1)
a) The modal class is the class interval with the highest frequency.
b) To find the median height, arrange the lengths of the leaves in ascending order and find the value in the middle. If there is an even number of measurements, take the average of the two middle values.
2) To find the length of AC, use the cosine rule: AC = sqrt(AB^2 + BC^2 - 2*AB*BC*cos(angle ABC)).
To find the area of the triangle, use the formula: Area = (1/2)*AB*BC*sin(angle ABC).
3) Let the length of the hall be x meters. The width of the hall is x - 16 meters.
The area of the rectangle is length * width, so x(x - 16) = 132.
Solve this quadratic equation to find the length of the hall, and then calculate the perimeter using the formula: Perimeter = 2 * (length + width).
4) a)
i) The original price per chair is 16200/y.
ii) The price per chair after the discount is (16200/y) - 60.
b)
i) Solve (16200/y) = ((16200/y) - 60) * (y + 3) to find the number of chairs originally intended to buy.
ii) Substitute the value of y into (16200/y) - 60 to find the price per chair after the discount.
5) The surface area of a cylinder is given by the formula: A = 2πr(r + h), where r is the radius and h is the height.
We are given the surface area of the larger can (480cm^2) and the heights of the two cans (8cm and 10cm). Use these values to find the radius of the larger can, and then calculate the surface area of the smaller can using the same formula.
6) Let the present ages of the father and son be F and S respectively.
Three years ago, the father's age was F - 3 and the son's age was S - 3.
According to the given information, F - 3 = 3(S - 3) and F + S + (F - 3) + (S - 3) = 76. Solve these two equations simultaneously to find their present ages.
7) The perimeter of an isosceles triangle is equal to the sum of the lengths of all three sides.
Use the given information to calculate the length of the two equal sides of the triangle, and then multiply that length by 2 to find the length of the base.
Finally, add the lengths of all three sides to find the length of the wire.
8) Simplify the expression by combining like terms and factoring the numerator and denominator.
9) To find the coordinates of point P where the line intersects the x-axis, set y = 0 and solve for x.
To find the coordinates of point Q where the line intersects the y-axis, set x = 0 and solve for y.
10) Since line P is parallel to line Q, they have the same gradient (slope).
Use the point-slope form of the equation of a line to find the equation of line Q, given its gradient and a point it passes through.
11a) To find the equation of the perpendicular bisector of the line joining points A and B, find the midpoint of AB and then calculate the negative reciprocal of the slope of AB. Use the point-slope form to write the equation in the given form.
b) Substitute the coordinates of C into the equation of the perpendicular bisector to find the point of intersection.
c) Use the slope of the line through points (2,4) and (-3,1) to find the angle it makes with the positive x-axis.
12) Simplify the expression by combining like terms and finding a common denominator.
13) Solve the equation by cross-multiplication.
14) Solve the equation by simplifying the expressions on both sides and finding a common denominator.
15) Rearrange the equation to have sin^2θ and cosθ on one side. Use the identity sin^2θ + cos^2θ = 1 to further simplify the equation. Then, solve for sinθ and cosθ using the quadratic formula. Finally, use the inverse sine and inverse cosine functions to find the angles in degrees.