National Junior College H2 Math papers, with video solutions

Without using a calculator

1. Solve the inequality \(\frac{2{{x}^{2}}+5x+9}{x-7}\le x-4\).[4]
2. Hence, solve the inequality \(\frac{2+5{{e}^{x}}+9{{e}^{2x}}}{7{{e}^{x}}-1}\ge 4{{e}^{x}}-1\).[2]

A group of 8 people consists of 4 married couples.

1. The group stands in line. Find the number of different possible orders in which no two men stand next to each other.

   [2]

2. The group stands in a circle. Find the number of different possible orders in which each man stands next to his wife.

   [2]

3. The group forms a committee consisting of two teams of four people each. Find the number of ways that the committee can be formed such that neither team consists of only men or women.

   [2]

1. Using a graphical method, solve the inequalities $-3<\left| {{x}^{2}}-1 \right|-\left| x+3 \right|<3$.

   [3]

2. Without using a calculator, solve the inequality $\frac{{{x}^{2}}+1}{x+3}\le 1$.

   [4]

A game is played with a number of cards each which has single letter printed on it. A player has ten cards, which are lettered $A,C,E,E,E,F,L,L,W$ and $X$ respectively.

1. A code is any arrangement of theses ten letters, for example, $LXCELWFEEA$.
   1. Find the total number of different codes that the player can make.

      [1]

   2. Find the number of codes which begin and end with $E$ and in which the two letters $L$ are consecutive.

      [2]

   3. Find the number of codes such that the letter $L$’s are separated by at least two of the remaining letters.

      [3]

2. A second player chooses three of the first player’s ten cards at random, and takes them from the first player. Calculate the probability that
   1. the three cards chosen will all carry the letter $E$ or $L$,

      [2]

   2. the first player’s remaining seven cards will all carry different letters.

      [2]

Write down the real root of

${{w}^{3}}+8=0$,

Where $w$ is a complex number.

Hence, find the other roots of ${{w}^{3}}+8=0$ in exact form.

Deduce the roots of ${{w}^{4}}+{{w}^{3}}+8w+8=0$.

Determine the roots of $z$ such that ${{z}^{4}}-\mathbf{i}{{z}^{3}}+8\mathbf{i}z+8=0$.