solve the inequality
$$| x - 0.2 | < 0.03$$ (ii) Hence, find all integers \(n\) such that $$\left| 0.95 ^ { n } - 0.2 \right| < 0.03$$
OCR C3 Q2
5 marks Standard +0.3
2.
\includegraphics[max width=\textwidth, alt={}]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-1_474_823_685_482}
The diagram shows the curve with equation \(y = x \sqrt { 2 - x } , 0 \leq x \leq 2\).
Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
OCR C3 Q3
6 marks Standard +0.3
3. Solve, for \(0 \leq y \leq 360\), the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$
OCR C3 Q4
7 marks Standard +0.3
  1. A curve has the equation \(x = y \sqrt { 1 - 2 y }\).
    1. Show that
    $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point \(A\) on the curve has \(y\)-coordinate - 1 .
  2. Show that the equation of tangent to the curve at \(A\) can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where \(p\) and \(q\) are integers to be found.
OCR C3 Q5
7 marks Moderate -0.3
5. The function \(f\) is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\). The function g is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
  3. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.
OCR C3 Q6
8 marks Moderate -0.5
6. Find the value of each of the following integrals in exact, simplified form.
  1. \(\quad \int _ { - 1 } ^ { 0 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\)
  2. \(\int _ { 2 } ^ { 4 } \frac { 3 x ^ { 2 } - 2 } { x } \mathrm {~d} x\)
OCR C3 Q7
10 marks Standard +0.3
7 $$f ( x ) = 2 + \cos x + 3 \sin x$$
  1. Express \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = a + b \cos ( x - c )$$ where \(a , b\) and \(c\) are constants, \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\).
  3. Use Simpson's rule with four strips, each of width 0.5 , to find an approximate value for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$
OCR C3 Q8
11 marks Moderate -0.3
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.
OCR C3 Q9
13 marks Standard +0.3
9. \includegraphics[max width=\textwidth, alt={}, center]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-3_501_1111_877_413} The diagram shows a graph of the temperature of a room, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes.
The temperature is controlled by a thermostat such that when the temperature falls to \(12 ^ { \circ } \mathrm { C }\), a heater is turned on until the temperature reaches \(18 ^ { \circ } \mathrm { C }\). The room then cools until the temperature again falls to \(12 ^ { \circ } \mathrm { C }\). For \(t\) in the interval \(10 \leq t \leq 60 , T\) is given by $$T = 5 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are constants.
Given that \(T = 18\) when \(t = 10\) and that \(T = 12\) when \(t = 60\),
  1. show that \(k = 0.0124\) to 3 significant figures and find the value of \(A\),
  2. find the rate at which the temperature of the room is decreasing when \(t = 20\). The temperature again reaches \(18 ^ { \circ } \mathrm { C }\) when \(t = 70\) and the graph for \(70 \leq t \leq 120\) is a translation of the graph for \(10 \leq t \leq 60\).
  3. Find the value of the constant \(B\) such that for \(70 \leq t \leq 120\) $$T = 5 + B \mathrm { e } ^ { - k t }$$
OCR C3 Q1
5 marks Standard +0.3
1.
\includegraphics[max width=\textwidth, alt={}]{5e6a37a1-c51f-4637-aaae-48da6ab3eca0-1_305_606_219_539}
The diagram shows the curve with equation \(y = \ln ( 2 + \cos x ) , x \geq 0\).
The smallest value of \(x\) for which the curve meets the \(x\)-axis is \(a\) as shown.
  1. Find the value of \(a\).
  2. Use Simpson's rule with four strips of equal width to estimate the area of the region bounded by the curve in the interval \(0 \leq x \leq a\) and the coordinate axes.
OCR C3 Q2
6 marks Standard +0.3
2. 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 \(\operatorname { gf } ( x ) = \frac { 1 } { 2 }\).
OCR C3 Q3
6 marks Standard +0.3
3. Find the coordinates of the stationary points of the curve with equation $$y = \frac { x - 1 } { x ^ { 2 } - 2 x + 5 }$$
OCR C3 Q4
8 marks Standard +0.3
  1. (i) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\).
Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(ii) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.
OCR C3 Q5
8 marks Moderate -0.3
5. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .
  1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
  2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).
OCR C3 Q6
8 marks Standard +0.8
6. (i) Express \(3 \cos x ^ { \circ } + \sin x ^ { \circ }\) in the form \(R \cos ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(ii) Using your answer to part (a), or otherwise, solve the equation $$6 \cos ^ { 2 } x ^ { \circ } + \sin 2 x ^ { \circ } = 0$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place where appropriate.
OCR C3 Q7
9 marks Standard +0.3
7. The finite region \(R\) is bounded by the curve with equation \(y = x + \frac { 2 } { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
  1. Find the exact area of \(R\). The region \(R\) is rotated completely about the \(x\)-axis.
  2. Find the volume of the solid formed, giving your answer in terms of \(\pi\).
OCR C3 Q8
11 marks Standard +0.3
8. The population in thousands, \(P\), of a town at time \(t\) years after \(1 ^ { \text {st } }\) January 1980 is modelled by the formula $$P = 30 + 50 \mathrm { e } ^ { 0.002 t }$$ Use this model to estimate
  1. the population of the town on \(1 ^ { \text {st } }\) January 2010,
  2. the year in which the population first exceeds 84000 . The population in thousands, \(Q\), of another town is modelled by the formula $$Q = 26 + 50 \mathrm { e } ^ { 0.003 t }$$
  3. Show that the value of \(t\) when \(P = Q\) is a solution of the equation $$t = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t } \right)$$
  4. Use the iterative formula $$t _ { n + 1 } = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t _ { n } } \right)$$ with \(t _ { 0 } = 50\) to find \(t _ { 1 } , t _ { 2 }\) and \(t _ { 3 }\) and hence, the year in which the populations of these two towns will be equal according to these models.
OCR C3 Q9
11 marks Standard +0.3
9. \includegraphics[max width=\textwidth, alt={}, center]{5e6a37a1-c51f-4637-aaae-48da6ab3eca0-3_727_1022_244_342} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the axes at \(( p , 0 )\) and \(( 0 , q )\) and the lines \(x = 1\) and \(y = 2\) are asymptotes of the curve.
  1. Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 2 \mathrm { f } ( x + 1 )\). Given also that $$\mathrm { f } ( x ) \equiv \frac { 2 x - 1 } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq 1$$
  2. find the values of \(p\) and \(q\),
  3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
OCR C3 Q1
5 marks Moderate -0.3
1. $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$ Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = - 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C3 Q2
7 marks Standard +0.3
2. \includegraphics[max width=\textwidth, alt={}, center]{b124d427-1f9b-4770-95bb-ed79bae5b4fb-1_460_805_587_486} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \ln 3 x\).
  1. Express the equation of the curve in the form \(x = \mathrm { f } ( y )\). The shaded region is bounded by the curve, the coordinate axes and the line \(y = 1\).
  2. Find, in terms of \(\pi\) and e, the volume of the solid formed when the shaded region is rotated through four right angles about the \(y\)-axis.
OCR C3 Q3
8 marks Standard +0.3
3. (i) Use the identity for \(\sin ( A + B )\) to show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$ (ii) Hence find, in terms of \(\pi\), the solutions of the equation $$\sin 3 x - \sin x = 0$$ for \(x\) in the interval \(0 \leq x < 2 \pi\).
OCR C3 Q4
9 marks Moderate -0.3
4. 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 the graph meets the axes, sketch the graph of \(y = | \mathrm { f } ( x ) |\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
  2. Find \(\mathrm { fg } ( \mathrm { a } )\) in terms of \(a\).
  3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$