Question
Why is it that (xy)-1 = x-1y-1?
Answer:
We know that any number raised to -1 is its reciprocal.
Therefore, we have (xy)-1 = 1/xy which is equal to (1/x)*(1/y).
We see that the result is the product of the reciprocals of x and y and therefore we can also write it as x-1y-1.
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Question
How many minutes past 11AM is it if two hours ago it was three times as many minutes, past 8AM?
Answer:
It is 30 minutes past 11AM.
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Question
List the ways in which 7 / 12 can be written as a sum of two fractions in lowest terms such that the denominators of the two fractions are different and neither of them is more than 12.
Answer:
1/2+1/12; 1/3+1/4; 1/6+5/12.
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Question
By using four 3's with the mathematical operators ( + , - , / , * ) obtain the results 0, 1, 2 and 3.
Answer:
The results are given below.
3+3-3-3=0
(3/3)+3-3=1
(3/3)+(3/3)=2
(3+3+3)/3=3
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Question
By using four 4's along with any mathematical operator, make a total of 420.
Answer:
444 - 4!
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Question
Find the values of sin(x/2), cos(x/2) and tan(x/2) if tan(x)=-4/3.
Answer:
sin(x/2)=2√5/5, cos(x/2)=√5/5 and tan(x/2)=2.
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Question
Prove that (cos2θ)(3+cos4θ) = 4(cos8θ-sin8θ).
Answer:
The proof is given below.
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Question
If Set A has 3 elements and Set B has 6 elements, prove that the minimum and maximum number of elements in A∪B is 6 and 9 respectively.
Answer:
The proof is given below.
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Question
If n(U)=500, n(F)=285, n(H)=195, n(B)=115, n(F∩B)=45, n(F∩H)=70, n(H∩B)=50, n(H∪B∪F)’=50, then find n(H∩B∩F) and n(H-B-F) + n(B-F-H) + n(F-B-H).
Answer:
Values of n(H∩B∩F) and n(H-B-F) + n(B-F-H) + n(F-B-H) are 20 and 325 respectively.
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Question
If n(M∪B)=25, n(M)=12, n(M-B)=8, then find n(B-M) and n(B∩M).
Answer:
Values of n(B-M) and n(B∩M) are 13 and 4 respectively.
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Question
If n(U)=10000, n(A)=4000, n(B)=2000, n(C)=1000, n(A∩C)=400, n(A∩B)=500, n(B∩C)=300, n(A∩C∩B)=200, then find n(A-B-C) and n(A∪B∪C)’.
Answer:
Values of n(A-B-C) and n(A∪B∪C)’ are 3300 and 4000 respectively.
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Question
If If n(U)=200, n(M)=120, n(P)=90, n(C)=70, n(M∩C)=50, n(M∩P)=40, n(P∩C)=30, n(P∪C∪M)’=20, then find n(M∩C∩P).
Answer:
Value of n(M∩C∩P) is 20.
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Question
If n(U)=60, n(C)=25, n(T)=20, n(C)=11, n(T∩C)=10, then find n(T∪C)’.
Answer:
Value of n(T∪C)’ is 25.
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Question
If n(U)=25, n(M)=15, n(P)=12, n(C)=11, n(M∩C)=5, n(M∩P)=9, n(P∩C)=4, n(M∩C∩P)=3, then find n(C-P-M), n(M-C-P), n(P-M-C), n((P∩C)-M), n((P∩M)-C), n(C-P-M) + n(M-C-P) + n(P-M-C), n(P∪C∪M), n(P∪C∪M)’.
Answer:
The answers are given below.
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Question
If |(z+i5)/(z-i5)| = 1, then prove z is purely real.
Answer:
The proof is given below.
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Question
Calculate the area under the curve f(x) = e-x bounded by the positive x and y axes.
Answer:
The area is 1 square units.
We have to calculate area under the curve bounded by the positive x and y axes.
So, x varies from 0 to ∞.
Therefore, we have to find ∫e-xdx when x varies from 0 to ∞.
To find the integral of e-x, we find a function whose derivative is e-x. Thus, we get ∫e-xdx = -e-x.
Therefore, the area under the curve
= -e-∞ - (-e0)
= 0 + 1
= 1 sq. unit
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Question
Two APs have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms. Why?
Answer:
The reason is given below.
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Tutor AV
Question
The location of a dot P at a given time t in the xy plane is given by (x,y) = (t−sint,1−cost). What is the distance traveled by P in the interval 0≤t≤2π?
Answer:
The distance traveled by the point is 2π units.
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Tutor AV
Question
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Answer:
He take 89 minutes or 1 hour and 29 minutes to count the whole sum.
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Tutor AV
Question
A tank, internally measuring 150cm x 120cm x 110cm, has 129600 cubic cm of water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorbs one-seventeenth of its own volume of water. How many bricks can be put in it without overflowing the water, each brick having dimensions 22.5cm x 7.5cm x 6.5cm?
Answer:
1792 bricks can be put in the tank.
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