Question 3 - A-Level Maths

OCR MEI C1 2007 January — Question 3 3 marks

Exam Board	OCR MEI
Module	C1 (Core Mathematics 1)
Year	2007
Session	January
Marks	3
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Solving quadratics and applications
Type	Rearranging formula - variable appears multiple times
Difficulty	Easy -1.8 This is a linear rearrangement, not quadratic—simply collecting terms with 'a' and factoring gives a = 2c/(f-2). It's purely algebraic manipulation with no problem-solving required, easier than typical A-level questions which involve calculus or genuine quadratic solving.
Spec	1.02c Simultaneous equations: two variables by elimination and substitution

3 Make $a$ the subject of the equation $$2 a + 5 c = a f + 7 c$$

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Question 3:

Simplify $(m^2-1)^2 - (m^2+1)^2$, showing your method.

Hence, given the right-angled triangle in Fig. 10, express $p$ in terms of $m$, simplifying your answer.

[4]

Fig. 10 shows a right-angled triangle with sides $m^2+1$, $m^2-1$, and $p$.

Section B (36 marks)

Question 11:

There is an insert for use in this question.

The graph of $y = x + \frac{1}{x}$ is shown on the insert. The lowest point on one branch is $(1, 2)$. The highest point on the other branch is $(-1, -2)$.

(i) Use the graph to solve the following equations, showing your method clearly.

(A) $x + \frac{1}{x} = 4$ [2]

(B) $2x + \frac{1}{x} = 4$ [4]

(ii) The equation $(x-1)^2 + y^2 = 4$ represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the $y$-axis. [2]

(iii) State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve $y = x + \frac{1}{x}$ but does not intersect with the other. [4]

Question 3:

Simplify $(m^2-1)^2 - (m^2+1)^2$, showing your method.

Hence, given the right-angled triangle in Fig. 10, express $p$ in terms of $m$, simplifying your answer.

[4]

Fig. 10 shows a right-angled triangle with sides $m^2+1$, $m^2-1$, and $p$.

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Section B (36 marks)

Question 11:

There is an insert for use in this question.

The graph of $y = x + \frac{1}{x}$ is shown on the insert. The lowest point on one branch is $(1, 2)$. The highest point on the other branch is $(-1, -2)$.

(i) Use the graph to solve the following equations, showing your method clearly.

(A) $x + \frac{1}{x} = 4$ [2]

(B) $2x + \frac{1}{x} = 4$ [4]

(ii) The equation $(x-1)^2 + y^2 = 4$ represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the $y$-axis. [2]

(iii) State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve $y = x + \frac{1}{x}$ but does not intersect with the other. [4]

Show LaTeX source

3 Make $a$ the subject of the equation

$$2 a + 5 c = a f + 7 c$$

\hfill \mbox{\textit{OCR MEI C1 2007 Q3 [3]}}

This paper (13 questions)

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Q1 3 Q2 3 Q3 3 Q4 3 Q5 3 Q6 4 Q7 4 Q8 4 Q9 5 Q10 4 Q11 12 Q12 12 Q13 12

OCR MEI C1 2007 January — Question 3 3 marks

Question 3:

Simplify \((m^2-1)^2 - (m^2+1)^2\), showing your method.

Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer.

[4]

Fig. 10 shows a right-angled triangle with sides \(m^2+1\), \(m^2-1\), and \(p\).

Section B (36 marks)

Question 11:

There is an insert for use in this question.

The graph of \(y = x + \frac{1}{x}\) is shown on the insert. The lowest point on one branch is \((1, 2)\). The highest point on the other branch is \((-1, -2)\).

(i) Use the graph to solve the following equations, showing your method clearly.

(A) \(x + \frac{1}{x} = 4\) [2]

(B) \(2x + \frac{1}{x} = 4\) [4]

(ii) The equation \((x-1)^2 + y^2 = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis. [2]

(iii) State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac{1}{x}\) but does not intersect with the other. [4]

This paper (13 questions)