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Delhi: State Level Solutions (MAT) - 2015-16
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Page 1
NTSE-I (2015)
(For Class X Students)
(MAT)
Time: 45 Minutes Max Marks: 50
INSTRUCTIONS TO CANDIDATES
Read the following instructions carefully before you open the Question Booklet.
1. Use blue/black ball point pen only.
2. Write your Roll No. very clearly (only one digit in on block) on this booklet and on the ANSWER
SHEET.
3. This test consists of 50 questions of one mark each. All the questions are COMPULSORY.
4. Answer to each question by filling the correct alternative among the four choices on the answer sheet.
Example:
Q.No.
Q.No.
1
1
Alternatives
Alternatives
2 4
1
1 2 3 ? 4
Correct way:
Wrong way:
5. Separate sheet has been provided for rough work in this test booklet.
Page 2
NTSE-I (2015)
(For Class X Students)
(MAT)
Time: 45 Minutes Max Marks: 50
INSTRUCTIONS TO CANDIDATES
Read the following instructions carefully before you open the Question Booklet.
1. Use blue/black ball point pen only.
2. Write your Roll No. very clearly (only one digit in on block) on this booklet and on the ANSWER
SHEET.
3. This test consists of 50 questions of one mark each. All the questions are COMPULSORY.
4. Answer to each question by filling the correct alternative among the four choices on the answer sheet.
Example:
Q.No.
Q.No.
1
1
Alternatives
Alternatives
2 4
1
1 2 3 ? 4
Correct way:
Wrong way:
5. Separate sheet has been provided for rough work in this test booklet.
NTSE
STAGE I (MAT)
HINTS & SOLUTIONS
1. 2
Sol.
22
p q 2pq ??
22
p q 2pq 0 ? ? ?
? ?
2
p q 0 ??
p = q
23 7
pq
qp
? ? ? ?
?
? ? ? ?
? ? ? ?
= 1 + 1 = 2
2. 4
Sol. Let distance be d.
?
d d 36
4 5 60
??
d 36
20 60
?
d = 12 km
? Actual time to reach on time =
12
1
4
? = 2 hrs
? Required speed = 6 km/hr
3. 2
Sol. 3A = 4B, 2C = 3B
A 4 8 B 2 6
,
B 3 6 C 3 9
? ? ? ?
? A : B : C = 8 : 6 : 9
4. 2
Sol. HCF =
? ?
? ?
hcf 6,4,2
lcm 5,15,5
= 2/15
5. 3
Sol.
3
10
7290 x 1
100
??
??
??
??
3
9
7290 x
10
??
?
??
??
7290 1000
x
729
?
?
x = Rs 10000
6. 4
Sol.
11
2 3 5 2 3 5
?
? ? ? ?
=
? ? ? ?
2 3 5 2 3 5
2 5 3 2 5 3
? ? ? ? ?
? ? ? ?
Page 3
NTSE-I (2015)
(For Class X Students)
(MAT)
Time: 45 Minutes Max Marks: 50
INSTRUCTIONS TO CANDIDATES
Read the following instructions carefully before you open the Question Booklet.
1. Use blue/black ball point pen only.
2. Write your Roll No. very clearly (only one digit in on block) on this booklet and on the ANSWER
SHEET.
3. This test consists of 50 questions of one mark each. All the questions are COMPULSORY.
4. Answer to each question by filling the correct alternative among the four choices on the answer sheet.
Example:
Q.No.
Q.No.
1
1
Alternatives
Alternatives
2 4
1
1 2 3 ? 4
Correct way:
Wrong way:
5. Separate sheet has been provided for rough work in this test booklet.
NTSE
STAGE I (MAT)
HINTS & SOLUTIONS
1. 2
Sol.
22
p q 2pq ??
22
p q 2pq 0 ? ? ?
? ?
2
p q 0 ??
p = q
23 7
pq
qp
? ? ? ?
?
? ? ? ?
? ? ? ?
= 1 + 1 = 2
2. 4
Sol. Let distance be d.
?
d d 36
4 5 60
??
d 36
20 60
?
d = 12 km
? Actual time to reach on time =
12
1
4
? = 2 hrs
? Required speed = 6 km/hr
3. 2
Sol. 3A = 4B, 2C = 3B
A 4 8 B 2 6
,
B 3 6 C 3 9
? ? ? ?
? A : B : C = 8 : 6 : 9
4. 2
Sol. HCF =
? ?
? ?
hcf 6,4,2
lcm 5,15,5
= 2/15
5. 3
Sol.
3
10
7290 x 1
100
??
??
??
??
3
9
7290 x
10
??
?
??
??
7290 1000
x
729
?
?
x = Rs 10000
6. 4
Sol.
11
2 3 5 2 3 5
?
? ? ? ?
=
? ? ? ?
2 3 5 2 3 5
2 5 3 2 5 3
? ? ? ? ?
? ? ? ?
=
? ?
? ? ? ?
22
2 2 5
2 5 3
?
??
=
? ?
2 2 5
7 2 10 3
?
??
=
? ?
2 2 5
4 2 10
?
?
=
? ?
? ?
2 2 5
2 2 2 5
?
?
=
1
2
7. 4
Sol.
2x y x y 3/2
3 3 27 3
??
? ? ?
3
2x y x y
2
? ? ? ?
3
2x y
2
??
And x + y =
3
2
x = 1, y = ½
?
x y 1/2
3 3 3
?
??
8. 1
Sol. Speed of A = a m/s
Speed of B = b m/s
?
100 100
5
ba
?? and
80 100
ba
?
=
a 5 4a
b
b 4 5
? ? ?
100 100
5
4a
a
5
??
Solving, we get a = 5 m/s
9. 3
Sol. 11 11 11 11... x ??
2
x 11x ?
2
x 11x 0 ??
? ? x x 11 0 ??
x 0, x 11 ? ? ?
10. 3
Sol. Since there are only odd multiples of 5, unit digit in the product will be 5.
11. 1
Sol. Final change = reduction by
2
12
%
100
??
??
??
??
= reduction by 1.44%
Page 4
NTSE-I (2015)
(For Class X Students)
(MAT)
Time: 45 Minutes Max Marks: 50
INSTRUCTIONS TO CANDIDATES
Read the following instructions carefully before you open the Question Booklet.
1. Use blue/black ball point pen only.
2. Write your Roll No. very clearly (only one digit in on block) on this booklet and on the ANSWER
SHEET.
3. This test consists of 50 questions of one mark each. All the questions are COMPULSORY.
4. Answer to each question by filling the correct alternative among the four choices on the answer sheet.
Example:
Q.No.
Q.No.
1
1
Alternatives
Alternatives
2 4
1
1 2 3 ? 4
Correct way:
Wrong way:
5. Separate sheet has been provided for rough work in this test booklet.
NTSE
STAGE I (MAT)
HINTS & SOLUTIONS
1. 2
Sol.
22
p q 2pq ??
22
p q 2pq 0 ? ? ?
? ?
2
p q 0 ??
p = q
23 7
pq
qp
? ? ? ?
?
? ? ? ?
? ? ? ?
= 1 + 1 = 2
2. 4
Sol. Let distance be d.
?
d d 36
4 5 60
??
d 36
20 60
?
d = 12 km
? Actual time to reach on time =
12
1
4
? = 2 hrs
? Required speed = 6 km/hr
3. 2
Sol. 3A = 4B, 2C = 3B
A 4 8 B 2 6
,
B 3 6 C 3 9
? ? ? ?
? A : B : C = 8 : 6 : 9
4. 2
Sol. HCF =
? ?
? ?
hcf 6,4,2
lcm 5,15,5
= 2/15
5. 3
Sol.
3
10
7290 x 1
100
??
??
??
??
3
9
7290 x
10
??
?
??
??
7290 1000
x
729
?
?
x = Rs 10000
6. 4
Sol.
11
2 3 5 2 3 5
?
? ? ? ?
=
? ? ? ?
2 3 5 2 3 5
2 5 3 2 5 3
? ? ? ? ?
? ? ? ?
=
? ?
? ? ? ?
22
2 2 5
2 5 3
?
??
=
? ?
2 2 5
7 2 10 3
?
??
=
? ?
2 2 5
4 2 10
?
?
=
? ?
? ?
2 2 5
2 2 2 5
?
?
=
1
2
7. 4
Sol.
2x y x y 3/2
3 3 27 3
??
? ? ?
3
2x y x y
2
? ? ? ?
3
2x y
2
??
And x + y =
3
2
x = 1, y = ½
?
x y 1/2
3 3 3
?
??
8. 1
Sol. Speed of A = a m/s
Speed of B = b m/s
?
100 100
5
ba
?? and
80 100
ba
?
=
a 5 4a
b
b 4 5
? ? ?
100 100
5
4a
a
5
??
Solving, we get a = 5 m/s
9. 3
Sol. 11 11 11 11... x ??
2
x 11x ?
2
x 11x 0 ??
? ? x x 11 0 ??
x 0, x 11 ? ? ?
10. 3
Sol. Since there are only odd multiples of 5, unit digit in the product will be 5.
11. 1
Sol. Final change = reduction by
2
12
%
100
??
??
??
??
= reduction by 1.44%
12. 2
Sol. Let the remainder be ax + b
? ? f 1 2 a b ? ? ?
? ? f 1 0 a b ? ? ? ? ?
? 2b = 2
b = 1, a = 1
? Remainder = x + 1
13. 3
Sol.
x 7 2x 10
2x 3x
??
? {basic proportionality theorem}
? 3x + 21 = 4x + 20
? x = 1
14. 2
Sol.
(77 – x)%
(66 – x)% x%
English Math
13%
13 77 x x 66 x 100 ? ? ? ? ? ?
156 – x = 100
x = 56
? 56% of total = 784
Total =
784 100
1400
56
?
?
15. 2
Sol.
160 160 160 160
...
2 7 7 12 12 17 27 32
? ? ?
? ? ? ?
=
5 5 5
32 ...
2 7 7 12 27 32
??
? ? ?
??
? ? ?
??
=
1 1 1 1 1 1
32 ...
2 7 7 12 27 32
??
? ? ? ? ? ?
??
??
=
11
32
2 32
??
?
??
??
=
16 1
32
32
? ??
??
??
= 15
16. 1
Page 5
NTSE-I (2015)
(For Class X Students)
(MAT)
Time: 45 Minutes Max Marks: 50
INSTRUCTIONS TO CANDIDATES
Read the following instructions carefully before you open the Question Booklet.
1. Use blue/black ball point pen only.
2. Write your Roll No. very clearly (only one digit in on block) on this booklet and on the ANSWER
SHEET.
3. This test consists of 50 questions of one mark each. All the questions are COMPULSORY.
4. Answer to each question by filling the correct alternative among the four choices on the answer sheet.
Example:
Q.No.
Q.No.
1
1
Alternatives
Alternatives
2 4
1
1 2 3 ? 4
Correct way:
Wrong way:
5. Separate sheet has been provided for rough work in this test booklet.
NTSE
STAGE I (MAT)
HINTS & SOLUTIONS
1. 2
Sol.
22
p q 2pq ??
22
p q 2pq 0 ? ? ?
? ?
2
p q 0 ??
p = q
23 7
pq
qp
? ? ? ?
?
? ? ? ?
? ? ? ?
= 1 + 1 = 2
2. 4
Sol. Let distance be d.
?
d d 36
4 5 60
??
d 36
20 60
?
d = 12 km
? Actual time to reach on time =
12
1
4
? = 2 hrs
? Required speed = 6 km/hr
3. 2
Sol. 3A = 4B, 2C = 3B
A 4 8 B 2 6
,
B 3 6 C 3 9
? ? ? ?
? A : B : C = 8 : 6 : 9
4. 2
Sol. HCF =
? ?
? ?
hcf 6,4,2
lcm 5,15,5
= 2/15
5. 3
Sol.
3
10
7290 x 1
100
??
??
??
??
3
9
7290 x
10
??
?
??
??
7290 1000
x
729
?
?
x = Rs 10000
6. 4
Sol.
11
2 3 5 2 3 5
?
? ? ? ?
=
? ? ? ?
2 3 5 2 3 5
2 5 3 2 5 3
? ? ? ? ?
? ? ? ?
=
? ?
? ? ? ?
22
2 2 5
2 5 3
?
??
=
? ?
2 2 5
7 2 10 3
?
??
=
? ?
2 2 5
4 2 10
?
?
=
? ?
? ?
2 2 5
2 2 2 5
?
?
=
1
2
7. 4
Sol.
2x y x y 3/2
3 3 27 3
??
? ? ?
3
2x y x y
2
? ? ? ?
3
2x y
2
??
And x + y =
3
2
x = 1, y = ½
?
x y 1/2
3 3 3
?
??
8. 1
Sol. Speed of A = a m/s
Speed of B = b m/s
?
100 100
5
ba
?? and
80 100
ba
?
=
a 5 4a
b
b 4 5
? ? ?
100 100
5
4a
a
5
??
Solving, we get a = 5 m/s
9. 3
Sol. 11 11 11 11... x ??
2
x 11x ?
2
x 11x 0 ??
? ? x x 11 0 ??
x 0, x 11 ? ? ?
10. 3
Sol. Since there are only odd multiples of 5, unit digit in the product will be 5.
11. 1
Sol. Final change = reduction by
2
12
%
100
??
??
??
??
= reduction by 1.44%
12. 2
Sol. Let the remainder be ax + b
? ? f 1 2 a b ? ? ?
? ? f 1 0 a b ? ? ? ? ?
? 2b = 2
b = 1, a = 1
? Remainder = x + 1
13. 3
Sol.
x 7 2x 10
2x 3x
??
? {basic proportionality theorem}
? 3x + 21 = 4x + 20
? x = 1
14. 2
Sol.
(77 – x)%
(66 – x)% x%
English Math
13%
13 77 x x 66 x 100 ? ? ? ? ? ?
156 – x = 100
x = 56
? 56% of total = 784
Total =
784 100
1400
56
?
?
15. 2
Sol.
160 160 160 160
...
2 7 7 12 12 17 27 32
? ? ?
? ? ? ?
=
5 5 5
32 ...
2 7 7 12 27 32
??
? ? ?
??
? ? ?
??
=
1 1 1 1 1 1
32 ...
2 7 7 12 27 32
??
? ? ? ? ? ?
??
??
=
11
32
2 32
??
?
??
??
=
16 1
32
32
? ??
??
??
= 15
16. 1
Sol. By rule of allegation
3
4
150
x
120
150 120 3
120 x 4
?
?
?
?
30 3
120 x 4
?
?
? 120 – x = 40
? x = 80
17. 2
Sol.
8x
3 / 5 2 / 5
?
x = 16/3 hours
= 5 hours 20 minutes
18. 1
Sol.
1 1 1
x 1 1 1 1
x x 1 x 2
?? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ??
??
? ? ? ? ? ? ??
=
x 1 x 2 x 3
x . . 1
x x 1 x 2
? ? ? ??
?
??
??
??
=
x3
x1
x
? ??
?
??
??
=
x 3 x
x
x
?? ??
??
??
= 3
19. 3
Sol. tan x = ½
? ?
2sin x
1
cos x 2sin x
cos x
dividing numerator and denominator by cos x
sin x
cos x sin x
1
cos x
1 2tan x
1 tan x
?
?
?
?
?
?
?
?
=
11
1
1
2
?
?
= 4
20. 4
Sol. EC = BC
?BCE = 90 + 60 = 150
o
? CEB = ?CBE = 15
o
? x = 45
o
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