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08%20-%20Differential%20Calculus

# 08%20-%20Differential%20Calculus - 08 DIFFERENTIAL CALCULUS...

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08 - DIFFERENTIAL CALCULUS Page 1 ( Answers at the end of all questions ) ( 1 ) + + + 1 sec n 1 .......... n 4 sec n 2 n 1 sec n 1 lim 2 2 2 2 2 2 2 n is ( a ) 1 sec 2 1 ( b ) 1 cosec 2 1 ( c ) tan 1 ( d ) 2 1 tan 1 [ AIEEE 2005 ] ( 2 ) The normal to the curve x = a ( cos θ + θ sin θ ), y = a ( sin θ - θ cos θ ) at any point θ is such that ( a ) it passes through the origin ( b ) it makes angle 2 π + θ with the X-axis ( c ) it passes through ( a 2 π , - a ) ( d ) it is at a constant distance from the origin [ AIEEE 2005 ] ( 3 ) A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched ? Interval Function Interval Function ( a ) ( - , ) x 3 - 3x 2 + 3x + 3 ( b ) [ 2, ) 2x 3 - 3x 2 + 12x + 6 ( c ) ( - , 3 1 ) 3x 3 - 2x 2 + 1 ( d ) ( - , - 4 ) x 3 - 6x 2 + 6 [ AIEEE 2005 ] ( 4 ) Let α and β be the distinct roots of the equation ax 2 + bx + c = 0. Then 2 x ( ) c bx ax ( cos 1 lim ) 2 x α α + + - - is equal to ( a ) 2 a 2 ( α - β ) 2 ( b ) 0 ( c ) - 2 a 2 ( α - β ) 2 ( d ) 2 1 ( α - β ) 2 [ AIEEE 2005 ] ( 5 ) Suppose f ( x ) is differentiable at x = 1 and ) h 1 ( f h 1 lim 0 h + = 5, then f ’ ( 1 ) equals ( a ) 3 ( b ) 4 ( c ) 5 ( d ) 6 [ AIEEE 2005 ]

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08 - DIFFERENTIAL CALCULUS Page 2 ( Answers at the end of all questions ) ( 6 ) Let f be differentiable for al x. If f ( 1 ) = - 2 and f ’ ( x ) 2 for x [ 1, 6 ], then ( a ) f ( 6 ) 8 ( b ) f ( 6 ) < 8 ( c ) f ( 6 ) < 5 ( d ) f ( 6 ) = 5 [ AIEEE 2005 ] ( 7 ) If f is a real valued differentiable function satisfying l f ( x ) - f ( y ) l ( x - y ) 2 , x, y R and f ( 0 ) = 0, then f ( 1 ) equals ( a ) - 1 ( b ) 0 ( c ) 2 ( d ) 1 [ AIEEE 2005 ] ( 8 ) A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm 3 / min. When the thickness of ice is 5 cm, then the rate at which thickness of ice decreases in cm / min is ( a ) π 36 1 ( b ) π 8 1 1 ( c ) π 4 5 1 ( d ) π 6 5 [ AIEEE 2005 ] ( 9 ) Let f : R R be a differentiable function having f ( 2 ) = 6, f ’ ( 2 ) = 48 1 . Then dt 2 x 4t lim ) x f( 6 3 2 x - equals ( a ) 24 ( b ) 36 ( c ) 12 ( d ) 18 [ AIEEE 2005 ] ( 10 ) If the equation a n x n + a n - 1 x n - 1 + ….. + a 1 x = 0, a 1 0, n 2 has a positive root x = α , then the equation n a n x n - 1 + ( n - 1 ) a n - 1 x n - 2 + …. + a 1 = 0 has a positive root which is ( a ) greater than α ( b ) smaller than α ( c ) greater than or equal to α ( d ) equal to α [ AIEEE 2005 ] ( 11 ) If x 2 2 x x b x a 1 lim + + = e 2 , then the values of a and b are ( a ) a R, b R ( b ) a = 1, b R ( c ) a R, b = 2 ( d ) a = 1, b = 2 [ AIEEE 2004 ]
08 - DIFFERENTIAL CALCULUS Page 3 ( Answers at the end of all questions ) ( 12 ) Let f ( x ) = π - - 4x x tan 1 , x 4 π , x π 2 0, . If f ( x ) is continuous in π 2 0, , then f  π 4 is ( a ) 1 ( b ) 2 1 ( c ) - 2 1 ( d ) - 1 [ AIEEE 2004 ] ( 13 ) If x = + + ..... y e y e , x > 0, then dx dy is ( a ) x 1 x + ( b ) x 1 ( c ) x x 1 - ( d ) x x 1 + [ AIEEE 2004 ] ( 14 ) A point on the parabola y 2 = 18x at which the ordinate increases at twice the rate of the abscissa is ( a ) ( 2, 4 ) ( b ) ( 2, - 4 ) ( c ) 2 9 , 8 9 - ( d ) 2 9 , 8 9 [ AIEEE 2004 ] ( 15 ) A function y = f ( x ) has a second order derivative f ” ( x ) = 6 ( x - 1 ).

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08%20-%20Differential%20Calculus - 08 DIFFERENTIAL CALCULUS...

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