SSCE HSC Mathematics Extension 2 Integration by parts: The two non-parallel vectors **u

Question

The two non-parallel vectors **u** and **v** satisfy  $ \lambda \mathbf{u} + \mu \mathbf{v} = \mathbf{0} $ for some real numbers $ \lambda $ and $ \mu $.

(i) Show that $ \lambda = \, \, \mu = 0 $.

(ii) The two non-parallel vectors **u** and **v** satisfy $ \lambda_1 \mathbf{u} + \mu_1 \mathbf{v} = \lambda_2 \mathbf{u} + \mu_2 \mathbf{v} $ for some real numbers $ \lambda_1 $, $ \lambda_2 $, $ \mu_1 $, and $ \mu_2 $.  
Using part (i), or otherwise, show that $ \lambda_1 = \lambda_2 $ and $ \mu_1 = \mu_2 $.

The diagram below shows the tetrahedron with vertices A, B, C, and S.  
The point K is defined by $ \mathbf{SK} = \frac{1}{4} \mathbf{SB} + \frac{3}{4} \mathbf{SC} $, as shown in the diagram.  
The point L is the point of intersection of the straight lines SK and BC.  
(iii) Using part (ii), or otherwise, determine the position of L by showing that $ \mathbf{BL} = \frac{4}{7} \mathbf{BC} $.

(iv) The point P is defined by $ \mathbf{AP} = -6 \mathbf{AB} - 8 \mathbf{AC} $.  
Does P lie on the line AL? Justify your answer.

Let $ J_n = \int_0^1 r^n e^x \, dx $, where n is a non-negative integer.

(i) Show that $ J_0 = 1 - \frac{1}{e} $.

(ii) Show that $ J_n = \frac{n}{n + 1} $ for $ n \geq 1 $.

(iii) Show that $ J_n = \frac{n}{n!} \left( 1 - \frac{1}{e} \right) $.

(iv) Using parts (i) and (iii), show by mathematical induction, or otherwise, that for all n $ \geq 0 $,  $ J_n = n! \cdot \int_0^1 \frac{1}{n!} \, e^{-x} \, dx $.

(v) Using parts (ii) and (iv) prove that $ \lim_{n \to \infty} J_n = 0 $.

The two non-parallel vectors u and v satisfy \( \lambda \mathbf{u} + \mu \mathbf{v} = \mathbf{0} \) for some real numbers \( \lambda \) and \( \mu \) - HSC - SSCE Mathematics Extension 2 - Question 14 - 2022 - Paper 1

Worked Solution & Example Answer:The two non-parallel vectors u and v satisfy \( \lambda \mathbf{u} + \mu \mathbf{v} = \mathbf{0} \) for some real numbers \( \lambda \) and \( \mu \) - HSC - SSCE Mathematics Extension 2 - Question 14 - 2022 - Paper 1

Show that \( \lambda = 0 \) and \( \mu = 0 \).

Using part (i), show that \( \lambda_1 = \lambda_2 \) and \( \mu_1 = \mu_2 \).

Determine the position of L by showing that \( \mathbf{BL} = \frac{4}{7} \mathbf{BC} \).

Does P lie on the line AL? Justify your answer.

Show that \( J_0 = 1 - \frac{1}{e} \).

Show that \( J_n = \frac{n}{n + 1} \) for \( n \geq 1 \).

Show that \( J_n = \frac{n}{n!} \left( 1 - \frac{1}{e} \right) \).

Using parts (i) and (iii), show by mathematical induction that for all n \( \geq 0 \), \( J_n = n! \cdot \int_0^1 \frac{1}{n!} e^{-x} \, dx \).

Using parts (ii) and (iv), prove that \( \lim_{n \to \infty} J_n = 0 \).

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The two non-parallel vectors **u** and **v** satisfy \( \lambda \mathbf{u} + \mu \mathbf{v} = \mathbf{0} \) for some real numbers \( \lambda \) and \( \mu \) - HSC - SSCE Mathematics Extension 2 - Question 14 - 2022 - Paper 1

Worked Solution & Example Answer:The two non-parallel vectors **u** and **v** satisfy \( \lambda \mathbf{u} + \mu \mathbf{v} = \mathbf{0} \) for some real numbers \( \lambda \) and \( \mu \) - HSC - SSCE Mathematics Extension 2 - Question 14 - 2022 - Paper 1