Questions — OCR C3 (285 questions)

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OCR C3 Specimen Q5
5 At time \(t\) minutes after an oven is switched on, its temperature \(\theta ^ { \circ } \mathrm { C }\) is given by $$\theta = 200 - 180 \mathrm { e } ^ { - 0.1 t }$$
  1. State the value which the oven's temperature approaches after a long time.
  2. Find the time taken for the oven's temperature to reach \(150 ^ { \circ } \mathrm { C }\).
  3. Find the rate at which the temperature is increasing at the instant when the temperature reaches \(150 ^ { \circ } \mathrm { C }\).
OCR C3 Specimen Q6
6 The function f is defined by $$\mathrm { f } : x \mapsto 1 + \sqrt { } x \quad \text { for } x \geqslant 0$$
  1. State the domain and range of the inverse function \(\mathrm { f } ^ { - 1 }\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. By considering the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), show that the solution to the equation $$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$ is \(x = \frac { 1 } { 2 } ( 3 + \sqrt { } 5 )\).
OCR C3 Specimen Q7
7
  1. Write down the formula for \(\tan 2 x\) in terms of \(\tan x\).
  2. By letting \(\tan x = t\), show that the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ becomes $$3 t ^ { 4 } - 8 t ^ { 2 } - 3 = 0$$
  3. Hence find all the solutions of the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ which lie in the interval \(0 \leqslant x \leqslant 2 \pi\).
OCR C3 Specimen Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_476_608_287_756} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. The point \(P\) on the curve is the point at which the gradient takes its maximum value. Show that the tangent at \(P\) passes through the point \(( 0 , - 1 )\).
OCR C3 Specimen Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_424_707_1260_724} The diagram shows the curve \(y = \tan ^ { - 1 } x\) and its asymptotes \(y = \pm a\).
  1. State the exact value of \(a\).
  2. Find the value of \(x\) for which \(\tan ^ { - 1 } x = \frac { 1 } { 2 } a\). The equation of another curve is \(y = 2 \tan ^ { - 1 } ( x - 1 )\).
  3. Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of \(a\).
  4. Verify by calculation that the value of \(x\) at the point of intersection of the two curves is 1.54 , correct to 2 decimal places. Another curve (which you are not asked to sketch) has equation \(y = \left( \tan ^ { - 1 } x \right) ^ { 2 }\).
  5. Use Simpson's rule, with 4 strips, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( \tan ^ { - 1 } x \right) ^ { 2 } \mathrm {~d} x\).
OCR C3 Q1
  1. Show that
$$\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x = \ln 3$$
OCR C3 Q2
  1. Find the set of values of \(x\) such that
$$| 3 x + 1 | \leq | x - 2 |$$
OCR C3 Q3
  1. Find all values of \(\theta\) in the interval \(- 180 < \theta < 180\) for which
$$\tan ^ { 2 } \theta ^ { \circ } + \sec \theta ^ { \circ } = 1$$
OCR C3 Q4
  1. Solve each equation, giving your answers in exact form.
    1. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
    2. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
    3. (i) Prove, by counter-example, that the statement
      "cosec \(\theta - \sin \theta > 0\) for all values of \(\theta\) in the interval \(0 < \theta < \pi\) " is false.
    4. Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that
    $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.
OCR C3 Q6
6. 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 \(C\).
OCR C3 Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{6cdbc2bc-8863-4003-a218-44552d75d137-2_556_777_246_468} The diagram 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 the curve with equation \(y = 1 + 2 \mathrm { f } ( x )\). 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.
OCR C3 Q8
8. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$
  1. State the range of f .
  2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3$$
  4. Evaluate fg(-3).
  5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x )$$
OCR C3 Q9
  1. 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 between - 2 and - 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.
OCR C3 Q1
  1. (i) Differentiate \(x ^ { 3 } \ln x\) with respect to \(x\).
    (ii) Given that
$$x = \frac { y + 1 } { 3 - 2 y }$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).
OCR C3 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-1_638_677_749_443} The diagram 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. Write down the coordinates of \(A\) and \(B\). 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 \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).
OCR C3 Q3
3. 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 \(\mathrm { gf } ( 1 )\).
  2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
  3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$
OCR C3 Q4
  1. (i) Use the identity for \(\cos ( A + B )\) to prove that
$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for \(\cos x \neq 0\), $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\) for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$
OCR C3 Q5
  1. (i) Show that the equation
$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for \(x\) in the interval \(0 \leq x \leq \pi\).
OCR C3 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
  1. Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
  2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).
OCR C3 Q7
7. (i) Sketch on the same diagram the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$
OCR C3 Q8
  1. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , x \neq 0\).
    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 .
  2. 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 \(x _ { 0 } = - 1\).
  3. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.
OCR C3 2009 January Q1
1 Find
  1. \(\int 8 \mathrm { e } ^ { - 2 x } \mathrm {~d} x\),
  2. \(\int ( 4 x + 5 ) ^ { 6 } \mathrm {~d} x\).
OCR C3 2009 January Q2
2
  1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Deduce an approximation to \(\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x\).
OCR C3 2009 January Q3
3
  1. Express \(2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta }\) in terms of \(\sec \theta\).
  2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$
OCR C3 2009 January Q4
4 For each of the following curves, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the exact \(x\)-coordinate of the stationary point:
  1. \(y = \left( 4 x ^ { 2 } + 1 \right) ^ { 5 }\),
  2. \(y = \frac { x ^ { 2 } } { \ln x }\).