Questions — OCR (4907 questions)

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OCR C1 Q7
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A (5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
OCR C1 Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.
  1. Find an equation for \(l\). [5]
The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,
  1. find the coordinates of \(B\). [6]
OCR C1 Q9
13 marks Moderate -0.3
The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
  2. Find the exact coordinates of the stationary point of \(C\). [5]
  3. Determine the nature of the stationary point. [2]
  4. Sketch the curve \(C\). [2]
OCR C1 Q2
5 marks Standard +0.3
Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [5]
OCR C1 Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]
OCR C1 Q4
6 marks Moderate -0.8
The curve \(C\) has the equation \(y = (x - a)^2\) where \(a\) is a constant. Given that $$\frac{dy}{dx} = 2x - 6,$$ \begin{enumerate}[label=(\roman*)] \item find the value of \(a\), [4] \item describe fully a single transformation that would map \(C\) onto the graph of \(y = x^2\). [2]
OCR C1 Q5
7 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\). \begin{enumerate}[label=(\roman*)] \item Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [4] \item Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR C1 Q6
10 marks Moderate -0.8
\begin{enumerate}[label=(\alph*)] \item Given that \(y = 2^x\), find expressions in terms of \(y\) for
  1. \(2^{x+2}\), [2]
  2. \(2^{3-x}\). [2]
\item Show that using the substitution \(y = 2^x\), the equation $$2^{x+2} + 2^{3-x} = 33$$ can be rewritten as $$4y^2 - 33y + 8 = 0.$$ [2] \item Hence solve the equation $$2^{x+2} + 2^{3-x} = 33.$$ [4]
OCR C1 Q7
11 marks Moderate -0.8
The point \(A\) has coordinates \((4, 6)\). Given that \(OA\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). [4]
Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\). \begin{enumerate}[label=(\roman*)] \setcounter{enumi}{1} \item Find the coordinates of \(B\). [2] \item Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 Q8
12 marks Moderate -0.3
  1. Express \(3x^2 - 12x + 11\) in the form \(a(x + b)^2 + c\). [4]
  2. Sketch the curve with equation \(y = 3x^2 - 12x + 11\), showing the coordinates of the minimum point of the curve. [3]
Given that the curve \(y = 3x^2 - 12x + 11\) crosses the \(x\)-axis at the points \(A\) and \(B\), \begin{enumerate}[label=(\roman*)] \setcounter{enumi}{2} \item find the length \(AB\) in the form \(k\sqrt{3}\). [5]
OCR C1 Q9
13 marks Moderate -0.3
A curve has the equation \(y = x^3 - 5x^2 + 7x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. [4]
The point \(P\) on the curve has coordinates \((3, 3)\).
  1. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [6]
The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  1. Show that triangle \(OQR\), where \(O\) is the origin, has area \(28\frac{1}{8}\). [3]
OCR C2 Q1
6 marks Easy -1.2
A sequence \(S\) has terms \(u_1, u_2, u_3, \ldots\) defined by $$u_n = 3n - 1,$$ for \(n \geqslant 1\).
  1. Write down the values of \(u_1, u_2\) and \(u_3\), and state what type of sequence \(S\) is. [3]
  2. Evaluate \(\sum_{n=1}^{100} u_n\). [3]
OCR C2 Q2
7 marks Moderate -0.3
\includegraphics{figure_2} A sector \(OAB\) of a circle of radius \(r\) cm has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is 36 cm² (see diagram).
  1. Write down two equations involving \(r\) and \(\theta\). [2]
  2. Hence show that \(r = 6\), and state the value of \(\theta\). [2]
  3. Find the area of the segment bounded by the arc \(AB\) and the chord \(AB\). [3]
OCR C2 Q3
7 marks Moderate -0.8
  1. Find \(\int (2x + 1)(x + 3) \, dx\). [4]
  2. Evaluate \(\int_0^9 \frac{1}{\sqrt{x}} \, dx\). [3]
OCR C2 Q4
8 marks Standard +0.3
\includegraphics{figure_4} In the diagram, \(ABCD\) is a quadrilateral in which \(AD\) is parallel to \(BC\). It is given that \(AB = 9\), \(BC = 6\), \(CA = 5\) and \(CD = 15\).
  1. Show that \(\cos BCA = -\frac{1}{3}\), and hence find the value of \(\sin BCA\). [4]
  2. Find the angle \(ADC\) correct to the nearest \(0.1°\). [4]
OCR C2 Q5
8 marks Moderate -0.3
The cubic polynomial \(f(x)\) is given by $$f(x) = x^3 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 1)\) is a factor of \(f(x)\) and that the remainder when \(f(x)\) is divided by \((x - 3)\) is 16.
  1. Find the values of \(a\) and \(b\). [5]
  2. Hence verify that \(f(2) = 0\), and factorise \(f(x)\) completely. [3]
OCR C2 Q6
8 marks Moderate -0.8
  1. Find the binomial expansion of \(\left(x^2 + \frac{1}{x}\right)^3\), simplifying the terms. [4]
  2. Hence find \(\int \left(x^2 + \frac{1}{x}\right)^3 dx\). [4]
OCR C2 Q7
7 marks Moderate -0.8
  1. Evaluate \(\log_3 15 + \log_3 20 - \log_3 12\). [3]
  2. Given that \(y = 3 \times 10^{2x}\), show that \(x = a \log_{10}(by)\), where the values of the constants \(a\) and \(b\) are to be found. [4]
OCR C2 Q8
9 marks Moderate -0.3
The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9. The amount pumped in 2001 was 100 000 barrels.
  1. Calculate the amount pumped in 2004. [2]
It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.
  1. Calculate in which year the amount pumped will fall below 5000 barrels. [4]
  2. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production. [3]
OCR C2 Q9
12 marks Standard +0.2
    1. Write down the exact values of \(\cos \frac{1}{6}\pi\) and \(\tan \frac{1}{6}\pi\) (where the angles are in radians). Hence verify that \(x = \frac{1}{6}\pi\) is a solution of the equation $$2 \cos x = \tan 2x.$$ [3]
    2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2x.$$ [4]
    1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.) [4]
    2. State with a reason whether this approximation is an underestimate or an overestimate. [1]
OCR C2 2007 January Q1
4 marks Moderate -0.8
In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms. [4]
OCR C2 2007 January Q2
5 marks Easy -1.2
\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle, centre \(O\) and radius 8 cm. The angle \(AOB\) is \(46°\).
  1. Express \(46°\) in radians, correct to 3 significant figures. [2]
  2. Find the length of the arc \(AB\). [1]
  3. Find the area of the sector \(OAB\). [2]
OCR C2 2007 January Q3
5 marks Easy -1.2
  1. Find \(\int (4x - 5) dx\). [2]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = 4x - 5\). The curve passes through the point \((3, 7)\). Find the equation of the curve. [3]
OCR C2 2007 January Q4
6 marks Moderate -0.8
In a triangle \(ABC\), \(AB = 5\sqrt{2}\) cm, \(BC = 8\) cm and angle \(B = 60°\).
  1. Find the exact area of the triangle, giving your answer as simply as possible. [3]
  2. Find the length of \(AC\), correct to 3 significant figures. [3]
OCR C2 2007 January Q5
8 marks Moderate -0.8
    1. Express \(\log_3(4x + 7) - \log_3 x\) as a single logarithm. [1]
    2. Hence solve the equation \(\log_3(4x + 7) - \log_3 x = 2\). [3]
  2. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int_3^9 \log_{10} x \, dx,$$ giving your answer correct to 3 significant figures. [4]