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Definition of a Determinant

I: Let α=x+...

Question

I: Let $¯ ¯¯ ¯ α = (x + 4 y) a + (2 x + y + 1) b$ , $¯ ¯ ¯ β = y - 2 x + 2) a + (2 x - 3 y - 1) b$ where a and $b$ are nonzero, noncollinear vectors if 3 $¯ ¯¯ ¯ α = 2 ¯ ¯ ¯ β = x = 2, y = - 1$ .

II: Let $D, E, F$ be the middle points of the sides $B C . C A, A B$ respectively of $Δ A B C$ then $A D + B E + C F = 0$ .

A

Only I is true

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B

Only II is true

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C

Both I and II are true

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D

Neither I nor II are true

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Solution

The correct option is C Both I and II are true
I $3 \to α = 2 \to β$
$3 ((x + 4 y) a + (2 x + y + 1) b) = 2 ((y - 2 x + 2) a + (2 x - 3 y - 1) b$ )
$(3 x + 12 y) a + (6 x + 3 y + 3) b) = (2 y - 4 x + 4) a + (4 x - 6 y - 2) b$ )
$3 x + 12 y = 2 y - 4 x + 4$ ( a & b are not collinear )
$7 x + 10 y = 4$ --(1)
$6 x + 3 y + 3 = 4 x - 6 y - 2$
$2 x + 9 y = - 5$ --(2)
$x = 2 y = - 1$
$A D + B E + C F = \to D + \to E + \to F - \to A - \to B - \to C$
$= \to a + \to b + \to c - \to a - \to b - \to c$
$= 0$

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Similar questions

a, b

be two noncollinear vectors. If

A = (x + 4 y) a + (2 x + y + 1) b

B = (y - 2 x + 2) a + (2 x - 3 y - 13) b

3 A = 2 B

(x, y)

=

Q. Let a,b be two noncoffinear vectors. If

¯ ¯¯¯¯¯¯ ¯ O A = (x + 4 y) ¯ ¯ ¯ a + (2 x + y + 1) ¯ ¯ b, ¯ ¯¯¯¯¯¯ ¯ O B = (y - 2 x + 2) ¯ ¯ ¯ a + (2 x - 3 y - 1) ¯ ¯ b

3 ¯ ¯¯¯¯¯¯ ¯ O A = 2 ¯ ¯¯¯¯¯¯ ¯ O B,

(x, y) =

Statements: U # X, X * D, D @ Y
Conclusions:
I. U @ Y
II. U * Y

I .

3 x - 5 y = - 1

x - y = - 1

x = - 2, y = - 1

I I .

2 x + 3 y = 9, 3 x + 4 y = 5 \Rightarrow x = - 21, y = 17

I I I .

\frac{2 x}{a} + \frac{y}{b} = 2, \frac{x}{a} - \frac{y}{b} = 4 \Rightarrow x = 2 a, y = 2 b

If $y = f (x),$ where $f : A \to B$ is a function in $x,$ then which of the following statements is true?

(i) Every element of $A$ needs to have an image.

(ii) $x \in A$ must be related to one and only one value of $y$ of $B .$

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