Mechanics 1 Modelling with Friction Question 11

A chute at a water sports centre has been designed so that swimmers first slide down a steep part which is 10 m long and at an angle of 40° to the horizontal. They then come to a 20 m section with a gentler slope, 11° to the horizontal, where they travel at constant speed.

(i) Find the coefficient of friction between a swimmer and the chute.

(ii) Find the acceleration of a swimmer on the steep part.

(iii) Find the speed at the end of the chute of a swimmer who starts at rest.

(You may assume that no speed is lost at the point where the slope changes.)

An alternative design of chute has the same starting and finishing points but has a constant gradient.

(iv) With what speed do swimmers arrive at the end of this chute?

Solution:

Mechanics 1 Modelling with Friction Question 9

In each of the following situations a brick is about to slide down a rough inclined plane. Find the unknown quantity.
(i) The plane is inclined at 30° to the horizontal and the brick has mass 2 kg: find μ.
(ii) The brick has mass 4 kg and the coefficient of friction is 0.7: find the angle of the slope.
(iii) The plane is at 65° to the horizontal and the brick has mass 5 kg: find μ.
(iv) The brick has mass 6 kg and μ is 1.2: find the angle of slope.

Solution:

Tough Integration Question 3

Let S = .

Using the substitution x = cos² ϴ , show that .

Hence, find the exact value of S in terms of π .

(Note: Check the notes on the next page about definite integrals for integration using substitution)

Solution:

Tough Integration Question 2

Let R =  (4x^2 + 9x - 8) / (2x^2 + 3x - 2 ) . By expressing R in partial fraction, show that 

Solution:

Tough Integration Question 1

Let R = (4x^2 + 9x - 8) / (x^2 + 5x + 2 ). By expressing R in partial fraction,

and solve

Solution:

9709/Oct Nov/2002/3/Q6

Let f(x) = (6+7x) / [(2-x)(1+x^2)]
(i) Express f(x) in partial fractions.
(ii) Show that, when x is sufficiently small for x^4 and higher powers to be neglected, f(x) = 3 + 5x - 1/2 x^2 - 15/4x^3

Solution:
i)

ii)

Reference: PYQ - Oct/Nov 2002 Paper 3 Q6

9709/MayJune/2002/3/Q6

Let f(x) = (4x) / [(3x+1)(x+1)^2]
(i) Express f(x) in partial fractions.
(ii) Hence show that

Solution:

Reference: PYQ - May/June 2002 Paper 3 Q6

9709/MayJune/2002/3/Q3

The polynomial x^4 + 4^2 + x + a is denoted by p(x). It is given that (x^2 + x + 2) is a factor of p(x). Find the value of a and the other quadratic factor of p(x).

Solution:

Reference: PYQ - May/June 2002 Paper 3 Q3

9709/MayJune/2009/3/Q1

Solve the equation ln (2 + e^-x) = 2, giving your answer correct to 2 decimal places.

Solution:

Reference: PYQ - May/June 2009 Paper 3 Q1

9709/OctNov/2008/2/Q3

The variables x and y satisfy the equation y = A(b^−x), where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 1.3) and (1.6, 0.9), as shown in the diagram. Find the values of A and b, correct to 2 decimal places.

Solution:

Reference: PYQ - Oct/Nov 2008 Paper 2 Q3

Logarithms and Exponentials tricky question

Find how many terms there are in these geometric sequences.

(i) –1, 2, –4, 8, …, –16 777 216

Solution: