Elitmus previous questions set - 5

Elitmus previous questions set - 5

1. Two identical cubes if one of them is painted pink on its 4 sides and blue on the remaining two side then how many faces painted pink to other cube so that probability of getting the same color is 1/3 when we roll both the cubes.

Explanation:
First cube has got 4 pink sides and 2 black sides.
Let the other cube got x sides pink and (6 - x) sides black.
Now when we roll both the dice, we can either pink on both cubes or black on both cubes.
Probability = 46×x6+26×6−x6=13${\displaystyle \frac{4}{6}}\times {\displaystyle \frac{x}{6}}+{\displaystyle \frac{2}{6}}\times {\displaystyle \frac{6-x}{6}}={\displaystyle \frac{1}{3}}$
=4x+12−2x36=13$={\displaystyle \frac{4x+12-2x}{36}}={\displaystyle \frac{1}{3}}$
⇒x=0$\Rightarrow x=0$
So second cube should not have any pink faces at all.

2. In a right angled triangle, two sides are consecutive whole number in which one side is hypotenuse. what could be the possible length of third side?
1. 360
2. 361
3. 362
4. none of these
Answer: 2
Explanation:
Pythagorean triplets are generated with each "odd number" greater than 1 by using a formula.
If n is an odd number, then Pythagorean triplet = n,n2−12,n2+12$n,{\displaystyle \frac{n^2-1}{2}},{\displaystyle \frac{n^2+1}{2}}$.
Here 361 is an odd number.  So the triplet is 361, 65160, 65161.

3. Heinz produces tomato puree by boiling tomato juice. The tomato puree has only 20% water while the tomato juice has 90% water. How many liters of tomato puree will be obtained from 20 litres of tomato juice?
a. 2 liters
b. 2.4 liters
c. 2.5 liters
d. 6 liters
Answer:
Explanation:
In each of the solutions, there is a pure tomato component and some water.  So while boiling, water evaporates but tomato not. So we equate tomato part in the both equations.
⇒$\Rightarrow$ 10%(20) = 80%(x)
⇒$\Rightarrow$ x = 2.5 liters.

4. x,y are odd and z is even then ((x^2+y^2)z^2)/8 is
a. even
b. odd
c. either even or odd
d. fraction
Explanation: c
As x, y are odd x2+y2$x^2+y^2$ is always even. Now if z is a multiple of 4, then z2$z^2$ is divisible by 8, then the equation is even.  if z is a not a multiple of 4, but only a multiple of 2, then z2$z^2$ is not completely divisible as it contains only two 2's and other two is cancelled in x2+y2$x^2+y^2$ which results in an odd number.
(32+52)428=34×168=34×2${\displaystyle \frac{\left(3^2+5^2\right)4^2}{8}}={\displaystyle \frac{34\times 16}{8}}=34\times 2$
(32+52)628${\displaystyle \frac{\left(3^2+5^2\right)6^2}{8}}$ = 34×368=17×9${\displaystyle \frac{34\times 36}{8}}=17\times 9$

5. In the formula of converting temperature from Celsius to Fahrenheit F=9/5C+32, How many integer values(not fractional) of F will be there that lies between 100 to 200 for integer values of C.
Explanation:
F=95C+32$F={\displaystyle \frac{9}{5}}C+32$
As F needs to be integer, then C should be a multiple of 5. First integer value of F for C = 5 is 41, next value for C =10 is 50 and so on.
The values of F are in A.P with common difference of 9. They are in the format of 41 + 9n.
The first value of F which is greater than 100 is for n = 7 which is 104.
The last value of F which is less than 200 is for n = 17, which is 194.
Total values are 194−1049+1${\displaystyle \frac{194-104}{9}}+1$ = 11

6. The product of digit is a Factor of a two digit number.  Number of such digit are:
a. 3
b. 5
c. 9
d. 27
Answer: b
Explanation:
Let the number be xy. So 10x+yxy=k${\displaystyle \frac{10x+y}{xy}}=k$.  Here k is some interzer.
⇒$\Rightarrow$ 10x + y = kxy
⇒$\Rightarrow$ x(10 - ky) = - y
⇒$\Rightarrow$ x(ky - 10) = y
So x is a factor of y. The possibilities are,
11, 1×1=1; 12 , 1×2=2; 15 ,1×5=5; 24 ,2×4=8; 36 , 3×6=18

7. Data sufficiency question.
is x>y ?
1. 5x+15y=40
2.7x+21y=56 
Explanation:
Statement 1 has three solutions,(8, 0), (5, 1), (2, 2) but we cannot say precisely about the relationship
Statement 2 has three solutions, (8, 0), (5, 1), (2, 2) but we cannot say about the relationship.
So data insufficient.