Let f:R→R be given by
Then which of the following options is/are correct?
Let fprime(x)=2+sin4πx192x3forallx∈Rwithf(21)=0.Ifm≤∫211f(x)dx≤M, then the possible values of mandM are (a)m=13,M=24 (b) m=41,M=21(c)m=−11,M=0 (d) m=1,M=12
Let u^=u1i^+u2j^+u3k^ be a unit vector in be a unit vector in R3andw^=61(i^+j^+2k^).Given that there exists vector v^ in R3 such that ∣u^×v∣=1andw^.(u^×v)=1. Which of the following statement(s) is(are) correct?
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8:15is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60
Let F1(x1,0) and F2(x2,0), for x1<0 and x2>0, be the foci of the ellipse 9x2+8y2=1 Suppose a parabola having vertex at the origin and focus at F2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is
Ifα=∫01(e9x+3tan(−1)x)(1+x212+9x2)dxwherηn−1takes only principal values, then the value of ((log)e∣1+α∣−43π)is