If the system of equations 2 x - y + z = 0 , x - 2 y + z = 0 and a x - y + 2 z = 0 has infinitely many solutions and f ( x ) be a continuous function such that f ( 5 + x ) + f ( x ) = 2 x R , then 0 - 2 a f ( x ) ? d x is equal to:

Question

If the system of equations $2 x - y + z = 0, x - 2 y + z = 0$ and $a x - y + 2 z = 0$ has infinitely many solutions and $f (x)$ be a continuous function such that $f (5 + x) + f (x) = 2 \forall x \in R$ , then $\int_{0}^{- 2 a} f (x) d x$ is equal to:

Smart Achievers · Accepted Answer

(1)

$Δ = 0, a = 5$

$\int_{0}^{- 10} f (x) d x = \int_{0}^{5} f (x) d x + \int_{- 5}^{10} f (x + 5) d x + \int_{- 85}^{10} f (x) d x = 2 \int_{- 5}^{- 10} d x = - 10$

Solution

Q.
If the system of equations $2 x - y + z = 0, x - 2 y + z = 0$ and $a x - y + 2 z = 0$ has infinitely many solutions and $f (x)$ be a continuous function such that $f (5 + x) + f (x) = 2 \forall x \in R$ , then $\int_{0}^{- 2 a} f (x) d x$ is equal to:

1 $- 10$

2 $- 4 a$

3 $\int_{0}^{a} [x] d x [where [x] denotes G.I.F of x]$

4 $2 a$

Ans.
(1)

$Δ = 0, a = 5$

$\int_{0}^{- 10} f (x) d x = \int_{0}^{5} f (x) d x + \int_{- 5}^{10} f (x + 5) d x + \int_{- 85}^{10} f (x) d x = 2 \int_{- 5}^{- 10} d x = - 10$

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Solution

Q. If the system of equations 2x-y+z=0, x-2y+z=0 and ax-y+2z=0 has infinitely many solutions and f(x) be a continuous function such that f(5+x)+f(x)=2∀x∈R, then ∫0-2af(x) dx is equal to:

1 -10

2 -4a

3 ∫0a[x] dx [where [x] denotes G.I.F of x]

4 2a

Ans. (1) Δ=0, a=5 ∫0-10f(x) dx=∫05f(x) dx+∫-510f(x+5) dx+∫-8510f(x) dx=2∫-5-10dx=-10