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South Africa
11) Show that \( \sin (\theta+x)+\sin (\theta-x)=2 \sin \theta \cos x \) (2) Hence show that \( \sin 43^{\circ}+\sin 17^{\circ}=\cos 13^{\circ} \).
Trigonometry Feb 20, 2025
\( \begin{array}{ll}\text { (1) } \cos 5 x-\cos 15 x=2 \sin -10 x \sin 5 x & \text { (2) } \frac{\sin 3 x+\sin x}{2}=\sin 2 x \cos x \\ \sin 4 \theta-\sin 2 \theta \\ \text { 3) } 4 \theta+\cos 2 \theta & \tan \theta\end{array} \)
Trigonometry Feb 20, 2025
Write the following as a single trigonometric ratio: (1) \( \frac{1}{2} \cos 10^{\circ}-\frac{\sqrt{3}}{2} \sin 10^{\circ} \)
Trigonometry Feb 20, 2025
If \( \tan \mathrm{A}=\frac{3}{\sqrt{40}} \) and \( 0^{\circ}<\hat{\mathrm{A}}<90^{\circ} \). determine the values of the following with tho aid of a stecteh and without using a calculator. Leave your answers in surd form, if necessary. \( 9.1 .1 \quad \cos \mathrm{~A} \) \( 9.1 .2 \sin \left(180^{\circ}+\mathrm{A}\right) \)
Trigonometry Feb 20, 2025
Calculate the following without the use of a calculato (1) \( \sin 75^{\circ} \) (3) \( \sin 50^{\circ} \cos 10^{\circ}+\cos 50^{\circ} \sin 10^{\circ} \)
Trigonometry Feb 20, 2025
QUESTION 4 Ball \( \mathbf{A} \) is projected vertically upwards at a velocity of \( 16 \mathrm{~m} \cdot \mathrm{~s}^{-1} \) from the ground. Ignors the effects of air resistance. Use the ground as zero reference. 4.1 Calculate the time taken by ball \( \mathbf{A} \) to return to the ground. 4.2 Sketch a velocity-time graph for ball \( \mathbf{A} \). Show the following on the graph: (a) Initial velocity of ball \( \mathbf{A} \) (b) Time taken to reach the highest point of the motion (c) Time taken to return to the ground ONE SECOND after ball \( \mathbf{A} \) is projected upwards, a second ball, B, is thrown vertically downwards at a velocity of \( 9 \mathrm{~m} \cdot \mathrm{~s}^{-1} \) from a balcony 30 m above the ground. Refer to the diagram. 4.3 Calculate how high above the ground ball \( \mathbf{A} \) will be at the instant the two balls pass each other.
Physics Feb 20, 2025
Multiplication and Division of Exponents (Theory on page 37) Exercise 2: Simplify the following as far as possible and write your answers with positive exponent \( \begin{array}{lll}\text { a) } \frac{2^{x} \cdot 4^{x-1}}{2 \cdot 8^{x}} & \text { e) } \frac{2^{2 x+1} \cdot 3^{2 x+3}}{6^{2 x}} & \text { i) } \frac{\left(x y^{2}\right)^{n}\left(x^{3}\right)^{n-1}}{\left(y^{3}\right)^{n+1}\left(x^{2}\right)^{2 n-1}} \\ \text { b) } \frac{2^{2 n+1} 2^{2 n-2} \cdot 2}{2^{4 n-2}} & \text { f) } \frac{3^{y} \cdot 15^{y-1} \cdot 5^{2 y}}{125^{y} \cdot 9^{y-1}} & \text { j) } \frac{6^{n} \cdot 9^{n-1} \cdot 2 \cdot 3^{1-n}}{18^{n+1}} \\ \text { c) } \frac{5^{2 x-1} \cdot 25^{x+1} \cdot 5}{25^{2 x}} & \text { g) } \frac{4^{n-1} \cdot 24^{-2 n} \cdot 4}{\sigma^{n} \cdot 3^{-n}} & \text { k) }\left(\frac{2}{3}\right)^{2 x+2 y}\left(\frac{9}{4}\right)^{-x-y}\end{array} \) d) \( \frac{9^{x-1} \cdot 12^{x+1}}{27^{x} 4^{x-1}} \)
Algebra Feb 20, 2025
Exercise 1: Simplify the following as far as possible and write your a) \( 5^{0} \) g) \( 23^{-2} \) m) \( x, x, x \) 5) \( \left(\frac{-1}{8}\right)^{-2} \) b) \( 5 x^{0} \) (b) \( (2.3)^{-2} \) (a)) \( 2 x \times 3 x \) t) \( \left(\frac{x+y}{3}\right)^{-1} \) c) \( (5 x)^{0} \) i) \( \frac{1}{3^{-2}} \) o) \( \left(x^{3} y^{2}\right)\left(x^{2} y^{3}\right) \) u) \( \frac{-4 x^{-2}}{2 x^{-5}} \) d) \( 2^{-2} \) j) \( \frac{1}{2.3^{-2}} \) (p) \( \left(x^{3} y^{2}\right)^{-1}\left(x^{2} y^{3}\right) \) c) \( 5^{-2} \) (6) \( \frac{1}{2 x^{-2} y^{3}} \) (4) \( \left[\left(x^{3} y^{2}\right)\left(x^{2} y^{3}\right)\right]^{-1} \) \( 3^{-2} \) 1) \( \left(\frac{x^{2}}{y}\right)^{-2} \) (r) \( \frac{2}{\left(\frac{1}{8}\right)^{-1}} \)
Algebra Feb 20, 2025
If it is gien that \( x \) is a nattial number solve for \( x^{2}-3 x \leqslant 4 \) and \( -4 x+3<-2 \)
Algebra Feb 20, 2025
ACTIVITY 4 DETERMINE ORIGINAL VALUE 1. The value of an item after \( 15 \% \) VAT was added is R 155,50 . What was the value of the item before VAT was added? 2. During 2022 the price of an article increased by \( 7,5 \% \) to R 3350,00 . What was the price of the article before the increase? 3. Sello received a \( 5,5 \% \) increase in his salary. His salary after the increase was R 35500 . What was his salary before the increase? 4. Due to inflation, the price of bread increased by \( 4,5 \% \) to R 16,99 . Calculate the price of the bread before the increase due to inflation. 5. House prices in KwaZulu-Natal increased on average by \( 18 \% \) from 2006 to 2007 . If a house cost R 680000 in 2008 , how much would that same house have cost in 2006 ? A bicycle costs R 755,00 including VAT. If VAT is \( 15 \% \) how much VAT is included in the price of the bicycle?
Arithmetic Feb 20, 2025
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