1. In an examination consisting of 75 questions, one mark is given for every correct answer and ¼ mark is deducted for every wrong answer. A student attempts all the 75 questions and scores a total of 50 marks. Find the number of questions he marked wrong.

a. 20

b. 25

c. 30

d. 50

## EXPLANATION

Let’s denote the number of correct answers as (x) and the number of incorrect answers as (y).

Given that one mark is given for every correct answer and (1/4) mark is deducted for every wrong answer, the total marks obtained can be expressed as:

[ x – y/4} = 50 ]

Now, we know that the total number of questions attempted is 75:

[ x + y = 75 ]

We have a system of two equations:

[ x – y/4 = 50 ]

[ x + y = 75 ]

Let’s solve these equations to find the values of (x) and (y). First, multiply the first equation by 4 to eliminate the fraction:

[ 4x – y = 200 ]

Now, add this equation to the second equation:

[ (4x – y) + (x + y) = 200 + 75 ]

Combine like terms:

[ 5x = 275 ]

Divide by 5:

[ x = 55 ]

Now that we know (x), substitute it back into one of the original equations. Let’s use the second equation:

[ x + y = 75 ]

[ 55 + y = 75 ]

Subtract 55 from both sides:

[ y = 20 ]

So, the student marked 20 questions wrong.

2. How many 4 letter words, with or without meaning, can be formed out of the letters of the word ‘LOGARITHMS’, if repetition of letters is not allowed?

a. 40

b. 400

c. 5040

d. 2520

## EXPLANATION

The word “ LOGARITHMS” has 10 different letters and we have to form words using any 4 letters from the 10 given letters.

Therefore,

Total no. of words that can be formed

^{10}P_{4} = 10! / (10-4)! = 5040

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