Question Paper of Probability/Statistics/Proof

Question 1:

$E v a l u a t e t h e a r i t h m e t i c s e r i e s 3 + 7 + 11 + 15 + \cdot \cdot \cdot + 4003 . A b a g c o n t a i n s 8 b l u e m a r b l e s a n d 4 y e l l o w m a r b l e s . T w o m a r b l e s a r e s e l e c t e d a t r a n d o m w i t h o u t r e p l a c e m e n t . (i) D r a w a t r e e d i a g r a m t o s h o w a l l p o s s i b l e o u t c o m e s . I n c l u d e t h e p r o b a b i l i t y o n e a c h b r a n c h . (i i) W h a t i s t h e p r o b a b i l i t y t h a t t h e t w o m a r b l e s a r e o f d i f f e r e n t c o l o u r s ?$

Question 2:

$P r o v e \int_{e}^{e^{2}} \frac{1 + \ln x}{x \ln x} d x = 1 + \ln 2$

Question 3:

$D o n a l d a n d B e l l a a r e p l a y i n g a g a m e o f t e n n i s . T h e y h a v e 8 u n u s e d b a l l s a n d 5 u s e d b a l l s . I f t h e y c h o s e t w o b a l l s t o p l a y a m a t c h, f i n d t h e p r o b a b i l i t y t h a t t h e y c h o o s e t w o u n u s e d b a l l s . A f t e r t h e m a t c h, t h e b a l l s a r e r e t u r n e d t o t h e b o x a n d I v a n k a a n d J a c o b r a n d o m l y c h o o s e t w o b a l l s t o p l a y t h e i r m a t c h . W h a t i s t h e p r o b a b i l i t y t h a t t h e y p l a y w i t h u n u s e d b a l l s ?$

Question 4:

$A C o n c o r d e n g i n e u s e s f u e l a t t h e r a t e o f R l i t r e s p e r s e c o n d . T h e r a t e o f t h e f u e l u s e s e c o n d s a f t e r t h e e n g i n e s t a r t s o p e r a t i o n i s g i v e n b y R = 20 + \frac{15}{1 + t} . W h a t i s R w h e n t = 9 ? W h a t h a p p e n s t o R a s t a p p r o a c h e s ? J u s t i f y y o u r a n s w e r . D r a w a s k e t c h o f R a s a f u n c t i o n o f t . C a l c u l a t e t h e t o t a l a m o u n t o f p e t r o l u s e d d u r i n g t h e f i r s t 9 s e c o n d s . G i v e y o u r a n s w e r c o r r e c t t o t h e n e a r e s t l i t r e .$

Question 5:

$P r o v e t h a t \frac{2^{2 x} + 1}{2^{x}} i s a n e v e n f u n c t i o n .$

Question 6:

$The fourth term of an arithmetic series is 27 and the seventh term is 12 .What is the common difference? (A) 5 (B) - 5 (C) 13 (D) - 13$

Question 7:

$A cupboard contains 7 white mugs and 4 black mugs. A mug is drawn at random from this cupboard, and then returned to the cupboard after its colour has been noted. A second mug is then drawn at random from the cupboard. What is the probability both mugs are the same colour? (A) \frac{28}{121} (B) \frac{49}{121} (C) \frac{56}{121} (D) \frac{65}{121}$

Question 8:

$T h e p r o b a b i l i t y t h a t t h e r e w i l l b e r a i n i n S y d n e y t h i s w e e k e n d i s \frac{1}{3} . T h e p r o b a b i l i t y t h a t t h e r e w i l l b e r a i n i n N e w c a s t l e t h i s w e e k e n d i s \frac{2}{5} . W h a t i s t h e p r o b a b i l i t y t h a t t h e r e w i l l b e n o r a i n i n S y d n e y a n d n o r a i n i n N e w c a s t l e t h i s w e e k e n d ? (A) \frac{2}{3} (B) \frac{3}{5} (C) \frac{2}{5} (D) \frac{2}{15}$

Question 9:

$U a m, H a r r y a n d Z a c h w o r k i n d e p e n d e n t l y o n a p r o b l e m . I f t h e r e s p e c t i v e p r o b a b i l i t i e s t h a t t h e y w i l l s o l v e i t a r e \frac{3}{4}, \frac{1}{2} a n d \frac{2}{5}, f i n d t h e p r o b a b i l i t y t h a t t h e p r o b l e m w i l l b e s o l v e d .$

Question 10:

$In the diagram, the points A, B, C, D, E, F and G lie on the curve y = f (x) . Point B,D and F are stationary points C,D and E are points of inflection.$

$Which point corresponds to the description y > 0, \frac{d y}{d x} = 0, \frac{d^{2} y}{d x^{2}} < 0 ? (A) B (B) C (C) D (D) F$