Question Paper of Problem Solving

Question 1:

$T h e p e r c e n t a g e o f C a r b o n i n a n o r g a n i s m f a l l s e x p o n e n t i a l l y a f t e r t h e d e a t h o f t h e o r g a n i s m . A f t e r 1845 y e a r s 80 % o f t h e o r i g i n a l c o n c e n t r a t i o n o f C a r b o n r e m a i n s . U s e t h e e q u a t i o n C = C_{o} e^{- k t} t o r e p r e s e n t t h e e x p o n e n t i a l f a l l o f C a r b o n . F i n d t h e e x a c t v a l u e o f k . A n a r t w o r k c o n t a i n i n g t h i s o r g a n i s m h a s 65 % o f t h e o r i g i n a l c o n c e n t r a t i o n o f C a r b o n . H o w l o n g h a s t h i s o r g a n i s m b e e n d e a d ? G i v e t h e a n s w e r t o t h e n e a r e s t y e a r . A s e a s p o n g e c o n t a i n i n g t h i s o r g a n i s m h a s b e e n d e a d f o r 12000 y e a r s . W h a t p e r c e n t a g e (t o 1 d e c i m a l p l a c e) o f t h e o r i g i n a l C a r b o n c o n c e n t r a t i o n d o e s i t h a v e ? T w o s a i l o r s a r e p a i d t o b r i n g a m o t o r b o a t b a c k t o a h a r b o u r f r o m a n i s l a n d, a t o t a l d i s \tan c e o f 1200 k m . T h e y a r e e a c h p a i d $ 25 p e r h o u r f o r t h e 2 t i m e s p e n t a t s e a . T h e b o a t u s e s f u e l a t a r a t e o f 20 + \frac{v^{2}}{10} l i t r e s p e r h o u r w h e r e 10 t h e s p e e d o f t h e b o a t i s v k m p e r h o u r . D i e s e l f u e l \cos t s $ 1.25 p e r l i t r e . S h o w t h a t t o b r i n g t h e b o a t b a c k f r o m t h e i s l a n d, t h e t o t a l \cos t ($ C) t o t h e o w n e r 1 s C = \frac{90000}{v} + 150 v F i n d t h e s p e e d t h a t m i n i m i s e s t h e \cos t a n d d e t e r m i n e t h i s \cos t, g i v i n g y o u r a n s w e r t o t h e n e a r e s t d o l l a r .$

Question 2:

ABCD is a parallelogram. DE bisects <ADC, EC bisects <DCB and BF bisects <ABC

(i) Prove $∆ D E C \equiv ∆ B F C$

(ii) Show that <DEC = 90°

(iii) If DE =6cm , EC=10 cm and EF= FC, find the length of BC.

Question 3:

A particle moving along the x axis starts at the origin. At time t seconds the particle has a displacement $x$ metres from the origin and is travelling with a velocity of $v m s^{- 1}$
The displacement is given by

$x = 4 t - 6 \ln (t + 1)$ .

(i) Find an expression for v the velocity of the particle.

(ii) Find the initial velocity of the particle.

(iii) Find when the particle comes to rest.

(iv) Find the distance travelled by the particle in the first 3 seconds.

(v) When is the acceleration of the particle positive?

Question 4:

$F i n d t h e v a l u e o f m f o r w h i c h t h e e q u a t i o n (m - 1) x^{2} + 3 x - 3 = 0 h a s o n e r o o t t w i c e t h e o t h e r T h e l i n e y = m x = b i s a \tan g e n t t o c u r v e y = x^{2} + 4 x + 2 a t x = - 3 . f i n d t h e v a l u e o f m . F i n d t h e a n g l e o f i n c l i n a t i o n t h a t t h e \tan g e n t m a k e w i t h t h e p o s i t i v e x - a x i s .$

Question 5:

$Simplify fully \sqrt{27} + \sqrt{48} - 2 \sqrt{75}$

Question 6:

A drug is used to control a medical condition. it is known that the quantity Q of the drug

remaining in the body after t hours satisfies the equation
$Q = Q_{o} e^{- k t}$ where Q0 and k are constants.

(i) An initial dose is administered and 4 hours later half the original dose remains.

Find the value of k.

(ii) What percentage of the initial dose remains after 6 hours?

Question 7:

$S i m p l i f y : 3 x - (8 - 13 x) . E x p r e s s \frac{4}{2 - \sqrt{5}} w i t h a r a t i o n a l d e n o m i n a t o r . S o l v e | 3 - 2 x | = 4 . F a c t o r i z e f u l l y 81 - 3 x^{3}$

Question 8:

A rectangular box with a square base and Q top is drawn below .

$T h e v o l u m e o f t h e b o x i s 500 c m^{3} S h o w t h a t t h e s u r f a c e a r e a (A) o f t h e b o x i s g i v e n b y A = x^{2} + \frac{2000}{x} . F i n d t h e l e a s t a r e a o f s h e e t m e t a l r e q u i r e d t o m a k e t h e b o x .$

Question 9:

$t h e n u m b e r o f a n i m a l s P o n a n i s l a n d a t t i m e t i s g i v e n t h a t p = 7000 e^{- k t} c o n s \tan t . o v e r t i m e t h e n u m b e r o f a n i m a l o n t h e i s l a n d i s (A) i n c r e a \sin g e x p o n e n t i a l l y (B) d e c r e a \sin g e x p o n e n t i a l l y (C) i n c r e a \sin g a t a c o n s \tan t r a t e (D) d e c r e a \sin g a t a c o n s \tan t r a t e$

Question 10:

$A n a i r l i n e c o m p a n y m a r k s t h e p r i c e o f a f l g h t a t $ 400, l e s s a g r o u p d i s c o u n t b a s e d o n t h e n u m b e r o f b o o k i n g s m a d e . T h e p r i c e R d o l l a r s f o r e a c h p e r s o n i n t h e g r o u p o f x p e o p l e i s R = 400 - 0.5 x . T h e \cos t o f r u n n i n g t h e f l i g h t i s a f i x e d \cos t o f $ 5000 p l u s $ 150 p e r p e r s o n . S h o w t h a t t h e p r o f i t o n a f l i g h t o f a g r o u p o f x p e o p l e i s (250 x - 0.5 x^{2} - 5000) d o l l a r s . H e n c e f i n d t h e r e q u i r e d g r o u p s i z e t o g a i n t h e m a x i m u m p r o f i t a n d f i n d t h i s p r o f i t . T w o p a r t i c l e s A a n d B a r e m o v i n g a l o n g t h e x - a x i s . T h e i r d i s p l a c e m e n t s f r o m t h e o r i g i n a r e g i v e n b y : x = \frac{- 1}{π} (1 + cosπ t) a n d x = t^{2} - 4 t r e s p e c t i v e l y . E x p r e s s t h e v e l o c i t i e s o f t h e t w o p a r t i c l e s i n t e r m s o f t i m e . O n t h e s a m e d i a g r a m s k e t c h t h e t w o v e l o c i t i e s . U s e t h a t s k e t c h t o s h o w t h a t t h e p a r t i c l e s h a v e t h e s a m e n o n - z e r o v e l o c i t i e s a t t w o o c c a s i o n s b y m a r k i n g t_{1} a n d t_{2} . S h o w t h a t t h e d i s \tan c e t r a v e l l e d b y t h e s e c o n d p a r t i c l e b e t w e e n t_{1} a n d t_{2} i s : ({t_{1}}^{2} + {t_{2}}^{2}) - 4 (t_{1} + t_{2}) + 8$