Rithvik's Blog GCSE

A treasure hunter discovers a mysterious ancient box with two compartments. Each compartment contains a mix of gold and silver coins. The first compartment has 3 gold and 7 silver coins, while the second compartment has 8 gold and 2 silver coins. The treasure hunter decides to randomly select one coin from each compartment. What is the probability that both coins selected are gold? Remember to use the principles of probability to calculate the likelihood of each event occurring independently and then combine them to find the total probability.

In a distant galaxy, an astronomer is charting the positions of stars using a coordinate system. A star is located at a point (8, 15) in this system. The astronomer wishes to calculate the distance of the star from the origin of the coordinate system and the angle it makes with the positive x-axis. Use Pythagoras' theorem to find the distance of the star from the origin and basic trigonometry to calculate the angle with the x-axis. Remember, in trigonometry, the tangent of the angle is the ratio of the y-coordinate to the x-coordinate of the point.

A space probe travels along a path described by the function � ( � ) = � � 3 + � � 2 + � � + � f ( x ) = a x 3 + b x 2 + c x + d , where � a , � b , � c , and � d are constants. The probe starts its journey at � = 0 x = 0 and reaches its furthest point at � = 4 x = 4 hours, where it then changes its course. Given that the probe's path reaches a maximum height at � = 2 x = 2 hours and the heights at � = 0 x = 0 , � = 2 x = 2 , and � = 4 x = 4 are 0 0 , 64 64 , and 0 0 meters respectively, determine the values of � a , � b , � c , and � d . Hint: Use the conditions given to set up equations and solve for the unknowns.