A can give B a 200 metre startup and C a 300 metre startup in a race of 1 Km so that the race ends in a dead heat. Out of the following options, identify the possible values of startup that B can give to C so that C wins the race?
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Solution
Ans: B, C, E
Parallelogram PQRS is cyclic if
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Solution
Ans: (C, D)
A parallelogram will be cyclic when it will be a rectangle, for that diagonals should be equal, or opposite angles should add to 180°, or the adjacent angles should be equal. Hence C,D. Other options are true for any parallelogram
p, q and r, are real numbers. Find the value of \(\sqrt{p^{-1}q}\sqrt{q^{-1}r}\sqrt{r^{-1}p}\)
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Solution
Ans: (C)
The given expression is \((p^{-1}q)^{\frac{1}{2}}(q^{-1}r)^{\frac{1}{2}}(r^{-1}p)^{\frac{1}{2}}\)
=\(\left ( \frac{q}{p} \right )^{\frac{1}{2}}\left ( \frac{r}{q} \right )^{\frac{1}{2}}\left ( \frac{p}{r} \right )^{\frac{1}{2}}=\left ( \frac{pqr}{pqr} \right )^{\frac{1}{2}}=1^{\frac{1}{2}}=1\)
A group 630 children is arranged in rows for a group photograph. Each row contains three fewer children than the row in front of it. Which of the following rows are possible if all of them must be in the rows?
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Solution
Ans: (B, C, D, E)
The number of children will be in AP with d = -3, Sn = 630
sn = n⁄2[2a+(n-1)d]
Put values of n as 6, 3, (all options) and check value of a to be an integer
A man rows a certain distance upstream and back again downstream to the same point in 6 hours. The speed of rowing in still water is 1.5 times the speed of stream. Find the ratio of time required for downstream journey to that for upstream journey?
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Solution
Ans: (C, D, E)
Distance traveled upstream = distance traveled downstream.
Let the speed of river is x, then the speed of boat in Still water is 1.5x
Hence the ratio of speed downstream to speed upstream will be
\(\frac{1.5 + 1}{1.5-1}=\frac{2.5}{0.5}\) = 5 : 1. Hence required ratio is 1 : 5 = 2 : 10 = 0.2
Michael said to his son, “At the time of your birth, I was as old as you are at present.” Michael is ‘x’ years old now and his son is ‘y’ years old now, then which two numbers can denote x and y?
Mark two correct options
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Solution
Ans: (A, E)
Son's present age be x.
∴ Michael's age x years ago was x years.
∴ Michael's present age = 2x = Y
14 × 2 =28, hence 14 and 28
(Since Michael was as old as his son at the time of the birth of his son, his son today must be half of his age.)
On her mother’s birthday, Ana hired a car and took her mother on a long drive. Ana was charged $4 per km for first 60 km, $5 per km for next 60 km and $8 for every 5 km for further journey. Balance amount 1th left with Ana was \(\frac{1}{4}^{th}\) of what she paid towards the 4 charges of hired car for traveling 320 km How much money did she initially have?
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Solution
Ans: (D)
Let Ana have $x initially.
Consider the breakup of her journey i.e. 320 km.
(60 × 4)+ (60 × 5)+(200 × 1⁄5 × 8)
= (240 +300 +320) = $860
∴ Ana pays $860 for the entire journey.
∴ According to the given condition
x = 860 + \(\frac{860}{4}\) = 860 + 215 = $1075
Two trains 280 meters and 220 meters long respectively, are going in same direction. The faster train takes 50 seconds to pass the other completely. Which of the following gives the speeds of the two trains?
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Solution
Ans: (B, D)
To cross the slower train, the faster train will have to cross length of the slower train as well as its own length i.e. 280 + 220 = 500m
In 50 seconds 500 meters are crossed
∴ In 3600 seconds 36000m are crossed.
∴ Speed of overtaking is 36 kmph.
Speed of overtaking = Speed of faster train - Speed of slower train.
Therefore 36 = Speed of faster train - Speed of slower train
A batsman has an average score of 49 in the first 9 innings. Which of the following runs, if scored in the 10th inning, would result in an average of 51 or above?
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Solution
Ans: Options 'A', 'B', 'D', 'E'
Total runs scored in 9 innings = 49 × 9 = 441
Runs scored in 10th inning = (51) (10) - 441 = 510 - 441 =69 Runs
For an average of 51 or greater after 10th innings, the runs that he must score in the 10th innings is 69 or more than that
A box contains 5 blocks numbered 1, 2, 3, 4, 5. John picks a block and puts it back. Lisa then picks a block. Which of the following will give the probability that the sum of the numbers they pick is even?
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Solution
Ans: (A, B, D)
The sum can be even if John picks an even number and Lisa also picks an even number OR John picks an odd number and Lisa also picks an odd number. Out of the given 5 numbers, probability of choosing an odd number is 3⁄5 and that of even number is 2⁄5.
∴ Case (I): Both pick even numbers.
P(I) = 2⁄5 × 2⁄5 =\(\frac{2}{25}\)
Case (II): Both pick odd numbers.
P(II) = 3⁄5 × 3⁄5 = \(\frac{9}{25}\)
∴ Require probability = \(\frac{4}{25}+\frac{9}{25}=\frac{13}{25}\)