AP EAMCET Engineering 23 Sep 2020 Shift 1 Paper

f′(x)=a‌sin‌x+b‌cos‌x ...(i)
Put x=0
f′(0)=a‌sin(0)+b‌cos(0)
4=a(0)+(b)(1)[∵f′(0)=4]
∴b=4
From Eq. (i),
f′(x)=a‌sin‌x+b‌cos‌x
Integrating on both sides.
∫f′(x)dx=∫(a‌sin‌x+b‌cos‌x)dx
f(x)=a(−cos‌x)+b(sin‌x)+c ....(ii)
Put x=0 in Eq. (ii),
f(0)=a(−cos‌0)+b‌sin(0)+C
3=−a+C‌‌[∵f(0)=3]
∴−a+c=3....(iii)
Put x=‌

π

2

in Eq. (ii)
f(‌

π

2

)=a(−cos‌

π

2

)+b(sin‌

π

2

)+C
5=a(0)+b(1)+c[∵f(‌

π

2

)=5]
b+c=5
4+c=5[∵b=4]
C=1
From Eq. (iii),
−a+1=3⇒a=−2
From Eq. (ii), f(x)=(−2)(−cos‌x)+4(sin‌x)+1
f(x)=2‌cos‌x+4‌sin‌x+1