Mathematics Test

If f(x) and g(x) are continuous functions satisfying f(x) = f(a – x) and g(x) + g(a – x) = 2, then what is \(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right){\rm{dx}}\) equal to?

Question 1/10

\({e^x}f'\left( x \right) + C\)

\({e^x}f\left( x \right) +C\)

\({e^x} + f\left( x \right) +C\)

None of these

Question 2/10

0

\(\rm \frac{1}{2}\)

1

2

Question 3/10

\(\dfrac{\pi}{2}\)

\(\dfrac{\pi}{4}\)

\(\dfrac{\pi}{8}\)

0

Question 4/10

1 - cos 1

1 - sin 1

1 + cos 1

None

Question 5/10

0

1

2

None of these.

Question 6/10

√2 log 2

-√2 log √2

-2 log √2

None

Question 7/10

\(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{g}}\left( {\rm{x}} \right){\rm{dx}}\)

\(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\)

\(2\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\)

0

Question 8/10

2sinx - ln(sec x + tan x) + C

2sinx + ln(sec x - tan x) + C

2sinx + ln(sec x + tan x) + C

2sinx - ln(sec x - tan x) + C

Question 9/10

\( C\ =\ \frac{\pi}{4} \ -\ 1\)

\( C\ =\ \frac{\pi}{4} \ +\ 1\)

\( C\ =\ 1\ -\ \frac{\pi}{4} \ \)

\( C\ =\ 1\ +\ \frac{\pi}{4} \ \)

Question 10/10

elog x (sin x - cos x) + C

(sin x - x cos x) + C

(x sin x + cos x) + C

(sin x + x cos x) - C

e^{log x} (sin x - cos x) + C