Question 1 A card is selected from a regular pack of 52 cards. Find the probability that it is neither black nor a picture card. There are 26 black cards, 6 black picture cards (J, Q, K) and 6 red picture cards. 6 black picture cards are included in 26 black cards. So the number […]

# Author Archives: iitutor

# Voronoi Diagrams

Introduction This diagram shows the locations of the three hospitals, $A$, $B$ and $C$ in a city. When a car accident occurs, it is important to locate the nearest hospital. How could we improve the diagram to make it easier to identify the closest hospital to any given location? If it is decided that a […]

# Trigonometric Ratios of Differences of Two Angles

Proof 1 \( \sin (\alpha – \beta) = \sin \alpha \cos \beta – \cos \alpha \sin \beta \) \( \begin{align} \angle RPN &= 90^{\circ} – \angle PNR \\ &= \angle RNB \\ &= \angle QON \\ &= \alpha \\ \sin(\alpha – \beta) &= \sin \angle MOP \\ &= \displaystyle \frac{MP}{OP} \\ &= \frac{MR-PR}{OP} \\ &= […]

# Trigonometric Ratios of Sums of Two Angles

Proof 1 \( \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \) \( \begin{align} \angle RPN &= 90^{\circ} – \angle PNR \\ &= \angle RNO \\ &= \angle RNO \\ &= \angle NOQ \\ &= \alpha \\ \sin (\alpha + \beta) &= \sin \angle AOC \\ &= \displaystyle \frac{MP}{OP} […]

# Long Division of Polynomials

Example 1 Divide \( x^3 + 3x^3 + 2x + 1 \) by \( x+2 \). Example 2 Perform a long division: \( (4x^4-6x^3+2x^2-3x+5) \div (2x+1) \).

# Differentiation and Displacement, Velocity and Acceleration

Distance Distance is the magnitude of the total movement from the start point or a fixed point. Displacement The displacement of a moving position relative to a fixed point. Displacement gives both the distance and direction that a particle is from a fixed point. For example, a particle moves \( 5 \) units forwards from […]

# Probability with Wins, Draws and Losses in Go Matches

In a Go match between the Amy team and Ben team, a game is played on each of board \( 1 \), board \( 2 \), board \( 3 \) and board \( 4 \). On each board, the probability that the Amy team wins is \( 0.2 \), the probability of a draw is […]

# Calculating Chi-Squared

\( 2 \times 2 \) Contingency Table The following shows the results of a survey of a sample of \( 200 \) randomly chosen adults classified according to gender and sport. These are called observed values or observed frequencies. $$ \begin{array}{|c|c|c|c|} \hline & \text{loves sport} & \text{hates sport} & \text{sum} \\ \hline \text{male} & 72 […]

# Drawing Venn Diagrams Effectively

Consider the following situation to illustrate through Venn diagrams. Two Circles There are \( 50 \) students in a certain high school. \( 16 \) study Physics, \( 13 \) study Chemistry, and \( 15 \) study both Physics and Chemistry. Illustrate this information on a Venn diagram. $$ \begin{align} a+b+c+d &= 50 \text{ total […]