1. The perimeter of a equilateral triangle and regular hexagon are equal. Find out the ratio of their areas?
a. 3:2b. 2:3
c. 1:6
d. 6:1
Correct Option: b
Explanation:Let the side of the equilateral triangle = $a$ units and side of the regular hexagon is $b$ units.
Given that, $3a = 6b$ $ \Rightarrow \dfrac{a}{b} = \dfrac{2}{1}$
Now ratio of the areas of equilateral triangle and hexagon = $ \dfrac{{\sqrt 3 }}{4}{a^2} : \dfrac{{3\sqrt 3 }}{2}{b^2}$
$ \Rightarrow \dfrac{{\sqrt 3 }}{4}{\left( 2 \right)^2} : \dfrac{{3\sqrt 3 }}{2}{\left( 1 \right)^2}$
$ \Rightarrow 2:3$
2. What is the remainder of (32^31^301) when it is divided by 9?
a. 3
b. 5
c. 2
d. 1
Correct option: bExplanation:
See solved example 6 here
$\dfrac{{{{32}^{{{31}^{301}}}}}}{9}$ = $\dfrac{{{5^{{{31}^{301}}}}}}{9}$
Euler totient theorem says that ${\left[ {\dfrac{{{a^{\phi (n)}}}}{n}} \right]_{{\mathop{\rm Re}\nolimits} m}} = 1$
$\phi (n) = n\left( {1 - \dfrac{1}{a}} \right)\left( {1 - \dfrac{1}{b}} \right)...$ here $n = {a^p}.{b^q}...$
Now $\phi (9) = 9\left( {1 - \dfrac{1}{3}} \right) = 6$
Therefore, ${5^6}$ when divided by 9 remainder 1.
Now $\dfrac{{{{31}^{301}}}}{6} = {1^{301}} = 1$
So ${{{31}^{301}}}$ can be written as 6k + 1
$\Rightarrow {5^{{{31}^{301}}}} = {\left( {{5^6}} \right)^K}{.5^1}$
$\dfrac{{{5^{{{31}^{301}}}}}}{9} = \dfrac{{{{\left( {{5^6}} \right)}^K}{{.5}^1}}}{9} = \dfrac{{{1^K}.5}}{9} = 5$
3. Which of the following numbers must be added to 5678 to give a reminder 35 when divided by 460?
a. 980b. 797
c. 955
d. 618
Correct option: b
Explanation:
Let $x$ be the number to be added to 5678.
When you divide 5678 + $x$ by 460 the remainder = 35.
Therefore, 5678 + $x$ = 460k + 35 here $k$ is some quotient.
$ \Rightarrow $ 5643 + $x$ should exactly divisible by 460.
Now from the given options x = 797.
4. A girl entered a store and bought x flowers for y dollars (x and y are integers). When she was about to leave, the clerk said, “If you buy 10 more flowers I will give you all for $\$$2, and you will save 80 cents a dozen”. The values of x and y are:
a. (15,1)b. (10,1)
c. (5,1)
d. Cannot be determined from the given information.
Correct option: c
Explanation:
Given she bought $x$ flowers for $y$ dollars.
So 1 flower cost = $\dfrac{y}{x}$
12 flowers or 1 dozen cost = $\dfrac{{12y}}{x}$
Again, $x$+10 cost = 2 dollars
1 flower cost = $\dfrac{2}{{10 + x}}$
12 flowers or 1 dozen cost = $\dfrac{{2 \times 12}}{{10 + x}} = \dfrac{{24}}{{10 + x}}$
Given that this new dozen cost is 80 cents or 4/5 dollar less than original cost.
$ \Rightarrow \dfrac{{12y}}{x} - \dfrac{{24}}{{10 + x}} = \dfrac{4}{5}$
From the given options, c satisfies this.
5. If a number is divided by 357 the remainder is 5, what will be the remainder if the number is divided by 17?
a. 9b. 3
c. 5
d. 7
Correct option: c
Explanation:
Let $'N'$ be the given number.
$N = 357k + 5$ = $17 \times 21k + 5$
If this number is divided by 17 remainder is 5 as 357k is exactly divided by 17.
6. In how many possible ways can write 3240 as a product of 3 positive integers a,b and c.
a. 450b. 420
c. 350
d. 320
Correct option:
Explanation:
$3450 = {2^3} \times {3^4} \times {5^1} = a \times b \times c$
We have to distribute three 2's to a, b, c in ${}^{3 + 3 - 1}{C_{3 - 1}} = {}^5{C_2} = 10$ ways
We have to distribute four 3's to a, b, c in ${}^{3 + 4 - 1}{C_{3 - 1}} = {}^6{C_2} = 15$ ways
We have to distribute one 5 to a, b, c in 3 ways.
Total ways = $10 \times 15 \times 3 = 450$ ways.
7. On door A - It leads to freedomOn door B - It leads to Ghost houseOn door C - door B leads to Ghost houseThe statement written on one of the doors is wrong.Identify which door leads to freedom.
a. Ab. B
c. C
d. None
Correct option: c
Explanation:
Case 1: A, B are true. In this case, Statement C also correct. So contradiction.
Case 2: B, C are true. In this case, B leads to ghost house and C confirms it. Now A is wrong. So door A does not lead to freedom. So Door C leads to freedom.
8. In the given figure, If the sum of the values along each side is equal. Find the possible values a, b, c, d, e, and f.
a. 9, 7, 20, 16, 6, 38
b. 4, 9, 10, 13, 16, 38
c. 4, 7, 20, 13, 6, 38
d. 4, 7, 20, 16, 6, 33
Correct option: c
Explanation:
From the above table, 42 + a + b = 47 + e. Therefore, a + b = 5 + e. Option a, b ruled out.
47 + e = 15 + f. Therefore, 32 + e = f. Option d ruled out.
4 men throw a die each simultaneously. Find the probability that at least 2 people get the same number
a. 5/18
b. 13/18
c. 1/36
d. 1/2
9. 70, 54, 45, 41……. What is the next number in the given series?
a. 35b. 36
c. 38
d. 40
Correct option: d
Explanation:
Consecutive squares are subtracted from the numbers.
70 - 54 = 16
54 - 45 = 9
45 - 41 = 4
So next we have to subtract 1. So answer = 41 - 1 = 40
10. How many positive integers less than 500 can be formed using the numbers 1,2,3,and 5 for digits, each digit being used only once.
a. 52b. 68
c. 66
d. 34
Correct option:
Explanation:
Single digit number = 4
Double digit number = 4$\times$3 = 12
Three digit numbers = 3$\times$3$\times$2= 18 ($\because$ If Hundred's place is 5, then the number is greater than 500)
Total = 34.
