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Q u a n t i t a t i v e A b i l i t y S A 1 . A r p i t a a n d N i k i t a , w o r k i n g t o g e t h e r , c a n c o m p l e t e a n a s s i g n e d j o b i n 1 2 d a y s . I f A r p i t a w o r k s i n i t i a l l y t o c o m p l e t e 4 0 % o f t h e j o b , a n d t h e r e m a i n i n g j o b i s c o m p l e t e d b y N i k i t a a l o n e , i t t a k e s 2 4 d a y s t o c o m p l e t e t h e j o b . T h e p o s s i b l e n u m b e r o f d a y s t h a t N i k i t a r e q u i r e s t o c o m p l e t e t h e e n t i r e j o b , w o r k i n g a l o n e , i s 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f b i o l o g i s t s i n t e a m E i s _ _ _ _ _ 3 . I f a , b , c a r e t h r e e d i s t i n c t n a t u r a l n u m b e r s , a l l l e s s t h a n 1 0 0 , s u c h t h a t , t h e n t h e m a x i m u m p o s s i b l e v a l u e o f b i s _ _ _ _ _ _ 4 . E i g h t t e a m s p a r t i c i p a t e i n a t o u r n a m e n t w h e r e e a c h t e a m p l a y s a g a i n s t e v e r y o t h e r t e a m e x a c t l y o n c e . I n a p a r t i c u l a r y e a r , o n e t e a m g o t s u s p e n d e d a f t e r p l a y i n g 3 m a t c h e s , d u e t o a d i s c i p l i n a r y i s s u e . T h e o r g a n i s e r s d e c i d e d t o p r o c e e d , n o n e t h e l e s s , w i t h t h e r e m a i n i n g m a t c h e s . T h e t o t a l n u m b e r o f m a t c h e s t h a t w e r e p l a y e d i n t h e t o u r n a m e n t t h a t y e a r i s 5 . I f t h e s u m o f t h e r s t 2 1 t e r m s o f t h e s e q u e n c e : i s , t h e n t h e v a l u e o f i s 6 . T h e E n g l i s h a n d M a t h e x a m s w e r e c o n d u c t e d s e p a r a t e l y f o r a c l a s s o f 1 2 0 s t u d e n t s . T h e n u m b e r o f s t u d e n t s w h o d i d n o t a p p e a r f o r t h e E n g l i s h e x a m i s t w i c e t h a t o f t h o s e w h o d i d n o t a p p e a r f o r t h e M a t h e x a m . T h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e M a t h e x a m i s t w i c e t h a t o f t h o s e w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m . I f t h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e E n g l i s h e x a m i s t w i c e t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e M a t h e x a m , t h e n t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m i s _ _ _ _ _ _ _ _ 7 . I f s u c h t h a t , t h e n e q u a l s _ _ _ _ _ _ _ _ _ _ _ 8 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . | a - b | + | b - c |=| c - a |
ln , ln , ln , ln , ........
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Page 2
Q u a n t i t a t i v e A b i l i t y S A 1 . A r p i t a a n d N i k i t a , w o r k i n g t o g e t h e r , c a n c o m p l e t e a n a s s i g n e d j o b i n 1 2 d a y s . I f A r p i t a w o r k s i n i t i a l l y t o c o m p l e t e 4 0 % o f t h e j o b , a n d t h e r e m a i n i n g j o b i s c o m p l e t e d b y N i k i t a a l o n e , i t t a k e s 2 4 d a y s t o c o m p l e t e t h e j o b . T h e p o s s i b l e n u m b e r o f d a y s t h a t N i k i t a r e q u i r e s t o c o m p l e t e t h e e n t i r e j o b , w o r k i n g a l o n e , i s 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f b i o l o g i s t s i n t e a m E i s _ _ _ _ _ 3 . I f a , b , c a r e t h r e e d i s t i n c t n a t u r a l n u m b e r s , a l l l e s s t h a n 1 0 0 , s u c h t h a t , t h e n t h e m a x i m u m p o s s i b l e v a l u e o f b i s _ _ _ _ _ _ 4 . E i g h t t e a m s p a r t i c i p a t e i n a t o u r n a m e n t w h e r e e a c h t e a m p l a y s a g a i n s t e v e r y o t h e r t e a m e x a c t l y o n c e . I n a p a r t i c u l a r y e a r , o n e t e a m g o t s u s p e n d e d a f t e r p l a y i n g 3 m a t c h e s , d u e t o a d i s c i p l i n a r y i s s u e . T h e o r g a n i s e r s d e c i d e d t o p r o c e e d , n o n e t h e l e s s , w i t h t h e r e m a i n i n g m a t c h e s . T h e t o t a l n u m b e r o f m a t c h e s t h a t w e r e p l a y e d i n t h e t o u r n a m e n t t h a t y e a r i s 5 . I f t h e s u m o f t h e r s t 2 1 t e r m s o f t h e s e q u e n c e : i s , t h e n t h e v a l u e o f i s 6 . T h e E n g l i s h a n d M a t h e x a m s w e r e c o n d u c t e d s e p a r a t e l y f o r a c l a s s o f 1 2 0 s t u d e n t s . T h e n u m b e r o f s t u d e n t s w h o d i d n o t a p p e a r f o r t h e E n g l i s h e x a m i s t w i c e t h a t o f t h o s e w h o d i d n o t a p p e a r f o r t h e M a t h e x a m . T h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e M a t h e x a m i s t w i c e t h a t o f t h o s e w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m . I f t h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e E n g l i s h e x a m i s t w i c e t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e M a t h e x a m , t h e n t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m i s _ _ _ _ _ _ _ _ 7 . I f s u c h t h a t , t h e n e q u a l s _ _ _ _ _ _ _ _ _ _ _ 8 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . | a - b | + | b - c |=| c - a |
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- T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f t e a m s h a v i n g m o r e g e o l o g i s t s t h a n b i o l o g i s t s i s _ _ _ _ _ _ 9 . I f a n d a r e t h e r s t t h r e e t e r m s o f a n a r i t h m e t i c p r o g r e s s i o n , t h e n t h e s u m o f t h e n e x t t h r e e t e r m s o f t h e p r o g r e s s i o n i s 1 0 . A c i r c l e o f r a d i u s 1 3 c m t o u c h e s t h e a d j a c e n t s i d e s A B a n d B C o f a s q u a r e A B C D a t M a n d N , r e s p e c t i v e l y . I f A B = 1 8 c m a n d t h e c i r c l e i n t e r s e c t s t h e o t h e r t w o s i d e s C D a n d D A a t P a n d Q , r e s p e c t i v e l y , t h e n t h e a r e a , i n s q . c m , o f t r i a n g l e P M D i s 1 1 . M o n i c a , w h o i s 1 8 y e a r s o l d , i s o n e - t h i r d t h e a g e o f h e r f a t h e r . T h e a g e a t w h i c h s h e w i l l b e h a l f t h e a g e o f h e r f a t h e r i s _ _ _ _ 1 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e m e d i a n n u m b e r o f b i o l o g i s t s a c r o s s 5 t e a m s i s _ _ _ _ _ _ 1 3 . I f m a n d n a r e t w o p o s i t i v e i n t e g e r s s u c h t h a t , t h e n t h e m i n i m u m p o s s i b l e v a l u e o f m + n i s 1 4 . T h e n u m b e r o f f a c t o r s o f t h a t a r e p e r f e c t s q u a r e s i s 1 5 . I f t h e p o l y n o m i a l l e a v e s a r e m a i n d e r 3 w h e n d i v i d e d b y , a n d a r e m a i n d e r 2 w h e n d i v i d e d b y , t h e n e q u a l s Q u a n t i t a t i v e A b i l i t y M C Q 1 6 . G i v e n t h a t , t h e v a l u e o f i s A log ( x -
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Page 3
Q u a n t i t a t i v e A b i l i t y S A 1 . A r p i t a a n d N i k i t a , w o r k i n g t o g e t h e r , c a n c o m p l e t e a n a s s i g n e d j o b i n 1 2 d a y s . I f A r p i t a w o r k s i n i t i a l l y t o c o m p l e t e 4 0 % o f t h e j o b , a n d t h e r e m a i n i n g j o b i s c o m p l e t e d b y N i k i t a a l o n e , i t t a k e s 2 4 d a y s t o c o m p l e t e t h e j o b . T h e p o s s i b l e n u m b e r o f d a y s t h a t N i k i t a r e q u i r e s t o c o m p l e t e t h e e n t i r e j o b , w o r k i n g a l o n e , i s 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f b i o l o g i s t s i n t e a m E i s _ _ _ _ _ 3 . I f a , b , c a r e t h r e e d i s t i n c t n a t u r a l n u m b e r s , a l l l e s s t h a n 1 0 0 , s u c h t h a t , t h e n t h e m a x i m u m p o s s i b l e v a l u e o f b i s _ _ _ _ _ _ 4 . E i g h t t e a m s p a r t i c i p a t e i n a t o u r n a m e n t w h e r e e a c h t e a m p l a y s a g a i n s t e v e r y o t h e r t e a m e x a c t l y o n c e . I n a p a r t i c u l a r y e a r , o n e t e a m g o t s u s p e n d e d a f t e r p l a y i n g 3 m a t c h e s , d u e t o a d i s c i p l i n a r y i s s u e . T h e o r g a n i s e r s d e c i d e d t o p r o c e e d , n o n e t h e l e s s , w i t h t h e r e m a i n i n g m a t c h e s . T h e t o t a l n u m b e r o f m a t c h e s t h a t w e r e p l a y e d i n t h e t o u r n a m e n t t h a t y e a r i s 5 . I f t h e s u m o f t h e r s t 2 1 t e r m s o f t h e s e q u e n c e : i s , t h e n t h e v a l u e o f i s 6 . T h e E n g l i s h a n d M a t h e x a m s w e r e c o n d u c t e d s e p a r a t e l y f o r a c l a s s o f 1 2 0 s t u d e n t s . T h e n u m b e r o f s t u d e n t s w h o d i d n o t a p p e a r f o r t h e E n g l i s h e x a m i s t w i c e t h a t o f t h o s e w h o d i d n o t a p p e a r f o r t h e M a t h e x a m . T h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e M a t h e x a m i s t w i c e t h a t o f t h o s e w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m . I f t h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e E n g l i s h e x a m i s t w i c e t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e M a t h e x a m , t h e n t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m i s _ _ _ _ _ _ _ _ 7 . I f s u c h t h a t , t h e n e q u a l s _ _ _ _ _ _ _ _ _ _ _ 8 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . | a - b | + | b - c |=| c - a |
ln , ln , ln , ln , ........
b
a
b b
a
b
2
a
b
2
b
a
ln
b
n
a
m
m + n
A = [
2
4
n
1
] A =
3
27 [
4
p
q
r
] p + q + r
- T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f t e a m s h a v i n g m o r e g e o l o g i s t s t h a n b i o l o g i s t s i s _ _ _ _ _ _ 9 . I f a n d a r e t h e r s t t h r e e t e r m s o f a n a r i t h m e t i c p r o g r e s s i o n , t h e n t h e s u m o f t h e n e x t t h r e e t e r m s o f t h e p r o g r e s s i o n i s 1 0 . A c i r c l e o f r a d i u s 1 3 c m t o u c h e s t h e a d j a c e n t s i d e s A B a n d B C o f a s q u a r e A B C D a t M a n d N , r e s p e c t i v e l y . I f A B = 1 8 c m a n d t h e c i r c l e i n t e r s e c t s t h e o t h e r t w o s i d e s C D a n d D A a t P a n d Q , r e s p e c t i v e l y , t h e n t h e a r e a , i n s q . c m , o f t r i a n g l e P M D i s 1 1 . M o n i c a , w h o i s 1 8 y e a r s o l d , i s o n e - t h i r d t h e a g e o f h e r f a t h e r . T h e a g e a t w h i c h s h e w i l l b e h a l f t h e a g e o f h e r f a t h e r i s _ _ _ _ 1 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e m e d i a n n u m b e r o f b i o l o g i s t s a c r o s s 5 t e a m s i s _ _ _ _ _ _ 1 3 . I f m a n d n a r e t w o p o s i t i v e i n t e g e r s s u c h t h a t , t h e n t h e m i n i m u m p o s s i b l e v a l u e o f m + n i s 1 4 . T h e n u m b e r o f f a c t o r s o f t h a t a r e p e r f e c t s q u a r e s i s 1 5 . I f t h e p o l y n o m i a l l e a v e s a r e m a i n d e r 3 w h e n d i v i d e d b y , a n d a r e m a i n d e r 2 w h e n d i v i d e d b y , t h e n e q u a l s Q u a n t i t a t i v e A b i l i t y M C Q 1 6 . G i v e n t h a t , t h e v a l u e o f i s A log ( x -
3
2
1), log (2 x +
3
2
1) log (6 x +
3
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3)
7 m + 11 n = 200
3 ×
5
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8
7
2
a x +
2
b x + 5 x - 1
x + 1 2 b - 4 a
1 + +
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6
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B C D 1 7 . I f , w h i c h o f t h e f o l l o w i n g i s t r u e ? A i s p r o p o r t i o n a l t o B i s p r o p o r t i o n a l t o C i s p r o p o r t i o n a l t o D i s p r o p o r t i o n a l t o 1 8 . I f a r e t h e r o o t s o f t h e e q u a t i o n , t h e m t h e v a l u e o f i s A 0 B 2 C 8 D 4 1 9 . S u p p o s e a , b a n d c a r e t h r e e r e a l n u m b e r s s u c h t h a t M a x ( a , b , c ) + M i n ( a , b , c ) = 1 5 , a n d M e d i a n ( a , b , c ) - M e a n ( a , b , c ) = 2 . T h e n t h e m e d i a n o f a , b a n d c i s A 1 1 B 1 0 . 5 C 1 0 D 9 . 5 2 0 . I f t h e n x i s : A 4 B 1 6 C 2 D 8 2 1 . A n a t u r a l n u m b e r n l i e s b e t w e e n 1 0 0 a n d 4 0 0 , a n d t h e s u m o f i t s d i g i t s i s 1 0 . T h e p r o b a b i l i t y t h a t n i s d i v i s i b l e b y 4 , i s A
6
p
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y = a + b log x
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1 2 8
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8
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7
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1
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a +
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log 5 log (1 + log (1 + 2 log x)) =
25
[
3 3 2
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Page 4
Q u a n t i t a t i v e A b i l i t y S A 1 . A r p i t a a n d N i k i t a , w o r k i n g t o g e t h e r , c a n c o m p l e t e a n a s s i g n e d j o b i n 1 2 d a y s . I f A r p i t a w o r k s i n i t i a l l y t o c o m p l e t e 4 0 % o f t h e j o b , a n d t h e r e m a i n i n g j o b i s c o m p l e t e d b y N i k i t a a l o n e , i t t a k e s 2 4 d a y s t o c o m p l e t e t h e j o b . T h e p o s s i b l e n u m b e r o f d a y s t h a t N i k i t a r e q u i r e s t o c o m p l e t e t h e e n t i r e j o b , w o r k i n g a l o n e , i s 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f b i o l o g i s t s i n t e a m E i s _ _ _ _ _ 3 . I f a , b , c a r e t h r e e d i s t i n c t n a t u r a l n u m b e r s , a l l l e s s t h a n 1 0 0 , s u c h t h a t , t h e n t h e m a x i m u m p o s s i b l e v a l u e o f b i s _ _ _ _ _ _ 4 . E i g h t t e a m s p a r t i c i p a t e i n a t o u r n a m e n t w h e r e e a c h t e a m p l a y s a g a i n s t e v e r y o t h e r t e a m e x a c t l y o n c e . I n a p a r t i c u l a r y e a r , o n e t e a m g o t s u s p e n d e d a f t e r p l a y i n g 3 m a t c h e s , d u e t o a d i s c i p l i n a r y i s s u e . T h e o r g a n i s e r s d e c i d e d t o p r o c e e d , n o n e t h e l e s s , w i t h t h e r e m a i n i n g m a t c h e s . T h e t o t a l n u m b e r o f m a t c h e s t h a t w e r e p l a y e d i n t h e t o u r n a m e n t t h a t y e a r i s 5 . I f t h e s u m o f t h e r s t 2 1 t e r m s o f t h e s e q u e n c e : i s , t h e n t h e v a l u e o f i s 6 . T h e E n g l i s h a n d M a t h e x a m s w e r e c o n d u c t e d s e p a r a t e l y f o r a c l a s s o f 1 2 0 s t u d e n t s . T h e n u m b e r o f s t u d e n t s w h o d i d n o t a p p e a r f o r t h e E n g l i s h e x a m i s t w i c e t h a t o f t h o s e w h o d i d n o t a p p e a r f o r t h e M a t h e x a m . T h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e M a t h e x a m i s t w i c e t h a t o f t h o s e w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m . I f t h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e E n g l i s h e x a m i s t w i c e t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e M a t h e x a m , t h e n t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m i s _ _ _ _ _ _ _ _ 7 . I f s u c h t h a t , t h e n e q u a l s _ _ _ _ _ _ _ _ _ _ _ 8 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . | a - b | + | b - c |=| c - a |
ln , ln , ln , ln , ........
b
a
b b
a
b
2
a
b
2
b
a
ln
b
n
a
m
m + n
A = [
2
4
n
1
] A =
3
27 [
4
p
q
r
] p + q + r
- T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f t e a m s h a v i n g m o r e g e o l o g i s t s t h a n b i o l o g i s t s i s _ _ _ _ _ _ 9 . I f a n d a r e t h e r s t t h r e e t e r m s o f a n a r i t h m e t i c p r o g r e s s i o n , t h e n t h e s u m o f t h e n e x t t h r e e t e r m s o f t h e p r o g r e s s i o n i s 1 0 . A c i r c l e o f r a d i u s 1 3 c m t o u c h e s t h e a d j a c e n t s i d e s A B a n d B C o f a s q u a r e A B C D a t M a n d N , r e s p e c t i v e l y . I f A B = 1 8 c m a n d t h e c i r c l e i n t e r s e c t s t h e o t h e r t w o s i d e s C D a n d D A a t P a n d Q , r e s p e c t i v e l y , t h e n t h e a r e a , i n s q . c m , o f t r i a n g l e P M D i s 1 1 . M o n i c a , w h o i s 1 8 y e a r s o l d , i s o n e - t h i r d t h e a g e o f h e r f a t h e r . T h e a g e a t w h i c h s h e w i l l b e h a l f t h e a g e o f h e r f a t h e r i s _ _ _ _ 1 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e m e d i a n n u m b e r o f b i o l o g i s t s a c r o s s 5 t e a m s i s _ _ _ _ _ _ 1 3 . I f m a n d n a r e t w o p o s i t i v e i n t e g e r s s u c h t h a t , t h e n t h e m i n i m u m p o s s i b l e v a l u e o f m + n i s 1 4 . T h e n u m b e r o f f a c t o r s o f t h a t a r e p e r f e c t s q u a r e s i s 1 5 . I f t h e p o l y n o m i a l l e a v e s a r e m a i n d e r 3 w h e n d i v i d e d b y , a n d a r e m a i n d e r 2 w h e n d i v i d e d b y , t h e n e q u a l s Q u a n t i t a t i v e A b i l i t y M C Q 1 6 . G i v e n t h a t , t h e v a l u e o f i s A log ( x -
3
2
1), log (2 x +
3
2
1) log (6 x +
3
2
3)
7 m + 11 n = 200
3 ×
5
5 ×
8
7
2
a x +
2
b x + 5 x - 1
x + 1 2 b - 4 a
1 + +
2
2
1
+
3
2
1
+
4
2
1
...... =
6
p
2
1 + +
3
2
1
+
5
2
1
+
7
2
1
......
-
6
p
2
1
B C D 1 7 . I f , w h i c h o f t h e f o l l o w i n g i s t r u e ? A i s p r o p o r t i o n a l t o B i s p r o p o r t i o n a l t o C i s p r o p o r t i o n a l t o D i s p r o p o r t i o n a l t o 1 8 . I f a r e t h e r o o t s o f t h e e q u a t i o n , t h e m t h e v a l u e o f i s A 0 B 2 C 8 D 4 1 9 . S u p p o s e a , b a n d c a r e t h r e e r e a l n u m b e r s s u c h t h a t M a x ( a , b , c ) + M i n ( a , b , c ) = 1 5 , a n d M e d i a n ( a , b , c ) - M e a n ( a , b , c ) = 2 . T h e n t h e m e d i a n o f a , b a n d c i s A 1 1 B 1 0 . 5 C 1 0 D 9 . 5 2 0 . I f t h e n x i s : A 4 B 1 6 C 2 D 8 2 1 . A n a t u r a l n u m b e r n l i e s b e t w e e n 1 0 0 a n d 4 0 0 , a n d t h e s u m o f i t s d i g i t s i s 1 0 . T h e p r o b a b i l i t y t h a t n i s d i v i s i b l e b y 4 , i s A
6
p
12
p
2
8
p
2
y = a + b log x
e
y- a
1
x
b
y - a x
b
e
y
x
b
log y
e
x
a , a , ......., a
1 2 8
x +
8
x +
7
..... + x + 1 = 0 a +
1
2025
a +
2
2025
.... + a
8
2025
log 5 log (1 + log (1 + 2 log x)) =
25
[
3 3 2
]
2
1
4
1
B C D 2 2 . I n t r i a n g l e A B C , A B = A C = x , a n d t h e c i r c u m r a d i u s i s e q u a l t o y . T h e n e q u a l s A B C D 2 3 . I f i s a q u a d r a t i c e q u a t i o n i n x , s u c h t h a t o n e o f i t s r o o t s i s p t i m e s t h e o t h e r , a n d p , k a r e p o s i t i v e r e a l n u m b e r s , t h e n k e q u a l s A B C D 2 4 . L e t A ( 1 , 3 ) a n d B ( 5 , 1 ) b e t w o p o i n t s . I f a l i n e w i t h s l o p e m i n t e r s e c t s A B a t a n a n g l e o f , t h e n t h e p o s s i b l e v a l u e s o f m a r e A B C D 2 5 . L e t P ( x ) b e a q u a d r a t i c p o l y n o m i a l s u c h t h a t L e t P ( 0 ) = 2 a n d P ( 1 ) + P ( 2 ) + P ( 3 ) = 1 4 . T h e n P ( 4 ) e q u a l s A - 1 4 B 3 0
27
7
3
1
9
2
? A B C = ?
y
x
sin ?
cos ?
2 cos ?
2 sin ?
8 x -
2
2 k x + k = 0
( p + )
p
1
2( p + )
p
1
2 + ( p
p
1
)
2
+ ( p
p
1
)
2
45
°
7,
7
1
3,
3
1
-3,
3
1
5, -
5
1
=
|
|
| |
|
P (0)
P (0)
P (1)
P (2)
|
|
| |
|
0
Page 5
Q u a n t i t a t i v e A b i l i t y S A 1 . A r p i t a a n d N i k i t a , w o r k i n g t o g e t h e r , c a n c o m p l e t e a n a s s i g n e d j o b i n 1 2 d a y s . I f A r p i t a w o r k s i n i t i a l l y t o c o m p l e t e 4 0 % o f t h e j o b , a n d t h e r e m a i n i n g j o b i s c o m p l e t e d b y N i k i t a a l o n e , i t t a k e s 2 4 d a y s t o c o m p l e t e t h e j o b . T h e p o s s i b l e n u m b e r o f d a y s t h a t N i k i t a r e q u i r e s t o c o m p l e t e t h e e n t i r e j o b , w o r k i n g a l o n e , i s 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f b i o l o g i s t s i n t e a m E i s _ _ _ _ _ 3 . I f a , b , c a r e t h r e e d i s t i n c t n a t u r a l n u m b e r s , a l l l e s s t h a n 1 0 0 , s u c h t h a t , t h e n t h e m a x i m u m p o s s i b l e v a l u e o f b i s _ _ _ _ _ _ 4 . E i g h t t e a m s p a r t i c i p a t e i n a t o u r n a m e n t w h e r e e a c h t e a m p l a y s a g a i n s t e v e r y o t h e r t e a m e x a c t l y o n c e . I n a p a r t i c u l a r y e a r , o n e t e a m g o t s u s p e n d e d a f t e r p l a y i n g 3 m a t c h e s , d u e t o a d i s c i p l i n a r y i s s u e . T h e o r g a n i s e r s d e c i d e d t o p r o c e e d , n o n e t h e l e s s , w i t h t h e r e m a i n i n g m a t c h e s . T h e t o t a l n u m b e r o f m a t c h e s t h a t w e r e p l a y e d i n t h e t o u r n a m e n t t h a t y e a r i s 5 . I f t h e s u m o f t h e r s t 2 1 t e r m s o f t h e s e q u e n c e : i s , t h e n t h e v a l u e o f i s 6 . T h e E n g l i s h a n d M a t h e x a m s w e r e c o n d u c t e d s e p a r a t e l y f o r a c l a s s o f 1 2 0 s t u d e n t s . T h e n u m b e r o f s t u d e n t s w h o d i d n o t a p p e a r f o r t h e E n g l i s h e x a m i s t w i c e t h a t o f t h o s e w h o d i d n o t a p p e a r f o r t h e M a t h e x a m . T h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e M a t h e x a m i s t w i c e t h a t o f t h o s e w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m . I f t h e n u m b e r o f s t u d e n t s w h o p a s s e d t h e E n g l i s h e x a m i s t w i c e t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e M a t h e x a m , t h e n t h e n u m b e r o f s t u d e n t s w h o a p p e a r e d b u t f a i l e d t h e E n g l i s h e x a m i s _ _ _ _ _ _ _ _ 7 . I f s u c h t h a t , t h e n e q u a l s _ _ _ _ _ _ _ _ _ _ _ 8 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . | a - b | + | b - c |=| c - a |
ln , ln , ln , ln , ........
b
a
b b
a
b
2
a
b
2
b
a
ln
b
n
a
m
m + n
A = [
2
4
n
1
] A =
3
27 [
4
p
q
r
] p + q + r
- T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e n u m b e r o f t e a m s h a v i n g m o r e g e o l o g i s t s t h a n b i o l o g i s t s i s _ _ _ _ _ _ 9 . I f a n d a r e t h e r s t t h r e e t e r m s o f a n a r i t h m e t i c p r o g r e s s i o n , t h e n t h e s u m o f t h e n e x t t h r e e t e r m s o f t h e p r o g r e s s i o n i s 1 0 . A c i r c l e o f r a d i u s 1 3 c m t o u c h e s t h e a d j a c e n t s i d e s A B a n d B C o f a s q u a r e A B C D a t M a n d N , r e s p e c t i v e l y . I f A B = 1 8 c m a n d t h e c i r c l e i n t e r s e c t s t h e o t h e r t w o s i d e s C D a n d D A a t P a n d Q , r e s p e c t i v e l y , t h e n t h e a r e a , i n s q . c m , o f t r i a n g l e P M D i s 1 1 . M o n i c a , w h o i s 1 8 y e a r s o l d , i s o n e - t h i r d t h e a g e o f h e r f a t h e r . T h e a g e a t w h i c h s h e w i l l b e h a l f t h e a g e o f h e r f a t h e r i s _ _ _ _ 1 2 . F i v e t e a m s - A , B , C , D , a n d E - e a c h c o n s i s t i n g o f 1 5 m e m b e r s , a r e g o i n g o n e x p e d i t i o n s t o v e d i f f e r e n t l o c a t i o n s . E a c h t e a m i n c l u d e s m e m b e r s f r o m t h r e e d i f f e r e n t s k i l l s e t s : b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s . H o w e v e r , t h e n u m b e r o f m e m b e r s f r o m e a c h s k i l l s e t v a r i e s b y t e a m , a n d e a c h m e m b e r h a s o n l y o n e s p e c i a l i t y . T h e t o t a l n u m b e r o f b i o l o g i s t s , g e o l o g i s t s , a n d e x p l o r e r s i s e q u a l . T h e f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e - E v e r y t e a m h a s a t l e a s t 2 m e m b e r s f r o m e a c h s k i l l s e t . - T e a m s C a n d D h a v e 6 b i o l o g i s t s e a c h , a n d T e a m A h a s 6 g e o l o g i s t s . - E v e r y t e a m e x c e p t A h a s m o r e b i o l o g i s t s t h a n e x p l o r e r s . - T h e n u m b e r o f e x p l o r e r s i n e a c h t e a m i s d i s t i n c t a n d d e c r e a s e s i n t h e o r d e r A , B , C , D , a n d E . T h e m e d i a n n u m b e r o f b i o l o g i s t s a c r o s s 5 t e a m s i s _ _ _ _ _ _ 1 3 . I f m a n d n a r e t w o p o s i t i v e i n t e g e r s s u c h t h a t , t h e n t h e m i n i m u m p o s s i b l e v a l u e o f m + n i s 1 4 . T h e n u m b e r o f f a c t o r s o f t h a t a r e p e r f e c t s q u a r e s i s 1 5 . I f t h e p o l y n o m i a l l e a v e s a r e m a i n d e r 3 w h e n d i v i d e d b y , a n d a r e m a i n d e r 2 w h e n d i v i d e d b y , t h e n e q u a l s Q u a n t i t a t i v e A b i l i t y M C Q 1 6 . G i v e n t h a t , t h e v a l u e o f i s A log ( x -
3
2
1), log (2 x +
3
2
1) log (6 x +
3
2
3)
7 m + 11 n = 200
3 ×
5
5 ×
8
7
2
a x +
2
b x + 5 x - 1
x + 1 2 b - 4 a
1 + +
2
2
1
+
3
2
1
+
4
2
1
...... =
6
p
2
1 + +
3
2
1
+
5
2
1
+
7
2
1
......
-
6
p
2
1
B C D 1 7 . I f , w h i c h o f t h e f o l l o w i n g i s t r u e ? A i s p r o p o r t i o n a l t o B i s p r o p o r t i o n a l t o C i s p r o p o r t i o n a l t o D i s p r o p o r t i o n a l t o 1 8 . I f a r e t h e r o o t s o f t h e e q u a t i o n , t h e m t h e v a l u e o f i s A 0 B 2 C 8 D 4 1 9 . S u p p o s e a , b a n d c a r e t h r e e r e a l n u m b e r s s u c h t h a t M a x ( a , b , c ) + M i n ( a , b , c ) = 1 5 , a n d M e d i a n ( a , b , c ) - M e a n ( a , b , c ) = 2 . T h e n t h e m e d i a n o f a , b a n d c i s A 1 1 B 1 0 . 5 C 1 0 D 9 . 5 2 0 . I f t h e n x i s : A 4 B 1 6 C 2 D 8 2 1 . A n a t u r a l n u m b e r n l i e s b e t w e e n 1 0 0 a n d 4 0 0 , a n d t h e s u m o f i t s d i g i t s i s 1 0 . T h e p r o b a b i l i t y t h a t n i s d i v i s i b l e b y 4 , i s A
6
p
12
p
2
8
p
2
y = a + b log x
e
y- a
1
x
b
y - a x
b
e
y
x
b
log y
e
x
a , a , ......., a
1 2 8
x +
8
x +
7
..... + x + 1 = 0 a +
1
2025
a +
2
2025
.... + a
8
2025
log 5 log (1 + log (1 + 2 log x)) =
25
[
3 3 2
]
2
1
4
1
B C D 2 2 . I n t r i a n g l e A B C , A B = A C = x , a n d t h e c i r c u m r a d i u s i s e q u a l t o y . T h e n e q u a l s A B C D 2 3 . I f i s a q u a d r a t i c e q u a t i o n i n x , s u c h t h a t o n e o f i t s r o o t s i s p t i m e s t h e o t h e r , a n d p , k a r e p o s i t i v e r e a l n u m b e r s , t h e n k e q u a l s A B C D 2 4 . L e t A ( 1 , 3 ) a n d B ( 5 , 1 ) b e t w o p o i n t s . I f a l i n e w i t h s l o p e m i n t e r s e c t s A B a t a n a n g l e o f , t h e n t h e p o s s i b l e v a l u e s o f m a r e A B C D 2 5 . L e t P ( x ) b e a q u a d r a t i c p o l y n o m i a l s u c h t h a t L e t P ( 0 ) = 2 a n d P ( 1 ) + P ( 2 ) + P ( 3 ) = 1 4 . T h e n P ( 4 ) e q u a l s A - 1 4 B 3 0
27
7
3
1
9
2
? A B C = ?
y
x
sin ?
cos ?
2 cos ?
2 sin ?
8 x -
2
2 k x + k = 0
( p + )
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FAQs on IPMAT 2025: Past Year Question Paper and Solution - 100 DILR Questions for CAT Preparation
| 1. What is IPMAT and what is its significance in management education? |  |
Ans. The Integrated Program in Management Aptitude Test (IPMAT) is an entrance examination conducted for admission to integrated management programs offered by select institutions. It is significant because it provides a gateway for students to pursue a five-year integrated management course, combining undergraduate and postgraduate studies in management, thereby enabling early entry into the field of business education.
| 2. What subjects are typically covered in the IPMAT exam? | |
Ans. The IPMAT exam generally includes sections on quantitative ability, verbal ability, and logical reasoning. These subjects assess a candidate's mathematical skills, language proficiency, and reasoning capabilities, which are crucial for success in management studies.
| 3. How can candidates effectively prepare for the IPMAT exam? | |
Ans. Candidates can prepare for the IPMAT exam by following a structured study plan that includes understanding the exam pattern, practicing previous years' question papers, taking mock tests, and focusing on their weak areas. Regular reading and vocabulary enhancement can also significantly improve performance in the verbal ability section.
| 4. What is the selection process after the IPMAT exam? | |
Ans. After the IPMAT exam, the selection process typically includes a personal interview and a written ability test. Candidates who score above the cutoff marks in the exam are invited to participate in these additional assessment rounds, which help evaluate their communication skills and overall suitability for the management program.
| 5. What are the common career prospects for students who complete the integrated management program? | |
Ans. Students who complete the integrated management program often have diverse career opportunities available to them. They can pursue roles in various sectors such as finance, marketing, operations, consulting, and entrepreneurship. Many graduates also opt for higher studies or specialized certifications to enhance their career prospects further.
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Sep 30, 2025
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