49th special BCS
Subject: Mathematics
Demo question from lecture -1
1. a,b∈R Trichotomy Property-
(A) a<b a>b
(B) a≤b a≥b
(C) a<b, a=b, a>b
(D) a<b b<c, a<c
Correct Answer: (C)
2. a(b+c)=ab+ac
(A) Commutative Property
(B) Associative Property
(C) Distributive Property
(D) Identity Property
Correct Answer: (C)
3. x<y c<0
(A) xc<yc
(B) xc>yc
(C) x+c<y+c
(D) x−c>y−c
Correct Answer: (B)
4. (addition) (identity)
(A) 1
(B) -1
(C) 0
(D)
Correct Answer: (C)
5.
(A) Closure Property
(B) Commutative Property
(C) Inverse Property
(D) Identity Property
Correct Answer: (A)
6. a,b,c∈R a≤b, b≤c a≤c
(A) Trichotomy Property
(B) Transitive Property
(C) Commutative Property
(D) Completeness Property
Correct Answer: (B)
7. a,b∈R ab=0
(A) a=0 b=0
(B) a=0 b=0
(C) a b
(D) a+b=0
Correct Answer: (B)
8. x>y
(A) x+2>y+2
(B) x−3>y−3
(C) 2x>2y
(D) x2>y2
Correct Answer: (D)
9.x<0 y<0 , xy-
(A) xy<0
(B) xy>0
(C) xy=0
(D)
Correct Answer: (B)
10.∣x−a∣<δ
(A) x a- δ-
(B) x a- δ-
(C) x- a+δ
(D) x- a−δ
Correct Answer: (B)
11. a,b c a+bc —
(A)
(B)
(C) b=0
(D)
Correct Answer: (C)
12. R- (multiplication) (identity)
(A) 0
(B) 1
(C) -1
(D) e
Correct Answer: (B)
13. a,b ∣∣a∣−∣b∣∣≤∣a−b∣
(A) Triangle Inequality-
(B)
(C)
(D) Chebyshev's Inequality
Correct Answer: (A)
14. 2, 4, 8 AM GM-
(A) 14/3−4
(B) 4−14/3
(C) 0
(D) 2
Correct Answer: (A)
15. a,b,c (Geometric Progression)
AM GM-
(A) AM>GM
(B) AM=GM
(C) AM<GM
(D) AM,GM a=b=c
Correct Answer: (D)
16. 1, 2, 3, 4 Root-Mean-Square (RMS) মান কত?
(A) 2.5
(B) 7.5
(C) 30
(D) 10
Correct Answer: (B)
17. Mean Inequality Chain (RMS-AM-GM-HM)
(A)
(B)
(C) (Non-negative real numbers)
(D)
Correct Answer: (C)
18. AM 13 GM 12,
(A) 10, 16
(B) 9, 18
(C) 8, 18
(D) 9, 16
Correct Answer: (C)
19. a,b,c , AM-GM
(a+b−c)(b+c−a)(c+a−b)≤abc।
(A)
(B)
(C)
(D)
Correct Answer: (C)
20. 1 49- AM, GM HM
(A) AM=25, GM=7, HM=1.96
(B) AM=25, GM=7, HM=13.7
(C) AM=25, GM=7, HM=1.38
(D) AM=50, GM=7, HM=1.96
Correct Answer: (A)
21. x,y (x+y)2≥4xy
(A) AM-GM
(B) Cauchy-Schwarz
(C) Triangle Inequality
(D) AM-HM
Correct Answer: (A)
22. AM GM 2 HM
GM 1.6
(A) 4, 16
(B) 2, 8
(C) 5, 20
(D) 1, 9
Correct Answer: (A)
23. Chebyshev's Sum Inequality- sequence {ai} {bi}-
(A) non-increasing, non-decreasing
(B) non-increasing non-decreasing
(C)
(D)
Correct Answer: (B)
24. {ai} {bi non-increasing sequence Chebyshev's
Sum Inequality
(A) (∑ai)(∑bi)≤n∑aibi
(B) (∑ai)(∑bi)≥n∑aibi
(C) (∑ai)(∑bi)=n∑aibi
(D) (∑ai)(∑bi)≤∑aibi
Correct Answer: (A)
25. sequence {ai} non-increasing {bi} non-decreasing
Chebyshev's Sum Inequality-
(A) (∑ai)(∑bi)≤n∑aibi
(B) (∑ai)(∑bi)≥n∑aibi
(C)
(D) (∑ai)(∑bi)=n∑aibi
Correct Answer: (B)
26. Chebyshev's inequality- Bienaymé-Chebyshev
inequality Bienaymé
(A) Chebyshev-
(B) Chebyshev-
(C) Chebyshev- formulate
(D) Chebyshev-
Correct Answer: (C)
27. a1≥a2≥⋯≥an b1≤b2≤⋯≤bn , ∑aibi (∑ai)(∑bi)-
(A) n∑aibi≥(∑ai)(∑bi)
(B) n∑aibi≤(∑ai)(∑bi)
(C) n∑aibi=(∑ai)(∑bi)
(D)
Correct Answer: (B)
28. Chebyshev's Sum Inequality
(A) a1=a2=⋯=an b1=b2=⋯=bn
(B) sequence
(C) sequence
(D) sequence
Correct Answer: (A)
29. a1=3,a2=2,a3=1 b1=1,b2=2,b3=3 ∑aibi
(A) 10
(B) 12
(C) 14
(D) 15
Correct Answer: (A)
30. a,b,c , ba+cb+ac≥3
(A) AM-GM
(B) Cauchy-Schwarz
(C) Chebyshev
(D) AM-HM
Correct Answer: (A)
