Commerce Exam > Commerce Notes > Mathematics (Maths) Class 11 > Past Year Paper, Maths(Set - 1), 2018, Class 11
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Page 1 C B S E C l a s s – X I M A T H E M A T I C S T i m e a l l o w e d : 3 h o u r s , M a x i m u m M a r k s : 1 0 0 G e n e r a l I n s t r u c t i o n s : a ) A l l q u e s t i o n s a r e c o m p u l s o r y . b ) T h e q u e s t i o n p a p e r c o n s i s t s o f 2 6 q u e s t i o n s d i v i d e d i n t o t h r e e s e c t i o n s A , B a n d C . S e c t i o n A c o m p r i s e s o f 6 q u e s t i o n s o f o n e m a r k e a c h , S e c t i o n B c o m p r i s e s o f 1 3 q u e s t i o n s o f f o u r m a r k s e a c h a n d S e c t i o n C c o m p r i s e s o f 7 q u e s t i o n s o f s i x m a r k s e a c h . c ) A l l q u e s t i o n s i n S e c t i o n A a r e t o b e a n s w e r e d i n o n e w o r d , o n e s e n t e n c e o r a s p e r t h e e x a c t r e q u i r e m e n t o f t h e q u e s t i o n . d ) U s e o f c a l c u l a t o r s i s n o t p e r m i t t e d . S e c t i o n A 1 . F i n d t h e n u m b e r o f s u b s e t s o f a s e t A c o n t a i n i n g 1 0 e l e m e n t s . S o l : N u m b e r o f s u b s e t s 1 0 C 0 + 1 0 C 1 + 1 0 C 2 + 1 0 C 3 + 1 0 C 4 + 1 0 C 5 + 1 0 C 6 + 1 0 C 7 + 1 0 C 8 + 1 0 C 9 + 1 0 C 1 0 = 2 1 0 2 . H o w m a n y w a y s c a n y o u c h o o s e o n e o r m o r e s t u d e n t s f r o m 3 s t u d e n t s ? S o l : 3 C 1 + 3 C 2 + 3 C 3 + = 2 3 - 1 = 7 3 . I n H o w m a n y w a y s c a n o n e c h o o s e 3 c a r d s f r o m a p a c k o f 5 2 c a r d s i n s u c c e s s i o n ( 1 ) w i t h r e p l a c e m e n t ( 2 ) w i t h o u t r e p l a c e m e n t ? S o l : ( 1 ) E a c h c a r d c a n b e d r a w n i n 5 2 w a y s a n d s o t h e t o t a l n u m b e r o f w a y s 5 2 5 2 5 2 = 5 2 3 ( 2 ) I f t h e r e i s n o r e p l a c e m e n t t h e f i r s t c a r d c a n b e d r a w n i n 5 2 w a y s , t h e s e c o n d b y 5 1 w a y s a n d t h e t h i r d b y 5 0 w a y s . H e n c e t h e t o t a l n u m b e r o f w a y s i s 5 2 5 1 5 0 = 1 3 2 6 0 0 Page 2 C B S E C l a s s – X I M A T H E M A T I C S T i m e a l l o w e d : 3 h o u r s , M a x i m u m M a r k s : 1 0 0 G e n e r a l I n s t r u c t i o n s : a ) A l l q u e s t i o n s a r e c o m p u l s o r y . b ) T h e q u e s t i o n p a p e r c o n s i s t s o f 2 6 q u e s t i o n s d i v i d e d i n t o t h r e e s e c t i o n s A , B a n d C . S e c t i o n A c o m p r i s e s o f 6 q u e s t i o n s o f o n e m a r k e a c h , S e c t i o n B c o m p r i s e s o f 1 3 q u e s t i o n s o f f o u r m a r k s e a c h a n d S e c t i o n C c o m p r i s e s o f 7 q u e s t i o n s o f s i x m a r k s e a c h . c ) A l l q u e s t i o n s i n S e c t i o n A a r e t o b e a n s w e r e d i n o n e w o r d , o n e s e n t e n c e o r a s p e r t h e e x a c t r e q u i r e m e n t o f t h e q u e s t i o n . d ) U s e o f c a l c u l a t o r s i s n o t p e r m i t t e d . S e c t i o n A 1 . F i n d t h e n u m b e r o f s u b s e t s o f a s e t A c o n t a i n i n g 1 0 e l e m e n t s . S o l : N u m b e r o f s u b s e t s 1 0 C 0 + 1 0 C 1 + 1 0 C 2 + 1 0 C 3 + 1 0 C 4 + 1 0 C 5 + 1 0 C 6 + 1 0 C 7 + 1 0 C 8 + 1 0 C 9 + 1 0 C 1 0 = 2 1 0 2 . H o w m a n y w a y s c a n y o u c h o o s e o n e o r m o r e s t u d e n t s f r o m 3 s t u d e n t s ? S o l : 3 C 1 + 3 C 2 + 3 C 3 + = 2 3 - 1 = 7 3 . I n H o w m a n y w a y s c a n o n e c h o o s e 3 c a r d s f r o m a p a c k o f 5 2 c a r d s i n s u c c e s s i o n ( 1 ) w i t h r e p l a c e m e n t ( 2 ) w i t h o u t r e p l a c e m e n t ? S o l : ( 1 ) E a c h c a r d c a n b e d r a w n i n 5 2 w a y s a n d s o t h e t o t a l n u m b e r o f w a y s 5 2 5 2 5 2 = 5 2 3 ( 2 ) I f t h e r e i s n o r e p l a c e m e n t t h e f i r s t c a r d c a n b e d r a w n i n 5 2 w a y s , t h e s e c o n d b y 5 1 w a y s a n d t h e t h i r d b y 5 0 w a y s . H e n c e t h e t o t a l n u m b e r o f w a y s i s 5 2 5 1 5 0 = 1 3 2 6 0 0 4 . S t a t e t h e c o n d i t i o n u n d e r w h i c h t h e p r o d u c t o f t w o c o m p l e x n u m b e r s i s p u r e l y i m a g i n a r y . S o l : 1 . N o n e o f t h e f a c t o r s a r e z e r o 2 . F a c t o r s m u s t b e o f t h e f o r m w h e r e k i s a r e a l n u m b e r . 5 . I n a c i r c l e o f r a d i u s 1 u n i t w h a t i s t h e l e n g t h o f t h e a r c t h a t s u b m i t s a n a n g l e o f 2 r a d i a n s a t t h e c e n t r e . S o l : L e n g t h o f a r c = H e n c e l e n g t h o f a r c = = 2 u n i t s 6 . I s p o s i t i v e o r n e g a t i v e i f . S o l : 1 F u l l r o t a t i o n i s 5 0 0 r a d i a n s = r o t a t i o n s 7 9 f u l l r o t a t i o n s a n d 0 . 5 7 o f a r o t a t i o n T h e i n c o m p l e t e r o t a t i o n i s b e t w e e n o f a r o t a t i o n . H e n c e 5 0 0 r a d i a n s i s i n t h i r d q u a d r a n t . S o i s n e g a t i v e S e c t i o n B 7 . P r o v e b y m a t h e m a t i c a l i n d u c t i o n t h a t i s d i v i s i b l e b y 6 i f n i s a n a t u r a l n u m b e r . S o l : L e t T h e n = 6 a n d d i v i s i b l e b y 6 L e t i t b e d i v i s i b l e b y 6 f o r T h e n W h e r e i s a n i n t e g e r F o r t h e e x p r e s s i o n i s ( m + 1 ) ( m + 2 ) ( 2 m + 2 + 1 ) = ( m + 2 ) ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 3 m + 3 ) = m ( m + 1 ) ( 2 m + 1 ) + 6 ( m + 1 ) 2 = 6 k + 6 ( m + 1 ) 2 , T h i s i s d i v i s i b l e b y 6 . Page 3 C B S E C l a s s – X I M A T H E M A T I C S T i m e a l l o w e d : 3 h o u r s , M a x i m u m M a r k s : 1 0 0 G e n e r a l I n s t r u c t i o n s : a ) A l l q u e s t i o n s a r e c o m p u l s o r y . b ) T h e q u e s t i o n p a p e r c o n s i s t s o f 2 6 q u e s t i o n s d i v i d e d i n t o t h r e e s e c t i o n s A , B a n d C . S e c t i o n A c o m p r i s e s o f 6 q u e s t i o n s o f o n e m a r k e a c h , S e c t i o n B c o m p r i s e s o f 1 3 q u e s t i o n s o f f o u r m a r k s e a c h a n d S e c t i o n C c o m p r i s e s o f 7 q u e s t i o n s o f s i x m a r k s e a c h . c ) A l l q u e s t i o n s i n S e c t i o n A a r e t o b e a n s w e r e d i n o n e w o r d , o n e s e n t e n c e o r a s p e r t h e e x a c t r e q u i r e m e n t o f t h e q u e s t i o n . d ) U s e o f c a l c u l a t o r s i s n o t p e r m i t t e d . S e c t i o n A 1 . F i n d t h e n u m b e r o f s u b s e t s o f a s e t A c o n t a i n i n g 1 0 e l e m e n t s . S o l : N u m b e r o f s u b s e t s 1 0 C 0 + 1 0 C 1 + 1 0 C 2 + 1 0 C 3 + 1 0 C 4 + 1 0 C 5 + 1 0 C 6 + 1 0 C 7 + 1 0 C 8 + 1 0 C 9 + 1 0 C 1 0 = 2 1 0 2 . H o w m a n y w a y s c a n y o u c h o o s e o n e o r m o r e s t u d e n t s f r o m 3 s t u d e n t s ? S o l : 3 C 1 + 3 C 2 + 3 C 3 + = 2 3 - 1 = 7 3 . I n H o w m a n y w a y s c a n o n e c h o o s e 3 c a r d s f r o m a p a c k o f 5 2 c a r d s i n s u c c e s s i o n ( 1 ) w i t h r e p l a c e m e n t ( 2 ) w i t h o u t r e p l a c e m e n t ? S o l : ( 1 ) E a c h c a r d c a n b e d r a w n i n 5 2 w a y s a n d s o t h e t o t a l n u m b e r o f w a y s 5 2 5 2 5 2 = 5 2 3 ( 2 ) I f t h e r e i s n o r e p l a c e m e n t t h e f i r s t c a r d c a n b e d r a w n i n 5 2 w a y s , t h e s e c o n d b y 5 1 w a y s a n d t h e t h i r d b y 5 0 w a y s . H e n c e t h e t o t a l n u m b e r o f w a y s i s 5 2 5 1 5 0 = 1 3 2 6 0 0 4 . S t a t e t h e c o n d i t i o n u n d e r w h i c h t h e p r o d u c t o f t w o c o m p l e x n u m b e r s i s p u r e l y i m a g i n a r y . S o l : 1 . N o n e o f t h e f a c t o r s a r e z e r o 2 . F a c t o r s m u s t b e o f t h e f o r m w h e r e k i s a r e a l n u m b e r . 5 . I n a c i r c l e o f r a d i u s 1 u n i t w h a t i s t h e l e n g t h o f t h e a r c t h a t s u b m i t s a n a n g l e o f 2 r a d i a n s a t t h e c e n t r e . S o l : L e n g t h o f a r c = H e n c e l e n g t h o f a r c = = 2 u n i t s 6 . I s p o s i t i v e o r n e g a t i v e i f . S o l : 1 F u l l r o t a t i o n i s 5 0 0 r a d i a n s = r o t a t i o n s 7 9 f u l l r o t a t i o n s a n d 0 . 5 7 o f a r o t a t i o n T h e i n c o m p l e t e r o t a t i o n i s b e t w e e n o f a r o t a t i o n . H e n c e 5 0 0 r a d i a n s i s i n t h i r d q u a d r a n t . S o i s n e g a t i v e S e c t i o n B 7 . P r o v e b y m a t h e m a t i c a l i n d u c t i o n t h a t i s d i v i s i b l e b y 6 i f n i s a n a t u r a l n u m b e r . S o l : L e t T h e n = 6 a n d d i v i s i b l e b y 6 L e t i t b e d i v i s i b l e b y 6 f o r T h e n W h e r e i s a n i n t e g e r F o r t h e e x p r e s s i o n i s ( m + 1 ) ( m + 2 ) ( 2 m + 2 + 1 ) = ( m + 2 ) ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 3 m + 3 ) = m ( m + 1 ) ( 2 m + 1 ) + 6 ( m + 1 ) 2 = 6 k + 6 ( m + 1 ) 2 , T h i s i s d i v i s i b l e b y 6 . 8 . S o l v e . S o l : L e t T h e n , S o l v i n g t h i s q u a d r a t i c F i r s t v a l u e o f t i s r e j e c t e d a s s h o u l d l i e b e t w e e n G e n e r a l s o l u t i o n i s 9 . F o r w h a t v a l u e s o f m 2 x 2 + 2 ( m + 1 ) x + 4 = 0 w i l l h a v e e x a c t l y o n e z e r o . S o l : W h e n T h e g i v e n e q u a t i o n r e d u c e s t o a f i r s t d e g r e e a n d i t w i l l h a v e o n l y o n e s o l u t i o n A l s o w h e n t h e d i s c r i m i n a n t i s z e r o i t w i l l h a v e o n l y o n e s o l u t i o n D i s c r i m i n a n t i s O n s i m p l i f y i n g a n d s o l v i n g , H e n c e t h e t h r e e v a l u e s o f f o r w h i c h t h e e q u a t i o n w i l l h a v e o n l y o n e s o l u t i o n i s 1 0 . T h r e e n u m b e r s a r e i n A P . A n o t h e r 3 n u m b e r s a r e i n G P . T h e s u m o f f i r s t t e r m o f t h e A P a n d t h e f i r s t t e r m o f t h e G P i s 8 5 , t h e s u m o f s e c o n d t e r m o f A P a n d t h e s e c o n d t e r m o f t h e G P i s 7 6 a n d t h a t o f t h e 3 r d t e r m o f A P a n d 3 r d t e r m o f G P i s 8 4 . T h e s u m o f t h e A P i s 1 2 6 . F i n d e a c h t e r m o f A P a n d G P . S o l : Page 4 C B S E C l a s s – X I M A T H E M A T I C S T i m e a l l o w e d : 3 h o u r s , M a x i m u m M a r k s : 1 0 0 G e n e r a l I n s t r u c t i o n s : a ) A l l q u e s t i o n s a r e c o m p u l s o r y . b ) T h e q u e s t i o n p a p e r c o n s i s t s o f 2 6 q u e s t i o n s d i v i d e d i n t o t h r e e s e c t i o n s A , B a n d C . S e c t i o n A c o m p r i s e s o f 6 q u e s t i o n s o f o n e m a r k e a c h , S e c t i o n B c o m p r i s e s o f 1 3 q u e s t i o n s o f f o u r m a r k s e a c h a n d S e c t i o n C c o m p r i s e s o f 7 q u e s t i o n s o f s i x m a r k s e a c h . c ) A l l q u e s t i o n s i n S e c t i o n A a r e t o b e a n s w e r e d i n o n e w o r d , o n e s e n t e n c e o r a s p e r t h e e x a c t r e q u i r e m e n t o f t h e q u e s t i o n . d ) U s e o f c a l c u l a t o r s i s n o t p e r m i t t e d . S e c t i o n A 1 . F i n d t h e n u m b e r o f s u b s e t s o f a s e t A c o n t a i n i n g 1 0 e l e m e n t s . S o l : N u m b e r o f s u b s e t s 1 0 C 0 + 1 0 C 1 + 1 0 C 2 + 1 0 C 3 + 1 0 C 4 + 1 0 C 5 + 1 0 C 6 + 1 0 C 7 + 1 0 C 8 + 1 0 C 9 + 1 0 C 1 0 = 2 1 0 2 . H o w m a n y w a y s c a n y o u c h o o s e o n e o r m o r e s t u d e n t s f r o m 3 s t u d e n t s ? S o l : 3 C 1 + 3 C 2 + 3 C 3 + = 2 3 - 1 = 7 3 . I n H o w m a n y w a y s c a n o n e c h o o s e 3 c a r d s f r o m a p a c k o f 5 2 c a r d s i n s u c c e s s i o n ( 1 ) w i t h r e p l a c e m e n t ( 2 ) w i t h o u t r e p l a c e m e n t ? S o l : ( 1 ) E a c h c a r d c a n b e d r a w n i n 5 2 w a y s a n d s o t h e t o t a l n u m b e r o f w a y s 5 2 5 2 5 2 = 5 2 3 ( 2 ) I f t h e r e i s n o r e p l a c e m e n t t h e f i r s t c a r d c a n b e d r a w n i n 5 2 w a y s , t h e s e c o n d b y 5 1 w a y s a n d t h e t h i r d b y 5 0 w a y s . H e n c e t h e t o t a l n u m b e r o f w a y s i s 5 2 5 1 5 0 = 1 3 2 6 0 0 4 . S t a t e t h e c o n d i t i o n u n d e r w h i c h t h e p r o d u c t o f t w o c o m p l e x n u m b e r s i s p u r e l y i m a g i n a r y . S o l : 1 . N o n e o f t h e f a c t o r s a r e z e r o 2 . F a c t o r s m u s t b e o f t h e f o r m w h e r e k i s a r e a l n u m b e r . 5 . I n a c i r c l e o f r a d i u s 1 u n i t w h a t i s t h e l e n g t h o f t h e a r c t h a t s u b m i t s a n a n g l e o f 2 r a d i a n s a t t h e c e n t r e . S o l : L e n g t h o f a r c = H e n c e l e n g t h o f a r c = = 2 u n i t s 6 . I s p o s i t i v e o r n e g a t i v e i f . S o l : 1 F u l l r o t a t i o n i s 5 0 0 r a d i a n s = r o t a t i o n s 7 9 f u l l r o t a t i o n s a n d 0 . 5 7 o f a r o t a t i o n T h e i n c o m p l e t e r o t a t i o n i s b e t w e e n o f a r o t a t i o n . H e n c e 5 0 0 r a d i a n s i s i n t h i r d q u a d r a n t . S o i s n e g a t i v e S e c t i o n B 7 . P r o v e b y m a t h e m a t i c a l i n d u c t i o n t h a t i s d i v i s i b l e b y 6 i f n i s a n a t u r a l n u m b e r . S o l : L e t T h e n = 6 a n d d i v i s i b l e b y 6 L e t i t b e d i v i s i b l e b y 6 f o r T h e n W h e r e i s a n i n t e g e r F o r t h e e x p r e s s i o n i s ( m + 1 ) ( m + 2 ) ( 2 m + 2 + 1 ) = ( m + 2 ) ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 3 m + 3 ) = m ( m + 1 ) ( 2 m + 1 ) + 6 ( m + 1 ) 2 = 6 k + 6 ( m + 1 ) 2 , T h i s i s d i v i s i b l e b y 6 . 8 . S o l v e . S o l : L e t T h e n , S o l v i n g t h i s q u a d r a t i c F i r s t v a l u e o f t i s r e j e c t e d a s s h o u l d l i e b e t w e e n G e n e r a l s o l u t i o n i s 9 . F o r w h a t v a l u e s o f m 2 x 2 + 2 ( m + 1 ) x + 4 = 0 w i l l h a v e e x a c t l y o n e z e r o . S o l : W h e n T h e g i v e n e q u a t i o n r e d u c e s t o a f i r s t d e g r e e a n d i t w i l l h a v e o n l y o n e s o l u t i o n A l s o w h e n t h e d i s c r i m i n a n t i s z e r o i t w i l l h a v e o n l y o n e s o l u t i o n D i s c r i m i n a n t i s O n s i m p l i f y i n g a n d s o l v i n g , H e n c e t h e t h r e e v a l u e s o f f o r w h i c h t h e e q u a t i o n w i l l h a v e o n l y o n e s o l u t i o n i s 1 0 . T h r e e n u m b e r s a r e i n A P . A n o t h e r 3 n u m b e r s a r e i n G P . T h e s u m o f f i r s t t e r m o f t h e A P a n d t h e f i r s t t e r m o f t h e G P i s 8 5 , t h e s u m o f s e c o n d t e r m o f A P a n d t h e s e c o n d t e r m o f t h e G P i s 7 6 a n d t h a t o f t h e 3 r d t e r m o f A P a n d 3 r d t e r m o f G P i s 8 4 . T h e s u m o f t h e A P i s 1 2 6 . F i n d e a c h t e r m o f A P a n d G P . S o l : A . P a - d , a , a + d G . P b / g , b , b g W h e n g = 2 4 2 - d + = 8 5 d = - 2 6 a = 4 2 , d = - 2 6 , g = 2 , b = 3 4 1 1 . I f f i n d i n t e r m s o f . S o l : = 1 2 . I f P r o v e t h a t Page 5 C B S E C l a s s – X I M A T H E M A T I C S T i m e a l l o w e d : 3 h o u r s , M a x i m u m M a r k s : 1 0 0 G e n e r a l I n s t r u c t i o n s : a ) A l l q u e s t i o n s a r e c o m p u l s o r y . b ) T h e q u e s t i o n p a p e r c o n s i s t s o f 2 6 q u e s t i o n s d i v i d e d i n t o t h r e e s e c t i o n s A , B a n d C . S e c t i o n A c o m p r i s e s o f 6 q u e s t i o n s o f o n e m a r k e a c h , S e c t i o n B c o m p r i s e s o f 1 3 q u e s t i o n s o f f o u r m a r k s e a c h a n d S e c t i o n C c o m p r i s e s o f 7 q u e s t i o n s o f s i x m a r k s e a c h . c ) A l l q u e s t i o n s i n S e c t i o n A a r e t o b e a n s w e r e d i n o n e w o r d , o n e s e n t e n c e o r a s p e r t h e e x a c t r e q u i r e m e n t o f t h e q u e s t i o n . d ) U s e o f c a l c u l a t o r s i s n o t p e r m i t t e d . S e c t i o n A 1 . F i n d t h e n u m b e r o f s u b s e t s o f a s e t A c o n t a i n i n g 1 0 e l e m e n t s . S o l : N u m b e r o f s u b s e t s 1 0 C 0 + 1 0 C 1 + 1 0 C 2 + 1 0 C 3 + 1 0 C 4 + 1 0 C 5 + 1 0 C 6 + 1 0 C 7 + 1 0 C 8 + 1 0 C 9 + 1 0 C 1 0 = 2 1 0 2 . H o w m a n y w a y s c a n y o u c h o o s e o n e o r m o r e s t u d e n t s f r o m 3 s t u d e n t s ? S o l : 3 C 1 + 3 C 2 + 3 C 3 + = 2 3 - 1 = 7 3 . I n H o w m a n y w a y s c a n o n e c h o o s e 3 c a r d s f r o m a p a c k o f 5 2 c a r d s i n s u c c e s s i o n ( 1 ) w i t h r e p l a c e m e n t ( 2 ) w i t h o u t r e p l a c e m e n t ? S o l : ( 1 ) E a c h c a r d c a n b e d r a w n i n 5 2 w a y s a n d s o t h e t o t a l n u m b e r o f w a y s 5 2 5 2 5 2 = 5 2 3 ( 2 ) I f t h e r e i s n o r e p l a c e m e n t t h e f i r s t c a r d c a n b e d r a w n i n 5 2 w a y s , t h e s e c o n d b y 5 1 w a y s a n d t h e t h i r d b y 5 0 w a y s . H e n c e t h e t o t a l n u m b e r o f w a y s i s 5 2 5 1 5 0 = 1 3 2 6 0 0 4 . S t a t e t h e c o n d i t i o n u n d e r w h i c h t h e p r o d u c t o f t w o c o m p l e x n u m b e r s i s p u r e l y i m a g i n a r y . S o l : 1 . N o n e o f t h e f a c t o r s a r e z e r o 2 . F a c t o r s m u s t b e o f t h e f o r m w h e r e k i s a r e a l n u m b e r . 5 . I n a c i r c l e o f r a d i u s 1 u n i t w h a t i s t h e l e n g t h o f t h e a r c t h a t s u b m i t s a n a n g l e o f 2 r a d i a n s a t t h e c e n t r e . S o l : L e n g t h o f a r c = H e n c e l e n g t h o f a r c = = 2 u n i t s 6 . I s p o s i t i v e o r n e g a t i v e i f . S o l : 1 F u l l r o t a t i o n i s 5 0 0 r a d i a n s = r o t a t i o n s 7 9 f u l l r o t a t i o n s a n d 0 . 5 7 o f a r o t a t i o n T h e i n c o m p l e t e r o t a t i o n i s b e t w e e n o f a r o t a t i o n . H e n c e 5 0 0 r a d i a n s i s i n t h i r d q u a d r a n t . S o i s n e g a t i v e S e c t i o n B 7 . P r o v e b y m a t h e m a t i c a l i n d u c t i o n t h a t i s d i v i s i b l e b y 6 i f n i s a n a t u r a l n u m b e r . S o l : L e t T h e n = 6 a n d d i v i s i b l e b y 6 L e t i t b e d i v i s i b l e b y 6 f o r T h e n W h e r e i s a n i n t e g e r F o r t h e e x p r e s s i o n i s ( m + 1 ) ( m + 2 ) ( 2 m + 2 + 1 ) = ( m + 2 ) ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( m + 2 ) = m ( m + 1 ) ( 2 m + 1 ) + 2 ( m + 1 ) ( 3 m + 3 ) = m ( m + 1 ) ( 2 m + 1 ) + 6 ( m + 1 ) 2 = 6 k + 6 ( m + 1 ) 2 , T h i s i s d i v i s i b l e b y 6 . 8 . S o l v e . S o l : L e t T h e n , S o l v i n g t h i s q u a d r a t i c F i r s t v a l u e o f t i s r e j e c t e d a s s h o u l d l i e b e t w e e n G e n e r a l s o l u t i o n i s 9 . F o r w h a t v a l u e s o f m 2 x 2 + 2 ( m + 1 ) x + 4 = 0 w i l l h a v e e x a c t l y o n e z e r o . S o l : W h e n T h e g i v e n e q u a t i o n r e d u c e s t o a f i r s t d e g r e e a n d i t w i l l h a v e o n l y o n e s o l u t i o n A l s o w h e n t h e d i s c r i m i n a n t i s z e r o i t w i l l h a v e o n l y o n e s o l u t i o n D i s c r i m i n a n t i s O n s i m p l i f y i n g a n d s o l v i n g , H e n c e t h e t h r e e v a l u e s o f f o r w h i c h t h e e q u a t i o n w i l l h a v e o n l y o n e s o l u t i o n i s 1 0 . T h r e e n u m b e r s a r e i n A P . A n o t h e r 3 n u m b e r s a r e i n G P . T h e s u m o f f i r s t t e r m o f t h e A P a n d t h e f i r s t t e r m o f t h e G P i s 8 5 , t h e s u m o f s e c o n d t e r m o f A P a n d t h e s e c o n d t e r m o f t h e G P i s 7 6 a n d t h a t o f t h e 3 r d t e r m o f A P a n d 3 r d t e r m o f G P i s 8 4 . T h e s u m o f t h e A P i s 1 2 6 . F i n d e a c h t e r m o f A P a n d G P . S o l : A . P a - d , a , a + d G . P b / g , b , b g W h e n g = 2 4 2 - d + = 8 5 d = - 2 6 a = 4 2 , d = - 2 6 , g = 2 , b = 3 4 1 1 . I f f i n d i n t e r m s o f . S o l : = 1 2 . I f P r o v e t h a t S o l : 1 3 . F i n d t h e v a l u e o f . S o l :Read More
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FAQs on Past Year Paper, Maths(Set - 1), 2018, Class 11 - Mathematics (Maths) Class 11 - Commerce
| 1. What are the important topics to focus on in the Class 11 Maths exam? |  |
Ans. The important topics to focus on in the Class 11 Maths exam include Sets, Relations and Functions, Trigonometric Functions, Principle of Mathematical Induction, Linear Inequalities, Permutations and Combinations, Binomial Theorem, Sequences and Series, Straight Lines, Conic Sections, and Statistics.
| 2. How can I effectively prepare for the Class 11 Maths exam? | |
Ans. To effectively prepare for the Class 11 Maths exam, you can follow these steps:
1. Understand the concepts thoroughly by referring to your textbook and class notes.
2. Practice solving a variety of problems from each chapter to enhance your problem-solving skills.
3. Create a study schedule and allocate specific time for Maths practice.
4. Make use of reference books and online resources for additional practice and solving previous year question papers.
5. Seek help from your teachers or classmates in case you face any difficulties in understanding a particular topic.
| 3. Are there any recommended books for Class 11 Maths exam preparation? | |
Ans. Yes, there are several recommended books for Class 11 Maths exam preparation. Some popular ones include:
1. R.D. Sharma Mathematics for Class 11
2. NCERT Exemplar Problems: Solutions Mathematics Class 11
3. Mathematics for Class 11 by R.S. Aggarwal
4. All in One Mathematics Class 11 by Arihant Experts
5. Mathematics for Class 11 by Dhanpat Rai Publications
| 4. How can I improve my problem-solving speed in the Class 11 Maths exam? | |
Ans. To improve your problem-solving speed in the Class 11 Maths exam, you can follow these tips:
1. Practice regularly and solve a wide range of problems to enhance your familiarity with different types of questions.
2. Learn and apply various shortcuts and tricks for solving problems quickly.
3. Focus on understanding the underlying concepts and principles, as it will help you solve problems more efficiently.
4. Time yourself while solving practice papers or previous year question papers to identify areas where you need to improve.
5. Maintain a calm and composed mindset during the exam and avoid getting stuck on a single question for too long.
| 5. How can I score well in the Class 11 Maths exam? | |
Ans. To score well in the Class 11 Maths exam, you can follow these strategies:
1. Understand the syllabus and exam pattern thoroughly.
2. Practice regularly and solve a variety of problems from each chapter.
3. Revise the concepts frequently to ensure a strong foundation.
4. Solve previous year question papers to get acquainted with the exam pattern and identify areas of improvement.
5. Seek help from your teachers or classmates for any doubts or clarifications.
6. Manage your time effectively during the exam and attempt all the questions with accuracy.
7. Stay calm and confident during the exam to perform to the best of your abilities.
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