Elitmus previous questions - 2

Elitmus previous questions - 2

    

1. How many five digit numbers can be formed by using the digits 0,1,2,3,4,5 such that the number is divisible by 4?

Explanation:
If a number has to be divisible by 4, the last two digit of that number should be divisible by 4.
So _ _ _ x y.  Here xy should be a multiple of 4.
There are two cases:
Case 1: xy can be 04, 20 or 40
In this case the remaining 3 places can be filled in 4×3×2 = 24.  So total 24×3 = 72 ways.
Case 2: xy can be 12, 24, 32, 52.
In this case, left most place cannot be 0.  So left most place can be filled in 3 ways.  Number of ways are 3×3×2 = 18.  Total ways = 18×4 = 72.
Total ways = 144

2. Data sufficiency question:
There are six people. Each cast one vote in favor of other five. Who won the elections?
i)  4 older cast their vote in favor of the oldest candidate
ii) 2 younger cast their vote to the second oldest
Explanation:
Total possible votes are 6.  Of which 4 votes went to the oldest person.  So he must have won the election. Statement 1 is sufficient.

3. Ram draws a card randomly among card 1-23 and keep it back.  Then Sam draws a card among those.  What is the probability that Sam has drawn a card greater than ram.
Explanation:
If Ram draws 1, then Sam can draw anything from 2 to 23 = 22 ways
If Ram draws 2, then Sam can draw anything from 3 to 23 = 21 ways
. . . .
. . . .
If Ram draws 23, Sam has no option = 0 ways.
Total required ways = 22 + 21 + 20 + . . . . + 0 =  22×232$\frac{22\times 23}{2}$ = 253
Total ways of drawing two cards = 23×23
Required probability = 253529=1123${\displaystyle \frac{253}{529}}={\displaystyle \frac{11}{23}}$

4. Decipher the following multiplication table:
     M A D
          B E
-------------
     M A D
  R A E
-------------
  A M I D
-------------
Explanation:
It is clear that E = 1 as MAD×E=MAD
From the hundred's line, M + A = 10 + M or 1 + M + A = 10 + M
As A = 10 not possible, A = 9
So I = 0.
and From the thousand's line R + 1 = A. So R = 8.
     M 9 D
         B 1
-------------
     M 9 D
  8 9  1
-------------
  9 M 0 D
-------------
As B×D = 1, B and D takes 3, 7 in some order.
If B = 7 and D = 3, then M93×7 =   _51 is not satisfying. So B = 3 and D = 7.
     2 9 7
        3 1
-------------
     2 9 7
8  9  1
-------------
9 2 0 7
-------------

5. If log3N+log9N${\log }_{3}N+{\log }_{9}N$ is whole number, then how many numbers possible for N between 100 to 100?
Explanation:
log3N+log9N${\log }_{3}N+{\log }_{9}N$ = log3N+log32N${\log }_{3}N+{\log }_{3^2}N$ = log3N+12log3N${\log }_{3}N+{\displaystyle \frac{1}{2}}{\log }_{3}N$ =32log3N${\displaystyle \frac{3}{2}}{\log }_{3}N$
Now this value should be whole number.
Let 32log3N${\displaystyle \frac{3}{2}}{\log }_{3}N$ = w
⇒log3N=23w$\Rightarrow {\log }_{3}N={\displaystyle \frac{2}{3}}w$
N=3(23w)$N=3^{\left(\frac{2}{3}w\right)}$
As N is a positive integer, So for w = 0, 3, 6 we get N = 1, 9, 81.
Three values are possible.

6. If an ant moves a distance x,then turn left 120 degree and travel x distance again and turn right 120 dgree and travel, he is 9 inches from the starting point,  what is the value of x?
a.  3root(3)
b.  9root(3)
c.  3root(3)/2
d.  3root(3)/4
Explanation:

See the above diagram.  (A bit confusing though!!)
The ant starts from A travelled to B, turned 120 degrees left and travelled to C. and turned 120 degrees right and travelled to D.
From the diagram, ∠ACB=600$\mathrm{\angle }ACB={60}^0$.  As BC and AB are equal, ∠BCA=∠BAC=600$\mathrm{\angle }BCA=\mathrm{\angle }BAC={60}^0$.
So ΔABC$\mathrm{\Delta }ABC$ is equilateral.
Also ∠BCD=600$\mathrm{\angle }BCD={60}^0$.
As, ∠ACB=∠BCD$\mathrm{\angle }ACB=\mathrm{\angle }BCD$ equal, CB divides AD into two equal parts. Aso DE = 9/2.
In the triangle CED, Sin600=3√2=92x$Sin{60}^0={\displaystyle \frac{\sqrt{3}}{2}}={\displaystyle \frac{\frac{9}{2}}{x}}$
So x=93√$x={\displaystyle \frac{9}{\sqrt{3}}}$

7. If a⁴ +(1/a⁴)=119 then a power 3-(1/a³) = 
a. 32 
b. 39
c.  Data insufficient
d.  36
Explanation:
Given that a4+1a4=119$a^4+{\displaystyle \frac{1}{a^4}}=119$ , adding 2 on both sides, we get : (a2+1a2)2=121${\left(a^2+{\displaystyle \frac{1}{a^2}}\right)}^2=121$
⇒a2+1a2=11$\Rightarrow a^2+{\displaystyle \frac{1}{a^2}}=11$
Again, by subtracting 2 on both sides, we have, ⇒(a−1a)2=9$\Rightarrow {\left(a-{\displaystyle \frac{1}{a}}\right)}^2=9$
⇒a−1a=3$\Rightarrow a-{\displaystyle \frac{1}{a}}=3$
Now, ⇒a3−1a3$\Rightarrow a^3-{\displaystyle \frac{1}{a^3}}$ = (a−1a)(a2+1a2+1)$\left(a-{\displaystyle \frac{1}{a}}\right)\left(a^2+{\displaystyle \frac{1}{a^2}}+1\right)$ = 12×3 = 36

Amcat previous questions set - 2

Amcat previous questions set - 2

     

1. What is the largest power of 20 contained in 100!?

A. 56

B. 1

C. 24

D. 2

Answer: C

Explanation:

20 = 5×2^2.

Power of 2 in 100!= [1002]+[1004]$\left[{\displaystyle \frac{100}{2}}\right]+\left[{\displaystyle \frac{100}{4}}\right]$ + [1008]+[10016]$\left[{\displaystyle \frac{100}{8}}\right]+\left[{\displaystyle \frac{100}{16}}\right]$ + [10032]+[10064]$\left[{\displaystyle \frac{100}{32}}\right]+\left[{\displaystyle \frac{100}{64}}\right]$ = 97

Here [ ] is greatest integer function.  So only integer parts should be considered for positive numbers.
So highest power of 4 in 100! = (2)97=(22)48×2${\left(2\right)}^{97}={\left(2^2\right)}^{48}\times 2$ = 48 times.
Highest power of 5 in 100! = [1005]+[10025]$\left[{\displaystyle \frac{100}{5}}\right]+\left[{\displaystyle \frac{100}{25}}\right]$ = 24

so, Power of 20 in 100! = least of the two powers = 24

2. A seller uses 900 gm in place of 1 kg.Find his actual percentage of profit or loss when he sells his articles at 5% loss on cost price?

A. 5

B. 6

C. 5.55

D. 20

Answer: C

Explanation:
Let us assume one grams costs Rs.1.
He gives only 900 gm for 1 kg. So cost price = Rs.900
But he collects 5% less on the cost price.  So he collects Rs.950 from the customer. (∵$\because$ This
reduction should be calculated on Rs.1000 as the customer does not know about his cheating).

Profit % = 50900×100${\displaystyle \frac{50}{900}}\times 100$ = 5.55%

3. A coin is tossed thrice.  What is the probability that first toss of the coin lands head, second lands tail and third lands tail as well

A. 1/3

B. 1/ 6

C. 1/8

D. 1/10

Answer: C

Explanation:

Total ways when 3 coins are tossed = 23$2^3$ = 8

Now HTT is one of the way. So required probability = 1/8

Alternative method: 

If a coin is tossed, getting heads or tails has the probability = 1/2

We have 3 such cases here so = 1/2 × 1/2 × 1/2 = 1/8

4. If the LCM of a and (a>b) is 667 the the value of 7b – 5a is

Answer: 16
Explanation:
667 can be written as 29 × 23. So a = 29,  b = 23. 
7b – 5a = 7×29 – 5×23 = 16

5. The money at compound interest amount to thrice it self in 16 years.  So,in how many years it will be 27 times itself ?

Answer: 48 
Explanation:
Let P be the principal.
It will thrice itself in 16 years. So it becomes = 3×P = 3P.
After another 16 years it will be 3×3P = 9P
Another after 16 years it will be 3×9P = 27P
So it will take 48 years to become 27 times. 

6. INGENUITY (OPPOSITE)

A. skillfulness
B. cunning
C. inventive
D. dullness

Answer: 48 

Explanation:

Ingenuity means the quality of being original, clever.  It came from the root 'genuine' which is nobility.  So dullness can be opposite to this. 

7. PRANKISH (OPPOSITE)
A. whimsical
B. Machiavellian
C. impish
D. serious
Answer: 

Explanation: D

Prank means childish behaviour , to put pranks on others.  So opposite to this is serious.  Machiavellian means cunning, and impish means lack of seriousness. 

8. PERT (OPPOSITE)

A. impudent
B. brash
C. savvy
D. polite

Answer: D

Explanation:

Pert means shameless boldness.  Impudent also means the same.  Brash means the same as  impertinent, impudent.  Savvy means knowledgeable.  So Polite is opposite to pert. 

9. GAUNT (OPPOSITE)

A. emaciated
B. sombre
C. plump
D. piquant

Answer: C

Explanation:

Guant means lean and haggard, especially because of suffering, hunger, or age.  Opposite to this is plump - having a full rounded shape.  Emaciated means weak and thin.  Sombre means gloomy, dark and dull. Piquant has two meanings. 1. having sharp taste 2. interesting 

10. He protracted illness has reduced him ________ skeleton
A. THROUGH
B. TILL
C. TO
D. ROUND
E. FROM

Answer: C
Reduced is followed by the preposition to.