LEVEL I. A circular metal plate epands under heating so that its radius increase by %. Find the approimate increase in the area of the plate, if the radius of the plate before heating is 0cm.. The length of a rectangle is decreasing at the rate of cm/sec and width y is increasing at the rate of cm/sec. When = cm and y = 5 cm, find the rate of change of (i) the perimeter and (ii) the area of the rectangle. A ladder 6 cm long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of cm/sec. How fast is its height on the wall decreasing when the foot of the ladder is cm away from the wall?. Prove that the tangent to 9y = 6 which is perpendicular to the straight 5 + y 0 = 0 does not eist. 5. If, are the intercepts made on the aes by the tangent at any point of the curve = a cos, y bsin, prove that. a b 6. Show that the normal to the rectangular hyperbola y = c at the point t meets the curve again at the point t such that t t =. F 7. A(0, ), B, I K J are two points on the graph given by y = sin + cos. Prove that there eists a point P on the curve between A and B such that tangent at P is parallel to AB. Find the coordinates of P. 8. If f() differentiable in [, 5], then show that f (5) f () 8f (a).f (b), where a, b [, 5]. 9. Let the function f be a continuous in [a, b] and derivable in (a, b) show that there eist a number. c (a, b) such that c [f(a) f(b)] = f (c)[a b ]. 0. If a function f is continuous in [a, b] and differentiable in (a, b), where ab > 0, then prove that there f (b) f (a) eists at least one c (a, b) for which c f(c). b a. Find the condition for the polynomial equation f() = 0 to have a repeated real roots by using Rolle s theorem. Hence or otherwise prove that n r r! 0 can not have a repeated root. r0 HTTP://AKJ59.WORPRESS.COM
. If a, b, c be non-zero real numbers such that : z z 8 ( cos )( a b c) d = ( cos 8 )( a b c) d 0, then equation a + b 0 0 + c = 0 will have one root between 0 and other root between and.. Prove that if (n )a na < 0 then the roots of the equation n + a n +...+ a n + a n = 0 cannot be all real. (n ). Given that k a.n. a 0 (n ). a k 0. Where R k 0,,,...,n. Show that... a a k. a n n i0 i in in for any permutation a, a,...a of a, a,...a ) ( i0 i in 0 n. 0 has at least one real root in (, ) 5. Determine the intervals in which the function f() = /(-), increasing or decreasing. 6. Find the set of all values of a for which the function, f() = F I KJ a a 5 - + log 5 decreases for all real. 7. If f() = e - a e - + (a + ) - monotonically increases for every R then find the range of values of a. 8. Find the values of a for which the function f() = sin a sin sin a increase throughout the number line. 9. The interval to which b may belong so that the function, f() = b b b + 5 + 6, increases for all. 0. Let g () = f + f ( ) and f () < 0 (0, ). Find the intervals of increase and decrease of g().. Prove the inequality, tan tan Prove the following : for 0.. sin tan for 0.. < e - < - + for all 0. b g. F I for 0 KJ. log log for (, ) 5. log. (, ). e j for > 0. tan 6. e e HTTP://AKJ59.WORPRESS.COM
7. cos(sin ) > sin(cos ) for 0 < <. p q 8. Show that sin cos attains a maimum, when tan p q. 9. Find the least value of f() = a sec + b cosec, given ab 0. 0. Show that f() = m - n, R, m, n N, m, n >, has a point of maima at which the value is m n m n m n ( m n) ( ).. Find the value of p for which f() = + 6(p - ) + (p - ) + 0 has a positive point of maimum. f( ). Find the polynomial f() of degree 6, which satisfies Lim 0 F I K J / = e and has local maimum at = and local minimum at = 0 and.. Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.. A bo of maimum volume with top open is to be made by cutting out four equal squares from four corners of a square tin sheet of side length a ft, and then folding up the flaps. Find the side of the square cut off. 5. Show that the height of the closed cylinder of given surface and maimum volume, is equal to the diameter of its base. 6. Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm. 7. A cone is circumscribed about a sphere of radius r. Show that the volume of the cone is minimum F when its semi-vertical angle is, sin H G I K J. 8. If the sum of the lengths of the hypotenuse and another side of a right angled triangle is given, show that the area of the triangle is a maimum when the angle between these sides is /. 9. Prove that the area of right-angled triangle of given hypotenuse is maimum when the triangle is isosceles. 0. In constructing an A-C transformer it is important to insert into the coil a cross-shaped iron core of greatest possible surface area. Shown in the figure the cross-section of the core with appropriate dimensions. Find the most suitable and y if the radius of the coil is equal to a HTTP://AKJ59.WORPRESS.COM
LEVEL II. A particle moves so that the space described in time t is square root of a quadratic function of t. Then, prove that the acceleration of the particle varies as s.. A light shines from the top of a pole 50 ft high. A ball is drooped from the same height from a point 0 ft away from the light. How fast is the shadow of the ball moving along the ground / sec. later? [Assume the ball falls a distance s = 6 t ft in t sec.. An air force plane is ascending vertically at the rate of 00 km/h. If the radius of the earth is R Km, how fast the area of the earth, visible from the plane increasing at min after it started ascending. Take visible area A = R h R h where h is height of the plane in kms above the earth.. If and y be the intercepts on the aes of and y cut off by the tangent to the curve, n n F I y a K J F H G I b K J F, then show that a I n n n n b KJ F /( ) H G /( ) I y K J 5. Prove that the curves y = f(), (f() > 0) and y = f() sin, where f() is differentiable function, have common tangents at common points.. 6. A curve is given by the equations = at and y = at. A variable pair of perpendicular lines through the origin O meet the curve at P and Q. Show that the locus of the point of intersection of the tangents at P and Q is y = a - a. 7. Show that the condition, that the curves / + y / = c / and a where a, b and c R. y may touch, if c = a + b, b 8. For the function F() = z t dt, find the tangent lines which are parallel to the bisector of the angle in the first quadrant. 0 9. The tangent at a variable point P of the curve y = - meets it again at Q. Show that the locus of the middle point of PQ is y = - 9 + 8-8. 0. If a = -, b and f() = /, show that the conditions of Lagrange s mean value theorem are not satisfied in the interval [a, b], but the conclusion of the theorem is true if and only if b.. Let y = f() be differentiable in the closed interval [00, 00] and f(00) = f(00) = 0. Show that there eist a point on the curve y = f() at which the length of the subtangent is 00. HTTP://AKJ59.WORPRESS.COM
. Let f() be a differentiable function on [, ]. If f() = 0 and f() > 0 for all in (, ), prove that the equation r.f () f ( ) s f ()f ( ) has a solution in (, ) (r & s R).. Let f() and g() be differentiable functions such that f ().g() f ().g() for any real. Prove that between any two real solution of f() = 0, there is at least one real solution of g() = 0. Give one eample of such that a pair of solution.. Suppose f is continuous on [a, b] and differentiable on (a, b). Assume further that f(b) f(a) = b a. Prove that for every positive integer n, there eist distinct points c, c,...,c n in (a, b) such that f (c ) f (c )... f (c ) n. n F 5. Prove that f() = I K J is always an increasing function for every > 0. n n n 6. Prove that ( a b) a b for all a, b > 0 and 0 < n <. 7. Show that the function f() = + cos a is an increasing function and hence deduce that the equation + cos = a has no positive root for a < and has one positive root for a >. 8. Let ( ) = z f( t) dt and f() satisfies the following conditions a f( + y) = f() + f(y) + y -, y R and f ( 0) ( a a ) F I KJ a,, then prove that ( ) is entirely increasing. where a is constant and 9. Let f be differentiable at every value of and suppose that f() =, that f () 0 on (, ) and that f () 0 on (, ), show that f () for all R. 0. Find the range of parameter b, for which the function f() is entirely increasing or decreasing for all values of where, f () = ( bt b cos t)dt. 0. Let f () = [ c + (b ) c ] + (sin + cos ) d. If f () be an increasing function of R then find all possible values of b (if c R).. Find all the values of the parameter a for which the function ; f() = 8a - a sin 6-7 - sin 5 increases and has no critical points for all R. HTTP://AKJ59.WORPRESS.COM
. Find the set of all values of the parameter a for which the function f() = sin - 8(a + )sin + (a + 8a - ) increases for all R and has no critical points for all R. n. Prove that following inequality, 0,. ( ) 5. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi - vertical angle is 7 h tan. 6. A light hangs above the centre of a table of radius r ft. The illumination at any point on the table is directly proportional to the cosine of the angle of incidence. (i.e. the angle a ray of light makes with the normal )and is inversely proportional to the square of the distance from the light. How far the light be above the table in order to give the strongest illumination at the edges of the table. 7. A closed rectangular bo with a square base is to be made to contain 000 cubic feet. The cost of the material per square foot for the bottom is 5 paise, for the top 5 paise and for the sides 0 paise. The labor charges for making the bo are Rs. /-. Find the dimensions of the bo when the cost is minimum. 8. Find the cosine of the angle at the verte of an isosceles triangle having the greatest area for the given constant length L of the median drawn to its lateral side. 9. A segment of a line bisects a triangle ABC with sides a, b, c into two equal areas. Find the length of the shortest segment. 0. Show that the altitude of the greatest equilateral triangle that can be circumscribed about a given triangle ABC with its sides a, b, c is a b abcos C R ST F I K JU VW /.. The three sides of a trapezium are equal each being 6 cms long, find the area of the trapezium when it is maimum.. One corner of a long rectangular sheet of paper of width unit is folded over so as to reach the opposite edge of the sheet. Find the minimum length of the crease.. A ladder is to be carried in a horizontal position round a corner formed by two streets, a feet and b feet wide meeting at right angles. Prove that the length of the longest ladder that will pass round the corner without jamming is, (a / + b / ) /.. Two towns located on the same side of the river agree to construct a pumping station and filtration plant at the river s edge, to be used jointly to supply the towns with water. If the distance of the two from the river are a and b and the distance between them is c show that the pipe lines joining them to the pumping station is atleast as great as c ab. HTTP://AKJ59.WORPRESS.COM
5. A circle of radius unit touches positive -ais and positive y-ais at P and Q respectively. A variable line passing through origin intersects circle C in two points M and N. Find the equation of the line for which area of triangle MNQ is maimum. 6. A perpendicular is drawn from the centre to a tangent to an ellipse y. Find the greatest a b value of the intercept between the point of contact and the foot of the perpendicular. 7 In the graph of the function y = log, where [e.5, find the point P (, y) such that the segment of the tangent to the graph of the function at the point, intercepted between the point P and y-ais, is shortest. 8. A figure is bounded by the curves, y = +, y = 0, = 0 and =. At what point (a, b), a tangent should be drawn to the curve, y = + for it to cut off a trapezium of the greatest area from the figure. 9. A private telephone company serving a small community makes a profit of Rs..00 per subscriber, if it has 75 subscribers. It decides to reduce the rate by a fied sum for each subscriber over 75, thereby reducing the profit by one paise per subscriber. Thus there will be profit of Rs..99 on each the 76 subscriber, Rs..98 on each of 77 subscribers etc. What is the number of subscribers which will give the company the maimum profit? 0. Let f() = R S T log e( b b ),. Find all possible real values of b such that f() 8, has the smallest value at =. b g where 0, 9. Find the minimum value of F I KJ d i and R.. Let f be a function defined on an interval [a, b] what conditions could you place on f to guarantee f (b) f (a) that min f () ma f (), where min f () refer to the minimum and maimum (b a) values of f () on [a, b]. Give reason for your answer.. Find the greatest and the least values of the function f() defined as below. f() = minimum of t 8t 6t t t m ; r, maimum of t sin t ; t,. RST UVW HTTP://AKJ59.WORPRESS.COM
SET I. Angle of intersection of + y - 6 - y - 0 = 0 and y = - 5 is (A) (B) (D) 6. Total number of parallel tangents of f () = - + and f () = - - + is equal to (A) (B). Tangents are drawn to y = cos from the point P(0, 0). Points of contact of these tangents will always lie on (A) y (B) y + y = (D) - y =. The curve - y + c = 0 and y = will intersect orthogonally for (A) c (0, 6) (B) c (, ) c (, ) 5. If the line joining the points (0, ) and (5, -) is a tangent to the curve y = a, then (A) a (B) a a (D) a 6. If the line a + by + c = 0 is a tangent to the curve y + = 0 then (A) a > 0, b > 0 (B) a > 0, b < 0 a > 0, c > 0 (D) a > 0, c < 0 7. and are the side lengths of two variable squares S and S respectively. If 6 then rate of change of the area of S with respect to rate of change of the area of S when is equal to (A) (B) (D) 8. Total number of values of where the function f() = cos cos attains its maimum value is (A) (B) 9. If A + B = where A, B > 0, then minimum value of sec A + sec B is equal to (A) (B) 8 6 0. If = a is the point of local maima for y = f(), then which of the following is always true (A) f (A) 0 (B) f (A) 0, f (A) 0 f (A) 0, f (A) 0. Let f() = {}, where {.} denotes the fractional part. For f(), = 5 is (A) a point of local maima (B) a point of local minima neither a point of local minima nor maima (D) a stationary point HTTP://AKJ59.WORPRESS.COM
. f() = 6, then for f(), = is 7, (A) a point of local maima (B) a point of local minima neither a point of local minima nor maima (D) a stationary point cos, 0. f() =. Then = 0 will be point of local maima for f() if a, 0 (A) a (, ) (B) a (0, ) a 0 (D) a. f() =, 0, then (A) f() has no point of local maima f() has eactly one point of local minima (B) f() has no point local minima (D) f() has eactly two points of local minima 5. If f() = + a + b + c attains its local minima at certain negative real number then (A) a - b > 0, a < 0, b < 0 (B) a - b > 0, a < 0, b > 0 a - b > 0, a > 0, b < 0 (D) a - b > 0, a > 0, b > 0 6. Let f() = a + b + c + d, a 0. If and are the real and distinct roots of f () 0 then f() = 0 will have three real and distinct roots if (A). < 0 (B) f( ). f( ) > 0 f( ). f( ) < 0 (D). > 0 7. A rectangle is inscribed in an equilateral triangle of side length a units. Maimum area of this rectangle can be (A) a (B) a a (D) 8. If the equation a + b + c = 0 has its coefficients such that a + b + c = 0 where a, b, c R then the equation has at least one real root in the interval (A) (, ) (B) (, ), 9. If f () 0 R and g () = f( ) + f(6 ) then a (A) g() is an increasing in [, 0] (B) g() is an increasing in [, ) g() has a local minima at = (D) g() has a local maima at = 0. f() = log (log(log(cos t a))) dt. If f() is increasing for all real values of then 0 (A) a (, ) (B) a (, 5) a (, ) (D) a (5, ). Let P be a point on = y that is nearest to the point A(0, ) then coordinates of P are (A) (, ) (B) (0, 0), HTTP://AKJ59.WORPRESS.COM 8 (D) (, )
. Let f () 0 R and g() = f( - ) + f( + ). Then g() is increasing in (A) (, ) (B) (, 0) (, ). Let f :[0, ) [0, ) and g :[0, ) [0, ) be non-increasing and non-decreasing functions, h() = g(f()). If f and g are differentiable for all points in their respective domains and h(0) = 0 then h() will always be (A) an increasing function (B) a decreasing function identically zero. If y = 0 then minimum value of + y is equal to (A) 5 (B) 0 9 (D) 0 5. If 9 - > a has atleast one negative solution, where then complete set of values of a is 5 (A), 9 5 (B), 9 7, 9 6. f() be a differentiable function such that f () (log (log all values of then (cos / 5 5 (A) a (5, ) (B) a, a, 5 7 (D), 9. If f() is increasing for a))) 7. Let f() be a function such that () log (log (sin a)). If f() is decreasing for all real f / values of then (A) a (, ) (B) a (, ) a (, ) (D) a (, ) 8. Tangents are drawn to + y = 6 from the point P(0, h). These tangents meet the -ais at A and B. If the area of triangle PAB is minimum then (A) h (B) h 6 h 8 (D) h 9. Tangents PA and PB are drawn to y = - + from the point P, h. If the area of triangle PAB is maimum then (A) h (B) h h = - 0. The curve C : y = - cos, (0, ) and C : y = a will touch each other if (A) a (B) a a (D) a HTTP://AKJ59.WORPRESS.COM
SET II. The parabolas y = a and = by intersect orthogonally at point P(, y ) where.y 0 provided (A) b = a (B) b = a b = a. Two variable curves C : y = a( - b ) and C : = a(y - b ) where a is a given positive real number and b and b are variables, touch each other. Locus of the point of contact is (A) y = a (B) y = a y = a. Point on y = that is nearest to the circle + (y - ) =, is (A) (, -) (B) (, ) (9, 6) (D) (9, -6). The function f() = has (A) no point of local minima eactly one point of local minima (B) no point of local maima (D) eactly one point of local maima 5. The function f() = ( - ) n ( - + ), n N assumes a local minima at = then (A) n can be any odd number (B) n can only be an odd prime number n can be any even number (D) n can only be a multiple of four 6. tan, f (),, then (A) f() has no point of local maima f() has eactly one point of local maima (B) f() has no point of local minima (D) f() has eactly two points of local minima 7. f() = e.cos, [0, ]. The slope of tangent of the function is minimum for (A) (B) 8. If f() = a n b has etremes at = and = then (A) a, b 8 (B) a, b 8 a, b 8 (D) (D) a, b 8 9. Total number of critical points of f() = are equal to (A) (B) (D) 0. f () (t )cot t dt, (0, ).f () attains local maimum value at 0 HTTP://AKJ59.WORPRESS.COM
(A) (B) = (D) t. f () (e ) (t ) (sin t cos t)sin t dt,, then f() is 0 (A) Decreasing in, 0, Decreasing in,, Decreasing in, (B) Decreasing in,, Decreasing in Decreasing in,, Decreasing in 5,,, Decreasing in 5, (D) Decreasing in 0,, Decreasing in,, Decreasing in 5,. f : R R be a differentiable function R. If tangent drawn to the curve at any point (a, b) always lie below the curve then (A) f () 0,f () 0 (a, b) (B) f () 0,f () 0 (a, b) f () 0,f () 0 (a, b). A spherical balloon is pumped at the constant rate of m /min. The rate of increase of its surface area at certain instant is found to be 5 m /min. At this instant its radius is equal to (A) m 5 (B) m 5. The abscissa of points P and Q on the curve y = e + e such that tangents at P and Q make 60 0 with -ais (A) 7 n and n 5 6 m 5 (B) n 7 (D) m 5 7 n (D) 7 n 5. A lamp of negligible height, is placed on the ground m away from a wall. A man m tall is walking at a speed of m / sec from the lamp to the nearest point on the wall. When he is 0 midway between the lamp and the wall, the rate of change in the length of his shadow on the wall is HTTP://AKJ59.WORPRESS.COM
(A) 5 m / sec (B) m / sec m / sec (D) m / sec 5 5 6. Let f() and g() be real valued functions such that f(). g() =, y R. If f ( ) and f () g() g () eist for all values of, and f () and g () are never zero, then is equal to f () g() (A) g() f () (B) g() g() f () g() (D) f () f () 7. Consider the parabola y =. A (, ) and B (9, 6) be two fied points on the parabola. Let C be a moving point on the parabola between A and B such that the area of triangle ABC is maimum, then coordinate of C is (A), (B) (, ), (D), 8. The equation - + a = 0 will have eactly one real root if (A) (0, ) (B) (-, ) (, ) (, ) (D) (-, 0) 9. The inequality - > cot - is true in (A) [0, ] (B) (, 5) (, ] [5, ) (D) (-, ) 0. Total number of critical points of f() = ma. {sin, cos } (, ) is equal to (A) 5 (B) 7 (D). The equation - + [a] = 0, where [.] denotes the greatest integer function, will have real and distinct roots if (A) a (, ) (B) a (0, ) a (, ) (0, ) (D) a [, ). y = f() is parabola, having its ais parallel to y-ais. If the line y = touches this parabola at =, then (A) f () f (0) (B) f (0) f () f () f (0) (D) f (0) f (). Let g () 0 and f () 0 R then (A) g ( f ( + ) ) > g ( f ( - ) ) (B) f ( g ( - ) ) > f ( g ( + ) ) g ( f ( + ) ) < g ( f ( - ) ) (D) g ( g ( + ) ) < g ( g ( - ) ). If f () 0 R then for any two real numbers and, ) ( HTTP://AKJ59.WORPRESS.COM
(A) f () f ( ) f f () f ( f ) (B) (D) f () f ( ) f f () f ( ) f 5. Let f (sin ) 0 and f (sin ) 0 0, and g() = f(sin ) + f(cos ), then g() is decreasing in (A), (B) 0, 0, (D), 6 6. f() = sin - sin + sin + 5 0, then (A) f is increasing in 0, (B) f is decreasing in 0, f is increasing in 0, and decreasing in, (D) f is decreasing in 0, and increasing in, 7. Let f() = n(a a ), 0. Complete set of a such that f() has a local 8, minima at =, is (A) [-, ] (B) (, ) (, ) [, ] (D) (, ) (, ) 8. The equation + cos = a has eactly one positive root. Complete set of values of a is (A) (0, ) (B) (, ) (-, ) (D) (, ) 9. If the function f() = ( ).e has its local maima at = a then (A) a = (B) a = a = (D) a = y 0. If the curve and y = 6 intersect at right angle then a (A) a (B) a HTTP://AKJ59.WORPRESS.COM a (D) a
SET III. If f is twice differentiable at = a ; then which of the following is True (A) If f(a) is an etreme value of f(), then f (a) 0 (B) If f (a) 0, then f(a) is an etreme value of f() If f (a) 0 andf (a) 0 then function has a local minima at = a. The line a by + c = 0 is normal to the curve y = then which one of the following is/are is not true (A) a > 0, b > 0 (B) a < 0, b < 0 a > 0, b < 0 (D) a <, b >. Let p() = a 0 + a + a +...+ a n n be a polynomial in a real variable with 0 < a 0 < a < a <...<a n. The function p() has (A) Neither a maimum nor a minimum (B) only one maimum only one minimum. At = a, there is minimum for a given function f (), then (A) lim a f () = lim a f () (B) lim a f () > 0, lim a f () < 0 lim a f () < 0, lim a f () < 0 (D) nothing can be said 5. Let f be a twice differentiable function satisfying f() = e; f() = e ; f() = e, then which of the following is/are false (A) f() = e [, ] (B) f () e has atleast three solution in [, ] f () e has atleast two solution in [, ] (D) f () e has a solution in [0, ] 6. Let f() and g() are defined and differentiable for 0 and f( 0 ) = g( 0 ), f () > g () for > 0, then which of the following is/are not true (A) f() > g() for some > 0 (B) f() = g() for some > 0 f() > g() for all > 0 (D) f() > g() for no > 0 7. If the function f() increases in the interval (a, b) then the function () = [f()] (A) increases in (a, b) (B) decreases in (a, b) we cannot say that () increases or decreases in (a, b) HTTP://AKJ59.WORPRESS.COM
8. Let f() = sin, 0, then, (A) f() has local maima at = (B) f() has local minima at = f() does not have any local etrema at = (D) f() has a global minima at = 9. Among the following statements which one is/are true (A) n( ) in (0, ) (B) n( ) in (0, ) tan in (0, / ) (D) tan in (0, / ) 0. If a < b < c < d and R then the least value of the function, f() = a + b + c + d is (A) a + c b d (B) a + b + c + d c + d a b (D) a + b c d. Let f() be a differentiable function upto any order such that f ().f () 0 R. If and be the two consecutive real roots of f() = 0, then (A) f ( ) must be equal to zero for atleast one (, ) (B) f ( ) must be equal to zero for atleast one (, ) f () 0 (, ). Among the following statements which one is/are true (A) The cubic equation + + + 5 = 0 has three real roots. (B) The cubic equation + + + 5 = 0 has only one real root. The cubic equation + + + 5 = 0 has only real root, such that [ ] =. (D) The cubic equation + + + 5 = 0 has three real roots,,, such that [ ], [ ], [ ], (where [.] denotes the greatest integer function) a,. Let f() =. If f() has a local minima at =, then a is not, (A ) less than 5 (B) greater than or equal to 5 less than or equal to 5. A car is driven at speed of km/hr., where (0, 0) and its mileage is given by n(g()) f () e, where g() = g(), then the best economical speed is 50 (A) 70 km./hr. (B) 9 + e km./hr. 50 km./hr. (D) 59 + e km./hr. HTTP://AKJ59.WORPRESS.COM
W I Read the following passage and give the answer of question 5 to 7 If a function f() is : (a) continuous is closed interval [a, b], (b) differentiable in open interval (a, b), then eists at least one c between a and b such f (b) f (a) that f (c). b a 5. Suppose f() =, 0, then in the interval [, ], 0 (A) (B) (D) both LMVT and Rolle's theorem can be applied only LMVT can be applied only Rolle's theorem can be applied neither Rolle's theorem nor LMVT can be applied 6. By Lagrange's Mean Value Theorem, which of the following is true for > (A) + n < < + n (B) + n < < + n < + n < + n (D) + n < + n < 7. If f() and g() satisfy the conditions of Mean Value Theorem on the interval [a, b], then which of the following function satisfies the conditions of Rolle's Theorem on [a, b] (A) g(a) f() + g(b) g() (B) (g (a) + g(b)) f() + (f(a) + f(b)) g() (g(b) g(a)) f() + (f(a) f(b)) g() W II Read the following passage and answer the question 8 to question A conical vessel is to be prepared out of a circular sheet of copper of unit radius as shown in the figure where be the angle of the sector removed (i.e. AOB ), then 8. The volume of the vessel. ( If ) (A) (B) 6 9. The value of r for which volume is maimum (when is variable) (A) (B) 0. The value of for which volume is maimum (when is variable) (A) (B) HTTP://AKJ59.WORPRESS.COM
. The sectorial area is to be removed from the sheet so that vessel has the maimum volume, is (A) (B) W III Read the following passage and give the answer of question to Let f be continuous and differentiable on an interval I. Then f is increasing or decreasing on I if and only if f ( ) 0 or f ( ) 0 respectively for all in I. Answer the following questions from 5 to 8.. Let f ( ) cos. Then the equation f ( ) 0 has (A) Unique solution in ( 0, / 6) (B) Unique solution in ( / 6, / ) infinitely many solutions in ( 0, / ) (D) infinitely many solutions in ( 0, / ). Let f be continuous and differentiable function such that f () and f () have opposite signs everywnere. Then (A) f is increasing (B) f is decreasing f is increasing and decreasing (D) f is decreasing. Let f ( ) e ( )( ) d. Then f decreases in the interval : (A) (, ) (B) (, ) (, ) (D) (, ) W IV Consider the following function and answer the question 5 to question 9 f() = (a ) + 6a + a + 5. The value of a for which f() has eactly one point of local maima and one point of local minima (A) (, ) (9, ) (B) (, ] [9, ) [, 9] (D) (, 9) 6. The value of a for which f() has local minima at some negative real (A) (, ) (9, ) (B) (, ] [9, ) (0, ) (D) (, 9) 7. The values of a for which f() has local maima at some negative and local minima at positive real (A) (, 0] [9, ) (B) (9, ) (0, ] (D) (, 0) 8. The values of a for which f() has no point of etrema (A) [, 9] (B) (, 0) (D) (, 9) 9. The values of a for which f() is increasing in [, ) (A) [, 9] (B) (, 9) (, 9] (D) (, ] HTTP://AKJ59.WORPRESS.COM
W V Consider the following function and answer the question 0 to question Consider the curve = t, y = t t. If tangent at point ( t, t t ) inclined at an angle to positive ais and tangent at point P(, ) cuts the curve again at Q. 0. The curve is symmetrical about (A) y = 0 (B) y + = 0 y = 0 (D) = 0. tan sec is equal to (A) t (B) t t + t. The point Q will be (A), 9 (B) (, ) (, ). The angle between the tangents at P and Q will be (A) (B) 6 (D) W VI Read the following passage and give the answer of question to 8 A circular arc PQ of radius subtends an angle of radian at its centre O, ( 0 < < ) as shown in the figure. The point R is the intersection of the two tangents at points P and Q of the arc. Let us define the following function S() = area of the sector OPQ T() = area of the triangle PQR U() = area of the shaded region. S has value (A) (B) 8 5. The epression for T() is (A) sin tan sin (B) tan sin HTTP://AKJ59.WORPRESS.COM
6. If 0, then the function U() is (A) always increasing (B) always decreasing increases in0, and decrease in, (D) decreases in 0, and increases in, 7. For the domain 0, the root of the equation U() T() is (A) (B) U() 8. The value of the limit Lim, is equal to 0 T() (A) (B) W VII Consider the following function and answer the question 9 to question Suppose f () is continuous on an interval I, and a and b are two points of I. Then if y is a number 0 between f (a) and f (b), there eists a number c between a and b such that f (c) = y. In particular 0 if f (a) and f (b) possess opposite signs, then there eists atleast one solution of the equation f () = 0 in the open interval (a, b). 9. Let f () 5, then f() = 0 has (A) no real roots (B) at most one real roots eactly real roots (D) eactly real roots 0. f() = a + b + c and a + b + c = 0, then f() = 0 has (A) two real and distinct roots (B) two real and equal roots non real roots (D) real roots as well as non real roots depending upon a, b and c. Let f :[0, ) R be a continuous function such that f() + 0f() + 005f() = 0, then f () = 0 has always at least one real root in (A) [, ] (B) [, ] (, ) (D) (0, ) HTTP://AKJ59.WORPRESS.COM
LEVEL I ANSWER. 8 sq. cm. 0 cm /sec,cm /sec. cm / sec 5 7. F I, 6 K J 5. I in, 0 L N M O Q, D in [0, ), I in,, D in (, ] 6., P (, ) 7. a 0 8. [, ) 9. b ( 7, ) (, ) 0. g () is I in (0,/] and D in [/,) 9. (a + b). p < -, < p < -. ma. at a/6 6. r 8. - /5 5 + / 6 0. = a sin, y = a cos, where = 0.5 tan - LEVEL II. -500 ft/sec. 00 R ( R 5) 6. r 9.. km /h 8. y and feet 7. side 0 height 0 8. cos A = 0.8 ( c a b)( c a b) units 5. y =. 7 sq. cms 8. F 5, 6. a - b I K J y 7. when = e / 9. 96 or 96.. greatest =, least = 8 SET I. D. D. B. D 5. B 6. B 7. D 8. A 9. A 0. D HTTP://AKJ59.WORPRESS.COM
. B. C. D. C 5. D 6. C 7. D 8. A 9. D 0. D. C. C. C. B 5. D 6. D 7. B 8. D 9. D 0. A SET II. D. C. B. D 5. C 6. C 7. A 8. C 9. D 0. A. C. C. C. D 5. B 6. D 7. A 8. C 9. C 0. B. D. C. C. B 5. B 6. A 7. C 8. D 9. A 0. D SET III. AC. ABD. C. D 5. ABC 6. ABD 7. C 8. A 9. AC 0. C. B. BC. AC. C 5. A 6. B 7. C 8. C 9. C 0. B. B. B. D. C 5. A 6. C 7. D 8. A 9. C 0. C. B. A. D. C 5. B 6. A 7. C 8. C 9. C 0. A. D HTTP://AKJ59.WORPRESS.COM