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Hints offered by H Gilchrist, with video solutions by 'DLBmaths'
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Paper 1
Question 1
Hint 1: Use equations of uniform motion, find acceleration, find final velocity,
Hint 2: Work out further distance, stopping distance and total distance
Hint 3: and here is a video of the solution:
Question 2
2a) Hint 1: Use correct form of partial fractions: 3 distinct linear factors and process
2b) Hint 2: Integrate using partial fractions
2b) Hint 3: Recognise standard log integrals (take care of negatives)
2b) Hint 4: Use rules of logs to rewrite answer in required format
Hint 5: and here is a video of the solution:
Question 3
Hint 1: Use Newton's second law with frictional force to work out deceleration
Hint 2: Use equations of uniform motion to calculate velocity
Hint 3: Use conservation of momentum given that the bodies coalesce to find velocity
Hint 4: and here is a video of the solution:
Question 4
Hint 1: Differentiate using standard derivatives and chain rule
Hint 2: evaluate expression
Hint 3: and here is a video of the solution:
Question 5
Hint 1: Draw a diagram and resolve horizontally and vertically using F = mrw²
Hint 2: Find tan(θ) by getting ratio of horizontal and vertical
Hint 3: Use information about triangle side lengths to get expression for tan(θ), equate and rearrange
Hint 4: and here is a video of the solution:
Question 6
Hint 1: Express volume as integral by π times the integral of y² with respect to x
Hint 2: Integrate and evaluate
Hint 3: and here is a video of the solution:
Question 7
7a) Hint 1: Find ω
7a) Hint 2: Know x = a sin(ωt + α), and work out value of α when t=0 when at centre of oscillation (α = 0)
7a) Hint 3: Substitute and solve for first two times
7b) Hint 4: Know v is derivative of above, work out and substitute
7b) Hint 5: Interpret the sign of velocity information
Hint 6: and here is a video of the solution:
Question 8
Hint 1: Work out dx/dt and dy/dt
Hint 2: Evaluate for t = 3
Hint 3: Know how the speed relates to the magnitude of the above (using pythagoras' theorem)
Hint 4: and here is a video of the solution:
Question 9
9a) Hint 1: Draw a diagram and derive basic results for horizontal and vertical for acceleration, velocity and displacement
9a) Hint 2: Know that at range, y = 0 and find expression for t
9a) Hint 3: Substitute value of t into expression for x and rearrange as required
9b) Hint 4: Substitute both angles into range formula
9b) Hint 5: Note one is R, the other R+5
9b) Hint 6: Set up from above and solve for v
9b) Hint 7: Obtain range given angle is 35°
9b) Hint 8: Find the time of flight
9b) Hint 9: Find extra distance given tail wind and add to range above
Hint 10: and here is a video of the solution:
Question 10
10a) Hint 1: Derive expressions for mass and centre of mass of the circular lamina and the circular hole
10a) Hint 2: Derive expressions for mass and centre of mass of the semi-circular hole
10a) Hint 3: Take moments horizontally and solve to find x
10a) Hint 4: Take moments vertically and solve to find y
10b) Hint 5: Draw a diagram and interpret rotation
Hint 6: and here is a video of the solution:
Question 11
11a) Hint 1: Calculate displacement of A and B
11a) Hint 2: Calculate velocity given uniform motion
11b) Hint 3: Express displacement of A and B as functions of time
11b) Hint 4: Know that when they collide, i and j components are equal for the same value of t
11b) Hint 5: Find the displacement given the value of t above.
Hint 6: and here is a video of the solution:
Question 12
12a) Hint 1: Use Newton's second law parallel to wire
12a) Hint 2: Resolve perpendicular to cable and get expression for acceleration
12a) Hint 3: Use equations of uniform motion to calculate velocity
12b) Hint 4: Find total initial energy and final energy
12b) Hint 5: Use conservation of energy to form equation
12b) Hint 6: Substitute values and calculate angle
Hint 7: and here is a video of the solution:
Question 13
Hint 1: Find an expression for du
Hint 2: Evaluate new limits
Hint 3: Make the substitution
Hint 4: Simplify and do the integration and evaluate
Hint 5: and here is a video of the solution:
Question 14
Hint 1: Model elastic potential energy (PE) in rope
Hint 2: Equate PE and elastic PE at lowest point
Hint 3: Set up quadratic equation and solve
Hint 4: Interpret and select appropriate solution and complete
Hint 5: and here is a video of the solution:
Question 15
15a) Hint 1: Recognise second order homogeneous differential equation and set up auxiliary equation
15a) Hint 2: Solve quadratic equation to give general solution
15a) Hint 3: Substitute initial conditions for x and dx/dt to find constants
15b) Hint 4: Substitute for t and work out distance moved (remember to interpret x in the context)
Hint 5: and here is a video of the solution:
Question 16
16a) Hint 1: Sketch graph showing speed against time for both runners and annotate
16b) Hint 2: Find the displacement for P
16b) Hint 3: Find the displacement for Q
16b) Hint 4: Interpret what the displacements mean (diagram is helpful) and calculate distance
Hint 5: and here is a video of the solution:
Question 17
Hint 1: Use Newton's second law , choose appropriate expression for a
Hint 2: Choose a = dv/dt
Hint 3: Know Impulse is change in momentum
Hint 4: Solve differential equation by method of separation of variables, and integrate
Hint 5: use initial conditions t = 0 and v = I/m
Hint 6: Manipulate to complete proof
Hint 7: and here is a video of the solution: