1.06c - OCR Spec

Edexcel C3 Q7

10 marks Moderate -0.3

Given that $y = \log_a x$, $x > 0$, where $a$ is a positive constant,

1. express $x$ in terms of $a$ and $y$, [1]
2. deduce that $\ln x = y \ln a$. [1]
Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x \ln a}$. [2]

The curve $C$ has equation $y = \log_{10} x$, $x > 0$. The point $A$ on $C$ has $x$-coordinate 10. Using the result in part (b),

find an equation for the tangent to $C$ at $A$. [4]

The tangent to $C$ at $A$ crosses the $x$-axis at the point $B$.

Find the exact $x$-coordinate of $B$. [2]

OCR C3 Q4

6 marks Moderate -0.3

It is given that $y = 5^{x-1}$.

Show that $x = 1 + \frac{\ln y}{\ln 5}$. [2]
Find an expression for $\frac{dx}{dy}$ in terms of $y$. [2]
Hence find the exact value of the gradient of the curve $y = 5^{x-1}$ at the point $(3, 25)$. [2]

AQA Paper 3 2018 June Q7

5 marks Moderate -0.8

Given that $\log_a y = 2\log_a 7 + \log_a 4 + \frac{1}{2}$, find $y$ in terms of $a$. [4 marks]
When asked to solve the equation $$2\log_a x = \log_a 9 - \log_a 4$$ a student gives the following solution: $2\log_a x = \log_a 9 - \log_a 4$ $\Rightarrow 2\log_a x = \log_a \frac{9}{4}$ $\Rightarrow \log_a x^2 = \log_a \frac{9}{4}$ $\Rightarrow x^2 = \frac{9}{4}$ $\therefore x = \frac{3}{2}$ or $-\frac{3}{2}$ Explain what is wrong with the student's solution. [1 mark]

Edexcel AS Paper 1 Q5

4 marks Moderate -0.3

A student is asked to solve the equation $$\log_3 x - \log_3 \sqrt{x - 2} = 1$$ The student's attempt is shown $$\log_3 x - \log_3 \sqrt{x - 2} = 1$$ $$x - \sqrt{x - 2} = 3^1$$ $$x - 3 = \sqrt{x - 2}$$ $$(x - 3)^2 = x - 2$$ $$x^2 - 7x + 11 = 0$$ $$x = \frac{7 + \sqrt{5}}{2} \text{ or } x = \frac{7 - \sqrt{5}}{2}$$

Identify the error made by this student, giving a brief explanation. [1]
Write out the correct solution. [3]

OCR MEI AS Paper 2 2018 June Q1

2 marks Easy -2.0

Write down the value of (A) $\log_a (a^4)$, [1] (B) $\log_a \left(\frac{1}{a}\right)$. [1]

WJEC Unit 1 2024 June Q17

7 marks Moderate -0.3

A function $f$ is defined by $f(x) = \log_{10}(2 - x)$. Another function $g$ is defined by $g(x) = \log_{10}(5 - x)$. The diagram below shows a sketch of the graphs of $y = f(x)$ and $y = g(x)$. \includegraphics{figure_17}

The point $(c, 1)$ lies on $y = f(x)$. Find the value of $c$. [2]
A point P lies on $y = f(x)$ and has $x$-coordinate $\alpha$. Another point Q lies on $y = g(x)$ and also has $x$-coordinate $\alpha$. The distance between P and Q is 1.2 units. Find the value of $\alpha$, giving your answer correct to three decimal places. [5]

WJEC Unit 3 2018 June Q12

10 marks Moderate -0.8

Given that $f$ is a function,
1. state the condition for $f^{-1}$ to exist,
2. find $ff^{-1}(x)$. [2]
The functions $g$ and $h$, are given by $$g(x) = x^2 - 1,$$ $$h(x) = e^x + 1.$$
1. Suggest a domain for $g$ such that $g^{-1}$ exists.
2. Given the domain of $h$ is $(-\infty, \infty)$, find an expression for $h^{-1}(x)$ and sketch, using the same axes, the graphs of $h(x)$ and $h^{-1}(x)$. Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
3. Determine an expression for $gh(x)$ in its simplest form. [8]

SPS SPS FM 2019 Q8

7 marks Standard +0.3

Sketch the curve $y = 2^{2x+3}$, stating the coordinates of any points of intersection with the axes. [2] The point $P$ on the curve $y = 3^{3x+2}$ has $y$-coordinate equal to 180. Use logarithms to find the $x$-coordinate of $P$ correct to 3 significant figures. [2] The curves $y = 2^{2x+3}$ and $y = 3^{3x+2}$ intersect at the point $Q$. Show that the $x$-coordinate of $Q$ can be written as $$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$ [3]

SPS SPS SM 2020 October Q9

6 marks Standard +0.8

In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]

SPS SPS SM 2022 October Q4

8 marks Standard +0.3

Find the positive value of $x$ such that $$\log_x 64 = 2$$ [2]
Solve for $x$ $$\log_2(11 - 6x) = 2\log_2(x - 1) + 3$$ [6]

SPS SPS FM 2022 February Q5

11 marks Moderate -0.8

Sketch the curve $y = \left(\frac{1}{2}\right)^x$, and state the coordinates of any point where the curve crosses an axis. [3]
Use the trapezium rule, with 4 strips of width 0.5, to estimate the area of the region bounded by the curve $y = \left(\frac{1}{2}\right)^x$, the axes, and the line $x = 2$. [4]
The point $P$ on the curve $y = \left(\frac{1}{2}\right)^x$ has $y$-coordinate equal to $\frac{1}{6}$. Prove that the $x$-coordinate of $P$ may be written as $$1 + \frac{\log_{10} 3}{\log_{10} 2}.$$ [4]

SPS SPS SM Pure 2022 June Q15