Which of the following binary operations is cumulative on the set of integers?
a ∗ b = a + 2b
a ∗ b = a + b - ab
a ∗ b = a2 + b
a ∗ b = a(b+1)2
Correct answer is B
a∗b=a+b−ab
b∗a=b+a−ba
On the set of integers, the two above are cumulative as multiplication and addition are cumulative on the set of integers.
Express 5x−12(x−2)(x−3) in partial fractions
2x+2−3x−3
2x−2+3x−3
2x−3−3x−2
5x−3−4x−2
Correct answer is B
5x−12(x−2)(x−3)=Ax−2+Bx−3
= A(x−3)+B(x−2)(x−2)(x−3)
⟹5x−12=Ax−3A+Bx−2B
A+B=5...(i)
−(3A+2B)=−12⟹3A+2B=12...(ii)
From (i), A=5−B
3(5−B)+2B=12
15−3B+2B=12⟹B=3
A+3=5⟹A=2
5x−12(x−2)(x−3)=2x−2+3x−3
1x+2
x−1x+1
x−1x+2
1x−2
Correct answer is A
x2−1(x−1)(x+2)(x+1) = (x−1)(x+1)(x−1)(x+2)(x+1)
= 1x+2
The 4th term of an A.P. is 13 while the 10th term is 31. Find the 21st term.
175
85
64
45
Correct answer is C
a + 3d = 13
a + 9d = 31
6d = 18
= d = 3
a = 13 - 9
= 4
a + 20d = 4 + (20 x 3)
= 64
Solve the inequality (x - 3)(x - 4) ≤ 0
3 ≤ x ≤ 4
3 < x < 4
3 ≤ x < 4
3 < x ≤ 4
Correct answer is A
(x - 3)(x - 4) ≤ 0
Case 1 (+, -) = x - 3 ≥ 0, X - 4 ≥ 0
= X ≤ 3, x ≥ 4
= 3 < x ≥ 4 (solution)
Case 2 = (-, +) = x - 3 ≤ 0, x - 4 ≥ 0
= x ≤ 3, x ≥ 4
therefore = 3 ≤ x ≤ 4