Sum of three consecutive terms of an AP is 21. If the second number is reduced by 1, and the third increased by 1, then 3 consecutive terms form a GP. Indicate all the groups.
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Solution
Ans: A, B
Try the options A and B. Both satisfy the condition 7 - d, 6, s + d for GP
In GP, b2= ac
∴ 62 = (7 - d)(8 + d)
36 = 56 - d - d2
d2 + d - 20 = 0
⇒ (d + 5)(d - 4) = 0
d = -5 or d = 4
d = -5 gives 12,7,2
d = 4 gives 3,7,11
Suppose P is a prime number greater than 3, then P can be written in which of the following forms?
Indicate all such options
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Solution
Ans:A,B
5 is written in the form 6(1) - 1 = 6k - 1
7 can be written as 6 (1) + 1 = 6k + 1
11 can be written as 6 (2) - 1 = 6k - 1
13 can be written as 6 (2) + 1 = 6k + 1
461 + 462 + 463 + 464 + 465 is not divisible by? Indicate all such numbers
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Solution
Ans: Not divisible by option A, B, D, E
461(1 + 4 + 16 + 64 + 256)=461 × 341.
341 is divisible by 11.
∴ The given expression is only divisible by 11
The product of two prime numbers can be? Indicate all possible options
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Solution
Ans: A, B, E
The product of two prime numbers can be even, odd or composite. If the numbers are '2' and '5' product is 10 i.e. even
'3' and 7 product is 21 i.e. odd
Furthers 10,21 are composites numbers 700
As x increases from 73 to 74, which of the following must increase? Indicate all such options
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Solution
Ans:A,B
With respect to option 'A', as x increases from 73 to 74, the expression 3x -7 increases since 3x increases.
With respect to option 'B', as x increases, its reciprocal decreases, therefore value of expression increases as a whole.
With respect to option 'C', for integers greater than 1, X2 increases more rapidly than does x, so value of denominator increases, causing reciprocal to decrease.
Hence; options 'A' and '8' increase while 'C' decreases
The product of integers m, n, 0, p is 770; such that m < n < 0 < p and no integer is equal to 1. What are the possible values of m + n? [mpcheck] [mccheck]A. 12[/mccheck]
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Solution
Ans: Options 'B','D' and 'E'
Prime factors of 770 are 2, 5, 7,11
∴ Values of m, n, 0, p can be 2, 5, 7, 11 respectively.
However values can be negative also therefore
m = -11, n = -7,0 = -5, p =-2
∴ Possible values of m + n are
m + n = 2 + 5 = 7
m + n = -11 -7 = -18
-5 , -2 , 7, 11 is also one of the possible combinations so m + n = -7
The base of a right circular cylinder has a radius of at least 3 inches and at the most 5 inches and the cylinders have a volume of at least 112.5π cubic inches and at the most 189π cubic inches. Which of the following could be the possible heights of the cylinder? Indicate all such options
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Solution
Ans:A,B,C,D
To find out the maximum height, divide maximum volume by minimum radius.
\(\frac{189\pi }{\pi 3^{2}}\) = 21 inches.
To find out the minimum heights, divide minimum volume by maximum radius.
\(\frac{112.5\pi }{\pi (5)^{2}}\)= 41⁄2 Inches.
∴ The height of cylinder must lie between 41⁄2 and 21 inches.
Average of seven numbers is 8. In order to determine the seventh number, which of the following statements are required?
Indicate all those statements
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Solution
Ans:A,B,C
As average of seven numbers is 6, total of seven numbers is 7 × 8 = 56.
Taking option 'c' - Any of '4' numbers is 7.
∴ Total sum of '4' numbers = 7 × 4 = 28
Further, option A' → Fifth number is 6.
∴ Sum of 6th and 7th number = 56 - (28 + 6) = 22
Option 'B' gives the relation between 6th and 7th numbers ⇒
6th numbers + 7th number = 22
(x + 7) + x = 22
2x + 8 = 22
2x = 14
x=7
∴ 6th number = 7 + 7 ⇒ 7th Number = 8 = 14
Option 'D' doesn't serve any purpose
What are the values of x for which
\(\left ( \frac{16}{x}+64 \right )\left ( 9-x^{2} \right )=0\)Indicate all such values
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Solution
Ans: Option A, D, E
Either or the two brackets is zero.
∴ \(\frac{16}{x}\) + 64 = 0 ⇒ x = - 1⁄4
9 - x- 1 = 0 ⇒ x2 = 9 ⇒ x = ±3
∴ All values of x are - 1⁄4;+ 3,-3
If cd > 0, which of the following options can be true?
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Solution
Ans: Options A', 'B', 'D
If cd > 0, it implies either 'c' or 'd' both are negative or both are positive. ∴ options 'A' and 'B' can be true.
Further if both c and d both are negative, 'c' is also true.
Also, option 'En is always true, whether both are positive or negative.
But c⁄d < 0 can never be true.