At what time between 3 & 4 O clock, are both hands of a clock in a straight line?
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Solution
Ans: The hands will be in a straight line at \(16\frac{4}{11}\) minutes past and 49\(\frac{1}{11}\) minutes
Options 'A' and 'D
The hands of a clock are in a straight line when
a) the angle between them is zero degrees.
b) the angle between them is 180 degrees.
a) Angle between them is zero degree at 3:00 clock, the minute hand is on 12 and hour hand is on 3.
They are separated by 90° or 15 minute. To coincide, minute hand has to corner 15 minute.
In 60 minute; minute hand gains 55 minutes one hour hand
∴ to gain 15 minutes, It will take \(\frac{15\times 60}{55}=\frac{180}{11}=16\frac{4}{11}\) Minute
Hands will coincide at 16\(\frac{4}{11}\)Minutes
b) Angle between them is 180°
For the angle to be 180°, the minute hand will have to gain 45 minutes one hour hand
⇒\(\frac{45\times 60}{55}=\frac{9\times 60}{11}=\frac{540}{11}\)
= 49\(\frac{1}{11}\) Minutes.
Four integers consecutively lower than 77 and four consecutively higher than 77, are added together. The sum is divisible by which of the following options?
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Solution
Ans: D, F
The eight integers are 73, 74, 75, 76 and 78, 79, 80,81
Note that 73 + 81 = 2 × 77
74 + 80 = 2 × 77 and so on
∴ Addition gives 77 × 8
It is divisible by 7, 11, and 2
For which of the following values of k (k is a positive integer); 100k is a factor of 22.3.53?
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Solution
Ans: B, C, D, E
100 k is a factor of 23.3.53
∴ k is factor of =\(\frac{2^{2}.3.5^{3}}{100}=\frac{4\times 3\times 125}{100}=15\)
∴ k is a factor of 15.
Possible values are 1,3,5, 15
x⁄y = 2; xy = 1. Which of the following can be y the values of either x or y? Indicate all such values
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Solution
Ans:A,C,D,E
x = 2y, and xy = 1
⇒ 2y⇒ = 1 ⇒ y⇒ = 1⁄2
⇒ Y = ± 1⁄√2 ∴ x = ±√2
x and y are integers and x – 2y =5. Which of the following can be a value of X2– 4y2? Indicate all such integers.
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Solution
Ans: B, D
x2 - 4y2 = (x + 2y)(x - 2y) = 5(x + 2y)
Clearly x2 - 4y2 has to be a multiple of 5, hence, x2 - 4y2 cannot have the value -3, 14 and 51
The value of the expression can be 0 or 45.
3x + 4y is an odd number. Which of the following options cannot be true? Indicate all such options
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Solution
Ans: C, D
Since the sum 3x + 4y is odd it means one of two numbers is odd and other is even but 4y is always even irrespective of the value of y (because product of odd and even number is always even)
∴ y can be odd or even.
Since 4y is even, 3x has to be compulsorily odd.
∴ x always has to be odd.
Therefore, x is odd, y can be odd/ even.
As x increases from 165 to 166 which of the following must increase? Indicate all such expressions.
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Solution
Ans: A, B increases
Consider option 'A'
As x increases from 165 to 166, the expression 2x - 5 increases as x increases. Consider option 'B'
As x increases, its reciprocal decreases but this decreased value is subtracted from 1 so the value of the expression as a whole increases.
Consider. 'C' for integers greater than 1, X2 increases more rapidly than does x, so the value of x2 - x in denominator increases.
So the value of reciprocal decreases.
p, r, s, t is an arithmetic sequence. Indicate all the following sequences that are in arithmetic sequence.
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Solution
Ans: A, B
Since p, r, s, t, u are in AP common difference d = r - p = s - r = t - s = u - t.
⇒ Consider 'A'
Common difference = 2r - 2p = 2( r - p) = 2d
As d is a constant, 2d is also a constant.
∴ A' is also an A.P with common difference 2d.
⇒ Consider 'B'
Common difference = (r - 3) - (p - 3) = r - 3 - p + 3
=r-p=d
∴ 'B' is also an A.P
⇒ Consider 'C
p2,r2,s2, t2,u2
Hear difference between 2 consecutive terms is not constant
e.g. If p, q, r, s, t, u are 1,2,3,4,5
The p2,q2,r2,s2,t2,u2 are 1,4,9,16,25
⇒ Difference is not constant.
n3 is odd, indicate all the options that are true
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Solution
Ans: C, D
n3 is odd ⇒ n is odd ⇒ 'D' is true
n2 is odd ⇒ '8' is false
Also, (n - 1) and n + 1 both are even, so the product is even ⇒ 'C' is also true
A, B, C are in GP; then which of the following are in GP
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Solution
Ans: A, B, C
e.g. 2, 4, 8 are in GP (common multiple 2)
A) 22,42,82 i.e. 4, 16,64 are in GP
B) 23,43,83 i.e. 8,16,64 are in GP
C) 22 + 42,(2×4)+(4×8); 42 + 82
i.e. 20, 40, 80 are in GP