Problem 77 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In a certain test there are \(n\) questions. In this test \(2^{k}\) students gave wrong answers to at least ( \(n-k)\) questions, where \(k=0,1,2, \ldots, n .\) If the total number of wrong answers is 4095 , then value of \(n\) is (A) 11 (B) 12 (C) 13 (D) 15

Step-by-Step Solution

Answer

The value of \(n\) is 12.

Step 1: Understanding the setup

In the test with \(n\) questions, the number of students who gave wrong answers to at least \(n-k\) questions is \(2^k\) for \(k = 0, 1, 2, \ldots, n\). We are given that the total number of wrong answers is 4095. We need to find the value of \(n\).

Step 2: Express total wrong answers mathematically

The total number of wrong answers is the sum of all students who answered incorrectly to at least \((n-k)\) questions multiplied by \((n-k)\). This can be expressed as a sum: \[\sum_{k=0}^{n} 2^k \cdot (n-k)\].

Step 3: Simplify the expression

First note that each term \(2^k \cdot (n-k)\) can be expanded and rearranged using properties of summation. Simplifying, it becomes easier to express it by separating constants and power sums.

Step 4: Apply standard summation forms

The term can be expanded to: \(n\sum_{k=0}^{n} 2^k - \sum_{k=0}^{n} k \cdot 2^k\). Use the formula for the sum of a geometric series, \(\sum_{k=0}^{n} 2^k = 2^{n+1} - 1\), and known results for the weighted sum \(\sum_{k=0}^{n} k \cdot x^k = \frac{x(x^{n}-nx+n)}{(x-1)^2}\).

Step 5: Derive the formula and solve for n

Using \(x=2\), find \(\sum_{k=0}^{n} k \cdot 2^k = 2 ((2^n)(n-1)+2)\), simplifying and substituting these into the original wrong answers equation: \(n(2^{n+1} - 1) - (2^{n+1}n - 2(n+1))\). Equate the simplified expression to 4095, solve it straightforwardly substituting potential \(n\) values from the options and checking correctness.

Key Concepts

Mathematical Summation

Understanding mathematical summation is crucial when dealing with problems involving sequences and series. In this context, summation notation helps us express the sum of terms in a concise form. In the test question scenario, we have a series of students who answered questions wrong, expressed as a sum: \(\sum_{k=0}^{n} 2^k \cdot (n-k)\). Here, \(2^k\) represents the number of students, and \((n-k)\) refers to the number of wrong answers. Simplifying such sums often requires recognizing patterns or applying known summation formulas.

• Basic Summation Notation: The symbol \(\sum\) denotes 'sum' and is used to add up terms systematically.
• Using Indices: The index \(k\) controls how many times the addition occurs, from \(k=0\) to \(k=n\).
• Simplification: Separating constants and using properties of exponents can help us simplify complex summations.

In the context of our problem, understanding and manipulating these sums is key to deriving the correct answer for \(n\).

Geometric Series

Geometric series play a significant role in simplifying problems involving exponential growth, like the doubling nature in this exercise's student responses. A geometric series is a series with a constant ratio between successive terms. For instance, \(2^0, 2^1, 2^2, \ldots , 2^n\) represents a geometric series with a common ratio of 2.

• Sum of a Geometric Series: The formula \(\sum_{k=0}^{n} 2^k = 2^{n+1} - 1\) provides the sum of all terms in the series.
• Application: Recognizing the characteristics of geometric series allows quick computation of prolonged sums.
• Use in Problem Solving: Good application of these sums in the exercise helps to evaluate expressions quickly and accurately.

This geometric series concept aids in formulating concise expressions for the number of students, helping solve for \(n\) accurately.

Problem Solving in Algebra

Algebraic problem-solving often involves setting up equations derived from given conditions, simplifying them, and finding unknowns. In our example, each student's wrong answers depend on \(n-k\) terms, leading us to an expression derived using algebraic manipulation.

• Formulating Equations: Our task starts with setting up the equation \(n(2^{n+1} - 1) - (2^{n+1}n - 2(n+1)) = 4095\), which depicts total wrong answers.
• Simplification: This involves utilizing standard algebraic methods like removing common factors or using known sums.
• Substitution and Verification: Substituting possible \(n\) values from given options into the simplified expression gives us the correct solution.

Algebra is invaluable for transforming the problem's language into solvable equations and determining the value of \(n\) through strategic manipulation.