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List IList IIP.The number of polynomials f(x) with non-negative integer coefficients of degree 2, satisfying f(0)=0 and 10f(x) dx=1, is 1.8Q.The number of points in the interval [13,13] at which f(x)=sin(x2)+cos(x2) attains its maximum value, is2.2R.223x21+ex dx equals 3.4S.⎜ ⎜ 1/21/2cos2xlog(1+x1x) dx⎟ ⎟ 1/20cos2xlog(1+x1x) dx equals 4.0

Which of the following option is correct?

A
(P)(3),(Q)(2)(R)(4),(S)(1)
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B
(P)(2),(Q)(3)(R)(4),(S)(1)
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C
(P)(3),(Q)(2)(R)(1),(S)(4)
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D
(P)(2),(Q)(3)(R)(1),(S)(4)
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Solution

The correct option is D (P)(2),(Q)(3)(R)(1),(S)(4)
(P)
f(0)=0
Let f(x)=ax2+bx
10f(x) dx=a3+b2=1a=0,b=2;a=3,b=0
Two such polynomials is possible.

(Q)
f(x)=sin(x2)+cos(x2)
x2[0,13]
Maximum value of f(x)=2 and its attans at
sin(x2)=cos(x2)
tan(x2)=1
x2=π4,9π4<13x=±π2,±3π2

Hence, there are four points in x[13,13]

(R)
I=223x21+ex dxI=223x21+ex dx2I=223x2(1+ex)1+ex dx2I=223x2 dxI=8

(S)
(1/21/2cos2xlog(1+x1x) dx)(1/20cos2xlog(1+x1x) dx)
I1=1/21/2cos2xlog(1+x1x) dxI1=1/21/2cos2xlog(1x1+x) dx2I1=0

Hence,
(P)(2),(Q)(3)(R)(1),(S)(4)

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