Question Paper of Problem Solving

Question 1:

$J o h n r u n s a m a r a t h o n, w h i c h i s a d i s \tan c e o f 42 k m . H e r u n s t h e f i r s t k i l o m e t r e i n 4 m i n u t e s a n d f r o m t h e r e o n, h e t a k e s 5 s e c o n d s l o n g e r t o r u n e a c h s u c c e s s i v e k i l o m e t r e . (i . e t h e 2 n d k i l o m e t r e t a k e s 4 m i n u t e s a n d 5 s e c o n d s . (i) H o w l o n g d o e s h e t a k e t o r u n t h e 16 t h k i l o m e t r e ? (i i) H o w l o n g d o e s i t t a k e t o r u n t h e e n t i r e m a r a t h o n ? (i i i) H o w f a r h a d h e r u n a f t e r 2 h o u r s 36 m i n u t e s a n d 15 s e c o n d s ?$

Question 2:

On their son Geoffrey's $11^{t h}$ birthday, Mr. and Mrs. shum deposited $600 into an account earning 5% p.a. interest compound annually. They will continue to deposit $600 on each of his successive birthdays, up to and including his $21^{s t}$ , giving him the accumulated fund as a present on his $21^{s t}$ birthday.

i) Show that the amount of Geoffrey's $21^{s t}$ birthday present was $8524 (to the nearest dollar)

ii) What single deposit on Geoffrey's $11^{t h}$ birthday would have, under the same conditions, provided the same $21^{s t}$ birthday present?

Question 3:

$s i m p l i f y \frac{1 - \cos^{2} α}{\sin α \cos α}$

Question 4:

$T h e v e l o c i t y o f p a r t i c l e t r a v e l i n g a l o n g t h e x - a x i s i s g i v e n b y v = 2 - \frac{10}{t + 2} m s^{- 1} w h e r e t i s t h e t i m e i n s e c o n d s . W h e n i s t h e p a r t i c l e s t a t i o n a r y ? W h a t h a p p e n s t o t h e v e l o v i t y a s t \to \infty ? F i n d t h e a c c e l e r a t i o n w h e n t h e p a r t i c l e i s s t a t i o n a r y . F i n d t h e d i s \tan c e t r a v e l l e d i n t h e f i r s t f i v e s e c o n d s i n e x a c t f o r m .$

Question 5:

$N e c t a r i o s i n v e s t s $ 50 000 i n a n a c c o u n t w h i c h e a r n s 8 % i n t e r e s t, c o m p o u n d e d a n n u a l l y . H e i n t e n d s t o w i t h d r a w $ M a t t h e e n d o f e a c h y e a r, i m m e d i a t e l y a f t e r t h e i n t e r e s t h a s b e e n p a i d . H e w i s h e s t o b e a b l e t o d o t h i s f o r e x a c t l y 20 y e a r s, s o t h a t t h e a c c o u n t w i l l t h e n b e e m p t y . i) W r i t e a n e x p r e s s i o n i n t e r m s o f M f o r t h e a m o u n t s t i l l i n t h e a c c o u n t i m m e d i a t e l y a f t e r h e h a s m a d e h i s f i r s t w i t h d r a w a l ? i i) W r i t e a n e x p r e s s i o n i n t e r m s o f M f o r t h e a m o u n t o f m o n e y i n t h e a c c o u n t i m m e d i a t e l y a f t e r h i s 20 t h w i t h d r a w a l . i i i) C a l c u l a t e t h e v a l u e o f M w h i c h l e a v e s h i s a c c o u n t e m p t y a f t e r t h e 20 t h w i t h d r a w a l . i v) S u p p o s e N e c t a r i o s w i s h e d t o b e a b l e t o w i t h d r a w $ 8000 p e r y e a r f o r t h e 20 y e a r s . F i n d, t o t h e n e a r e s t p e r c e n t, t h e i n t e r e s t r a t e h e w o u l d t h e n n e e d t o e a r n o n h i s a c c o u n t .$

$T h e w o r l d' s p o p u l a t i o n P i s i n c r e a \sin g a t a n i n c r e a \sin g r a t e o v e r t i m e t . W h a t d o e s t h i s s a y a b o u t t h e s i g n o f \frac{d P}{d t} a n d \frac{d^{2} P}{d t^{2}}$

Question 6:

A seedling with height 46 mm is planted and its growth tracked.

Its height after one week was 59.8 mm and after a further week its height was recorded as 77.74 mm.

Its height after 6 weeks

The week in which it's height is first recorded as over 1 metre

Question 7:

$A r a f f l e c o n s i s t s o f t w e n t y t i c k e t s i n w h i c h t h e r e a r e t w o p r i z e s . M i k e b u y s f i v e t i c k e t s . F i r s t p r i z e i s t w o m o v i e v o u c h e r s a n d s e c o n d p r i z e i s o n e m o v i e v o i i c h e r . T h e p r o b a b i l i t y t h a t M i k e w i n s a t l e a s t o n e m o v i e v o u c h e r i s (A) \frac{17}{38} (B) \frac{27}{76} (C) \frac{7}{16} (D) \frac{5}{20}$

Question 8:

$v = 6 \cos 2 t i s t h e e q u a t i o n o f a p a r t i c l e ’ s v e l o c i t y i n m / s, w h e r e t i s i n s e c o n d s . T h e p a r t i c l e w a s i n i t i a l l y 2 m e t r e s t o t h e l e f t o f t h e o r i g i n . F i n d t h e e q u a t i o n s f o r t h e d i s p l a c e m e n t a n d a c c e l e r a t i o n o f t h e p a r t i c l e . W h e n d o e s t h e p a r t i c l e f i r s t r e a c h t h e o r i g i n ? A n s w e r c o r r e c t t o 2 d e c i m a l p l a c e s . W h e n d o e s t h e p a r t i c l e f i r s t c o m e t o r e s t . F i n d t h e t o t a l d i s \tan c e t r a v e l l e d b y t h e p a r t i c l e d u r i n g t h e p e r i o d 0 \leq t \leq \frac{π}{2}$

Question 9:

$A b a c t e r i a c u l t u r e o f N b a c t e r i a i s i n c r e a \sin g e x p o n e n t i a l l y s o t h a t a f t e r 10 m i n u t e s \frac{d N}{d t} = k N . I f t h e n u m b e r o f b a c t e r i a i n c r e a s e s f r o m 100 t o 400 i n 2 m i n u t e s, f i n d : T h e v a l u e o f c o n s \tan t k . T h e n u m b e r o f b a c t e r i a p r e s e n t a f t e r 10 m i n u t e s . W h e n t h e r e w i l l b e 1000 b a c t e r i a .$

Question 10:

$F o r w h a t v a l u e s o f k, w i l l k x^{2} + 5 x + k b e p o s i t i v e d e f i n i t e ?$