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Page 1 Q u e s t i o n 1 T h e p o r t i o n o f t h e l i n e 4 x + 5 y = 2 0 i n t h e f i r s t q u a d r a n t i s t r i s e c t e d b y t h e l i n e s L 1 a n d L 2 p a s s i n g t h r o u g h t h e o r i g i n . T h e t a n g e n t o f a n a n g l e b e t w e e n t h e l i n e s L 1 a n d L 2 i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 1 ] O p t i o ns : A . 8 / 5 B . 2 5 / 4 1 C . 2 / 5 D . 3 0 / 4 1 A n s w e r : D So l u t i o n : Page 2 Q u e s t i o n 1 T h e p o r t i o n o f t h e l i n e 4 x + 5 y = 2 0 i n t h e f i r s t q u a d r a n t i s t r i s e c t e d b y t h e l i n e s L 1 a n d L 2 p a s s i n g t h r o u g h t h e o r i g i n . T h e t a n g e n t o f a n a n g l e b e t w e e n t h e l i n e s L 1 a n d L 2 i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 1 ] O p t i o ns : A . 8 / 5 B . 2 5 / 4 1 C . 2 / 5 D . 3 0 / 4 1 A n s w e r : D So l u t i o n : Q u e s t i o n 2 L e t R b e t h e i n t e r i o r r e g i o n b e t w e e n t h e l i n e s 3 x - y + 1 = 0 a n d x + 2 y - 5 = 0 c o n t a i n i n g t h e o r i g i n . T h e s e t o f a l l v a l u e s o f a , f o r w h i c h t h e p o i n t s ( a 2 , a + 1 ) l i e i n R , i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O p t i o ns : A . B . C . D . A n s w e r : B S o l u t i o n : Page 3 Q u e s t i o n 1 T h e p o r t i o n o f t h e l i n e 4 x + 5 y = 2 0 i n t h e f i r s t q u a d r a n t i s t r i s e c t e d b y t h e l i n e s L 1 a n d L 2 p a s s i n g t h r o u g h t h e o r i g i n . T h e t a n g e n t o f a n a n g l e b e t w e e n t h e l i n e s L 1 a n d L 2 i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 1 ] O p t i o ns : A . 8 / 5 B . 2 5 / 4 1 C . 2 / 5 D . 3 0 / 4 1 A n s w e r : D So l u t i o n : Q u e s t i o n 2 L e t R b e t h e i n t e r i o r r e g i o n b e t w e e n t h e l i n e s 3 x - y + 1 = 0 a n d x + 2 y - 5 = 0 c o n t a i n i n g t h e o r i g i n . T h e s e t o f a l l v a l u e s o f a , f o r w h i c h t h e p o i n t s ( a 2 , a + 1 ) l i e i n R , i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O p t i o ns : A . B . C . D . A n s w e r : B S o l u t i o n : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Q u e s t i o n 3 L e t A a n d B b e t w o f i n i t e s e t s w i t h m a n d n e l e m e n t s r e s p e c t i v e l y . T h e t o t a l n u m b e r o f s u b s e t s o f t h e s e t A i s 5 6 m o r e t h a n t h e t o t a l n u m b e r o f s u b s e t s o f B. T h e n t h e d i s t a n c e o f t h e p o i n t P ( m , n ) f r o m t h e p o i n t Q ( - 2 , - 3 ) i s [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O pt i o ns : A . 1 0 B . 6 C . 4 D . 8 Page 4 Q u e s t i o n 1 T h e p o r t i o n o f t h e l i n e 4 x + 5 y = 2 0 i n t h e f i r s t q u a d r a n t i s t r i s e c t e d b y t h e l i n e s L 1 a n d L 2 p a s s i n g t h r o u g h t h e o r i g i n . T h e t a n g e n t o f a n a n g l e b e t w e e n t h e l i n e s L 1 a n d L 2 i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 1 ] O p t i o ns : A . 8 / 5 B . 2 5 / 4 1 C . 2 / 5 D . 3 0 / 4 1 A n s w e r : D So l u t i o n : Q u e s t i o n 2 L e t R b e t h e i n t e r i o r r e g i o n b e t w e e n t h e l i n e s 3 x - y + 1 = 0 a n d x + 2 y - 5 = 0 c o n t a i n i n g t h e o r i g i n . T h e s e t o f a l l v a l u e s o f a , f o r w h i c h t h e p o i n t s ( a 2 , a + 1 ) l i e i n R , i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O p t i o ns : A . B . C . D . A n s w e r : B S o l u t i o n : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Q u e s t i o n 3 L e t A a n d B b e t w o f i n i t e s e t s w i t h m a n d n e l e m e n t s r e s p e c t i v e l y . T h e t o t a l n u m b e r o f s u b s e t s o f t h e s e t A i s 5 6 m o r e t h a n t h e t o t a l n u m b e r o f s u b s e t s o f B. T h e n t h e d i s t a n c e o f t h e p o i n t P ( m , n ) f r o m t h e p o i n t Q ( - 2 , - 3 ) i s [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O pt i o ns : A . 1 0 B . 6 C . 4 D . 8 A n s w e r : A So l u t i o n : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Q u e s t i o n 4 I f t h e s u m o f s q u a r e s o f a l l r e a l v a l u e s o f a , fo r w h i c h t h e l i n e s 2 x - y + 3 = 0 , 6 x + 3 y + 1 = 0 a n d a x + 2 y - 2 = 0 d o n o t f o r m a t r i a n g l e i s p , t h e n t h e g r e a t e s t i n t e g e r l e s s t h a n o r e q u a l t o p i s . . . . . . . . [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] A n s w e r : 3 2 So l u t i o n : Page 5 Q u e s t i o n 1 T h e p o r t i o n o f t h e l i n e 4 x + 5 y = 2 0 i n t h e f i r s t q u a d r a n t i s t r i s e c t e d b y t h e l i n e s L 1 a n d L 2 p a s s i n g t h r o u g h t h e o r i g i n . T h e t a n g e n t o f a n a n g l e b e t w e e n t h e l i n e s L 1 a n d L 2 i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 1 ] O p t i o ns : A . 8 / 5 B . 2 5 / 4 1 C . 2 / 5 D . 3 0 / 4 1 A n s w e r : D So l u t i o n : Q u e s t i o n 2 L e t R b e t h e i n t e r i o r r e g i o n b e t w e e n t h e l i n e s 3 x - y + 1 = 0 a n d x + 2 y - 5 = 0 c o n t a i n i n g t h e o r i g i n . T h e s e t o f a l l v a l u e s o f a , f o r w h i c h t h e p o i n t s ( a 2 , a + 1 ) l i e i n R , i s : [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O p t i o ns : A . B . C . D . A n s w e r : B S o l u t i o n : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Q u e s t i o n 3 L e t A a n d B b e t w o f i n i t e s e t s w i t h m a n d n e l e m e n t s r e s p e c t i v e l y . T h e t o t a l n u m b e r o f s u b s e t s o f t h e s e t A i s 5 6 m o r e t h a n t h e t o t a l n u m b e r o f s u b s e t s o f B. T h e n t h e d i s t a n c e o f t h e p o i n t P ( m , n ) f r o m t h e p o i n t Q ( - 2 , - 3 ) i s [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] O pt i o ns : A . 1 0 B . 6 C . 4 D . 8 A n s w e r : A So l u t i o n : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Q u e s t i o n 4 I f t h e s u m o f s q u a r e s o f a l l r e a l v a l u e s o f a , fo r w h i c h t h e l i n e s 2 x - y + 3 = 0 , 6 x + 3 y + 1 = 0 a n d a x + 2 y - 2 = 0 d o n o t f o r m a t r i a n g l e i s p , t h e n t h e g r e a t e s t i n t e g e r l e s s t h a n o r e q u a l t o p i s . . . . . . . . [ 2 7 - J a n - 2 0 2 4 S h i f t 2 ] A n s w e r : 3 2 So l u t i o n : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Q u e s t i o n 5 I n a ? A B C , s u p p o s e y = x i s t h e e q u a t i o n o f t h e b i s e c t o r o f t h e a n g l e B a n d t h e e q u a t i o n o f t h e s i d e A C i s 2 x - y = 2 . I f 2 A B = B C a n d t h e p o i n t A a n d B a r e r e s p e c t i v e l y ( 4 , 6 ) a n d ( a , ß ) , t h e n a + 2 ß i s e q u a l t o [ 2 9 - J a n - 2 0 2 4 S h i f t 1 ] O p t i o ns : A . 4 2 B . 3 9 C . 4 8 D . 4 5 A n s w e r : A So l u t i o n :Read More
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