Questions - OCR MEI C1 - A-Level Maths

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Find algebraically the coordinates of the points of intersection of the curve $y = 4x^2 + 24x + 31$ and the line $x + y = 10$. [5]
Express $4x^2 + 24x + 31$ in the form $a(x + b)^2 + c$. [4]
For the curve $y = 4x^2 + 24x + 31$,
1. write down the equation of the line of symmetry, [1]
2. write down the minimum $y$-value on the curve. [1]

OCR MEI C1 2011 June Q12

12 marks Moderate -0.8

\includegraphics{figure_12} Fig. 12 shows the graph of $y = \frac{4}{x^2}$.

On the copy of Fig. 12, draw accurately the line $y = 2x + 5$ and hence find graphically the three roots of the equation $\frac{4}{x^2} = 2x + 5$. [3]
Show that the equation you have solved in part (i) may be written as $2x^3 + 5x^2 - 4 = 0$. Verify that $x = -2$ is a root of this equation and hence find, in exact form, the other two roots. [6]
By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation $x^3 + 2x^2 - 4 = 0$. [3]

OCR MEI C1 2011 June Q13

13 marks Moderate -0.3

\includegraphics{figure_13} Fig. 13 shows the circle with equation $(x - 4)^2 + (y - 2)^2 = 16$.

Write down the radius of the circle and the coordinates of its centre. [2]
Find the $x$-coordinates of the points where the circle crosses the $x$-axis. Give your answers in surd form. [4]
Show that the point A $(4 + 2\sqrt{2}, 2 + 2\sqrt{2})$ lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it. Find the equations of both these tangents. [7]

OCR MEI C1 2012 June Q1

3 marks Easy -1.2

Find the equation of the line with gradient $-2$ which passes through the point $(3, 1)$. Give your answer in the form $y = ax + b$. Find also the points of intersection of this line with the axes. [3]

OCR MEI C1 2012 June Q2

3 marks Easy -1.8

Make $b$ the subject of the following formula. $$a = \frac{3}{5}b^2c$$ [3]

OCR MEI C1 2012 June Q3

4 marks Easy -1.8

Evaluate $\left(\frac{1}{5}\right)^{-2}$. [2]
Evaluate $\left(\frac{8}{27}\right)^{\frac{2}{3}}$. [2]

OCR MEI C1 2012 June Q4

3 marks Easy -1.2

Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]

OCR MEI C1 2012 June Q5

5 marks Moderate -0.8

Simplify $\frac{10\sqrt{6}}{3}{\sqrt{24}}$. [3]
Simplify $\frac{1}{4 - \sqrt{5}} + \frac{1}{4 + \sqrt{5}}$. [2]

OCR MEI C1 2012 June Q6

5 marks Moderate -0.8

Evaluate $^5C_3$. [1]
Find the coefficient of $x^3$ in the expansion of $(3 - 2x)^5$. [4]

OCR MEI C1 2012 June Q7

4 marks Moderate -0.5

Find the set of values of $k$ for which the graph of $y = x^2 + 2kx + 5$ does not intersect the $x$-axis. [4]

OCR MEI C1 2012 June Q8

5 marks Standard +0.3

The function $f(x) = x^4 + bx + c$ is such that $f(2) = 0$. Also, when $f(x)$ is divided by $x + 3$, the remainder is $85$. Find the values of $b$ and $c$. [5]

OCR MEI C1 2012 June Q9

4 marks Moderate -0.8

Simplify $(n + 3)^2 - n^2$. Hence explain why, when $n$ is an integer, $(n + 3)^2 - n^2$ is never an even number. Given also that $(n + 3)^2 - n^2$ is divisible by $9$, what can you say about $n$? [4]

OCR MEI C1 2012 June Q10

11 marks Moderate -0.3

\includegraphics{figure_10} Fig. 10 is a sketch of quadrilateral ABCD with vertices A $(1, 5)$, B $(-1, 1)$, C $(3, -1)$ and D $(11, 5)$.

Show that $AB = BC$. [3]
Show that the diagonals AC and BD are perpendicular. [3]
Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

OCR MEI C1 2012 June Q11

12 marks Moderate -0.3

A cubic curve has equation $y = f(x)$. The curve crosses the $x$-axis where $x = -\frac{1}{2}$, $-2$ and $5$.

Write down three linear factors of $f(x)$. Hence find the equation of the curve in the form $y = 2x^3 + ax^2 + bx + c$. [4]
Sketch the graph of $y = f(x)$. [3]
The curve $y = f(x)$ is translated by $\begin{pmatrix} 0 \\ -8 \end{pmatrix}$. State the coordinates of the point where the translated curve intersects the $y$-axis. [1]
The curve $y = f(x)$ is translated by $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$ to give the curve $y = g(x)$. Find an expression in factorised form for $g(x)$ and state the coordinates of the point where the curve $y = g(x)$ intersects the $y$-axis. [4]

OCR MEI C1 2012 June Q12

13 marks Moderate -0.3

\includegraphics{figure_12} Fig. 12 shows the graph of $y = \frac{-1}{x - 3}$.

Draw accurately, on the copy of Fig. 12, the graph of $y = x^2 - 4x + 1$ for $-1 < x < 5$. Use your graph to estimate the coordinates of the intersections of $y = \frac{-1}{x - 3}$ and $y = x^2 - 4x + 1$. [5]
Show algebraically that, where the curves intersect, $x^3 - 7x^2 + 13x - 4 = 0$. [3]
Use the fact that $x = 4$ is a root of $x^3 - 7x^2 + 13x - 4 = 0$ to find a quadratic factor of $x^3 - 7x^2 + 13x - 4$. Hence find the exact values of the other two roots of this equation. [5]