Question Paper of Problem Solving

Question 1:

$K a n d o, t h e m a t h e m a t i c a l k a n g a r o o a l w a y s h o p s (i . e j u m p s) a c c o r d i n g i n m a t h e m a t i c a l r u l e s, D n e d a y, K a n d o d e c i d e s t o g o h o p p i n g a c c o r d i n g t o t h e F o l l o w i n g r u l e s T h e l e n g t h a f o d d n u m b e r h o p e (1 s t, 3 r d, 5 t h h o p e t c) . I n m e t r e s t o g i v e n b y t h e a r i t h m e t i c s e r i e s, t_{n} = 4 (n - 1) w h e r e n = 1, 3, 5 i s a n o d d n u m b e r T h e l e n g t h o f e v e n n u m b e r h o p s (2 n d, 4 t h, 6 t h h o p e t c) i n m e t e r s, i s g i v e n b y t h e g e o m e t r i c s e r i e s T_{N} = \frac{192}{63} {(\frac{1}{2})}^{\frac{N - 2}{2}} I f t h e l e n g t h o f a h o p i s n e g a t i v e a c c o r d i n g t o t h e r e l e v a n t s e r i e s K a n d o h a s t h e p r e s c r i b e d d i s \tan c e b a c k w a r d s (i) W r i t e d o w n t h e f i r s t t e r m a n d c o m m o n d i f f e r e n c e f o r t h e s e r i e s (i i) W r i t e d o w n t h e f i r s t t e r m a n d c o m m o n r a t i o f o r t h e s e r i e s T_{n} (i i i) F i n d w h e r e k a n d o i s r e l a t i v e t o h e r s t a r t i n g p o i n t a f t e r 12 h o p s (i v) F i n d t h e t o t a l d i s \tan c e t r a v e l l e d b a c k w a r d s i n t h e f i r s t 16 h o p s$

Question 2:

$E x a c t l y 12 y e a r s a g o, P a u l t o o k o u t a m o r t g a g e o f $ 500 000 t o b u y a h o u s e . T h e l o a n w a s t a k e n o v e r 25 y e a r s a t 12 % p . a . w i t h i n t e r e s t c o m p o u n d i n g m o n t h l y a n d P a u l m a k e s m o n t h l y r e p a y m e n t s . P a u l h a s j u s t w o n a l o t t e r y p r i z e o f $ 400 000 . S h o w t h a t t h e p r i z e i s i n s u f f i c i e n t t o p a y o u t t h e r e m a i n i n g d e b t . H o w m a n y p a y m e n t s w i l l s t i l l b e r e q u i r e d t o p a y o f f t h e d e b t ?) (Y o u m a y a s s u m e t h a t P a u l p u t s t h e e n t i r e p r i z e i n t o p a y i n g o f f t h e d e b t .$

Question 3:

A particle moves in a straight line so that its displacement, in meters, is given by $x = 4 t^{3} - 12 t^{2}$ where t is the time in seconds. At what time after t = 0 is the particle at rest?

(A) t = 1

(B) t = 2

(D) t = 4

Question 4:

What are the solutions of $x^{2}$ < 9?

Question 5:

Find the limiting sum of the geometric series $3 + \frac{1}{2} + \frac{1}{12} + . . . .$

Question 6:

A particle is moving along the x-axis. The displacement of the particle after t seconds is given by x = $t^{2}$ - 3 $t$ metres.

Question 7:

The diagram shows a sketch of the gradient function $y = f^{'} (x)$

passing through the points A, B, C and D.

which point represents the horizontal point of inflection of the curve $y = f (x) ?$

Question 8:

Solve |2x + 3 | < 9 .

Question 9:

$A n A u s t r a l i a n a i r l i n e r e s t r i c t s t h e s i z e o f c a b i n b a g g a g e a l l o w e d o n t o 6 a i r c r a f t . F o r b a g s i n t h e r e c o m m e n d e d s h a p e o f a r e c \tan g u l a r p r i s m, t h e s u m o f t h e l e n g t h, b r e a d t h a n d h e i g h t s h o u l d b e e q u a l t o 1 15 c m . S a n d y b u y s a b a g i n t h e r e c o m m e n d e d s h a p e, w i t h a s q u a r e b a s e o f l e n g t h x c m a n d h e i g h t h c m, t o t a k e o n t o t h e a i r c r a f t .$

$(i) S h o w t h a t t h e v o l u m e i n c u b i c c e n t i m e t e r s i s g i v e n b y V = 115 x^{2} - 2 x^{3} (i i) H e n c e f i n d t h e m a x i m u m v o l u m e .$

Question 10:

n the Kurramurra Theatre, rows of seats are labelled alphabetically, starting with Row A. Row A has seats numbered 1 to 27, Row B has seats numbered 1 to 29, Row C has seats numbered 1 to 31, and so on until the last row of the theatre, Row M.

i) How many seats are in Row M?

ii) What is the total number of seats in Kurramurra Theatre?