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.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOgywQZmCf9qKvDYuB0GmYxDYzLmrq4xeaAFVIZtIGkGxfKEz0iSKaYlpdN9uxVtyOoH1M9TCfll77CE_pvTz_iU48aQ0ckYTTJiYRmG3N4Hd5ycFLA143idatZFAYuSIM2296m5tnH7ta/s1600/Untitled.png)
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.