JEE Mathematics Miscellaneous Problems MCQs Set 03

Answer: (c) 4096

Explanation:
1. Let f(x) = (1 + x + x² + x³)⁷. The sum of coefficients of powers that are multiples of 4 is \( \frac14[f(1)+f(i)+f(-1)+f(-i)] \).
2. f(1) = 4⁷ = 16384, while 1 + i + i² + i³ = 0 gives f(i) = f(−1) = f(−i) = 0.
3. Sum = 16384/4 = 4096.

Teacher's Note:
1. R
2. o
3. o
4. t
5. s
6.
7. o
8. f
9.
10. u
11. n
12. i
13. t
14. y
15.
16. p
17. i
18. c
19. k
20.
21. o
22. u
23. t
24.
25. e
26. v
27. e
28. r
29. y
30.
31. f
32. o
33. u
34. r
35. t
36. h
37.
38. c
39. o
40. e
41. f
42. f
43. i
44. c
45. i
46. e
47. n
48. t
49. .
50.
51. T
52. h
53. r
54. e
55. e
56.
57. o
58. f
59.
60. t
61. h
62. e
63.
64. f
65. o
66. u
67. r
68.
69. v
70. a
71. l
72. u
73. e
74. s
75.
76. v
77. a
78. n
79. i
80. s
81. h
82.
83. h
84. e
85. r
86. e
87. .

Answer: (a) 6885

Explanation:
1. Boys get 16 and girls get 8 apples.
2. Boys: \( \binom{18}2=153 \) ways; girls: \( \binom{10}2=45 \) ways.
3. Total = 153 × 45 = 6885.

Teacher's Note:
1. S
2. p
3. l
4. i
5. t
6.
7. t
8. h
9. e
10.
11. t
12. o
13. t
14. a
15. l
16.
17. u
18. s
19. i
20. n
21. g
22.
23. t
24. h
25. e
26.
27. r
28. a
29. t
30. i
31. o
32.
33. 2
34.
35. :
36.
37. 1
38. ,
39.
40. t
41. h
42. e
43. n
44.
45. u
46. s
47. e
48.
49. s
50. t
51. a
52. r
53. s
54. -
55. a
56. n
57. d
58. -
59. b
60. a
61. r
62. s
63.
64. f
65. o
66. r
67.
68. e
69. a
70. c
71. h
72.
73. g
74. r
75. o
76. u
77. p
78. .

Answer: (b) 10

Explanation:
1. The three conditions give a·b + a·c = 0, b·c + b·a = 0, c·a + c·b = 0, so a·b = b·c = c·a = 0.
2. Then \( |\vec a+\vec b+\vec c|^2=9+16+25=50 \).
3. So \( \sqrt2\cdot\sqrt{50}=10 \).

Teacher's Note:
1. A
2. d
3. d
4. i
5. n
6. g
7.
8. a
9. n
10. d
11.
12. s
13. u
14. b
15. t
16. r
17. a
18. c
19. t
20. i
21. n
22. g
23.
24. t
25. h
26. e
27.
28. t
29. h
30. r
31. e
32. e
33.
34. c
35. o
36. n
37. d
38. i
39. t
40. i
41. o
42. n
43. s
44.
45. f
46. o
47. r
48. c
49. e
50. s
51.
52. e
53. v
54. e
55. r
56. y
57.
58. d
59. o
60. t
61.
62. p
63. r
64. o
65. d
66. u
67. c
68. t
69.
70. t
71. o
72.
73. z
74. e
75. r
76. o
77. .

Answer: (c) 32

Explanation:
1. Intersections satisfy \( x^2+(2-m)x-4=0 \) with discriminant \( D=(2-m)^2+16 \).
2. Area between line and parabola = \( \frac{D^{3/2}}6 \), least when m = 2: \( \frac{64}6=\frac{32}3 \).
3. Three times this is 32.

Teacher's Note:
1. T
2. h
3. e
4.
5. a
6. r
7. e
8. a
9.
10. u
11. n
12. d
13. e
14. r
15.
16. a
17.
18. c
19. h
20. o
21. r
22. d
23.
24. o
25. f
26.
27. a
28.
29. p
30. a
31. r
32. a
33. b
34. o
35. l
36. a
37.
38. w
39. i
40. t
41. h
42.
43. l
44. e
45. a
46. d
47. i
48. n
49. g
50.
51. c
52. o
53. e
54. f
55. f
56. i
57. c
58. i
59. e
60. n
61. t
62.
63. 1
64.
65. i
66. s
67.
68. D
69. ^
70. (
71. 3
72. /
73. 2
74. )
75. /
76. 6
77. .
78.
79. M
80. i
81. n
82. i
83. m
84. i
85. s
86. e
87.
88. D
89. .

Answer: (b) 3

Explanation:
1. By the property \( \int_0^4g(x)dx=\int_0^4g(4-x)dx \): \( I=\int_0^4f(x)\varphi(4-x)\,dx \) using f(4 − x) = f(x).
2. Adding: \( 2I=\int_0^4f(x)[\varphi(x)+\varphi(4-x)]dx=3\cdot2=6 \).
3. So I = 3.

Teacher's Note:
1. U
2. s
3. e
4.
5. t
6. h
7. e
8.
9. s
10. u
11. b
12. s
13. t
14. i
15. t
16. u
17. t
18. i
19. o
20. n
21.
22. x
23.
24. t
25. o
26.
27. 4
28.
29. −
30.
31. x
32.
33. a
34. n
35. d
36.
37. a
38. d
39. d
40.
41. t
42. h
43. e
44.
45. t
46. w
47. o
48.
49. f
50. o
51. r
52. m
53. s
54.
55. o
56. f
57.
58. t
59. h
60. e
61.
62. i
63. n
64. t
65. e
66. g
67. r
68. a
69. l
70. .

Answer: (b) 45

Explanation:
1. x − [x] is the fractional part, which has period 1 and mean ½.
2. Over 90 unit intervals the integral is 90 × ½ = 45.

Teacher's Note:
1. T
2. h
3. e
4.
5. i
6. n
7. t
8. e
9. g
10. r
11. a
12. l
13.
14. o
15. f
16.
17. t
18. h
19. e
20.
21. f
22. r
23. a
24. c
25. t
26. i
27. o
28. n
29. a
30. l
31.
32. p
33. a
34. r
35. t
36.
37. o
38. v
39. e
40. r
41.
42. o
43. n
44. e
45.
46. u
47. n
48. i
49. t
50.
51. i
52. n
53. t
54. e
55. r
56. v
57. a
58. l
59.
60. i
61. s
62.
63. 1
64. /
65. 2
66. .

Answer: (b) 10

Explanation:
1. Differentiating, f′(x) = 1/f(x), so \( f\,f'=1 \) and \( f^2=2x+c \).
2. f(a) = 0 gives c = −2a. Also \( \int_a^1\frac{dx}{f}=f(1)-f(a)=f(1)=\sqrt2 \), so 2 − 2a = 2 and a = 0.
3. Therefore f² = 2x and f(50) = √100 = 10.

Teacher's Note:
1. C
2. o
3. n
4. v
5. e
6. r
7. t
8. i
9. n
10. g
11.
12. t
13. h
14. e
15.
16. i
17. n
18. t
19. e
20. g
21. r
22. a
23. l
24.
25. e
26. q
27. u
28. a
29. t
30. i
31. o
32. n
33.
34. t
35. o
36.
37. a
38.
39. d
40. i
41. f
42. f
43. e
44. r
45. e
46. n
47. t
48. i
49. a
50. l
51.
52. e
53. q
54. u
55. a
56. t
57. i
58. o
59. n
60.
61. f
62. ·
63. f
64. ′
65.
66. =
67.
68. 1
69.
70. m
71. a
72. k
73. e
74. s
75.
76. i
77. t
78.
79. e
80. a
81. s
82. y
83.
84. t
85. o
86.
87. s
88. o
89. l
90. v
91. e
92. .

Answer: (c) 4

Explanation:
1. Rotate axes: u = (x + y)/√2, v = (y − x)/√2. Then C becomes the square |u| ≤ 1, |v| ≤ 1, the side is u = 1 and the origin is unchanged.
2. Nearer to the side than the origin: \( 1-u\lt\sqrt{u^2+v^2} \), which simplifies to \( u\gt\frac{1-v^2}{2} \).
3. Area \( =\int_{-1}^{1}\left(1-\frac{1-v^2}2\right)dv=\int_{-1}^1\frac{1+v^2}2dv=\frac43 \), so three times the area is 4.

Teacher's Note:
1. R
2. o
3. t
4. a
5. t
6. i
7. n
8. g
9.
10. t
11. h
12. e
13.
14. a
15. x
16. e
17. s
18.
19. t
20. u
21. r
22. n
23. s
24.
25. t
26. h
27. e
28.
29. s
30. q
31. u
32. a
33. r
34. e
35.
36. i
37. n
38. t
39. o
40.
41. a
42. n
43.
44. a
45. x
46. i
47. s
48. -
49. p
50. a
51. r
52. a
53. l
54. l
55. e
56. l
57.
58. o
59. n
60. e
61. ,
62.
63. a
64. n
65. d
66.
67. t
68. h
69. e
70.
71. b
72. o
73. u
74. n
75. d
76. a
77. r
78. y
79.
80. i
81. n
82. t
83. o
84.
85. a
86.
87. p
88. a
89. r
90. a
91. b
92. o
93. l
94. a
95.
96. t
97. h
98. a
99. t
100.
101. i
102. s
103.
104. e
105. a
106. s
107. y
108.
109. t
110. o
111.
112. i
113. n
114. t
115. e
116. g
117. r
118. a
119. t
120. e
121. .

Answer: (b) 45000

Explanation:
1. Modulo 4: 2^{a₁} is 1, 2, 0 (for a₁ = 0, 1, ≥2); 3^{a₂} is 1 or 3 (even or odd a₂); 5^{a₃} ≡ 1; 7^{a₄} is 1 or 3; 9^{a₅} ≡ 1.
2. Sum ≡ 2^{a₁} + 3^{a₂} + 7^{a₄} + 2 (mod 4). Enumerating all cases for (a₁, a₂, a₄) gives 450 valid triples out of 1000.
3. a₃ and a₅ are free (10 × 10 = 100 choices), so the count is 450 × 100 = 45000.

Teacher's Note:
1. R
2. e
3. d
4. u
5. c
6. e
7.
8. e
9. a
10. c
11. h
12.
13. p
14. o
15. w
16. e
17. r
18.
19. m
20. o
21. d
22. u
23. l
24. o
25.
26. 4
27.
28. a
29. n
30. d
31.
32. c
33. o
34. u
35. n
36. t
37.
38. t
39. u
40. p
41. l
42. e
43. s
44. .
45.
46. T
47. h
48. e
49.
50. d
51. i
52. g
53. i
54. t
55. s
56.
57. a
58. ₃
59.
60. a
61. n
62. d
63.
64. a
65. ₅
66.
67. a
68. r
69. e
70.
71. f
72. r
73. e
74. e
75. .

Answer: (b) 20

Explanation:
1. a = 1, and f(x) − cx has roots 1, 2, 3, so \( f(x)=(x-1)(x-2)(x-3)(x-r)+cx \).
2. The even part of f is \( x^4+(11+6r)x^2+6r \); at x = i this is \( 1-(11+6r)+6r=-10 \).
3. So f(i) + f(−i) = −20 and k = 20.

Teacher's Note:
1. T
2. h
3. e
4.
5. o
6. d
7. d
8.
9. p
10. a
11. r
12. t
13.
14. c
15. a
16. n
17. c
18. e
19. l
20. s
21.
22. i
23. n
24.
25. f
26. (
27. i
28. )
29.
30. +
31.
32. f
33. (
34. −
35. i
36. )
37. .
38.
39. T
40. h
41. e
42.
43. u
44. n
45. k
46. n
47. o
48. w
49. n
50.
51. r
52.
53. a
54. l
55. s
56. o
57.
58. c
59. a
60. n
61. c
62. e
63. l
64. s
65.
66. f
67. r
68. o
69. m
70.
71. t
72. h
73. e
74.
75. e
76. v
77. e
78. n
79.
80. p
81. a
82. r
83. t
84. .

Answer: (b) 10

Explanation:
1. The normal of the plane is along (3, 5, 7), and it must be perpendicular to the contained line's direction (10, −20, k).
2. 30 − 100 + 7k = 0, so k = 10.

Teacher's Note:
1. A
2.
3. l
4. i
5. n
6. e
7.
8. l
9. i
10. e
11. s
12.
13. i
14. n
15.
16. a
17.
18. p
19. l
20. a
21. n
22. e
23.
24. p
25. e
26. r
27. p
28. e
29. n
30. d
31. i
32. c
33. u
34. l
35. a
36. r
37.
38. t
39. o
40.
41. a
42.
43. g
44. i
45. v
46. e
47. n
48.
49. d
50. i
51. r
52. e
53. c
54. t
55. i
56. o
57. n
58.
59. o
60. n
61. l
62. y
63.
64. i
65. f
66.
67. i
68. t
69. s
70.
71. d
72. i
73. r
74. e
75. c
76. t
77. i
78. o
79. n
80.
81. i
82. s
83.
84. p
85. e
86. r
87. p
88. e
89. n
90. d
91. i
92. c
93. u
94. l
95. a
96. r
97.
98. t
99. o
100.
101. t
102. h
103. e
104.
105. n
106. o
107. r
108. m
109. a
110. l
111. .