Question Paper of Problem Solving

Question 1:

$Solve \frac{2 x - 1}{3} - \frac{x - 2}{2} = 5 .$

Question 2:

$W h e n R o b b y i s 3 m o n t h s o l d, h i s p a r e n t s d e c i d e t o m a k e a r e g u l a r d e p o s i t o f $ 500 e v e r y 3 m o n t h s, s t a r t i n g w i t h f i r s t o n e w h e n R o b b y i s 3 m o n t h s o l d i n a n a c c o u n t t h a t e a r n s i n t e r e s t o f 8 % p . a ., t h e i n t e r e s t b e i n g p a i d e v e r y 3 m o n t h s . S h o w t h a t t h e d a y a f t e r R o b b y' s 1 s t b i r t h d a y (a f t e r p a y m e n t i s m a d e), t h e v a l u e o f 2 t h e a c c o u n t i s g i v e n b y $ 2060.80 . H o w m u c h m o n e y w i l l b e i n t h e a c c o u n t t h e d a y w h e n R o b b y t u r n s 15 a f t e r t h e p a y m e n t i s m a d e ? N o m o r e p a y m e n t s a r e m a d e i n t o t h e a c c o u n t a f t e r R o b b y t u r n s 15 a n d n o w i t h d r a w a l s a r e m a d e . F i n d t h e a m o u n t i n t h e a c c o u n t o n R o b b y' s 16 t h b i r t h d a y . R o b b y d e c i d e s t h a t h e w i l l w i t h d r a w a r e g u l a r a m o u n t o f m o n e y f r o m t h i s a c c o u n t e a c h b i r t h d a y, s t a r t i n g w i t h h i s 16 t h b i r t h d a y . H e c a n n o t d e c i d e w h e t h e r h e s h o u l d w i t h d r a w $ 4000 o r $ 5000 e a c h b i r t h d a y . B y c o n s i d e r i n g t h e r e s u l t o f p a r t (i i i), c o m m e n t o n w h a t w i l l h a p p e n i n e a c h c a s e .$

Question 3:

$A s o c c e r g o a l m o u t h i s 8 m e t r e s w i d e . I f a p l a y e r i s s \tan d i n g i n f r o n t o f t h e g o a l m o u t h 4 m e t r e s f r o m o n e g o a l p o s t a n d 10 m e t r e s f r o m t h e o t h e r g o a l p o s t, t h r o u g h w h a t a n g l e m u s t s h e s h o o t t o b e s u c c e s s f u l i n s c o r i n g t h e g o a l ? (A n s w e r t o t h e n e a r e s t d e g r e e .)$

Question 4:

$N a t j u s t r e t i r e d w i t h $ 1, 000, 000 i n h e r b a n k a c c o u n t . T h i s a c c o u n t a t t r a c t s i n t e r e s t a t a r a t e o f 5 % p . a . c o m p o u n d e d a n n u a l l y . S h e i n t e n d s t o w i t h d r a w m o n e y i n e q u a l a m o u n t s e v e r y y e a r i m m e d i a t e l y a f t e r i n t e r e s t i s c o m p o u n d e d . L e t A_{n} b e t h e a m o u n t o f m o n e y i n h e r a c c o u n t a f t e r t h e n^{t h} w i t h d r a w a l o f $ W . i . W r i t e d o w n a n e x p r e s s i o n f o r t h e a m o u n t l e f t a f t e r t w o w i t h d r a w a l s . i i . S h o w t h a t W = 80242.59, i f N a t w a n t s t h e m o n e y t o l a s t e x a c t l y 20 w i t h d r a w a l s . i i i . A f t e r h e r 10^{t h} w i t h d r a w a l, t h e b a n k c h a n g e s i t s i n t e r e s t r a t e t o 3 % p . a . o n t h e r e m a i n i n g b a l a n c e . H o w m a n y f u r t h e r w i t h d r a w a l s c a n N a t t a k e i f W s t a y s t h e s a m e ? i v . W h a t s h o u l d N a t' s w i t h d r a w a l a m o u n t b e r i g h t f r o m t h e b e g i n n i n g i f s h e w i s h e s t o s t i l l m a k e t h e m o n e y l a s t 20 w i t h d r a w a l s, t a k i n g i n t o a c c o u n t t h e c h a n g e i n i n t e r e s t r a t e a f t e r 10 y e a r s ?$

Question 5:

$S o l v e 3 \log_{8} 2 = \log_{8} x - \log_{8} 3$

Question 6:

$A p a r t i c l e m o v e s a l o n g a s t r a i g h t l i n e s o t h a t i t s d i s \tan c e x m e t r e s f r o m a f i x e d p o i n t O i s g i v e n b y x = 4 - 3 t + 12 \ln (t + 3) w h e r e t h e t i m e i s m e a s u r e d i n s e c o n d s . W h a t i s t h e e x a c t i n i t i a l p o s i t i o n ? F i n d e x p r e s s i o n s f o r t h e v e l o c i t y a n d a c c e l e r a t i o n o f t h e p a r t i c l e a t t i m e . F i n d t h e t i m e w h e n t h e v e l o c i t y o f t h e p a r t i c l e i s z e r o . W h a t i s t h e e x a c t d i s \tan c e t h a t t h e p a r t i c l e h a s t r a v e l l e d i n t h e f i r s t 2 s e c o n d s ?$

Question 7:

$A n e w c a r h a s a c a s h p r i c e o f $ 50, 000 b u t i s b o u g h t u \sin g t i m e p a y m e n t . T h e i n t e r e s t r a t e i s 9 % p . a . a n d m o n t h l y r e p a y m e n t s R a r e m a d e f o r 4 y e a r s . A r e p a y m e n t R i s m a d e i m m e d i a t e l y a f t e r t h a t m o n t h' s i n t e r e s t i s c h a r g e d . L e t A b e t h e a m o u n t s t i l l o w i n g o n t h e l o a n a f t e r n m o n t h s . i) W r i t e a s i m p l e e x p r e s s i o n f o r A_{1} i i) S h o w t h a t A_{2} = 50000 {(1.0075)}^{2} — A (I + 1.0 075) i i i) W r i t e a n e x p r e s s i o n f o r A_{48} . S i m p l i f y y o u r a n s w e r . i v) S h o w t h a t J 2 = $ 1244 (t o t h e n e a r e s t d o l l a r) . v) F i n d t h e a m o u n t s t i l l o w i n g a f t e r 24 m o n t h s (t o t h e n e a r e s t d o l l a r) . v i) A f t e r 24 m o n t h s, t h e c a r b u y e r i n c r e a s e s h e r r e p a y m e n t s t o $ 2200 p e r m o n t h i n o r d e r t o p a y o f f t h e l o a n m o r e q u i c k l y .$

Question 8:

$A G e i g e r c o u n t e r i s t a k e n i n t o a r e g i o n a f t e r a n u c l e a r a c c i d e n t a n d g i v e s a r e a d i n g O n e y e a r l a t e r, t h e s a m e G e i g e r c o u n t e r g i v e s a r e a d i n g o f 35 000 . I t i s k n o w n t h a t r e a d i n g N i s g i v e n b y t h e f o r m u l a N = N_{o} e^{- k t} w h e r e N, a n d a r e c o n s \tan t s a n d i t i s t h e t i m e m e a s u r e d i n y e a r s . E v a l u a t e t h e c o n s \tan t s N, a n d k . I t i s k n o w n t h a t t h e r e g i o n w i l l b e c o m e s a f e w h e n t h e r e a d i n g r e a c h e s 400 : A f t e r h o w m a n y y e a r s w i l l t h e r e g i o n b e c o m e s a f e ? (A n s w e r t o t h e n e a r e s t w h o l e y e a r .)$

Question 9:

$E x p r e s s 2.5 x 10^{- 3} a s a f r a c t i o n i n s i m p l i f i e d f o r r n . S o l v e 2^{x + 1} = 128 G r a p h o n a n u m b e r l i n e t h e s o l u t i o n o f l 3 x - 1 l < 2$

Question 10:

A total of 300 tickets are sold in a raffle which has 3 prizes.

There are 100 red, 100 green and 100 blue tickets. At the drawing of the raffle, winning tickets are NOT replaced before the next draw.

What is the probability that each of the 3 winning tickets

is red?

What is the probability that at least one of the winning tickets is not red? 1

What is the probability that there is one winning ticket of each colour?