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G e n e r a l 	 I n s t r u c t i o n s :
• 	 T h i s 	 q u e s t i o n 	 p a p e r 	 h a s 	 3 1 	 q u e s t i o n s 	 .
• 	 S e c t i o n 	 A 	 c o n s i s t s 	 o f 	 4 	 q u e s t i o n s 	 o f 	 1 	 m a r k 	 e a c h .
• 	 S e c t i o n 	 B 	 c o n s i s t s 	 o f 	 6 	 q u e s t i o n s 	 o f 	 2 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 C 	 c o n s i s t s 	 o f 	 1 0 	 q u e s t i o n s 	 o f 	 3 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 D 	 c o n s i s t s 	 o f 	 1 1 	 q u e s t i o n s 	 o f 	 4 	 m a r k s 	 e a c h .
• 	 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 	 t o 	 a t t e m p t .
• 	 A t t e m p t 	 t h e 	 p a p e r 	 n e a t l y 	 a n d 	 l e g i b l y .
S E C T I O N - A
1 . 	 F i n d 	 t h e 	 e s t i m a t e d 	 d i f f e r e n c e 	 o f 	 4 8 9 3 4 8 	 a n d 	 4 8 3 6 5 .
A n s . 	 4 8 9 3 4 8 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 4 9 0 0 0 0
4 8 3 6 5 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 5 0 0 0 0
E s t i m a t e d 	 d i f f e r e n c e 	 = 	 4 4 0 0 0 0
2 . 	 I n 	 a 	 p i c t o g r a p h , 	 i f 	 t h e 	 s y m b o l 	 	 r e p r e s e n t s 	 2 0 	 s t u d e n t s , 	 t h e n 	 h o w 	 m a n y 	 s t u d e n t s
w i l l 	 b e 	 r e p r e s e n t e d 	 b y 	 t h e 	 	 s y m b o l ?
A n s . 	 3 5 .
3 . 	 S o l v e 	 a n d 	 w r i t e 	 t h e 	 a n s w e r 	 i n 	 t h e 	 H i n d u 	 A r a b i c 	 N u m e r a l : 	 D 	 - 	 C D X L 	 = 	 _ _ _ _ _ _
A n s . 	 D 	 - 	 C D X L 	 = 	 5 0 0 	 – 	 4 4 0 	 = 	 6 0 	 = 	 L X
4 . 	 H o w 	 m a n y 	 d e g r e e s 	 a r e 	 t h e r e 	 i n 	 t h e 	 a n g l e 	 b e t w e e n 	 t h e 	 h a n d s 	 o f 	 a 	 c l o c k 	 w h e n 	 i t 	 i s
6 	 o ’ c l o c k ?
A n s . 	 1 8 0 	 d e g r e e s 	 = 	 s t r a i g h t 	 a n g l e
Page 2


G e n e r a l 	 I n s t r u c t i o n s :
• 	 T h i s 	 q u e s t i o n 	 p a p e r 	 h a s 	 3 1 	 q u e s t i o n s 	 .
• 	 S e c t i o n 	 A 	 c o n s i s t s 	 o f 	 4 	 q u e s t i o n s 	 o f 	 1 	 m a r k 	 e a c h .
• 	 S e c t i o n 	 B 	 c o n s i s t s 	 o f 	 6 	 q u e s t i o n s 	 o f 	 2 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 C 	 c o n s i s t s 	 o f 	 1 0 	 q u e s t i o n s 	 o f 	 3 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 D 	 c o n s i s t s 	 o f 	 1 1 	 q u e s t i o n s 	 o f 	 4 	 m a r k s 	 e a c h .
• 	 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 	 t o 	 a t t e m p t .
• 	 A t t e m p t 	 t h e 	 p a p e r 	 n e a t l y 	 a n d 	 l e g i b l y .
S E C T I O N - A
1 . 	 F i n d 	 t h e 	 e s t i m a t e d 	 d i f f e r e n c e 	 o f 	 4 8 9 3 4 8 	 a n d 	 4 8 3 6 5 .
A n s . 	 4 8 9 3 4 8 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 4 9 0 0 0 0
4 8 3 6 5 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 5 0 0 0 0
E s t i m a t e d 	 d i f f e r e n c e 	 = 	 4 4 0 0 0 0
2 . 	 I n 	 a 	 p i c t o g r a p h , 	 i f 	 t h e 	 s y m b o l 	 	 r e p r e s e n t s 	 2 0 	 s t u d e n t s , 	 t h e n 	 h o w 	 m a n y 	 s t u d e n t s
w i l l 	 b e 	 r e p r e s e n t e d 	 b y 	 t h e 	 	 s y m b o l ?
A n s . 	 3 5 .
3 . 	 S o l v e 	 a n d 	 w r i t e 	 t h e 	 a n s w e r 	 i n 	 t h e 	 H i n d u 	 A r a b i c 	 N u m e r a l : 	 D 	 - 	 C D X L 	 = 	 _ _ _ _ _ _
A n s . 	 D 	 - 	 C D X L 	 = 	 5 0 0 	 – 	 4 4 0 	 = 	 6 0 	 = 	 L X
4 . 	 H o w 	 m a n y 	 d e g r e e s 	 a r e 	 t h e r e 	 i n 	 t h e 	 a n g l e 	 b e t w e e n 	 t h e 	 h a n d s 	 o f 	 a 	 c l o c k 	 w h e n 	 i t 	 i s
6 	 o ’ c l o c k ?
A n s . 	 1 8 0 	 d e g r e e s 	 = 	 s t r a i g h t 	 a n g l e
S E C T I O N - B
5 . 	 F i n d 	 t h e 	 p r o d u c t 	 o f 	 t h e 	 p l a c e 	 v a l u e s 	 o f 	 t w o 	 7 s 	 i n 	 1 9 7 0 4 7 .
A n s . 	 P l a c e 	 v a l u e 	 o f 	 7 	 i n 	 1 9 7 0 4 7 	 i s 	 7 0 0 0 	 a n d 	 7
P r o d u c t 	 = 	 4 9 0 0 0
6 . 	 S i m p l i f y 	 u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y 	 o f 	 w h o l e 	 n u m b e r s : 	 8 3 5 	 x 	 1 0 5
A n s . 	 8 3 5 	 × 	 1 0 5
= 	 8 3 5 	 × 	 ( 1 0 0 	 + 	 5 )
= 	 8 3 5 	 × 	 1 0 0 	 + 	 8 3 5 	 × 	 5 	 ( u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y )
= 	 8 3 5 0 0 	 + 	 4 1 7 5 	 = 	 8 7 6 7 5
7 . 	 I s 	 2 8 	 a 	 p e r f e c t 	 n u m b e r ? 	 S h o w 	 t h e 	 w o r k i n g .
A n s . 	 F a c t o r s 	 o f 	 2 8 	 a r e 	 1 , 2 , 4 , 7 , 1 4 	 a n d 	 2 8
S u m 	 o f 	 f a c t o r s 	 = 	 1 	 + 	 2 	 + 	 4 	 + 	 7 	 + 	 1 4 	 + 	 2 8 	 = 	 5 6 	 w h i c h 	 i s 	 2 	 × 	 2 8
T h e r e f o r e , 	 Y e s 	 2 8 	 i s 	 a 	 p e r f e c t 	 n u m b e r
8 . 	 F i n d 	 t h e 	 d i a m e t e r 	 o f 	 a 	 c i r c l e 	 i f 	 i t s 	 r a d i u s 	 i s 	 4 . 5 	 c m .
A n s . 	 D i a m e t e r 	 = 	 2 	 x 	 4 . 5 	 c m 	 = 	 9 	 c m
9 . 	 T h e 	 f o l l o w i n g 	 a r e 	 t h e 	 m a r k s 	 s c o r e d 	 b y 	 2 0 	 s t u d e n t s 	 o f 	 V I 	 A 	 i n 	 M a t h e m a t i c s 	 e x a m :
5 7 , 	 6 5 , 	 8 0 , 	 5 7 , 	 8 0 , 	 6 5 , 	 8 5 , 	 3 4 , 	 5 7 , 	 4 1 , 	 5 4 , 	 8 0 , 	 4 1 , 	 5 4 , 	 3 4 , 	 4 1 , 	 8 5 , 	 5 7 , 	 8 0 , 	 6 5
P r e p a r e 	 a 	 f r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e .
A n s . 	 F r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e 	 n e a t l y 	 d r a w n
1 0 . 	 S t a t e 	 t h e 	 c o m m u t a t i v e 	 p r o p e r t y 	 o f 	 a d d i t i o n 	 o f 	 w h o l e 	 n u m b e r s . 	 G i v e 	 a n 	 e x a m p l e .
A n s . 	 a 	 + 	 b 	 = 	 b 	 + 	 a
F o r 	 e g : 	 3 	 + 5 	 = 	 5 	 + 	 3 	 = 	 8
S E C T I O N - C
Page 3


G e n e r a l 	 I n s t r u c t i o n s :
• 	 T h i s 	 q u e s t i o n 	 p a p e r 	 h a s 	 3 1 	 q u e s t i o n s 	 .
• 	 S e c t i o n 	 A 	 c o n s i s t s 	 o f 	 4 	 q u e s t i o n s 	 o f 	 1 	 m a r k 	 e a c h .
• 	 S e c t i o n 	 B 	 c o n s i s t s 	 o f 	 6 	 q u e s t i o n s 	 o f 	 2 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 C 	 c o n s i s t s 	 o f 	 1 0 	 q u e s t i o n s 	 o f 	 3 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 D 	 c o n s i s t s 	 o f 	 1 1 	 q u e s t i o n s 	 o f 	 4 	 m a r k s 	 e a c h .
• 	 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 	 t o 	 a t t e m p t .
• 	 A t t e m p t 	 t h e 	 p a p e r 	 n e a t l y 	 a n d 	 l e g i b l y .
S E C T I O N - A
1 . 	 F i n d 	 t h e 	 e s t i m a t e d 	 d i f f e r e n c e 	 o f 	 4 8 9 3 4 8 	 a n d 	 4 8 3 6 5 .
A n s . 	 4 8 9 3 4 8 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 4 9 0 0 0 0
4 8 3 6 5 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 5 0 0 0 0
E s t i m a t e d 	 d i f f e r e n c e 	 = 	 4 4 0 0 0 0
2 . 	 I n 	 a 	 p i c t o g r a p h , 	 i f 	 t h e 	 s y m b o l 	 	 r e p r e s e n t s 	 2 0 	 s t u d e n t s , 	 t h e n 	 h o w 	 m a n y 	 s t u d e n t s
w i l l 	 b e 	 r e p r e s e n t e d 	 b y 	 t h e 	 	 s y m b o l ?
A n s . 	 3 5 .
3 . 	 S o l v e 	 a n d 	 w r i t e 	 t h e 	 a n s w e r 	 i n 	 t h e 	 H i n d u 	 A r a b i c 	 N u m e r a l : 	 D 	 - 	 C D X L 	 = 	 _ _ _ _ _ _
A n s . 	 D 	 - 	 C D X L 	 = 	 5 0 0 	 – 	 4 4 0 	 = 	 6 0 	 = 	 L X
4 . 	 H o w 	 m a n y 	 d e g r e e s 	 a r e 	 t h e r e 	 i n 	 t h e 	 a n g l e 	 b e t w e e n 	 t h e 	 h a n d s 	 o f 	 a 	 c l o c k 	 w h e n 	 i t 	 i s
6 	 o ’ c l o c k ?
A n s . 	 1 8 0 	 d e g r e e s 	 = 	 s t r a i g h t 	 a n g l e
S E C T I O N - B
5 . 	 F i n d 	 t h e 	 p r o d u c t 	 o f 	 t h e 	 p l a c e 	 v a l u e s 	 o f 	 t w o 	 7 s 	 i n 	 1 9 7 0 4 7 .
A n s . 	 P l a c e 	 v a l u e 	 o f 	 7 	 i n 	 1 9 7 0 4 7 	 i s 	 7 0 0 0 	 a n d 	 7
P r o d u c t 	 = 	 4 9 0 0 0
6 . 	 S i m p l i f y 	 u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y 	 o f 	 w h o l e 	 n u m b e r s : 	 8 3 5 	 x 	 1 0 5
A n s . 	 8 3 5 	 × 	 1 0 5
= 	 8 3 5 	 × 	 ( 1 0 0 	 + 	 5 )
= 	 8 3 5 	 × 	 1 0 0 	 + 	 8 3 5 	 × 	 5 	 ( u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y )
= 	 8 3 5 0 0 	 + 	 4 1 7 5 	 = 	 8 7 6 7 5
7 . 	 I s 	 2 8 	 a 	 p e r f e c t 	 n u m b e r ? 	 S h o w 	 t h e 	 w o r k i n g .
A n s . 	 F a c t o r s 	 o f 	 2 8 	 a r e 	 1 , 2 , 4 , 7 , 1 4 	 a n d 	 2 8
S u m 	 o f 	 f a c t o r s 	 = 	 1 	 + 	 2 	 + 	 4 	 + 	 7 	 + 	 1 4 	 + 	 2 8 	 = 	 5 6 	 w h i c h 	 i s 	 2 	 × 	 2 8
T h e r e f o r e , 	 Y e s 	 2 8 	 i s 	 a 	 p e r f e c t 	 n u m b e r
8 . 	 F i n d 	 t h e 	 d i a m e t e r 	 o f 	 a 	 c i r c l e 	 i f 	 i t s 	 r a d i u s 	 i s 	 4 . 5 	 c m .
A n s . 	 D i a m e t e r 	 = 	 2 	 x 	 4 . 5 	 c m 	 = 	 9 	 c m
9 . 	 T h e 	 f o l l o w i n g 	 a r e 	 t h e 	 m a r k s 	 s c o r e d 	 b y 	 2 0 	 s t u d e n t s 	 o f 	 V I 	 A 	 i n 	 M a t h e m a t i c s 	 e x a m :
5 7 , 	 6 5 , 	 8 0 , 	 5 7 , 	 8 0 , 	 6 5 , 	 8 5 , 	 3 4 , 	 5 7 , 	 4 1 , 	 5 4 , 	 8 0 , 	 4 1 , 	 5 4 , 	 3 4 , 	 4 1 , 	 8 5 , 	 5 7 , 	 8 0 , 	 6 5
P r e p a r e 	 a 	 f r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e .
A n s . 	 F r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e 	 n e a t l y 	 d r a w n
1 0 . 	 S t a t e 	 t h e 	 c o m m u t a t i v e 	 p r o p e r t y 	 o f 	 a d d i t i o n 	 o f 	 w h o l e 	 n u m b e r s . 	 G i v e 	 a n 	 e x a m p l e .
A n s . 	 a 	 + 	 b 	 = 	 b 	 + 	 a
F o r 	 e g : 	 3 	 + 5 	 = 	 5 	 + 	 3 	 = 	 8
S E C T I O N - C
1 1 . 	 S u b t r a c t 	 2 5 2 7 0 5 3 	 f r o m 	 2 7 2 5 3 5 0 	 a n d 	 w r i t e 	 t h e 	 n u m b e r 	 s e n t e n c e .
A n s . 	 S u b t r a c t i o n 	 s h o w n
N u m b e r 	 s e n t e n c e : 	 2 7 2 5 3 5 0 	 – 	 2 5 2 7 0 5 3 	 = 	 1 9 8 2 9
1 2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 4 - d i g i t 	 n u m b e r 	 w h i c h 	 i s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 3 7 .
A n s . 	 G r e a t e s t 	 4 - d i g i t 	 n u m b e r = 	 9 9 9 9
D i v i d e 	 9 9 9 9 	 b y 	 3 7 	 R e m a i n d e r = 	 9
N o w , 	 r e q u i r e d 	 n u m b e r 	 = 	 9 9 9 9 	 – 	 9 	 = 	 9 9 9 0
1 3 . 	 I n 	 5 6 * 8 9 1 	 r e p l a c e 	 * 	 b y 	 ( a ) 	 b y 	 3 .
A n s . 	 A c c o r d i n g 	 t o 	 d i v i s i b i l i t y 	 t e s t 	 o f 	 3 , 	 S u m 	 o f 	 d i g i t s 	 = 	 5 	 + 	 6 	 + 	 * 	 + 	 8 	 + 	 9 	 + 	 1
= 	 2 9 	 + 	 *
( a ) 	 S m a l l e s t 	 d i g i t 	 = 	 1 	 ( b ) 	 L a r g e s t 	 d i g i t 	 = 	 7
1 4 . 	 F i n d 	 t h e 	 H C F 	 o f 	 3 8 5 	 a n d 	 6 2 1 . 	 A r e 	 t h e y 	 c o p r i m e
A n s . 	 H C F 	 o f 	 3 8 5 	 a n d 	 6 2 1 	 = 	 1
Y e s 	 t h e y 	 a r e 	 C o p r i m e
1 5 . 	 N a m e 	 a n y 	 s i x 	 a n g l e s 	 i n 	 t h e 	 g i v e n 	 f i g u r e :
A n s . 	 A B C , 	 B A C , 	 A C B , 	 B D C , 	 D C B , 	 	 D B C , 	 A B D 	 a n d 	 A C D 	 ( a t 	 l e a s t 	 6 )
1 6 . 	 D r a w 	 a 	 r o u g h 	 s k e t c h 	 o f 	 a 	 q u a d r i l a t e r a l 	 E F G H 	 a n d 	 n a m e
a . 	 a 	 p a i r 	 o f 	 o p p o s i t e 	 s i d e s .
b . 	 a 	 p a i r 	 o f 	 a d j a c e n t 	 a n g l e s
c . 	 t w o 	 d i a g o n a l s .
Page 4


G e n e r a l 	 I n s t r u c t i o n s :
• 	 T h i s 	 q u e s t i o n 	 p a p e r 	 h a s 	 3 1 	 q u e s t i o n s 	 .
• 	 S e c t i o n 	 A 	 c o n s i s t s 	 o f 	 4 	 q u e s t i o n s 	 o f 	 1 	 m a r k 	 e a c h .
• 	 S e c t i o n 	 B 	 c o n s i s t s 	 o f 	 6 	 q u e s t i o n s 	 o f 	 2 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 C 	 c o n s i s t s 	 o f 	 1 0 	 q u e s t i o n s 	 o f 	 3 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 D 	 c o n s i s t s 	 o f 	 1 1 	 q u e s t i o n s 	 o f 	 4 	 m a r k s 	 e a c h .
• 	 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 	 t o 	 a t t e m p t .
• 	 A t t e m p t 	 t h e 	 p a p e r 	 n e a t l y 	 a n d 	 l e g i b l y .
S E C T I O N - A
1 . 	 F i n d 	 t h e 	 e s t i m a t e d 	 d i f f e r e n c e 	 o f 	 4 8 9 3 4 8 	 a n d 	 4 8 3 6 5 .
A n s . 	 4 8 9 3 4 8 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 4 9 0 0 0 0
4 8 3 6 5 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 5 0 0 0 0
E s t i m a t e d 	 d i f f e r e n c e 	 = 	 4 4 0 0 0 0
2 . 	 I n 	 a 	 p i c t o g r a p h , 	 i f 	 t h e 	 s y m b o l 	 	 r e p r e s e n t s 	 2 0 	 s t u d e n t s , 	 t h e n 	 h o w 	 m a n y 	 s t u d e n t s
w i l l 	 b e 	 r e p r e s e n t e d 	 b y 	 t h e 	 	 s y m b o l ?
A n s . 	 3 5 .
3 . 	 S o l v e 	 a n d 	 w r i t e 	 t h e 	 a n s w e r 	 i n 	 t h e 	 H i n d u 	 A r a b i c 	 N u m e r a l : 	 D 	 - 	 C D X L 	 = 	 _ _ _ _ _ _
A n s . 	 D 	 - 	 C D X L 	 = 	 5 0 0 	 – 	 4 4 0 	 = 	 6 0 	 = 	 L X
4 . 	 H o w 	 m a n y 	 d e g r e e s 	 a r e 	 t h e r e 	 i n 	 t h e 	 a n g l e 	 b e t w e e n 	 t h e 	 h a n d s 	 o f 	 a 	 c l o c k 	 w h e n 	 i t 	 i s
6 	 o ’ c l o c k ?
A n s . 	 1 8 0 	 d e g r e e s 	 = 	 s t r a i g h t 	 a n g l e
S E C T I O N - B
5 . 	 F i n d 	 t h e 	 p r o d u c t 	 o f 	 t h e 	 p l a c e 	 v a l u e s 	 o f 	 t w o 	 7 s 	 i n 	 1 9 7 0 4 7 .
A n s . 	 P l a c e 	 v a l u e 	 o f 	 7 	 i n 	 1 9 7 0 4 7 	 i s 	 7 0 0 0 	 a n d 	 7
P r o d u c t 	 = 	 4 9 0 0 0
6 . 	 S i m p l i f y 	 u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y 	 o f 	 w h o l e 	 n u m b e r s : 	 8 3 5 	 x 	 1 0 5
A n s . 	 8 3 5 	 × 	 1 0 5
= 	 8 3 5 	 × 	 ( 1 0 0 	 + 	 5 )
= 	 8 3 5 	 × 	 1 0 0 	 + 	 8 3 5 	 × 	 5 	 ( u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y )
= 	 8 3 5 0 0 	 + 	 4 1 7 5 	 = 	 8 7 6 7 5
7 . 	 I s 	 2 8 	 a 	 p e r f e c t 	 n u m b e r ? 	 S h o w 	 t h e 	 w o r k i n g .
A n s . 	 F a c t o r s 	 o f 	 2 8 	 a r e 	 1 , 2 , 4 , 7 , 1 4 	 a n d 	 2 8
S u m 	 o f 	 f a c t o r s 	 = 	 1 	 + 	 2 	 + 	 4 	 + 	 7 	 + 	 1 4 	 + 	 2 8 	 = 	 5 6 	 w h i c h 	 i s 	 2 	 × 	 2 8
T h e r e f o r e , 	 Y e s 	 2 8 	 i s 	 a 	 p e r f e c t 	 n u m b e r
8 . 	 F i n d 	 t h e 	 d i a m e t e r 	 o f 	 a 	 c i r c l e 	 i f 	 i t s 	 r a d i u s 	 i s 	 4 . 5 	 c m .
A n s . 	 D i a m e t e r 	 = 	 2 	 x 	 4 . 5 	 c m 	 = 	 9 	 c m
9 . 	 T h e 	 f o l l o w i n g 	 a r e 	 t h e 	 m a r k s 	 s c o r e d 	 b y 	 2 0 	 s t u d e n t s 	 o f 	 V I 	 A 	 i n 	 M a t h e m a t i c s 	 e x a m :
5 7 , 	 6 5 , 	 8 0 , 	 5 7 , 	 8 0 , 	 6 5 , 	 8 5 , 	 3 4 , 	 5 7 , 	 4 1 , 	 5 4 , 	 8 0 , 	 4 1 , 	 5 4 , 	 3 4 , 	 4 1 , 	 8 5 , 	 5 7 , 	 8 0 , 	 6 5
P r e p a r e 	 a 	 f r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e .
A n s . 	 F r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e 	 n e a t l y 	 d r a w n
1 0 . 	 S t a t e 	 t h e 	 c o m m u t a t i v e 	 p r o p e r t y 	 o f 	 a d d i t i o n 	 o f 	 w h o l e 	 n u m b e r s . 	 G i v e 	 a n 	 e x a m p l e .
A n s . 	 a 	 + 	 b 	 = 	 b 	 + 	 a
F o r 	 e g : 	 3 	 + 5 	 = 	 5 	 + 	 3 	 = 	 8
S E C T I O N - C
1 1 . 	 S u b t r a c t 	 2 5 2 7 0 5 3 	 f r o m 	 2 7 2 5 3 5 0 	 a n d 	 w r i t e 	 t h e 	 n u m b e r 	 s e n t e n c e .
A n s . 	 S u b t r a c t i o n 	 s h o w n
N u m b e r 	 s e n t e n c e : 	 2 7 2 5 3 5 0 	 – 	 2 5 2 7 0 5 3 	 = 	 1 9 8 2 9
1 2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 4 - d i g i t 	 n u m b e r 	 w h i c h 	 i s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 3 7 .
A n s . 	 G r e a t e s t 	 4 - d i g i t 	 n u m b e r = 	 9 9 9 9
D i v i d e 	 9 9 9 9 	 b y 	 3 7 	 R e m a i n d e r = 	 9
N o w , 	 r e q u i r e d 	 n u m b e r 	 = 	 9 9 9 9 	 – 	 9 	 = 	 9 9 9 0
1 3 . 	 I n 	 5 6 * 8 9 1 	 r e p l a c e 	 * 	 b y 	 ( a ) 	 b y 	 3 .
A n s . 	 A c c o r d i n g 	 t o 	 d i v i s i b i l i t y 	 t e s t 	 o f 	 3 , 	 S u m 	 o f 	 d i g i t s 	 = 	 5 	 + 	 6 	 + 	 * 	 + 	 8 	 + 	 9 	 + 	 1
= 	 2 9 	 + 	 *
( a ) 	 S m a l l e s t 	 d i g i t 	 = 	 1 	 ( b ) 	 L a r g e s t 	 d i g i t 	 = 	 7
1 4 . 	 F i n d 	 t h e 	 H C F 	 o f 	 3 8 5 	 a n d 	 6 2 1 . 	 A r e 	 t h e y 	 c o p r i m e
A n s . 	 H C F 	 o f 	 3 8 5 	 a n d 	 6 2 1 	 = 	 1
Y e s 	 t h e y 	 a r e 	 C o p r i m e
1 5 . 	 N a m e 	 a n y 	 s i x 	 a n g l e s 	 i n 	 t h e 	 g i v e n 	 f i g u r e :
A n s . 	 A B C , 	 B A C , 	 A C B , 	 B D C , 	 D C B , 	 	 D B C , 	 A B D 	 a n d 	 A C D 	 ( a t 	 l e a s t 	 6 )
1 6 . 	 D r a w 	 a 	 r o u g h 	 s k e t c h 	 o f 	 a 	 q u a d r i l a t e r a l 	 E F G H 	 a n d 	 n a m e
a . 	 a 	 p a i r 	 o f 	 o p p o s i t e 	 s i d e s .
b . 	 a 	 p a i r 	 o f 	 a d j a c e n t 	 a n g l e s
c . 	 t w o 	 d i a g o n a l s .
A n s . 	 a . 	 A 	 p a i r 	 o f 	 o p p o s i t e 	 s i d e s 	 – 	 E F 	 a n d 	 G H
b . 	 A 	 p a i r 	 o f 	 a d j a c e n t 	 a n g l e s 	 - 	 E 	 a n d 	 G
c . 	 T w o 	 d i a g o n a l s 	 – 	 E G 	 a n d 	 F H
1 7 . 	 C l a s s i f y 	 t h e 	 t r i a n g l e s 	 o n 	 t h e 	 b a s i s 	 o f 	 a n g l e s .
A n s . 	 A c u t e 	 a n g l e d 	 t r i a n g l e , 	 O b t u s e 	 a n g l e d 	 t r i a n g l e 	 a n d 	 R i g h t 	 a n g l e d 	 t r i a n g l e
1 8 . 	 L o o k 	 a t 	 t h e 	 f i g u r e 	 b e l o w
a . 	 W r i t e 	 a n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 b .
b . 	 W r i t e 	 a n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 a .
c . 	 W r i t e 	 a 	 s e t 	 o f 	 c o l l i n e a r 	 p o i n t s .
A n s . 	 a . 	 A n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 b . - 	 Q S
b . 	 A n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 a . 	 – 	 P R
c . 	 A 	 s e t 	 o f 	 c o l l i n e a r 	 p o i n t s . 	 – 	 P , 	 Q 	 a n d 	 T 	 o r 	 R , 	 S 	 a n d 	 T
1 9 . 	 T h e 	 d a t a 	 g i v e n 	 b e l o w 	 p r o v i d e s 	 i n f o r m a t i o n 	 a b o u t 	 t h e 	 n u m b e r 	 o f 	 b o o k s 	 o n 	 d i f f e r e n t
s u b j e c t s 	 i n 	 a 	 l i b r a r y . 	 R e p r e s e n t 	 t h e 	 d a t a 	 u s i n g 	 a 	 p i c t o g r a p h 	 u s i n g 	 a 	 s u i t a b l e 	 s c a l e
	
S u b j e c t E n g l i s h H i n d i M a t h e m a t i c s S c i e n c e
N o . 	 o f 	 b o o k s 1 0 0 0 6 5 0 8 0 0 4 5 0
A n s . 	 P i c t o g r a p h 	 n e a t l y 	 d r a w n
S c a l e 	 s h o w n
2 0 . 	 T h e 	 c o l o u r s 	 o f 	 c a r s 	 p r e f e r r e d 	 b y 	 p e o p l e 	 l i v i n g 	 i n 	 a 	 l o c a l i t y 	 a r e 	 s h o w n 	 b y 	 t h e
Page 5


G e n e r a l 	 I n s t r u c t i o n s :
• 	 T h i s 	 q u e s t i o n 	 p a p e r 	 h a s 	 3 1 	 q u e s t i o n s 	 .
• 	 S e c t i o n 	 A 	 c o n s i s t s 	 o f 	 4 	 q u e s t i o n s 	 o f 	 1 	 m a r k 	 e a c h .
• 	 S e c t i o n 	 B 	 c o n s i s t s 	 o f 	 6 	 q u e s t i o n s 	 o f 	 2 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 C 	 c o n s i s t s 	 o f 	 1 0 	 q u e s t i o n s 	 o f 	 3 	 m a r k s 	 e a c h .
• 	 S e c t i o n 	 D 	 c o n s i s t s 	 o f 	 1 1 	 q u e s t i o n s 	 o f 	 4 	 m a r k s 	 e a c h .
• 	 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 	 t o 	 a t t e m p t .
• 	 A t t e m p t 	 t h e 	 p a p e r 	 n e a t l y 	 a n d 	 l e g i b l y .
S E C T I O N - A
1 . 	 F i n d 	 t h e 	 e s t i m a t e d 	 d i f f e r e n c e 	 o f 	 4 8 9 3 4 8 	 a n d 	 4 8 3 6 5 .
A n s . 	 4 8 9 3 4 8 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 4 9 0 0 0 0
4 8 3 6 5 	 r o u n d e d 	 o f f 	 t o 	 n e a r e s t 	 t e n 	 t h o u s a n d s 	 = 	 5 0 0 0 0
E s t i m a t e d 	 d i f f e r e n c e 	 = 	 4 4 0 0 0 0
2 . 	 I n 	 a 	 p i c t o g r a p h , 	 i f 	 t h e 	 s y m b o l 	 	 r e p r e s e n t s 	 2 0 	 s t u d e n t s , 	 t h e n 	 h o w 	 m a n y 	 s t u d e n t s
w i l l 	 b e 	 r e p r e s e n t e d 	 b y 	 t h e 	 	 s y m b o l ?
A n s . 	 3 5 .
3 . 	 S o l v e 	 a n d 	 w r i t e 	 t h e 	 a n s w e r 	 i n 	 t h e 	 H i n d u 	 A r a b i c 	 N u m e r a l : 	 D 	 - 	 C D X L 	 = 	 _ _ _ _ _ _
A n s . 	 D 	 - 	 C D X L 	 = 	 5 0 0 	 – 	 4 4 0 	 = 	 6 0 	 = 	 L X
4 . 	 H o w 	 m a n y 	 d e g r e e s 	 a r e 	 t h e r e 	 i n 	 t h e 	 a n g l e 	 b e t w e e n 	 t h e 	 h a n d s 	 o f 	 a 	 c l o c k 	 w h e n 	 i t 	 i s
6 	 o ’ c l o c k ?
A n s . 	 1 8 0 	 d e g r e e s 	 = 	 s t r a i g h t 	 a n g l e
S E C T I O N - B
5 . 	 F i n d 	 t h e 	 p r o d u c t 	 o f 	 t h e 	 p l a c e 	 v a l u e s 	 o f 	 t w o 	 7 s 	 i n 	 1 9 7 0 4 7 .
A n s . 	 P l a c e 	 v a l u e 	 o f 	 7 	 i n 	 1 9 7 0 4 7 	 i s 	 7 0 0 0 	 a n d 	 7
P r o d u c t 	 = 	 4 9 0 0 0
6 . 	 S i m p l i f y 	 u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y 	 o f 	 w h o l e 	 n u m b e r s : 	 8 3 5 	 x 	 1 0 5
A n s . 	 8 3 5 	 × 	 1 0 5
= 	 8 3 5 	 × 	 ( 1 0 0 	 + 	 5 )
= 	 8 3 5 	 × 	 1 0 0 	 + 	 8 3 5 	 × 	 5 	 ( u s i n g 	 d i s t r i b u t i v e 	 p r o p e r t y )
= 	 8 3 5 0 0 	 + 	 4 1 7 5 	 = 	 8 7 6 7 5
7 . 	 I s 	 2 8 	 a 	 p e r f e c t 	 n u m b e r ? 	 S h o w 	 t h e 	 w o r k i n g .
A n s . 	 F a c t o r s 	 o f 	 2 8 	 a r e 	 1 , 2 , 4 , 7 , 1 4 	 a n d 	 2 8
S u m 	 o f 	 f a c t o r s 	 = 	 1 	 + 	 2 	 + 	 4 	 + 	 7 	 + 	 1 4 	 + 	 2 8 	 = 	 5 6 	 w h i c h 	 i s 	 2 	 × 	 2 8
T h e r e f o r e , 	 Y e s 	 2 8 	 i s 	 a 	 p e r f e c t 	 n u m b e r
8 . 	 F i n d 	 t h e 	 d i a m e t e r 	 o f 	 a 	 c i r c l e 	 i f 	 i t s 	 r a d i u s 	 i s 	 4 . 5 	 c m .
A n s . 	 D i a m e t e r 	 = 	 2 	 x 	 4 . 5 	 c m 	 = 	 9 	 c m
9 . 	 T h e 	 f o l l o w i n g 	 a r e 	 t h e 	 m a r k s 	 s c o r e d 	 b y 	 2 0 	 s t u d e n t s 	 o f 	 V I 	 A 	 i n 	 M a t h e m a t i c s 	 e x a m :
5 7 , 	 6 5 , 	 8 0 , 	 5 7 , 	 8 0 , 	 6 5 , 	 8 5 , 	 3 4 , 	 5 7 , 	 4 1 , 	 5 4 , 	 8 0 , 	 4 1 , 	 5 4 , 	 3 4 , 	 4 1 , 	 8 5 , 	 5 7 , 	 8 0 , 	 6 5
P r e p a r e 	 a 	 f r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e .
A n s . 	 F r e q u e n c y 	 d i s t r i b u t i o n 	 t a b l e 	 n e a t l y 	 d r a w n
1 0 . 	 S t a t e 	 t h e 	 c o m m u t a t i v e 	 p r o p e r t y 	 o f 	 a d d i t i o n 	 o f 	 w h o l e 	 n u m b e r s . 	 G i v e 	 a n 	 e x a m p l e .
A n s . 	 a 	 + 	 b 	 = 	 b 	 + 	 a
F o r 	 e g : 	 3 	 + 5 	 = 	 5 	 + 	 3 	 = 	 8
S E C T I O N - C
1 1 . 	 S u b t r a c t 	 2 5 2 7 0 5 3 	 f r o m 	 2 7 2 5 3 5 0 	 a n d 	 w r i t e 	 t h e 	 n u m b e r 	 s e n t e n c e .
A n s . 	 S u b t r a c t i o n 	 s h o w n
N u m b e r 	 s e n t e n c e : 	 2 7 2 5 3 5 0 	 – 	 2 5 2 7 0 5 3 	 = 	 1 9 8 2 9
1 2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 4 - d i g i t 	 n u m b e r 	 w h i c h 	 i s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 3 7 .
A n s . 	 G r e a t e s t 	 4 - d i g i t 	 n u m b e r = 	 9 9 9 9
D i v i d e 	 9 9 9 9 	 b y 	 3 7 	 R e m a i n d e r = 	 9
N o w , 	 r e q u i r e d 	 n u m b e r 	 = 	 9 9 9 9 	 – 	 9 	 = 	 9 9 9 0
1 3 . 	 I n 	 5 6 * 8 9 1 	 r e p l a c e 	 * 	 b y 	 ( a ) 	 b y 	 3 .
A n s . 	 A c c o r d i n g 	 t o 	 d i v i s i b i l i t y 	 t e s t 	 o f 	 3 , 	 S u m 	 o f 	 d i g i t s 	 = 	 5 	 + 	 6 	 + 	 * 	 + 	 8 	 + 	 9 	 + 	 1
= 	 2 9 	 + 	 *
( a ) 	 S m a l l e s t 	 d i g i t 	 = 	 1 	 ( b ) 	 L a r g e s t 	 d i g i t 	 = 	 7
1 4 . 	 F i n d 	 t h e 	 H C F 	 o f 	 3 8 5 	 a n d 	 6 2 1 . 	 A r e 	 t h e y 	 c o p r i m e
A n s . 	 H C F 	 o f 	 3 8 5 	 a n d 	 6 2 1 	 = 	 1
Y e s 	 t h e y 	 a r e 	 C o p r i m e
1 5 . 	 N a m e 	 a n y 	 s i x 	 a n g l e s 	 i n 	 t h e 	 g i v e n 	 f i g u r e :
A n s . 	 A B C , 	 B A C , 	 A C B , 	 B D C , 	 D C B , 	 	 D B C , 	 A B D 	 a n d 	 A C D 	 ( a t 	 l e a s t 	 6 )
1 6 . 	 D r a w 	 a 	 r o u g h 	 s k e t c h 	 o f 	 a 	 q u a d r i l a t e r a l 	 E F G H 	 a n d 	 n a m e
a . 	 a 	 p a i r 	 o f 	 o p p o s i t e 	 s i d e s .
b . 	 a 	 p a i r 	 o f 	 a d j a c e n t 	 a n g l e s
c . 	 t w o 	 d i a g o n a l s .
A n s . 	 a . 	 A 	 p a i r 	 o f 	 o p p o s i t e 	 s i d e s 	 – 	 E F 	 a n d 	 G H
b . 	 A 	 p a i r 	 o f 	 a d j a c e n t 	 a n g l e s 	 - 	 E 	 a n d 	 G
c . 	 T w o 	 d i a g o n a l s 	 – 	 E G 	 a n d 	 F H
1 7 . 	 C l a s s i f y 	 t h e 	 t r i a n g l e s 	 o n 	 t h e 	 b a s i s 	 o f 	 a n g l e s .
A n s . 	 A c u t e 	 a n g l e d 	 t r i a n g l e , 	 O b t u s e 	 a n g l e d 	 t r i a n g l e 	 a n d 	 R i g h t 	 a n g l e d 	 t r i a n g l e
1 8 . 	 L o o k 	 a t 	 t h e 	 f i g u r e 	 b e l o w
a . 	 W r i t e 	 a n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 b .
b . 	 W r i t e 	 a n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 a .
c . 	 W r i t e 	 a 	 s e t 	 o f 	 c o l l i n e a r 	 p o i n t s .
A n s . 	 a . 	 A n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 b . - 	 Q S
b . 	 A n o t h e r 	 w a y 	 o f 	 n a m i n g 	 l i n e 	 a . 	 – 	 P R
c . 	 A 	 s e t 	 o f 	 c o l l i n e a r 	 p o i n t s . 	 – 	 P , 	 Q 	 a n d 	 T 	 o r 	 R , 	 S 	 a n d 	 T
1 9 . 	 T h e 	 d a t a 	 g i v e n 	 b e l o w 	 p r o v i d e s 	 i n f o r m a t i o n 	 a b o u t 	 t h e 	 n u m b e r 	 o f 	 b o o k s 	 o n 	 d i f f e r e n t
s u b j e c t s 	 i n 	 a 	 l i b r a r y . 	 R e p r e s e n t 	 t h e 	 d a t a 	 u s i n g 	 a 	 p i c t o g r a p h 	 u s i n g 	 a 	 s u i t a b l e 	 s c a l e
	
S u b j e c t E n g l i s h H i n d i M a t h e m a t i c s S c i e n c e
N o . 	 o f 	 b o o k s 1 0 0 0 6 5 0 8 0 0 4 5 0
A n s . 	 P i c t o g r a p h 	 n e a t l y 	 d r a w n
S c a l e 	 s h o w n
2 0 . 	 T h e 	 c o l o u r s 	 o f 	 c a r s 	 p r e f e r r e d 	 b y 	 p e o p l e 	 l i v i n g 	 i n 	 a 	 l o c a l i t y 	 a r e 	 s h o w n 	 b y 	 t h e
f o l l o w i n g 	 p i c t o g r a p h . 	 A n s w e r 	 t h e 	 q u e s t i o n s 	 t h a t 	 f o l l o w :
C o l o u r 	 o f 	 c a r N o . 	 o f 	 p e o p l e 	 ( 1 	 $ 	 = 	 1 5 	 p e o p l e )
B l a c k 	 	 $
S i l v e r 	
R e d 	 $
W h i t e 	 	
a . 	 F i n d 	 t h e 	 n u m b e r 	 o f 	 p e o p l e 	 p r e f e r r i n g 	 s i l v e r 	 c a r s .
b . 	 H o w 	 m a n y 	 l e s s 	 p e o p l e 	 p r e f e r 	 r e d 	 c a r s 	 t h a n 	 w h i t e 	 c a r s ?
c . 	 W h i c h 	 c o l o u r 	 i s 	 m o s t 	 l i k e d 	 b y 	 t h e 	 p e o p l e ?
A n s . 	 a . 	 - 	 6 0
b . 	 - 4 5
c . 	 - 	 W h i t e
S E C T I O N 	 D
2 1 . 	 U s i n g 	 t h e 	 d i g i t s 	 7 , 0 , 3 , 5 	 a n d 	 2 , 	 m a k e 	 a n y 	 f o u r 	 7 	 d i g i t 	 n u m b e r s . 	 I n s e r t 	 c o m m a s 	 u s i n g
I n t e r n a t i o n a l 	 s y s t e m 	 o f 	 n u m e r a t i o n . 	 A r r a n g e 	 t h e m 	 i n 	 a s c e n d i n g 	 o r d e r .
A n s . 	 A n y 	 f o u r 	 7 - 	 d i g i t 	 n u m b e r s 	 u s i n g 	 7 , 0 , 3 , 5 	 a n d 	 2
I n s e r t 	 c o m m a s 	 u s i n g 	 I n t e r n a t i o n a l 	 s y s t e m 	 o f 	 n u m e r a t i o n .
A s c e n d i n g 	 o r d e r
2 2 . 	 T h e 	 f o l l o w i n g 	 t a b l e 	 s h o w s 	 t h e 	 y e a r l y 	 p r o d u c t i o n 	 o f 	 c y c l e s 	 b y 	 a 	 c o m p a n y . 	 D r a w 	 a 	 b a r
g r a p h 	 f o r 	 t h e 	 g i v e n 	 d a t a .
A n s . 	 B a r 	 g r a p h 	 f o r 	 t h e 	 g i v e n 	 d a t a 	 n e a t l y 	 d r a w n .
T i t l e , 	 S c a l e , 	 L a b e l s
2 3 . 	 A 	 V o l v o 	 b u s 	 d r i v e r 	 o f f e r s 	 f r e e 	 s e r v i c e 	 t o 	 a 	 p i l g r i m a g e 	 ( h o l y 	 p l a c e ) 	 w h i c h 	 i s 	 2 1 k m
4 5 m 	 a w a y 	 f r o m 	 a 	 c i t y . 	 T h e 	 b u s 	 m a k e s 	 5 	 r o u n d 	 t r i p s 	 e v e r y 	 d a y . 	 H o w 	 m u c h 	 d i s t a n c e 	 d o e s
i t 	 c o v e r 	 i n 	 t h e 	 m o n t h 	 o f 	 A p r i l ? 	 W h a t 	 v a l u e 	 d o 	 w e 	 l e a r n 	 f r o m 	 t h i s ?

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FAQs on Class 6 Math: CBSE Past Year Paper - 5

1. What is the CBSE Class 6 Math Past Year Paper - 5?

Ans. The CBSE Class 6 Math Past Year Paper - 5 is a previous year's question paper for the Class 6 Math examination conducted by the Central Board of Secondary Education (CBSE) in India. It is used for practice and revision purposes by students preparing for their exams.

2. How can solving the CBSE Class 6 Math Past Year Paper - 5 benefit students?

Ans. Solving the CBSE Class 6 Math Past Year Paper - 5 can benefit students in several ways. It helps them understand the exam pattern, familiarize themselves with the types of questions asked, and assess their preparation level. By solving the paper, students can identify their strengths and weaknesses and focus on areas that need improvement.

3. Are the questions in the CBSE Class 6 Math Past Year Paper - 5 similar to the actual exam?

Ans. Yes, the questions in the CBSE Class 6 Math Past Year Paper - 5 are designed to be similar to the ones asked in the actual exam. These papers are created by experts who have a good understanding of the exam pattern and syllabus. However, it is important to note that the actual exam may have variations in terms of difficulty level and question distribution.

4. Can solving the CBSE Class 6 Math Past Year Paper - 5 guarantee good marks in the exam?

Ans. Solving the CBSE Class 6 Math Past Year Paper - 5 can certainly improve the chances of scoring well in the exam. It helps students become familiar with the question format, manage time effectively, and gain confidence. However, achieving good marks also depends on regular study, understanding concepts, and practicing additional questions beyond the past year paper.

5. Where can I find the CBSE Class 6 Math Past Year Paper - 5?

Ans. The CBSE Class 6 Math Past Year Paper - 5 can be found on various educational websites, online platforms, or in preparation books specifically designed for CBSE exams. These resources are easily accessible and often provide solutions or answer keys to help students evaluate their performance. Additionally, schools or coaching centers may also provide students with the past year papers for practice.

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