Questions — OCR (4907 questions)

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OCR C2 Q9
13 marks Standard +0.3
9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).
  1. State the gradient of \(C\) at \(P\).
  2. Find the \(x\)-coordinate of \(P\).
  3. Find an equation for \(C\).
  4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.
OCR C2 Q1
4 marks Easy -1.2
  1. Find
$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$
OCR C2 Q2
4 marks Standard +0.3
2. The diagram shows triangle \(P Q R\) in which \(P Q = x , P R = 7 - x , Q R = x + 1\) and \(\angle P Q R = 60 ^ { \circ }\). Using the cosine rule, find the value of \(x\).
OCR C2 Q3
7 marks Moderate -0.3
3. \includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-1_474_863_1283_520} The diagram shows the curve with equation \(y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 1\).
  1. Use the trapezium rule with four intervals, each of width 0.25 , to find an estimate for the area of the shaded region.
  2. State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.
OCR C2 Q4
8 marks Moderate -0.8
4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.
  1. Find the value of \(k\). Using this value of \(k\),
  2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375 ,
  3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$
OCR C2 Q5
8 marks Moderate -0.3
  1. (i) Given that
$$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0$$ (ii) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0$$
OCR C2 Q6
9 marks Moderate -0.8
6. $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
    2. Hence, or otherwise, solve the equation $$2 x ^ { 3 } + 3 x ^ { 2 } - 6 x - 8 = 0$$
OCR C2 Q7
10 marks Standard +0.3
  1. (i) Given that
$$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x$$ show that $$y = 2 x + 1$$ (ii) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\ & 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$
OCR C2 Q8
11 marks Moderate -0.3
  1. The first two terms of an arithmetic progression are \(( t - 1 )\) and \(\left( t ^ { 2 } - 5 \right)\) respectively, where \(t\) is a positive constant.
    1. Find and simplify expressions in terms of \(t\) for
      1. the common difference,
      2. the third term.
    Given also that the third term is 19 ,
  2. find the value of \(t\),
  3. show that the 10th term is 75,
  4. find the sum of the first 40 terms.
OCR C2 Q9
11 marks Standard +0.3
9. \includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-3_538_872_1790_447} The diagram shows the curves \(y = 2 x ^ { 2 } - 6 x - 3\) and \(y = 9 + 3 x - x ^ { 2 }\).
  1. Find the coordinates of the points where the two curves intersect.
  2. Find the area of the shaded region bounded by the two curves.
OCR C2 Q1
5 marks Moderate -0.8
  1. A sequence of terms is defined by
$$u _ { n } = 3 ^ { n } - 2 , \quad n \geq 1 .$$
  1. Write down the first four terms of the sequence. The same sequence can also be defined by the recurrence relation $$u _ { n + 1 } = a u _ { n } + b , \quad n \geq 1 , \quad u _ { 1 } = 1 ,$$ where \(a\) and \(b\) are constants.
  2. Find the values of \(a\) and \(b\).
OCR C2 Q2
5 marks Moderate -0.8
2. \includegraphics[max width=\textwidth, alt={}, center]{27703044-8bb3-4809-9454-ae6774fec060-1_485_808_973_520} The diagram shows the curve with equation \(y = \sqrt { 4 x - 1 }\).
  1. Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
  2. Explain briefly how you could use the trapezium rule to obtain a more accurate estimate of the area of the shaded region.
OCR C2 Q3
7 marks Moderate -0.3
3. (i) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(ii) Hence expand \(\left( 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C2 Q4
7 marks Moderate -0.3
4. $$\mathrm { f } ( x ) = \frac { 4 } { 2 + \sin x ^ { \circ } }$$
  1. State the maximum value of \(\mathrm { f } ( x )\) and the smallest positive value of \(x\) for which \(\mathrm { f } ( x )\) takes this value.
  2. Solve the equation \(\mathrm { f } ( x ) = 3\) for \(0 \leq x \leq 360\), giving your answers to 1 decimal place.
OCR C2 Q5
8 marks Moderate -0.3
5. (a) Given that \(t = \log _ { 3 } x\),
  1. write down an expression in terms of \(t\) for \(\log _ { 3 } x ^ { 2 }\),
  2. show that \(\log _ { 9 } x = \frac { 1 } { 2 } t\).
    (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4$$
OCR C2 Q6
8 marks Moderate -0.3
  1. Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 3 } - 4 } { x ^ { 3 } } , \quad x \neq 0$$ and that \(y = 0\) when \(x = - 1\), find the value of \(y\) when \(x = 2\).
OCR C2 Q7
9 marks Moderate -0.3
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
10 marks Standard +0.3
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
13 marks Moderate -0.3
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
4 marks Moderate -0.8
  1. Evaluate
$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$
OCR C2 Q2
5 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.3
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
9 marks Moderate -0.3
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
9 marks Moderate -0.5
  1. (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. \(( 1 - \sqrt { 2 } ) ^ { 8 }\)
    3. The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.