Datum / Date: SEPTEMBER 2024
Tyd / Time: 3 hours
Punte / Marks: 150
Graad / 12
Grade:
Vak / Subject: Technical Mathematics
Vraestel 2
/Paper:
Naam en Van / Name and
Surname:
General instructions for the writing of exams:
1. Quickly scan through your entire paper.
2. Read the question twice before you begin.
3. Make sure to answer ALL questions.
4. Write neatly and legibly.
If you struggle too much with a question, move on to the next one and return
5
once you have answered all the other questions.
6. If the question calls for a longer answer, plan before you start writing.
7. For any answer longer than 1-2 words, always write in full sentences.
8. Make sure every point you make refers to the central question.
9. Use correct grammar, spelling, and punctuation to the best of your ability.
If you make a mistake, cross it out clearly with a single, neat line and write
10.
the correct answer instead.
Number the answers correctly according to the numbering system used in
11.
this question paper.
12. Present your answers according to the instructions for each question.
Good luck with your exam!
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
, QUESTION 1
Consider a circle with the equation. x 2+ y 2=49 There is a
point D( x ,5) located on the circumference of this circle.
From a point E outside the circle, two tangents EF and EG
are drawn to the circle.
1.1. Determine the x−¿ coordinates of D. Leave your (3)
1 answer in simplified surd form.
1.1. (a) Write down the equations of EF and EG. (4)
2
(b) Hence, write down the coordinates of E. (2)
1.2 The equation of an ellipse is given by 16 x 2+ 49 y 2=784
1.2.1 Rewrite the equation into standard form. (2)
1.2.2 Hence, sketch the graph of the equation, (3)
clearly indicating the intercepts and the
shape
[13]
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
, QUESTION 2
The structure shown in the picture below, has sides in the form of
quadrilaterals. Quadrilaterals ABDF models a side view of the
building, in the Cartesian plane, with vertices A (−2 ; 8), B, D and F
(−8 ,−4 ).
The acute angle formed by the x−¿ axis and BD is 76 ° and ^
D=α . FD is
parallel to the x−¿ axis.
Determine:
2.1 The size of α (2)
2.2 The length of AF (leave the answer in simplified
surd form) (2)
2.3 The gradient of BD (rounded off to the nearest
integer) (2)
2.4 The coordinates of the midpoint of AF. (2)
2.5 Hence, the equation of the perpendicular bisector of
AF in the form y=… (5)
[13]
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
, QUESTION 3
3.1 Simplify
cos(180 °−x )tan (360 °−x ) (3)
3.1.1
sin x
sin 390° cos 160 ° +cos 200 ° tan 135° (5)
3.1.2
sec210 ° cos 340 °
3.2 2 2 1 2 (3)
Prove that cos β + tan β + 2
=sec β
cot β +1
[11]
QUESTION 4
4.1 1
Consider the following cos θ+ 1= and θ ∈[180 ° ;360 °] . Draw a triangle
2
on the Cartesian plane to represent θ in the correct quadrant and (4)
calculate all the sides of the triangle (in simplified root form)
4.2 Determine the following WITHOUT the use of a calculator.
(a) 9 √ 11(cot θ−cosec θ)
(3)
( sin θ∙ secθ )2
(b)
3 tan θ (3)
4.3 4 sin α cos3 α−sin(2 α )
Given: P ( α )=
sin 4 α
Determine:
4.3.1 P(22 °)
(2)
4.3.2 P(42° )
(2)
4.3.3 P(61 °)
(2)
4.3.4 Write down a conjecture about P( α)
(2)
[18
]
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
Tyd / Time: 3 hours
Punte / Marks: 150
Graad / 12
Grade:
Vak / Subject: Technical Mathematics
Vraestel 2
/Paper:
Naam en Van / Name and
Surname:
General instructions for the writing of exams:
1. Quickly scan through your entire paper.
2. Read the question twice before you begin.
3. Make sure to answer ALL questions.
4. Write neatly and legibly.
If you struggle too much with a question, move on to the next one and return
5
once you have answered all the other questions.
6. If the question calls for a longer answer, plan before you start writing.
7. For any answer longer than 1-2 words, always write in full sentences.
8. Make sure every point you make refers to the central question.
9. Use correct grammar, spelling, and punctuation to the best of your ability.
If you make a mistake, cross it out clearly with a single, neat line and write
10.
the correct answer instead.
Number the answers correctly according to the numbering system used in
11.
this question paper.
12. Present your answers according to the instructions for each question.
Good luck with your exam!
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
, QUESTION 1
Consider a circle with the equation. x 2+ y 2=49 There is a
point D( x ,5) located on the circumference of this circle.
From a point E outside the circle, two tangents EF and EG
are drawn to the circle.
1.1. Determine the x−¿ coordinates of D. Leave your (3)
1 answer in simplified surd form.
1.1. (a) Write down the equations of EF and EG. (4)
2
(b) Hence, write down the coordinates of E. (2)
1.2 The equation of an ellipse is given by 16 x 2+ 49 y 2=784
1.2.1 Rewrite the equation into standard form. (2)
1.2.2 Hence, sketch the graph of the equation, (3)
clearly indicating the intercepts and the
shape
[13]
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
, QUESTION 2
The structure shown in the picture below, has sides in the form of
quadrilaterals. Quadrilaterals ABDF models a side view of the
building, in the Cartesian plane, with vertices A (−2 ; 8), B, D and F
(−8 ,−4 ).
The acute angle formed by the x−¿ axis and BD is 76 ° and ^
D=α . FD is
parallel to the x−¿ axis.
Determine:
2.1 The size of α (2)
2.2 The length of AF (leave the answer in simplified
surd form) (2)
2.3 The gradient of BD (rounded off to the nearest
integer) (2)
2.4 The coordinates of the midpoint of AF. (2)
2.5 Hence, the equation of the perpendicular bisector of
AF in the form y=… (5)
[13]
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
, QUESTION 3
3.1 Simplify
cos(180 °−x )tan (360 °−x ) (3)
3.1.1
sin x
sin 390° cos 160 ° +cos 200 ° tan 135° (5)
3.1.2
sec210 ° cos 340 °
3.2 2 2 1 2 (3)
Prove that cos β + tan β + 2
=sec β
cot β +1
[11]
QUESTION 4
4.1 1
Consider the following cos θ+ 1= and θ ∈[180 ° ;360 °] . Draw a triangle
2
on the Cartesian plane to represent θ in the correct quadrant and (4)
calculate all the sides of the triangle (in simplified root form)
4.2 Determine the following WITHOUT the use of a calculator.
(a) 9 √ 11(cot θ−cosec θ)
(3)
( sin θ∙ secθ )2
(b)
3 tan θ (3)
4.3 4 sin α cos3 α−sin(2 α )
Given: P ( α )=
sin 4 α
Determine:
4.3.1 P(22 °)
(2)
4.3.2 P(42° )
(2)
4.3.3 P(61 °)
(2)
4.3.4 Write down a conjecture about P( α)
(2)
[18
]
Bl / Pg www.summariessa.co.za Opsteller / CHR SMI
Author:
Graad / Vak / TECHNICAL Datum / Date:
12 SEPT 2024
Grade: Subject: MATHEMATICS
