Questions — OCR C2 (296 questions)

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OCR C2 Q7
7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes. Assuming that the times he takes to complete successive papers form a geometric sequence,
  1. find, to the nearest minute, how long he will take to complete the fifth paper,
  2. show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
  3. find the least number of papers he must work through if he is to complete a paper in less than one hour.
OCR C2 Q8
8.
\includegraphics[max width=\textwidth, alt={}, center]{27703044-8bb3-4809-9454-ae6774fec060-3_501_492_242_607} The diagram shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(120 ^ { \circ }\) at the centre of the circle.
  1. Find the exact length of the major arc \(P Q\).
  2. Show that the perimeter of the shaded minor segment is given by \(k ( 2 \pi + 3 \sqrt { 3 } ) \mathrm { cm }\), where \(k\) is an integer to be found.
  3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle.
OCR C2 Q9
9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$
  1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
OCR C2 Q1
  1. Evaluate
$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$
OCR C2 Q2
2. The diagram shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\). Find
  1. the length \(B C\),
  2. the area of triangle \(A B C\).
OCR C2 Q3
3. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point (8, 7). Given that $$f ^ { \prime } ( x ) = 4 x ^ { \frac { 1 } { 3 } } - 5$$ find \(\mathrm { f } ( x )\).
OCR C2 Q4
4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\). Give your answers to 1 decimal place.
OCR C2 Q5
5. (i) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (ii) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$
OCR C2 Q6
\begin{enumerate} \setcounter{enumi}{5} \item (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
(b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
  1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
  2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\) \item The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.
OCR C2 Q8
8. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,
  1. find an expression for \(b\) in terms of \(a\). Given also that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
OCR C2 Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{33f9663f-26bb-445e-af6e-ca5ca927f7dd-3_638_757_1064_493} The diagram shows the curve with equation \(y = 5 + x - x ^ { 2 }\) and the normal to the curve at the point \(P ( 1,5 )\).
  1. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
  3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).
OCR C2 Q1
  1. Expand \(( 3 - 2 x ) ^ { 4 }\) in ascending powers of \(x\) and simplify each coefficient.
\includegraphics[max width=\textwidth, alt={}]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-1_558_830_395_461}
The diagram shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals, each of width 1 , to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).
OCR C2 Q3
3. (i) Given that $$5 \cos \theta - 2 \sin \theta = 0$$ show that \(\tan \theta = 2.5\)
(ii) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.
OCR C2 Q4
4. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
  2. \(\quad \log _ { 2 } ( \sqrt { x } )\).
    (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
OCR C2 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-2_439_807_242_534}
The diagram shows the sector \(O A B\) of a circle, centre \(O\), in which \(\angle A O B = 2.5\) radians. Given that the perimeter of the sector is 36 cm ,
  1. find the length \(O A\),
  2. find the perimeter and the area of the shaded segment.
OCR C2 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-2_503_839_1208_557} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).
  1. Show that \(a = 8\).
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.
OCR C2 Q7
7. (a) Evaluate $$\sum _ { r = 10 } ^ { 30 } ( 7 + 2 r )$$ (b) (i) Write down the formula for the sum of the first \(n\) positive integers.
(ii) Using this formula, find the sum of the integers from 100 to 200 inclusive.
(iii) Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3 .
OCR C2 Q8
8. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
  3. find the common ratio of the series,
  4. find the sum of the first eight terms of the series.
OCR C2 Q9
9. (i) Evaluate $$\int _ { 1 } ^ { 3 } ( 3 - \sqrt { x } ) ^ { 2 } d x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
(ii) The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 4 x + k$$ where \(k\) is a constant. Given that the curve passes through the points \(( 0 , - 2 )\) and \(( 2,18 )\), show that \(k = 2\) and find an equation for the curve.
OCR C2 Q1
  1. \(f ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is a remainder of - 35 ,
  1. find the value of the constant \(k\),
  2. find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 3 x - 2 )\).
OCR C2 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-1_554_848_685_461} The diagram shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).
Use the trapezium rule with three intervals, each of width 1 , to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
OCR C2 Q3
3. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.
OCR C2 Q4
4. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which $$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$ giving non-exact answers to 1 decimal place.
OCR C2 Q5
5. (i) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(ii) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(iii) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).
OCR C2 Q6
6. Evaluate
  1. \(\quad \int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\),
  2. \(\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 4 } } \mathrm {~d} x\).