If f x- x-3 2 x2-18 3 x3-81- x-5 2 x2-50 4 x3-500- 1 2 3 - then f1 - f3-f3 - f5-f5 - f1-A- f1B- f3C- f1-f3D- f1-f5

The correct option is B f (3)f(x) = 2(x 3)(x 5); 1 amp;x+3 amp;3(x^2+3x+9) 1 amp;x+5 amp;4(x^2+5x+25) 1 amp;1 amp;3 (Taking out (x 3),(x 5) and ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Properties of Determinants

If f x=|[ ...

Question

If $f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} x - 3 & 2 x^{2} - 18 & 3 x^{3} - 81 x - 5 & 2 x^{2} - 50 & 4 x^{3} - 500 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣$ then $f (1) . f (3) + f (3) . f (5) + f (5) . f (1) =$

f (1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

f (3)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

f (1) + f (3)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

f (1) + f (5)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B f (3)
$f (x) = 2 (x - 3) (x - 5); ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & x + 3 & 3 (x^{2} + 3 x + 9) 1 & x + 5 & 4 (x^{2} + 5 x + 25) 1 & 1 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣$
(Taking out (x-3),(x-5) and 2 from $I^{s t}$ row, $I I^{n d}$ row and $I I I^{r d}$ column respectively)
$f (x) = 2 (x - 3) (x - 5) ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & (x + 2) & 3 (x^{2} + 3 x + 8) 0 & 2 & x^{2} + 11 x + 73 1 & 1 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣, (R_{1} \to R_{1} - R_{3} a n d R_{2} \to R_{2} - R_{1}) = 2 (x - 3) (x - 5) [1 (x + 2) (x^{2} + 11 x + 73) - 6 (x^{2} + 3 x + 8)] = 2 (x^{2} - 8 x + 15) (x^{3} + 13 x^{2} + 95 x + 146 - 6 x^{2} - 18 x - 48) = 2 (x^{2} - 8 x + 15) (x^{3} + 7 x^{2} + 77 x + 98) = 2 (x^{5} - x^{4} + 36 x^{3} - 413 x^{2} + 371 x + 1470) f (1) = 2928, f (3) = 0, f (5) = 0 ∴ f (1) . f (3) + f (3) . f (5) + f (5) . f (1) = 0 + 0 + 0 = 0 = f (3) .$

Suggest Corrections

Similar questions

Q. If

f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} x - 3 & 2 x^{2} - 18 & 3 x^{3} - 81 x - 5 & 2 x^{2} - 50 & 4 x^{3} - 500 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣

then

f (1) . f (3) + f (3) . f (5) + f (5) . f (1) =

Q. If

f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} x - 3 & 2 x^{2} - 18 & 3 x^{3} - 81 x - 5 & 2 x^{2} - 50 & 4 x^{3} - 500 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣

then

f (1) . f (3) + f (3) . f (5) + f (5) . f (1) =

Q. For positive interger

n,

f (n) = {sin}^{n} θ + {cos}^{n} θ,

then

\frac{f (3) - f (5)}{f (5) - f (7)}

Let $F_{1}$ be the set of all parallelograms, $F_{2}$ the set of rectangles, $F_{3}$ the set of rhombuses, $F_{4}$ the set of squares and $F_{5}$ the set of trapeziums in a plane, then $F_{1}$ is equal to

Q. A function

f : [- 3, 7) \to R

is defined as follows

f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 4 x^{2} - 1; & - 3 \leq x < 2 3 x - 2; & 2 \leq x \leq 4 2 x - 3; & 4 < x < 7 \end{matrix}

(i)

f (5) + f (6)

(ii)

f (1) - f (3)

(iii)

f (- 2) - f (- 4)

(iv)

\frac{f (3) + f (1)}{2 f (6) - f (1)}

Join BYJU'S Learning Program

If f(x) = ∣∣ ∣ ∣∣x−32x2−183x3−81x−52x2−504x3−500123∣∣ ∣ ∣∣ then f(1).f(3)+f(3).f(5)+f(5).f(1) =

If $f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} x - 3 & 2 x^{2} - 18 & 3 x^{3} - 81 x - 5 & 2 x^{2} - 50 & 4 x^{3} - 500 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣$ then $f (1) . f (3) + f (3) . f (5) + f (5) . f (1) =$