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Let F R R Be Such That F 1 3 And F 1 6 Then

1) 1 2) e1/2 3) e2 4) e3 Solution: (3) e2 y = [f (1 + x) / f (1)]1/x log y = (1 / x) [log f (1 + x) - log f (1)] limx→0 log y =... View Article

1) (–1)K (K – 1)π 2) (–1)K–1 (K – 1) π 3) (–1)K K π 4) (–1)K – 1 K π Solution: (1) (–1)K (K – 1)π f (x) = [x] sin (π x) If x is... View Article

1) {–1, 1} 2) {-1, 0} 3) {0, 1} 4) {–1, 0, 1} Solution: (4) {–1, 0, 1} f is not differentiable at - 1, 0 and 1. Watch The... View Article

1) 22 2) 14 3) 18 4) 12 Solution: (1) 22 g (x) = (x2 + 2x + 3) f (x) g’ (x) = (x2 + 2x + 3) f’ (x) + (2x + 2) f (x) g’ (0) = 3... View Article

1) R - {0} 2) R - {1} 3) R - {- 1} 4) R - {- 1, 1} Solution: (4) R - {- 1, 1} f (x) = tan-1 x if |x| ≤ 1 = (1 / 2) [|x| - 1] if... View Article

1) 20 2) 22 3) 24 4) 26 Solution: (4) 26 P (x) = a + bx + cx2 + dx2 + ex4 P (2) = a + 2b + 4c + 8d + 16e - 1 = a + 2b + 4c +... View Article

1) tan (x + y) 2) loge (x + y) 3) -1 4) 1 Solution: (3) -1 sin (x + y) = loge (x + y) cos (x + y) (1 + (dy / dx)) = (1 / [x + y])... View Article

1) yy’’ - 2 (y’)2 + 1 = 0 2) yy’’ + (y’)2 + 1 = 0 3) yy’’ - (y’)2 - 1 = 0 4) yy’’ + 2 (y’)2 + 1 = 0 Solution: (2) yy’’ + (y’)2 + 1 =... View Article

1) onto if f is into 2) one − one if f is one – one 3) continuous if f is continuous 4) differentiable if f is differentiable... View Article

1) 0, 2 2) 1, 3 3) 3, 4 4) None of these Solution: (4) None of these f (x) = xn f’ (x) = nxn-1 f’ (a + b) = n(a + b)n-1... View Article

1) f’ (0) exists and = 0 2) f’ (0) does not exist 3) Lf’ (0) = - 1 4) Rf’ (0) = 1 Solution: (1) f’ (0) exists and = 0 Lf’ (0) = f’... View Article

1) c / fɸ 2) 2c / fɸ 3) 3c / fɸ 4) None of these Solution: (2) 2c / fɸ F (x) = f (x) ɸ (x) and f’ (x) ɸ’ (x) = c F’ (x) = f ‘(x)... View Article

1) –1 2) 0 3) 1 4) 2 Solution: (4) 2 f (x) = (x2 - 1) |x2 - 3x + 2| + cos |x| f (x) = g (x) + h (x) h (x) = cos |x| = {cosx,... View Article

1) h is discontinuous for all x 2) h is differentiable for all x 3) h’(x) = 2, for all x > 1 4) h is not differentiable at two values... View Article

1) 2 / e 2) 1 / e 3) 3 / e 4) None of these Solution: (2) 1 / e f (x) = cos-1 [(1 - (log x)2 / (1 + (log x)2)] Put logx = t f (x)... View Article

1) 0 only 2) –1,1 only 3) 1 only 4) –1 only Solution: (2) –1,1 only f (x) = 2 + √1 - x2, |x| ≤ 1 = 2e(1-x)^2, x < - 1 or x >... View Article

1) 1 2) 2 3) 3 4) 4 Solution: (1) 1 xexy = y + sin2 x 0 * e0 = y + sin2 0 0 = y + 0 y = 0 xexy = y + sin2 x x [exy [x (dy /... View Article

1) 16 2) 32 3) 64 4) None of these Solution: (2) 32 d [f2 (x) + g2 (x)] = 2 f (x) + f’ (x) + 2 g (x) g’ (x) f2 (x) + g2 (x) =... View Article

1) tanx 2) (3 / 2) x cos (x3) cosec (x2) 3) (2 / 3) sin x3 sin x2 4) None of these Solution: (2) (3 / 2) x cos (x3) cosec (x2) u... View Article

1) y’ (1) = 2 / 3 2) y’ (1) = 4 / 3 3) y’’ (1) = - 8 (22 / 7) 4) y’ (1) = - 4 / 3 Solution: (3) y’ (1) = - 8 (22 / 7) x3 - 2x2y2 +... View Article