Solve, in the set of real numbers, the equation $$2\log x = \log(x+20)$$

5. Domain: $x>0$. $2\log x=\log x^2$, so $x^2 = x+20$, i.e. $x^2-x-20=0$, $(x-5)(x+4)=0$. $x=-4$ is rejected ($x>0$). **$x=5$**.

Find the value of the real number $n$ if the polynomial $P(x)=x^3+nx^2+2x-5$ leaves a remainder of $3$ when divided by $x-2$.

-1. By the remainder theorem the remainder is $P(2)$: $8+4n+4-5=3 \Rightarrow 4n+7=3 \Rightarrow$ **$n=-1$**.

Determine the domain of definition of the function $$f(x)=\ln\left(\dfrac{x+1}{2-x}\right).$$ The domain of $f$ is:

A. $\left]-1,\,2\right[$. Need $\frac{x+1}{2-x}>0$. Sign table: numerator zero at $-1$, denominator zero at $2$. The quotient is positive only between the roots: $-1<x<2$.

All practice exams

Senior 6 (A-Level)MathematicsNESA formatMathematics II

S6 Mathematics II practice exam

Practise for the national exam the way it really feels. This paper follows the structure, question types, marks and timing of the NESA Senior 6 (A-Level) Mathematics exam.

These questions simulate the national exam. They are original Ganzaa questions written in the same format as the NESA papers — not the official NESA papers, and not endorsed by NESA. The official papers stay free on our past papers page.

A new paper every time

10 versions of every question, so each attempt is a fresh paper with the same structure.

Real exam conditions

180 minutes, 100 marks, the same sections and question types as the national exam.

Marked when you finish

Your score, the correct answer to every question and a clear explanation of why.

How the paper is organised

Section A

Attempt all questions. (55 marks)

55 marks

Section B

Attempt three questions only. (45 marks)

Answer any 3 of 5

45 marks

Before you start

This paper consists of two sections: A and B.
Section A: Attempt all questions. (55 marks)
Section B: Attempt only three questions. (45 marks)
Geometrical instruments and silent non-programmable calculators may be used.
Work out each answer on rough paper; enter the final answers here. Give decimal answers to the accuracy asked for.
The timer starts when you open the paper and keeps running if you leave.
Your answers save automatically — you can continue on another device before time runs out.
When time is up, the exam is submitted for you.

Sample questions

A taste of the paper. Every exam you sit has different questions in this style.

Sample 1 · Logarithmic equations
Solve, in the set of real numbers, the equation $2\log x = \log(x+20)$
Show the answer
Answer: 5
Domain: $x>0$ . $2\log x=\log x^2$ , so $x^2 = x+20$ , i.e. $x^2-x-20=0$ , $(x-5)(x+4)=0$ . $x=-4$ is rejected ( $x>0$ ). $x=5$ .
Sample 2 · Remainder and factor theorems
Find the value of the real number $n$ if the polynomial $P(x)=x^3+nx^2+2x-5$ leaves a remainder of $3$ when divided by $x-2$ .
Show the answer
Answer: -1
By the remainder theorem the remainder is $P(2)$ : $8+4n+4-5=3 \Rightarrow 4n+7=3 \Rightarrow$ $n=-1$ .
Sample 3 · Domain of functions
Determine the domain of definition of the function $f(x)=\ln\left(\dfrac{x+1}{2-x}\right).$
The domain of $f$ is:
1. A. $\left]-1,\,2\right[$
2. B. $\left]-\infty,-1\right[\cup\left]2,+\infty\right[$
3. C. $\left[-1,\,2\right[$
4. D. $\mathbb{R}\setminus\{2\}$
Show the answer
Answer: A. $\left]-1,\,2\right[$
Need $\frac{x+1}{2-x}>0$ . Sign table: numerator zero at $-1$ , denominator zero at $2$ . The quotient is positive only between the roots: $-1<x<2$ .

All practice exams

Senior 6 (A-Level)MathematicsNESA formatMathematics II

S6 Mathematics II practice exam

Practise for the national exam the way it really feels. This paper follows the structure, question types, marks and timing of the NESA Senior 6 (A-Level) Mathematics exam.

A new paper every time

10 versions of every question, so each attempt is a fresh paper with the same structure.

Real exam conditions

180 minutes, 100 marks, the same sections and question types as the national exam.

Marked when you finish

Your score, the correct answer to every question and a clear explanation of why.

How the paper is organised

Section A

Attempt all questions. (55 marks)

55 marks

Section B

Attempt three questions only. (45 marks)

Answer any 3 of 5

45 marks

Before you start

This paper consists of two sections: A and B.
Section A: Attempt all questions. (55 marks)
Section B: Attempt only three questions. (45 marks)
Geometrical instruments and silent non-programmable calculators may be used.
Work out each answer on rough paper; enter the final answers here. Give decimal answers to the accuracy asked for.
The timer starts when you open the paper and keeps running if you leave.
Your answers save automatically — you can continue on another device before time runs out.
When time is up, the exam is submitted for you.

Sample questions

A taste of the paper. Every exam you sit has different questions in this style.

Sample 1 · Logarithmic equations
Solve, in the set of real numbers, the equation $2\log x = \log(x+20)$
Show the answer
Answer: 5
Domain: $x>0$ . $2\log x=\log x^2$ , so $x^2 = x+20$ , i.e. $x^2-x-20=0$ , $(x-5)(x+4)=0$ . $x=-4$ is rejected ( $x>0$ ). $x=5$ .
Sample 2 · Remainder and factor theorems
Find the value of the real number $n$ if the polynomial $P(x)=x^3+nx^2+2x-5$ leaves a remainder of $3$ when divided by $x-2$ .
Show the answer
Answer: -1
By the remainder theorem the remainder is $P(2)$ : $8+4n+4-5=3 \Rightarrow 4n+7=3 \Rightarrow$ $n=-1$ .
Sample 3 · Domain of functions
Determine the domain of definition of the function $f(x)=\ln\left(\dfrac{x+1}{2-x}\right).$
The domain of $f$ is:
1. A. $\left]-1,\,2\right[$
2. B. $\left]-\infty,-1\right[\cup\left]2,+\infty\right[$
3. C. $\left[-1,\,2\right[$
4. D. $\mathbb{R}\setminus\{2\}$
Show the answer
Answer: A. $\left]-1,\,2\right[$
Need $\frac{x+1}{2-x}>0$ . Sign table: numerator zero at $-1$ , denominator zero at $2$ . The quotient is positive only between the roots: $-1<x<2$ .