MyQuestionIcon

1

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

Monotonically Increasing Functions

Prove ah + ...

Question

Prove $∣ ∣ ∣ ∣ \begin{matrix} a h + b g & g & a b + c h b f + b a & f & h b + b c a f + b c & d & b g + f c \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} a h + b g & a & h b f + b a & h & b a f + b c & g & f \end{matrix} ∣ ∣ ∣ ∣$

Open in App

Solution

We have
$L . H . S = ∣ ∣ ∣ ∣ \begin{matrix} a h + b g & g & a b + c h b f + b a & f & h b + b c a f + b c & c & b g + f c \end{matrix} ∣ ∣ ∣ ∣$

$= b ∣ ∣ ∣ ∣ \begin{matrix} a h + b g & g & a b f + b a & f & h a f + b c & c & b \end{matrix} ∣ ∣ ∣ ∣ + c ∣ ∣ ∣ ∣ \begin{matrix} a h + b g & g & h b f + b a & f & b a f + c & c & f \end{matrix} ∣ ∣ ∣ ∣$

In second determinant applying $C_{1} \to C_{1} - b C_{2} - a C_{3}$
$= ∣ ∣ ∣ ∣ \begin{matrix} a h + b g & b g & a b f + b a & b f & h a f + b c & b c & g \end{matrix} ∣ ∣ ∣ ∣ + c ∣ ∣ ∣ ∣ \begin{matrix} 0 & g & h 0 & f & b 0 & c & f \end{matrix} ∣ ∣ ∣ ∣$
In first determinant applying $C_{2} \to C_{2} - C_{1}$ then
$= ∣ ∣ ∣ ∣ \begin{matrix} a h + b g & - a g & a b f + b a & - b a & h a f + b c & - a f & g \end{matrix} ∣ ∣ ∣ ∣ + 0$ $= ∣ ∣ ∣ ∣ \begin{matrix} a b + b g & a & h b f + b a & h & b a f + b c & g & f \end{matrix} ∣ ∣ ∣ ∣$ $= R . H . S .$

Suggest Corrections

thumbs-up

0

Similar questions

△

~

△

△

AMT, MA = 6.3 cm, AT = 4.9 cm,

∠

°

\frac{MA}{HA} = \frac{7}{5}

△

Q. Given a parallelogram

A B C D

A H ⊥ D C, D C = 10 m, A H = 10 m

. Find area of parallelogram

A B C D .

A B

is a diameter of a circle.

P

is a point on the semi-circle

A P B

A H

B K

are perpendiculars from

A

B

respectively to the tangent at

P

A H + B K = A B

Q. If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that:

\frac{a}{h} + \frac{b}{k} = 1

△ A M T \sim △ A H E . I n △ A M T, A M = 6.3 c m, ∠ T A M = 50^{0}, A T = 5.6 c m, \frac{A M}{A H} = \frac{7}{5} . C o n s t r u c t △ A H E .

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Monotonicity

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Monotonically Increasing Functions

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon