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.

High-Yield Trend

Chapter Questions
4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Polynomials

High-Yield Trend

Chapter Questions 4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Chapter Questions
4 MCQs