Questions — OCR C3 (285 questions)

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OCR C3 Q1
  1. Evaluate
$$\int _ { 2 } ^ { 15 } \frac { 1 } { \sqrt [ 3 ] { 2 x - 3 } } d x$$
OCR C3 Q2
2.
\includegraphics[max width=\textwidth, alt={}]{49d985bf-7c94-4a54-88c1-c0084cd94000-1_563_833_532_513}
The diagram shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Find the volume of the solid formed when the shaded region is rotated through four right angles about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.
OCR C3 Q3
3. A curve has the equation \(y = ( 3 x - 5 ) ^ { 3 }\).
  1. Find an equation for the tangent to the curve at the point \(P ( 2,1 )\). The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  2. Find the coordinates of \(Q\).
OCR C3 Q4
4. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0
& \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$
OCR C3 Q5
  1. (i) Find the exact value of \(x\) such that
$$3 \tan ^ { - 1 } ( x - 2 ) + \pi = 0$$ (ii) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).
OCR C3 Q6
6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 3 x - 4 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \rightarrow \frac { 2 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$
  1. Evaluate fg(1).
  2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
  3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
OCR C3 Q7
7. (i) Express \(2 \sin x ^ { \circ } - 3 \cos x ^ { \circ }\) in the form \(R \sin ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(ii) Show that the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ can be written in the form $$2 \sin x ^ { \circ } - 3 \cos x ^ { \circ } = 1$$ (iii) Solve the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place.
OCR C3 Q8
8. The functions f and g are defined for all real values of \(x\) by $$\begin{aligned} & \mathrm { f } : x \rightarrow | x - 3 a |
& \mathrm { g } : x \rightarrow | 2 x + a | \end{aligned}$$ where \(a\) is a positive constant.
  1. Evaluate fg(-2a).
  2. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), showing the coordinates of any points where each graph meets the coordinate axes.
  3. Solve the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$
OCR C3 Q9
9.
\includegraphics[max width=\textwidth, alt={}]{49d985bf-7c94-4a54-88c1-c0084cd94000-3_485_945_1119_447}
The diagram shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).
  1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
  2. Show that \(1 < q < 2\).
  3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
  4. Use an iterative process based on the equation above with a starting value of 1.5 to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.
OCR C3 Q1
  1. Find the set of values of \(x\) such that
$$| 2 x - 3 | > | x + 2 |$$
OCR C3 Q2
  1. Find, to 2 decimal places, the solutions of the equation
$$3 \cot ^ { 2 } x - 4 \operatorname { cosec } x + \operatorname { cosec } ^ { 2 } x = 0$$ in the interval \(0 \leq x \leq 2 \pi\).
OCR C3 Q3
3. A curve has the equation \(x = y ^ { 2 } - 3 \ln 2 y\).
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { 2 y ^ { 2 } - 3 }$$
  2. Find an equation for the tangent to the curve at the point where \(y = \frac { 1 } { 2 }\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C3 Q4
4. (i) Use Simpson's rule with four intervals, each of width 0.25 , to estimate the value of the integral $$\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ (ii) Find the exact value of the integral $$\int _ { \frac { 1 } { 2 } } ^ { 1 } e ^ { 1 - 2 x } d x$$
OCR C3 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{14a2477a-c40e-4b4b-bc39-7100d1df9b4d-2_456_860_246_513}
The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 5\).
  1. Find the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
  2. Find the volume of the solid formed, giving your answer in the form \(k \pi \ln 2\).
OCR C3 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{14a2477a-c40e-4b4b-bc39-7100d1df9b4d-2_397_488_1299_632} The diagram shows a vertical cross-section through a vase.
The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60 ^ { \circ }\). When the depth of water in the vase is \(h \mathrm {~cm}\), the volume of water in the vase is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\). The vase is initially empty and water is poured in at a constant rate of \(120 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
  2. Find, to 2 decimal places, the rate at which \(h\) is increasing
    1. when \(h = 6\),
    2. after water has been poured in for 8 seconds.
OCR C3 Q7
7. (i) Prove that, for \(\cos x \neq 0\), $$\sin 2 x - \tan x \equiv \tan x \cos 2 x$$ (ii) Hence, or otherwise, solve the equation $$\sin 2 x - \tan x = 2 \cos 2 x$$ for \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\).
OCR C3 Q8
8. A rock contains a radioactive substance which is decaying. The mass of the rock, \(m\) grams, at time \(t\) years after initial observation is given by $$m = 400 + 80 \mathrm { e } ^ { - k t }$$ where \(k\) is a positive constant.
Given that the mass of the rock decreases by \(0.2 \%\) in the first 10 years, find
  1. the value of \(k\),
  2. the value of \(t\) when \(m = 475\),
  3. the rate at which the mass of the rock is decreasing when \(t = 100\).
OCR C3 Q9
9. \(\quad f ( x ) = 3 - e ^ { 2 x } , \quad x \in \mathbb { R }\).
  1. State the range of f .
  2. Find the exact value of \(\mathrm { ff } ( 0 )\).
  3. Define the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. Given that \(\alpha\) is the solution of the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\),
  4. explain why \(\alpha\) satisfies the equation $$x = \mathrm { f } ^ { - 1 } ( x )$$
  5. use the iterative formula $$x _ { n + 1 } = \mathrm { f } ^ { - 1 } \left( x _ { n } \right)$$ with \(x _ { 0 } = 0.5\) to find \(\alpha\) correct to 3 significant figures.
OCR C3 Q1
  1. The region bounded by the curve \(y = x ^ { 2 } - 2 x\) and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Find the volume of the solid formed, giving your answer in terms of \(\pi\).
OCR C3 Q2
2. (i) Solve the equation $$\ln ( 3 x + 1 ) = 2$$ giving your answer in terms of e.
(ii) Prove, by counter-example, that the statement $$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$ is false.
OCR C3 Q3
3. Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\quad \ln ( 3 x - 2 )\)
  2. \(\frac { 2 x + 1 } { 1 - x }\)
  3. \(x ^ { \frac { 3 } { 2 } } \mathrm { e } ^ { 2 x }\)
OCR C3 Q4
4. (i) 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.
(ii) 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.
OCR C3 Q5
5. 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 \(g f ( - 1 )\).
  3. Solve the equation $$\operatorname { fg } ( x ) = 17$$
OCR C3 Q6
  1. (i) Express \(4 \sin x + 3 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
    (ii) State the minimum value of \(4 \sin x + 3 \cos x\) and the smallest positive value of \(x\) for which this minimum value occurs.
    (iii) Solve the equation
$$4 \sin 2 \theta + 3 \cos 2 \theta = 2$$ for \(\theta\) in the interval \(0 \leq \theta \leq \pi\), giving your answers to 2 decimal places.
OCR C3 Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{039ebdba-4ad5-4974-9345-d66712fa0a08-3_401_712_228_479} The diagram shows the graph of \(y = \mathrm { f } ( x )\) which meets the coordinate axes at the points ( \(a , 0\) ) and ( \(0 , b\) ), where \(a\) and \(b\) are constants.
  1. Showing, in terms of \(a\) and \(b\), the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
    1. \(y = \mathrm { f } ^ { - 1 } ( x )\),
    2. \(y = 2 \mathrm { f } ( 3 x )\). Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9$$
  2. find the values of \(a\) and \(b\),
  3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.