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West Bengal: State Level Solutions (SAT) - 2015-16

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 Page 1


     
        
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015 - 16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – ANSWER KEY  
 
 
1 C  11 A 21 A  31  B 41  B 
2  C 12 D 22  C 32  C 42  D 
3  B 13 B 23  D 33  C 43  B 
4  D 14 A 24  A 34  B 44  C 
5  D 15 C 25  C 35  C 45  A 
6  A 16 C 26  A 36  D 46  C 
7  D 17 C 27  B 37  D 47  C 
8  B 18 A 28  C 38  D 48  D 
9  B 19 B 29  B 39  D 49  D 
10  D 20  D 30  D 40  B 50  D 
 
51  B 61 C  71  B 81 D  91  B 
52  C 62  A/B 72  A 82  A 92  D 
53  B 63  B 73  C 83  C 93  D 
54  A 64  D 74  D 84  B 94  C 
55  B 65  A 75  B 85  A 95  B 
56  D 66  C 76  D 86  C 96  B 
57  B 67  D 77  B 87  A 97  C 
58  D 68  B 78  B 88  A 98  D 
59  B 69  C 79  B 89  C 99  A 
60 B 70  C 80  A 90  A 100  D 
 
 
 
 
 
 
 
Page 2


     
        
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015 - 16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – ANSWER KEY  
 
 
1 C  11 A 21 A  31  B 41  B 
2  C 12 D 22  C 32  C 42  D 
3  B 13 B 23  D 33  C 43  B 
4  D 14 A 24  A 34  B 44  C 
5  D 15 C 25  C 35  C 45  A 
6  A 16 C 26  A 36  D 46  C 
7  D 17 C 27  B 37  D 47  C 
8  B 18 A 28  C 38  D 48  D 
9  B 19 B 29  B 39  D 49  D 
10  D 20  D 30  D 40  B 50  D 
 
51  B 61 C  71  B 81 D  91  B 
52  C 62  A/B 72  A 82  A 92  D 
53  B 63  B 73  C 83  C 93  D 
54  A 64  D 74  D 84  B 94  C 
55  B 65  A 75  B 85  A 95  B 
56  D 66  C 76  D 86  C 96  B 
57  B 67  D 77  B 87  A 97  C 
58  D 68  B 78  B 88  A 98  D 
59  B 69  C 79  B 89  C 99  A 
60 B 70  C 80  A 90  A 100  D 
 
 
 
 
 
 
 
         
     
       
 
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015-16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – SOLUTION  
 
 
MATHEMATICS 
 
1. Sol. 
? ? ? ?
2
2
2
2
2
1 1 1
f 2x 4x 1
x4 x
11
2x 4 1
4x
11
2x
4x
1
f x x
4
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
??
??
??
??
??
??
?? ??
??
??
??
??
?
 
2. Sol. 
? ? ? ?
x 2 x 3
x 2 x 3 0
x 3 x 1 0
x 3, x 1
x9
??
? ? ?
? ? ?
??
? ? ?
??
??
 
3. Sol. 
5 2 6 ? 
=
? ? ? ?
22
3 2 2 6
32
??
??
 
4. Sol. 
xy
ab
yx
x y x y
a 1 b 1
yx
11
1
a 1 1 b
??
??
? ? ? ?
??
??
 
5. Sol. 
 
? ? ? ?
? ? ? ?
22
22
2 x 2x 1 1 3 y 6y 9 9 31
2 x 1 2 3 y 3 27 31
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
 
Least value = 2 
Page 3


     
        
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015 - 16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – ANSWER KEY  
 
 
1 C  11 A 21 A  31  B 41  B 
2  C 12 D 22  C 32  C 42  D 
3  B 13 B 23  D 33  C 43  B 
4  D 14 A 24  A 34  B 44  C 
5  D 15 C 25  C 35  C 45  A 
6  A 16 C 26  A 36  D 46  C 
7  D 17 C 27  B 37  D 47  C 
8  B 18 A 28  C 38  D 48  D 
9  B 19 B 29  B 39  D 49  D 
10  D 20  D 30  D 40  B 50  D 
 
51  B 61 C  71  B 81 D  91  B 
52  C 62  A/B 72  A 82  A 92  D 
53  B 63  B 73  C 83  C 93  D 
54  A 64  D 74  D 84  B 94  C 
55  B 65  A 75  B 85  A 95  B 
56  D 66  C 76  D 86  C 96  B 
57  B 67  D 77  B 87  A 97  C 
58  D 68  B 78  B 88  A 98  D 
59  B 69  C 79  B 89  C 99  A 
60 B 70  C 80  A 90  A 100  D 
 
 
 
 
 
 
 
         
     
       
 
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015-16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – SOLUTION  
 
 
MATHEMATICS 
 
1. Sol. 
? ? ? ?
2
2
2
2
2
1 1 1
f 2x 4x 1
x4 x
11
2x 4 1
4x
11
2x
4x
1
f x x
4
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
??
??
??
??
??
??
?? ??
??
??
??
??
?
 
2. Sol. 
? ? ? ?
x 2 x 3
x 2 x 3 0
x 3 x 1 0
x 3, x 1
x9
??
? ? ?
? ? ?
??
? ? ?
??
??
 
3. Sol. 
5 2 6 ? 
=
? ? ? ?
22
3 2 2 6
32
??
??
 
4. Sol. 
xy
ab
yx
x y x y
a 1 b 1
yx
11
1
a 1 1 b
??
??
? ? ? ?
??
??
 
5. Sol. 
 
? ? ? ?
? ? ? ?
22
22
2 x 2x 1 1 3 y 6y 9 9 31
2 x 1 2 3 y 3 27 31
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
 
Least value = 2 
         
     
       
 
6. Sol. 
? ?
2
22
22
2
2r h r h
4r 4rh r
3r 4rh
r
4:3
h
? ? ?
??
?
?
 
7. Sol. 
y = c (cy + bz) + az 
(1 – c
2
)y = (bc + a) z ……….(1) 
Z = b(cy + bz) + ay 
(1 – b
2
) z = (bc + a) y ……….(2) 
2
2
2 2 2 2
1 c bc a
bc a 1b
1 b c b c
??
?
? ?
? ? ?
22
bc ?
2
2 2 2
abc abc a
a b c 1 2abc
???
? ? ? ? ?
 
8. Sol. 
x(x
3
 – 1) < 0 
x(x – 1) (x
2
 + x + 1) < 0 
x
2
 + x + 1 > 0 because discriminant is – ve 
x(x – 1) < 0  0 x 1 ? ? ? 
9. Sol. 
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
250 200 150 100
50 50 50 50
5 4 3 2
50 50 50 50
2 3 4 5
2 3 4 5
32 81 64 25
?
?
 
10. Sol. 
4 *5 + 5 * 6 
= (4 + 5 – 4 × 5) + (5 + 6 – 5 × 6) 
= - 30 
11. Sol. 
Let principal amount is A. then  
25
A1
A 10 1
2
1250 A 50,000
100 100
??
??
? ? ? ? 
12. Sol. 
Net cost price 
198 198
10 10
11
100 100
??
??
 
10 10
198 400
11 9
Selling price 2 198 396
loss 400 396 4
4
%loss 100 1
400
??
? ? ?
??
??
? ? ?
? ? ?
? ? ?
 
13. Sol. 
Let the price two years before be P.  
So, current price will be 
2
4
P1
100
??
??
??
??
 
Page 4


     
        
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015 - 16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – ANSWER KEY  
 
 
1 C  11 A 21 A  31  B 41  B 
2  C 12 D 22  C 32  C 42  D 
3  B 13 B 23  D 33  C 43  B 
4  D 14 A 24  A 34  B 44  C 
5  D 15 C 25  C 35  C 45  A 
6  A 16 C 26  A 36  D 46  C 
7  D 17 C 27  B 37  D 47  C 
8  B 18 A 28  C 38  D 48  D 
9  B 19 B 29  B 39  D 49  D 
10  D 20  D 30  D 40  B 50  D 
 
51  B 61 C  71  B 81 D  91  B 
52  C 62  A/B 72  A 82  A 92  D 
53  B 63  B 73  C 83  C 93  D 
54  A 64  D 74  D 84  B 94  C 
55  B 65  A 75  B 85  A 95  B 
56  D 66  C 76  D 86  C 96  B 
57  B 67  D 77  B 87  A 97  C 
58  D 68  B 78  B 88  A 98  D 
59  B 69  C 79  B 89  C 99  A 
60 B 70  C 80  A 90  A 100  D 
 
 
 
 
 
 
 
         
     
       
 
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015-16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – SOLUTION  
 
 
MATHEMATICS 
 
1. Sol. 
? ? ? ?
2
2
2
2
2
1 1 1
f 2x 4x 1
x4 x
11
2x 4 1
4x
11
2x
4x
1
f x x
4
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
??
??
??
??
??
??
?? ??
??
??
??
??
?
 
2. Sol. 
? ? ? ?
x 2 x 3
x 2 x 3 0
x 3 x 1 0
x 3, x 1
x9
??
? ? ?
? ? ?
??
? ? ?
??
??
 
3. Sol. 
5 2 6 ? 
=
? ? ? ?
22
3 2 2 6
32
??
??
 
4. Sol. 
xy
ab
yx
x y x y
a 1 b 1
yx
11
1
a 1 1 b
??
??
? ? ? ?
??
??
 
5. Sol. 
 
? ? ? ?
? ? ? ?
22
22
2 x 2x 1 1 3 y 6y 9 9 31
2 x 1 2 3 y 3 27 31
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
 
Least value = 2 
         
     
       
 
6. Sol. 
? ?
2
22
22
2
2r h r h
4r 4rh r
3r 4rh
r
4:3
h
? ? ?
??
?
?
 
7. Sol. 
y = c (cy + bz) + az 
(1 – c
2
)y = (bc + a) z ……….(1) 
Z = b(cy + bz) + ay 
(1 – b
2
) z = (bc + a) y ……….(2) 
2
2
2 2 2 2
1 c bc a
bc a 1b
1 b c b c
??
?
? ?
? ? ?
22
bc ?
2
2 2 2
abc abc a
a b c 1 2abc
???
? ? ? ? ?
 
8. Sol. 
x(x
3
 – 1) < 0 
x(x – 1) (x
2
 + x + 1) < 0 
x
2
 + x + 1 > 0 because discriminant is – ve 
x(x – 1) < 0  0 x 1 ? ? ? 
9. Sol. 
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
250 200 150 100
50 50 50 50
5 4 3 2
50 50 50 50
2 3 4 5
2 3 4 5
32 81 64 25
?
?
 
10. Sol. 
4 *5 + 5 * 6 
= (4 + 5 – 4 × 5) + (5 + 6 – 5 × 6) 
= - 30 
11. Sol. 
Let principal amount is A. then  
25
A1
A 10 1
2
1250 A 50,000
100 100
??
??
? ? ? ? 
12. Sol. 
Net cost price 
198 198
10 10
11
100 100
??
??
 
10 10
198 400
11 9
Selling price 2 198 396
loss 400 396 4
4
%loss 100 1
400
??
? ? ?
??
??
? ? ?
? ? ?
? ? ?
 
13. Sol. 
Let the price two years before be P.  
So, current price will be 
2
4
P1
100
??
??
??
??
 
         
     
       
 
2
4
P 1 6,76,000
100
??
? ? ? ?
??
??
 
14. Sol. 
OP
2
 = x
2
 + y
2
 
OR
2
 = (x – a)
2
 + (y – b)
2
 
OQ
2
 = (x – a)
2
 + y
2
 
OS
2
 = x
2
 + (y – b)
2
 
2 2 2 2
OP OR OQ OS ? ? ? ? 
? ? O x, y
? ? P o,o
? ? Q a ,o
? ? R a,b ? ? S o,b
 
15. Sol. 
?
? ?
Triangle 1
?
? ?
Triangle 2
 
So, both the triangles are similar and in case of similar triangles, area of triangles are in ratio of 
square of their heights  
2
11
2
2 2
hh 93
16 h 4 h
? ? ? ? 
16. Sol. 
Volume of tetrahedron = 
3
a
62
where 
a length of edge ? 
17. Sol. 
Volume of parallelopide = Number of coins × Volume of single coin 
2
11 9 6 n 1.5 0.25 ? ? ? ? ? ? ? ? 
18. Sol. 
sin
4
A = cos
2
A 
tan
2
A – tan
4
A 
24
24
22
44
sin A sin A
cos A cos A
sin A cos A
sin A cos A
??
??
 
22
22
22
cosec A sec A
cos A sin A
sin Acos A
??
?
 
4 2 2 2
2 2 2 2
sin A sin A sin A cos A
sin Acos A sin Acos A
1
??
??
??
 
19. Sol. 
22
sin x cos x
22 ? 
= 
2
2
sin x
sin x
2
2
2
? 
Applying A.M G.M ? 
Page 5


     
        
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015 - 16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – ANSWER KEY  
 
 
1 C  11 A 21 A  31  B 41  B 
2  C 12 D 22  C 32  C 42  D 
3  B 13 B 23  D 33  C 43  B 
4  D 14 A 24  A 34  B 44  C 
5  D 15 C 25  C 35  C 45  A 
6  A 16 C 26  A 36  D 46  C 
7  D 17 C 27  B 37  D 47  C 
8  B 18 A 28  C 38  D 48  D 
9  B 19 B 29  B 39  D 49  D 
10  D 20  D 30  D 40  B 50  D 
 
51  B 61 C  71  B 81 D  91  B 
52  C 62  A/B 72  A 82  A 92  D 
53  B 63  B 73  C 83  C 93  D 
54  A 64  D 74  D 84  B 94  C 
55  B 65  A 75  B 85  A 95  B 
56  D 66  C 76  D 86  C 96  B 
57  B 67  D 77  B 87  A 97  C 
58  D 68  B 78  B 88  A 98  D 
59  B 69  C 79  B 89  C 99  A 
60 B 70  C 80  A 90  A 100  D 
 
 
 
 
 
 
 
         
     
       
 
 
 
NATIONAL TALENT SEARCH EXAMINATION, 2015-16 
(STATE LEVEL) 
(FOR STUDENTS STUDYING IN CLASS X) 
SAT – SOLUTION  
 
 
MATHEMATICS 
 
1. Sol. 
? ? ? ?
2
2
2
2
2
1 1 1
f 2x 4x 1
x4 x
11
2x 4 1
4x
11
2x
4x
1
f x x
4
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
??
??
??
??
??
??
?? ??
??
??
??
??
?
 
2. Sol. 
? ? ? ?
x 2 x 3
x 2 x 3 0
x 3 x 1 0
x 3, x 1
x9
??
? ? ?
? ? ?
??
? ? ?
??
??
 
3. Sol. 
5 2 6 ? 
=
? ? ? ?
22
3 2 2 6
32
??
??
 
4. Sol. 
xy
ab
yx
x y x y
a 1 b 1
yx
11
1
a 1 1 b
??
??
? ? ? ?
??
??
 
5. Sol. 
 
? ? ? ?
? ? ? ?
22
22
2 x 2x 1 1 3 y 6y 9 9 31
2 x 1 2 3 y 3 27 31
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
 
Least value = 2 
         
     
       
 
6. Sol. 
? ?
2
22
22
2
2r h r h
4r 4rh r
3r 4rh
r
4:3
h
? ? ?
??
?
?
 
7. Sol. 
y = c (cy + bz) + az 
(1 – c
2
)y = (bc + a) z ……….(1) 
Z = b(cy + bz) + ay 
(1 – b
2
) z = (bc + a) y ……….(2) 
2
2
2 2 2 2
1 c bc a
bc a 1b
1 b c b c
??
?
? ?
? ? ?
22
bc ?
2
2 2 2
abc abc a
a b c 1 2abc
???
? ? ? ? ?
 
8. Sol. 
x(x
3
 – 1) < 0 
x(x – 1) (x
2
 + x + 1) < 0 
x
2
 + x + 1 > 0 because discriminant is – ve 
x(x – 1) < 0  0 x 1 ? ? ? 
9. Sol. 
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
250 200 150 100
50 50 50 50
5 4 3 2
50 50 50 50
2 3 4 5
2 3 4 5
32 81 64 25
?
?
 
10. Sol. 
4 *5 + 5 * 6 
= (4 + 5 – 4 × 5) + (5 + 6 – 5 × 6) 
= - 30 
11. Sol. 
Let principal amount is A. then  
25
A1
A 10 1
2
1250 A 50,000
100 100
??
??
? ? ? ? 
12. Sol. 
Net cost price 
198 198
10 10
11
100 100
??
??
 
10 10
198 400
11 9
Selling price 2 198 396
loss 400 396 4
4
%loss 100 1
400
??
? ? ?
??
??
? ? ?
? ? ?
? ? ?
 
13. Sol. 
Let the price two years before be P.  
So, current price will be 
2
4
P1
100
??
??
??
??
 
         
     
       
 
2
4
P 1 6,76,000
100
??
? ? ? ?
??
??
 
14. Sol. 
OP
2
 = x
2
 + y
2
 
OR
2
 = (x – a)
2
 + (y – b)
2
 
OQ
2
 = (x – a)
2
 + y
2
 
OS
2
 = x
2
 + (y – b)
2
 
2 2 2 2
OP OR OQ OS ? ? ? ? 
? ? O x, y
? ? P o,o
? ? Q a ,o
? ? R a,b ? ? S o,b
 
15. Sol. 
?
? ?
Triangle 1
?
? ?
Triangle 2
 
So, both the triangles are similar and in case of similar triangles, area of triangles are in ratio of 
square of their heights  
2
11
2
2 2
hh 93
16 h 4 h
? ? ? ? 
16. Sol. 
Volume of tetrahedron = 
3
a
62
where 
a length of edge ? 
17. Sol. 
Volume of parallelopide = Number of coins × Volume of single coin 
2
11 9 6 n 1.5 0.25 ? ? ? ? ? ? ? ? 
18. Sol. 
sin
4
A = cos
2
A 
tan
2
A – tan
4
A 
24
24
22
44
sin A sin A
cos A cos A
sin A cos A
sin A cos A
??
??
 
22
22
22
cosec A sec A
cos A sin A
sin Acos A
??
?
 
4 2 2 2
2 2 2 2
sin A sin A sin A cos A
sin Acos A sin Acos A
1
??
??
??
 
19. Sol. 
22
sin x cos x
22 ? 
= 
2
2
sin x
sin x
2
2
2
? 
Applying A.M G.M ? 
         
     
       
 
2
2
2
2
22
sin x
sin x
sin x
sin x
sin x cos x
2
2
2
2
2.
2
2
2 2 2 2
?
?
??
 
Least value = 22 
20. Sol. 
AC ? ? ? 
BD ? ? ? 
L.H.S = 
A C B D
tan tan tan tan
2 2 2 2
? 
 
C C D D
tan tan tan tan
2 2 2 2 2 2
C C D D
C tan C tan
2 2 2 2
2
?? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
 
PHYSICS 
21. Sol. 
2
2
1
E mv
2
1P
E
2m
P 2mE
?
??
?
 
22. Sol. 
21
21
12
Pv nRT
nR
vT
P
nR
tan
P
1
tan
P
tan tan
PP
??
?
??
??
? ? ?
? ? ?
??
 
 
 
 
 
2
?
1
?
1
P
2
P
V
T
 
23. Sol. 
Harmonics are multiple of fundamental frequencies. 
24. Sol. 
2
2
1
E k A
2
EA
?
?
 
25. Sol. 
Sinc = 
1
?
 
C = 45
o
 
26. Sol. 
In the absence of atmosphere, no scattering of light. 
27. Sol.
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West Bengal: State Level Solutions (SAT) - 2015-16 Notes

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West Bengal: State Level Solutions (SAT) - 2015-16 Class 10 Questions

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