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set up one question with solution for the following: mathematical knowledge, Routine procedures, Complex procedures and problem solving
Prices to a concert cost N$200 for an adult and N$150 for a child, the concert venue can
accommodate at most 240 people. The organizers are giving a N$20 discount to every adult
and N$10 discount for every child but do not want the total discounted amount to exceed
N$3200.
Draw a graph and use it to answer the following questions:
i. How many tickets altogether should the organisers sell in order to maximise their sales
amount?
ii. How many tickets of each type (adult and child) should they sell to get a maximum
sales amount?
iii. What is the maximum sales amount the organisers can make?
iv. If the profit is calculated as follows: profit = total sales – total discount amount. How
much profit will the organisers make from the maximum sales?
accommodate at most 240 people. The organizers are giving a N$20 discount to every adult
and N$10 discount for every child but do not want the total discounted amount to exceed
N$3200.
Draw a graph and use it to answer the following questions:
i. How many tickets altogether should the organisers sell in order to maximise their sales
amount?
ii. How many tickets of each type (adult and child) should they sell to get a maximum
sales amount?
iii. What is the maximum sales amount the organisers can make?
iv. If the profit is calculated as follows: profit = total sales – total discount amount. How
much profit will the organisers make from the maximum sales?
. (a) A number of particles with masses m1, m2, m3,··· ,mn are situated at the points with position vectors r1, r2, r3,··· ,rn relative to an origin O. The center of mass G of the particles is defined to be the point of space with position vector
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn .
Show that if a different origin O were used, this definition would still place G at the same point of space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.
ii. Find the angle between F1 and F4.
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn .
Show that if a different origin O were used, this definition would still place G at the same point of space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.
ii. Find the angle between F1 and F4.
. (a) A number of particles with masses m1, m2, m3,··· ,mn are situated at the points with position vectors r1, r2, r3,··· ,rn relative to an origin O. The center of mass G of the particles is defined to be the point of space with position vector
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn . Show that if a different origin Owere used, this definition would still place G at the same point of space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.
ii. Find the angle between F1 and F4.
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn . Show that if a different origin Owere used, this definition would still place G at the same point of space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.
ii. Find the angle between F1 and F4.
Show that the geometric mean between x and y is ±√xy and the common ratio r is r=^n+1√y/a
The sum of five numbers in arithmetic progression is 25 and of their square is 165. Find the numbers.
The sum of the first n terms of a series is 2n^2 -2. Find the nth term and show that the series is an arithmetic progression.
A number of particles with masses m1, m2, m3, · · · , mn are situated at the points with position vectors
r1, r2, r3, · · · , rn relative to an origin O. The center of mass G of the particles is defined to be the
point of space with position vector
R =
m1r1 + m2r2 + m3r3 + · · · + mnrn
m1 + m2 + m3 + · · · + mn
.
Show that if a different origin O0 were used, this definition would still place G at the same point of
space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted
by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 =
80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are
perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to
the nearest Newton.
ii. Find the angle between F1 and F4.
r1, r2, r3, · · · , rn relative to an origin O. The center of mass G of the particles is defined to be the
point of space with position vector
R =
m1r1 + m2r2 + m3r3 + · · · + mnrn
m1 + m2 + m3 + · · · + mn
.
Show that if a different origin O0 were used, this definition would still place G at the same point of
space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted
by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 =
80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are
perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to
the nearest Newton.
ii. Find the angle between F1 and F4.
The position vector r of a particle at time t is
r = a cos (ωt)i + b sin (ωt)j
where a, b, ∈ R and a 6= b.
(a) Find the velocity and acceleration vectors v and a respectively.
(b) Show that
ω
2
|v|
2 + |a|
2 = ω
4
(a
2 + b
2
),
and find the times at which the velocity and the acceleration vectors are perpendicular.
r = a cos (ωt)i + b sin (ωt)j
where a, b, ∈ R and a 6= b.
(a) Find the velocity and acceleration vectors v and a respectively.
(b) Show that
ω
2
|v|
2 + |a|
2 = ω
4
(a
2 + b
2
),
and find the times at which the velocity and the acceleration vectors are perpendicular.
(a) A particle P moves along the x-axis with constant acceleration a in the positive x-direction. Initially
P is at the origin and is moving with velocity u in the positive x-direction. Show that the velocity v
and displacement x of P at time t are given by
v = u + at, x = ut +
1
2
at2
,
and deduce that
v
2 = u
2 + 2ax.
(b) The trajectory of a charged particle moving in a magnetic field is given by
r = b cos (Ωt)i + b sin (Ωt)j + ctk,
where b, Ω and c are positive constants. Show that the particle moves with constant speed and find the
magnitude of its acceleration.
P is at the origin and is moving with velocity u in the positive x-direction. Show that the velocity v
and displacement x of P at time t are given by
v = u + at, x = ut +
1
2
at2
,
and deduce that
v
2 = u
2 + 2ax.
(b) The trajectory of a charged particle moving in a magnetic field is given by
r = b cos (Ωt)i + b sin (Ωt)j + ctk,
where b, Ω and c are positive constants. Show that the particle moves with constant speed and find the
magnitude of its acceleration.
