Polynomials

Step 1: Evaluate the determinant condition.

%5CDelta%20%3D%0Af(x)%5C%2Cf%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%0A-1%5Cbig%5Bf%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%2Bf(x)%5Cbig%5D%0A%3D0

%5CRightarrow%20f(x)f%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%0A-%20f%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%20-%20f(x)%3D0

Step 2: Rearrange the expression.

f(x)f%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%0A-%20f%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%0A-%20f(x)%3D0

Add 1 to both sides:

%5Cbig(f(x)-1%5Cbig)%5Cbig(f%5C!%5Cleft(%5Ctfrac%7B1%7D%7Bx%7D%5Cright)-1%5Cbig)%3D1

Step 3: Use the fact that $f(x)$ is a polynomial. Since

f(x)

is a polynomial,

f%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)

is also finite only if

f(x)%3Dax%2Bb

(a constant or linear polynomial).
Step 4: Assume $f(x)%3Dax%2Bb$ . Then:

f%5C!%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%3D%5Cfrac%7Ba%7D%7Bx%7D%2Bb

Substitute into:

%5Cbig(f(x)-1%5Cbig)%5Cbig(f%5C!%5Cleft(%5Ctfrac%7B1%7D%7Bx%7D%5Cright)-1%5Cbig)%3D1

(ax%2Bb-1)%5Cleft(%5Cfrac%7Ba%7D%7Bx%7D%2Bb-1%5Cright)%3D1

For this to be independent of

x

, we must have:

a%3Db-1

Step 5: Use the given value $f(2)%3D17$ .

f(2)%3D2a%2Bb%3D17

Substitute

a%3Db-1

2(b-1)%2Bb%3D17%0A%5CRightarrow%203b%3D19%0A%5CRightarrow%20b%3D%5Cfrac%7B19%7D%7B3%7D

a%3D%5Cfrac%7B16%7D%7B3%7D

Step 6: Find $f(5)$ .

f(5)%3D5a%2Bb%0A%3D5%5Cleft(%5Cfrac%7B16%7D%7B3%7D%5Cright)%2B%5Cfrac%7B19%7D%7B3%7D%0A%3D%5Cfrac%7B80%2B19%7D%7B3%7D%0A%3D%5Cfrac%7B99%7D%7B3%7D%0A%3D82

Hence, the required value is

%5Cboxed%7B82%7D

Step 1: Recall the order property of numbers. The symbols $%3E$ and $%3C$ are defined only for real numbers. Complex numbers $(a%20%2B%20ib)$ cannot, in general, be compared using inequalities.

Step 2: Condition for a complex number to be real. A complex number $a%20%2B%20ib$ is real only if its imaginary part is zero: $b%20%3D%200$ Similarly, $c%20%2B%20id$ is real only if: $d%20%3D%200$

Step 3: Apply the condition to the given inequality. The inequality $a%20%2B%20ib%3Ec%20%2B%20id$ is meaningful only when: $b%20%3D%200%20%5Cquad%20%5Ctext%7Band%7D%20%5Cquad%20d%20%3D%200$ Then it reduces to the real-number inequality: $a%3Ec$

Step 4: Identify the correct option. Option (B) satisfies the required condition.

About Polynomials - LPUNEST

Polynomials is a vital chapter for LPUNEST aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Polynomials PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Polynomials carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

About Polynomials - LPUNEST

Frequently Asked Questions

Why focus on Polynomials PYQs?

How to best use this analysis?

Polynomials

High-Yield Trend

Chapter Questions 4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

About Polynomials - LPUNEST

Frequently Asked Questions

Why focus on Polynomials PYQs?

How to best use this analysis?

Chapter Questions
4 MCQs