Question Paper of Hard Maths

Question 1:

$I n t h e d i a g r a m, ∠ B C A = ∠ C A F = 90 ° a n d A B = A E = E F . i) C o p y t h e d i a g r a m o n t o y o u r a n s w e r s h e e t . i i) P r o v e t h a t ∠ A B D = 2 ∠ D B C$

Question 2:

$A B C D i s a r h o m b u s . B D i s p r o d u c e d t o E s u c h t h a t A D = D E .$

$C o p y t h e d i a g r a m o n t o y o u r a n s w e r s h e e t . S h o w t h a t ∠ A B C = ∠ A E D 4 . S h o w t h a t D i s t h e m i d p o i n t o f B E g i v e n t h a t ∠ B A E = 90 .$

Question 3:

$A B C D i s a r e c \tan g l e i n w h i c h A B = 12 c m a n d A D = 8 c m . T h e p o i n t s P, Q, R l i e o n A B, B C a n d C D r e s p e c t i v e l y s o t h a t P B = Q C = D R = x c m . S h o w t h a t t h e a r e a o f t r i a n g l e P Q R i s g i v e n b y 2 A = x^{2} - 10 x + 48 . F i n d t h e l e a s t v a l u e o f t h i s a r e a . S h o w t h a t t h e g r e a t e s t v a l u e o f t h e a r e a o f t h i s t r i a n g l e i s 48 c m^{2} .$

Question 4:

$T h e v e l o c i t y - t i m e g r a p h o f a p a r t i c l e m o v i n g o n a s t r a i g h t l i n e i s s h o w n b e l o w, w h e r e v e l o c i t y i s g i v e n i n m s / a n d t i m e i n s e c o n d s :$

$H o w f a r h a s t h e p a r t i c l e t r a v e l l e d i n t h e t i m e i n t e r v a l b e t w e e n t = 6 a n d t = 14 ? W h a t w a s t h e a c c e l e r a t i o n w h e n t = 10 ?$

Question 5:

$A r e c \tan g u l a r b e a m o f w i d t h w e m a n d d e p t h, d e m i s c u t f r o m a c y l i n d r i c a l p i n e \log a s s h o w n . T h e d i a m e t e r o f t h e c r o s s s e c t i o n o f t h e \log (a n d h e n c e t h e d i a g o n a l o f t h e c r o s s s e c t i o n o f t h e b e a m) i s 15 c m . T h e s t r e n g t h S o f t h e b e a m i s p r o p o r t i o n a l t o t h e p r o d u c t o f i t s w i d t h a n d t h e s q u a r e o f i t s d e p t h s o t h a t S = k d^{2} w$

$(i) S h o w t h a t S = k (225 w - w^{3}) (i i) W h a t n u m e r i c a l d i m e n s i o n s w i l l g i v e a b e a m o f m a x i m u m s t r e n g t h ? (i i i) A s q u a r e b e a m w i t h d i a g o n a l 15 c m c o u l d h a v e b e e n c u t f r o m t h e \log . S h o w t h a t t h e r e c \tan g u l a r b e a m o f m a x i m u m s t r e n g t h i s m o r e t h a n 8 % s t r o n g e r t h a n t h i s s q u a r e b e a m .$

Question 6:

$I n t h e d i a g r a m b e l o w A E = E D = A D = D C, ∠ A D C = 90' a n d A E I I B C . ∠ B A C = 51' F i n d t h e s i z e o f ∠ E A B . G i v e r e a s o n s f o r y o u r a n s w e r F i n d t h e s i z e o f ∠ A B C . G i v e r e a s o n s f o r y o u r a n s w e r .$

Question 7:

$Andy and Mindy agree to meet for coffee at their favourite cafe. The cafe (C) is situated on a bearing of 110^{\circ} from Andy (A) . From Mindy (M), the bearing to the cafe is 168^{\circ} .$

$The distance between Andy and Mindy is 2 km . The distance from Andy to the cafe is 1.6 km . (i) F i n d t h e s i z e o f ∠ M C A (i i) W h a t i s t h e b e a r i n g f r o m A n d y t o M i n d y ? (i i i) C a l c u l a t e t h d i s \tan c e b e t w e e n M i n d y a n d t h e c a f e . G i v e y o u r a n s w e r c o r r e c t t o 1 d e c i m a l p l a c e s .$

Question 8:

$T w o c r u i s e s h i p s s e t s a i l f r o m S y d n e y H a r b o u r (S) . T h e E l v i s P r e s l e y T r i b u t e C r u i s e (E) s a i l s a t 18 k m / h o n a b e a r i n g o f 049 ° w h i l e t h e M i c h a e l J a c k s o n T r i b u t e C r u i s e (M) s a i l s a t 21 k m / h a l o n g a b e a r i n g o f 151 °$

$S h o w t h a t ∠ E S M = 102 ° . C a l c u l a t e t h e d i s \tan c e b e t w e e n t h e c r u i s e s h i p s t o t h e n e a r e s t k i l o m e t r e a f t e r 3 h o u r s .$

Question 9:

$J o e i s b u i l d i n g a s m a l l t o y b o x w i t h n o l i d . T h e b o x i s t w i c e a s l o n g a s i t i s w i d e . T h e b o x h a s a t o t a l e x t e r n a l s u r f a c e a r e a o f 3750 c m^{2}$

$S h o w t h a t t h e h e i g h t h o f t h e t o y b o x i s g i v e n b y h = \frac{625}{x} - \frac{x}{3} F i n d t h e d i m e n s i o n s o f t h e b o x w h i c h g i v e s a m a x i m u m v o l u m e . J o e d e c i d e s t h a t t h e h e i g h t o f t h e b o x w i l l b e 10 \frac{5}{6} c m . F i n d t h e n e w d i m e n s i o n s o f t h e b o x a n d h e n c e f i n d i t s v o l u m e i f t h e s u r f a c e a r e a i s t o r e m a i n a t 3750 c m^{2} .$

Question 10:

$Q R S i s a p a r a l l e \log r a m . P Q i s p r o d u c e d t o T s o t h a t Q T = Q R a n d P S i s p r o d u c e d t o U s o t h a t S U = P S . I t i s n o w d i s c o v e r e d t h a t T, R a n d U a r e c o l l i n e a r S h o w t h a t S U = R Q P r o v e P Q R S i s a r h o m b u s$