HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)

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1 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum respectively of the function f() = 9 a + a + respectively, then for the equality = to be true the value of 'a' must be 0 (B) (D) / Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is e (B) Q. The angle at which the curve y = Ke K intersects the y-ais is : e tan k (B) cot (k ) sec k e e (D) 8e (D) none n Q. {a, a,..., a,...} is a progression where a n =. The largest term of this progression is : n 00 a 6 (B) a 7 a 8 (D) none Q.5 The angle between the tangent lines to the graph of the function f () = ( t 5)dt at the points where the graph cuts the -ais is /6 (B) / / (D) / Q.6 The minimum value of the polynomial ( + ) ( + ) ( + ) is : 0 (B) 9/6 (D) / tan 6 Q.7 The minimum value of is : tan 0 (B) / (D) Q.8 The difference between the greatest and the least values of the function, f () = sin on, (B) 0 (D) Q.9 The radius of a right circular cylinder increases at the rate of 0. cm/min, and the height decreases at the rate of 0. cm/min. The rate of change of the volume of the cylinder, in cm /min, when the radius is cm and the height is cm is 8 (B) 5 5 (D) 5 Q.0 If a variable tangent to the curve y = c makes intercepts a, b on and y ais respectively, then the value of a b is 7 c (B) c 7 Q. Difference between the greatest and the least values of the function f () = (ln ) on [, e ] is (B) e e (D) 7 c (D) c 9

2 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q. Let f () = n tan, n N, where 0, r tan n r0 f () is bounded and it takes both of it's bounds and the range of f () contains eactly one integral point. (B) f () is bounded and it takes both of it's bounds and the range of f () contains more than one integral point. f () is bounded but minimum and maimum does not eists. (D) f () is not bounded as the upper bound does not eist. Q. If f () = + 7 then f () has a zero between = 0 and =. The theorem which best describes this, is Squeeze play theorem (B) Mean value theorem Maimum-Minimum value theorem (D) Intermediate value theorem sin for 0 Q. Consider the function f () = then the number of points in (0, ) where the 0 for 0 derivative f () vanishes, is 0 (B) (D) infinite Q.5 The sum of lengths of the hypotenuse and another side of a right angled triangle is given. The area of the triangle will be maimum if the angle between them is : 6 (B) (D) 5 Q.6 In which of the following functions Rolle s theorem is applicable? f() = 0, 0 on [0, ], sin, 0 (B) f() = 0, 0 on [, 0] f() = if on [,] (D) f() = 6 if,on [,] Q.7 Suppose that f (0) = and f ' () 5 for all values of. Then the largest value which f () can attain is 7 (B) 7 (D) 8 Q.8 The tangent to the graph of the function y = f() at the point with abscissa = a forms with the -ais an angle of / and at the point with abscissa = b at an angle of /, then the value of the integral, b f (). f () d is equal to a (B) 0 (D) [ assume f () to be continuous ] Q.9 Let C be the curve y = (where takes all real values). The tangent at A meets the curve again at B. If the gradient at B is K times the gradient at A then K is equal to (B) (D) /

3 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.0 The vertices of a triangle are (0, 0), (, cos ) and (sin, 0) where 0 < <. The maimum area for such a triangle in sq. units, is (B) (D) 6 Q. The subnormal at any point on the curve y n = a n + is constant for : n = 0 (B) n = n = (D) no value of n Q. Equation of the line through the point (/, ) and tangent to the parabola y = the curve y = is : + y 5 = 0 (B) + y = 0 y = 0 (D) none + and secant to Q. The lines y = and y = intersect the curve 5 + y + 5y = 0 at the points P and Q respectively. The tangents drawn to the curve at P and Q intersect each other at angle of 5º (B) are parallel to each other are perpendicular to each other (D) none of these Q. The least value of 'a' for which the equation, sin sin = a has atleast one solution on the interval (0, /) is : (B) 5 7 (D) 9 Q.5 If f() = Min f ( t) : 0 t ; 0 + and g() = [ ; g + g + g 5 has the value equal to : then 7 (B) 9 (D) 5 Q.6 Given : f () = / g () = tan [],, 0 0 h () = {} k () = log ( ) 5 then in [0, ] Lagranges Mean Value Theorem is NOT applicable to f, g, h (B) h, k f, g (D) g, h, k Q.7 Two curves C : y = and C : y = k, k R intersect each other at two different points. The tangent drawn to C at one of the points of intersection A (a,y ), (a > 0) meets C again at B(,y ) y y. The value of a is (B) (D) Q.8 f () = d then f is increasing in (0, ) and decreasing in (, 0) (B) increasing in (, 0) and decreasing in (0, increasing in (, (D) decreasing in (,

4 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.9 The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. The fraction of width folded over if the area of the folded part is minimum is : 5/8 (B) / / (D) /5 Q.0 A rectangle with one side lying along the -ais is to be inscribed in the closed region of the y plane bounded by the lines y = 0, y =, and y = 0. The largest area of such a rectangle is 5 8 (B) 5 5 (D) 90 Q. Which of the following statement is true for the function f () 0 It is monotonic increasing R 0 (B) f () fails to eist for distinct real values of f () changes its sign twice as varies from (, ) (D) function attains its etreme values at &, such that, > 0 Q. A closed vessel tapers to a point both at its top E and its bottom F and is fied with EF vertical when the depth of the liquid in it is cm, the volume of the liquid in it is, (5 ) cu. cm. The length EF is: 7.5 cm (B) 8 cm 0 cm (D) cm Q. Coffee is draining from a conical filter, height and diameter both 5 cms into a cylinderical coffee pot diameter 5 cm. The rate at which coffee drains from the filter into the pot is 00 cu cm /min. The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 0 cm, is 9 6 (B) 5 9 Q. Let f () and g () be two differentiable function in R and f () = 8, g () = 0, f () = 0 and g () = 8 then g ' () > f ' () (, ) (B) g ' () = f ' () for at least one (, ) g () > f () (, ) (D) g ' () = f ' () for at least one (, ) m Q.5 n Let m and n be odd integers such that o < m < n. If f() = for R, then f() is differentiable every where (B) f (0) eists f increases on (0, ) and decreases on (, 0) (D) f increases on R Q.6 A horse runs along a circle with a speed of 0 km/hr. A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse move along the fence at the moment when it covers /8 of the circle in km/hr is 0 (B)0 0 (D) 60 Q.7 Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-: If f : R R and c R is such that f is increasing in (c, c) and f is decreasing in (c, c + ) then f has a local maimum at c. Where is a sufficiently small positive quantity. Statement- : Let f : (a, b) R, c (a, b). Then f can not have both a local maimum and a point of inflection at = c. Statement- : The function f () = is twice differentiable at = 0. Statement- : Let f : [c, c + ] [a, b] be bijective map such that f is differentiable at c then f is also differentiable at f (c). FFTF (B) TTFT FTTF (D) TTTF 5 (D) 6 9

5 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.8 Let f : [, ] R be differentiable such that 0 f ' (t) for t [, 0] and f ' (t) 0 for t [0, ]. Then f () f ( ) (B) f () f ( ) f () f ( ) 0 (D) f () f ( ) 0 Q.9 A curve is represented by the equations, = sec t and y = cot t where t is a parameter. If the tangent at the point P on the curve where t = / meets the curve again at the point Q then PQ is equal to: 5 (B) Q.0 For all a, b R the function f () = a + b has : no etremum (B) eactly one etremum eactly two etremum (D) three etremum. (D) 5 Q. The set of values of p for which the equation ln p = 0 possess three distinct roots is 0, (B) (0, ) (,e) (D) (0,e) e Q. The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function f () = + 9 on the interval [, ]. If the difference between the first and the second term of the progression is equal to f ' (0) then the common ratio of the G.P. is / (B) / / (D) / Q. The lateral edge of a regular heagonal pyramid is cm. If the volume is maimum, then its height must be equal to : (B) (D) Q. The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle with the plane of the base. The value of for which the volume of the pyramid is greatest, is : (B) sin cot (D) Q.5 In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is l. The altitude of the prism for which the volume is greatest : (B) (D) Q.6 Let f () = 5 ( ) if if then the number of critical points on the graph of the function is (B) (D) Q.7 The curve y e y + = 0 has a vertical tangent at : (, ) (B) (0, ) (, 0) (D) no point Q.8 Number of roots of the equation. e = is : (B) 6 (D) zero Q.9 The point(s) at each of which the tangents to the curve y = cut off on the positive semi ais OX a line segment half that on the negative semi ais OY then the co-ordinates the point(s) is/ are given by : (, 9) (B) (, 5) (, ) (D) none

6 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.50 A curve with equation of the form y = a + b + c + d has zero gradient at the point (0, ) and also touches the -ais at the point (, 0) then the values of for which the curve has a negative gradient are > (B) < < (D) Q.5 Number of solution(s) satisfying the equation, = log ( + ) log is : (B) (D) none Q.5 Consider the function f () = cos sin, then identify the statement which is correct. f is neither odd nor even (B) f is monotonic decreasing at = 0 f has a maima at = (D) f has a minima at = Q.5 Consider the two graphs y = and y + y = 8. The absolute value of the tangent of the angle between the two curves at the points where they meet, is 0 (B) / (D) a b Q.5 The -intercept of the tangent at any arbitrary point of the curve = is proportional to: y square of the abscissa of the point of tangency (B) square root of the abscissa of the point of tangency cube of the abscissa of the point of tangency (D) cube root of the abscissa of the point of tangency. Q.55 For the cubic, f () = which one of the following statement, does not hold good? f () is non monotonic (B) increasing in (, ) (, ) and decreasing is (, ) f : R R is bijective (D) Inflection point occurs at = / Q.56 The function 'f' is defined by f() = p ( ) q for all R, where p,q are positive integers, has a maimum value, for equal to : pq p (B) 0 (D) p q p q Q.57 Let h be a twice continuously differentiable positive function on an open interval J. Let g() = lnh() for each J Suppose h '() > h''() h() for each J. Then g is increasing on J (B) g is decreasing on J g is concave up on J (D) g is concave down on J Q.58 Let f () = ( )(6 ) f has a local maima f has an inflection point 0 if if then at = (B) f has a local minima (D) f has a removable discontinuity Q.59 Let f () and g () be two continuous functions defined from R R, such that f ( ) > f ( ) and g ( ) < g ( ), >, then solution set of f g( ) is R (B) (, ) (D) R [, ] > f g( )

7 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.60 If f() = (t ) dt,, then global maimum value of f() is : (B) (D) 5 Q.6 A right triangle is drawn in a semicircle of radius / with one of its legs along the diameter. The maimum area of the triangle is (B) 6 (D) 8 Q.6 At any two points of the curve represented parametrically by = a ( cos t cos t) ; y = a ( sin t sin t) the tangents are parallel to the ais of corresponding to the values of the parameter t differing from each other by : / (B) / / (D) / Q.6 Let be the length of one of the equal sides of an isosceles triangle, and let be the angle between them. If is increasing at the rate (/) m/hr, and is increasing at the rate of /80 radians/hr then the rate in m /hr at which the area of the triangle is increasing when = m and = / 7 / (B) / 5 (D) / 5 5 Q.6 t If the function f () =, where 't' is a parameter has a minimum and a maimum then the range of values of 't' is (0, ) (B) (0, ) (, ) (D) (, ) Q.65 The least area of a circle circumscribing any right triangle of area S is : S (B) S S (D) S Q.66 A point is moving along the curve y = 7. The interval in which the abscissa changes at slower rate than ordinate, is (, ) (B) (, ) (, ) (D) (, ) (, ) Q.67 Let f () and g () are two function which are defined and differentiable for all 0. If f ( 0 ) = g ( 0 ) and f ' () > g ' () for all > 0 then f () < g () for some > 0 (B) f () = g () for some > 0 f () > g () only for some > 0 (D) f () > g () for all > 0 Q.68 P and Q are two points on a circle of centre C and radius, the angle PCQ being then the radius of the circle inscribed in the triangle CPQ is maimum when sin (B) sin 5 sin 5 (D) sin Q.69 The line which is parallel to -ais and crosses the curve y = at an angle of is y = / (B) = / y = / (D) y = / 5 Q.70 The function S() = t sin dt has two critical points in the interval [,.]. One of the critical 0 points is a local minimum and the other is a local maimum. The local minimum occurs at = (B) (D)

8 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.7 For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in km). If the speed of the current is steady at C km/hr then the most economical speed of the steamer going against the current will be.5 C (B).5 C.75C (D) C Q.7 Let f and g be increasing and decreasing functions, respectively from [0, ) to [0, ). Let h () = f [g ()]. If h (0) = 0, then h () h () is : always zero (B) strictly increasing always negative (D) always positive Q.7 The set of value(s) of 'a' for which the function f () = a + (a + ) + (a ) + possess a negative point of inflection. (, ) (0, ) (B) { /5 } (, 0) (D) empty set Q.7 A function y = f () is given by = t & y = increasing in (0, /) & decreasing in (/, ) (B) increasing in (0, ) increasing in (0, ) (D) decreasing in (0, ) Q.75 The set of all values of ' a ' for which the function, f () = (a a + ) cos t ( t ) for all t > 0 then f is : sin + (a ) + sin does not possess critical points is: [, ) (B) (0, ) (, ) (, ) (D) (, ) (, 5) Q.76 Read the following mathematical statements carefully: I. Adifferentiable function ' f ' with maimum at = c f ''(c) < 0. II. Antiderivative of a periodic function is also a periodic function. T T III. If f has a period T then for any a R. f ()d = f ( a)d 0 0 IV. If f () has a maima at = c, then 'f ' is increasing in (c h, c) and decreasing in (c, c + h) as h 0 for h > 0. Now indicate the correct alternative. eactly one statement is correct. (B) eactly two statements are correct. eactly three statements are correct. (D) All the four statements are correct. Q.77 If the point of minima of the function, f() = + a satisfy the inequality < 0, then 'a' must lie in the interval: 5 6, (B),, (D),, Q.78 The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is cm the altitude is 6 cm. When the radius is 6cm, the volume is increasing at the rate of Cu cm/sec. When the radius is 6cm, the volume is increasing at a rate of n cu. cm/sec. The value of 'n' is equal to: (B) 0 (D)

9 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.79 Two sides of a triangle are to have lengths 'a' cm & 'b' cm. If the triangle is to have the maimum area, then the length of the median from the verte containing the sides 'a' and 'b' is a b (B) a b a b (D) a Q.80 Let a > 0 and f be continuous in [ a, a]. Suppose that f ' () eists and f ' () for all ( a, a). If f (a) = a and f ( a) = a then f (0) equals 0 (B) equals / equals (D) is not possible to determine Q.8 The lines tangent to the curves y y + 5y = 0 and y y = 0 at the origin intersect at an angle equal to /6 (B) / / (D) / Q.8 The cost function at American Gadget is C() = ( in thousands of units and > 0) The production level at which average cost is minimum is (B) 5 (D) none Q.8 A rectangle has one side on the positive y-ais and one side on the positive - ais. The upper right hand verte on the curve y = n. The maimum area of the rectangle is e (B) e ½ (D) e ½ Q.8 A particle moves along the curve y = / in the first quadrant in such a way that its distance from the origin increases at the rate of units per second. The value of d/dt when = is (B) 9 b (D) none Q.85 Number of solution of the equation tan + = in 0, is 0 (B) (D) Q.86 Let f () = a b, where a and b are constants. Then at = 0, f () has a maima whenever a > 0, b > 0 (B) a maima whenever a > 0, b < 0 minima whenever a > 0, b > 0 (D) neither a maima nor minima whenever a > 0, b < 0 ln t Q.87 Let f () = t l n(t) dt ( > ) then t f () has one point of maima and no point of minima. (B) f ' () has two distinct roots f () has one point of minima and no point of maima (D) f () is monotonic Q.88 Consider f () = and g () = f () + b sin, < < then which of the following is correct? Rolles theorem is applicable to both f, g and b = / (B) LMVT is not applicable to f and Rolles theorem if applicable to g with b = / LMVT is applicable to f and Rolles theorem is applicable to g with b = (D) Rolles theorem is not applicable to both f, g for any real b.

10 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.89 Consider f () = t dt 0 t and g () = f () for, If P is a point on the curve y = g() such that the tangent to this curve at P is parallel to a chord joining the points, g 7 can't be found out (B), and (, g() ) of the curve, then the coordinates of the point P (, ) (D), 6 Q.90 The co-ordinates of the point on the curve 9y = where the normal to the curve makes equal intercepts with the aes is, (B), 8, (D) 6 6, Q.9 The angle made by the tangent of the curve = a (t + sint cost) ; y = a ( + sint) with the -ais at any point on it is t (B) sin t cos t Q.9 If f () = + + n t lnt t l dt, then f () increases in (D) sin t cos t (0, ) (B) (0, e ) (, ) no value (D) (, ) ln( ) Q.9 The function f () = ln(e ) is : increasing on [0, ) (B) decreasing on [0, ) increasing on [0, /e) & decreasing on [/e, ) (D) decreasing on [0, /e) & increasing on [/e, ) Directions for Q.9 to Q.96 Suppose you do not know the function f (), however some information about f () is listed below. Read the following carefully before attempting the questions (i) f () is continuous and defined for all real numbers (ii) f '( 5) = 0 ; f '() is not defined and f '() = 0 (iii) ( 5, ) is a point which lies on the graph of f () (iv) f ''() is undefined, but f ''() is negative everywhere else. (v) the signs of f '() is given below Q.9 On the possible graph of y = f () we have = 5 is a point of relative minima. (B) = is a point of relative maima. = is a point of relative minima. (D) graph of y = f () must have a geometrical sharp corner.

11 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.95 From the possible graph of y = f (), we can say that There is eactly one point of inflection on the curve. (B) f () increases on 5 < < and > and decreases on < < 5 and < <. The curve is always concave down. (D) Curve always concave up. Q.96 Possible graph of y = f () is (B) (D) Directions for Q.97 to Q.00 Consider the function f () = then Q.97 Domain of f () is (, 0) (0, ) (B) R { 0 } (, ) (0, ) (D) (0, ) Q.98 Which one of the following limits tends to unity? Lim f () (B) Lim f () 0 Q.99 The function f () has a maima but no minima has eactly one maima and one minima Lim f () (D) Lim f () (B) has a minima but no maima (D) has neither a maima nor a minima Q.00 Range of the function f () is (0, ) (B) (, e) (, ) (D) (, e) (e, ) Q.0 A cube of ice melts without changing shape at the uniform rate of cm /min. The rate of change of the surface area of the cube, in cm /min, when the volume of the cube is 5 cm, is (B) 6/5 6/6 (D) 8/5 Q.0 Let f () = and f ' (). for 6. The smallest possible value of f (6), is 9 (B) 5 (D) 9 Q.0 Which of the following si statements are true about the cubic polynomial P() = + +? (i) It has eactly one positive real root. (ii) It has either one or three negative roots. (iii) It has a root between 0 and. (iv) It must have eactly two real roots. (v) It has a negative root between and. (vi) It has no comple roots. only (i), (iii) and (vi) (B) only (ii), (iii) and (iv) only (i) and (iii) (D) only (iii), (iv) and (v)

12 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q.0 Given that f () is continuously differentiable on a b where a < b, f (a) < 0 and f (b) > 0, which of the following are always true? (i) f () is bounded on a b. (ii) The equation f () = 0 has at least one solution in a < < b. (iii) The maimum and minimum values of f () on a b occur at points where f ' (c) = 0. (iv) There is at least one point c with a < c < b where f ' (c) > 0. (v) There is at least one point d with a < d < b where f ' (c) < 0. only (ii) and (iv) are true (B) all but (iii) are true all but (v) are true (D) only (i), (ii) and (iv) are true Q.05 Consider the function f () = on the interval [ 6, 6]. The value of c that satisfies the conclusion of the mean value theorem, is 7/8 (B) 7/8 (D) 0 Q.06 Consider the curve represented parametrically by the equation = t t t and y = t + t 5 where t R. If H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then H = and V = (B) H = and V = H = and V = (D) H = and V = Q.07 At the point P(a, a n ) on the graph of y = n (n N) in the first quadrant a normal is drawn. The normal intersects the y-ais at the point (0, b). If Lim b, then n equals a0 (B) (D) Q.08 Suppose that f is a polynomial of degree and that f ''() 0 at any of the stationary point. Then f has eactly one stationary point. (B) f must have no stationary point. f must have eactly stationary points. (D) f has either 0 or stationary points. Q.09 Let f () = 8 for 0. Then intercept of the line that is tangent to the graph of f () is for 0 zero (B) (D) Q.0 Suppose that f is differentiable for all and that f '() for all. If f () = and f () = 8 then f () has the value equal to (B) 6 (D) 8 Q. There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per additional tree drops by 0 apples. Number of trees that should be added to the eisting orchard for maimising the output of the trees, is 5 (B) 0 5 (D) 0 Q. The ordinate of all points on the curve y = where the tangent is horizontal, is sin cos always equal to / (B) always equal to / / or / according as n is an even or an odd integer. (D) / or / according as n is an odd or an even integer.

13 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Select the correct alternatives : (More than one are correct) n y Q. The equation of the tangent to the curve a = (n N) at the point with abscissa equal to b 'a' can be : y a y = (B) b a = b n y a y = 0 (D) b a = 0 b Q. The function y = ( ) : is its own inverse (B) decreases for all values of has a graph entirely above -ais (D) is bound for all. Q.5 If y = is a tangent to the curve = Kt, y = K a b t, K > 0 then : a > 0, b > 0 (B) a > 0, b < 0 a < 0, b > 0 (D) a < 0, b < 0 Q.6 The etremum values of the function f() = sin cos, where R is : 8 (B) 8 (D) 8 Q.7 The function f() = t 0 dt is such that : it is defined on the interval [, ] (B) it is an increasing function it is an odd function (D) the point (0, 0) is the point of inflection Q.8 Let g() = f (/) + f ( ) and f () < 0 in 0 then g() : decreases in [0, /) (B) decreases in (/, ] increases in [0, /) (D) increases in (/, ] Q.9 The abscissa of the point on the curve y = a +, the tangent at which cuts off equal intersects from the co-ordinate aes is : ( a > 0) a a (B) Q.0 The function sin ( a ) has no maima or minima if : sin ( b) a (D) a b a = n, n I (B) b a = (n + ), n I b a = n, n I (D) none of these. Q. The co-ordinates of the point P on the graph of the function y = e where the portion of the tangent intercepted between the co-ordinate aes has the greatest area, is, e (B), (e, e e e ) (D) none Q. Let f() = ( ) n ( + + ) then f() has local etremum at = when : n = (B) n = (D) n = 6

14 HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA Q. For the function f() = ( ln 7) the point (, 7) is the point of inflection (B) = e / is the point of minima the graph is concave downwards in (0, ) (D) the graph is concave upwards in (, ) Q. The parabola y = + p + q cuts the straight line y = at a point with abscissa. If the distance between the verte of the parabola and the ais is least then : p = 0 & q = (B) p = & q = 0 least distance between the parabola and ais is (D) least distance between the parabola and ais is Q.5 The co-ordinates of the point(s) on the graph of the function, f() = where the tangent drawn cut off intercepts from the co-ordinate aes which are equal in magnitude but opposite in sign, is (, 8/) (B) (, 7/) (, 5/6) (D) none Q.6 On which of the following intervals, the function 00 + sin is strictly increasing. (, ) (B) (0, ) (/, ) (D) (0, /) Q.7 Let f() = 8 6 +, then f() = 0 has no root in (0,) (B) f() = 0 has at least one root in (0,) f (c) vanishes for some c (0,) (D) none Q.8 Equation of a tangent to the curve y cot = y tan at the point where the abscissa is is : + y = + (B) y = + = 0 (D) y = 0 Q.9 Let h () = f () {f ()} + {f ()} for every real number ' ', then : ' h ' is increasing whenever ' f ' is increasing (B) ' h ' is increasing whenever ' f ' is decreasing ' h ' is decreasing whenever ' f ' is decreasing (D) nothing can be said in general. Q.0 If the side of a triangle vary slightly in such a way that its circum radius remains constant, then, da d b d c cos A cos B cos C 6 R (B) R 0 (D) R(dA + db + dc) Q. In which of the following graphs = c is the point of inflection. (B) (D) Q. An etremum value of y = (t ) (t ) dt is : 0 5/6 (B) / (D)

15 ANSWER KEY Q. B Q. D Q. B Q. B Q.5 D Q.6 C Q.7 D Q.8 A Q.9 D Q.0 C Q. B Q. A Q. D Q. D Q.5 C Q.6 D Q.7 A Q.8 D Q.9 A Q.0 A Q. C Q. A Q. C Q. D Q.5 D Q.6 A Q.7 B Q.8 C Q.9 B Q.0 C Q. C Q. C Q. D Q. D Q.5 D Q.6 B Q.7 A Q.8 A Q.9 D Q.0 B Q. A Q. C Q. C Q. C Q.5 B Q.6 C Q.7 C Q.8 B Q.9 B Q.50 C Q.5 A Q.5 B Q.5 C Q.5 C Q.55 C Q.56 D Q.57 D Q.58 C Q.59 C Q.60 C Q.6 B Q.6 A Q.6 D Q.6 C Q.65 A Q.66 C Q.67 D Q.68 B Q.69 D Q.70 C Q.7 B Q.7 A Q.7 A Q.7 B Q.75 B Q.76 A Q.77 D Q.78 D Q.79 A Q.80 A Q.8 D Q.8 B Q.8 A Q.8 A Q.85 B Q.86 A Q.87 D Q.88 C Q.89 D Q.90 C Q.9 A Q.9 A Q.9 B Q.9 D Q.95 C Q.96 C Q.97 C Q.98 B Q.99 D Q.00 D Q.0 B Q.0 D Q.0 C Q.0 D Q.05 D Q.06 B Q.07 C Q.08 D Q.09 B Q.0 B Q. C Q. D Q. A,B Q. A,B Q.5 A,D Q.6 A,C Q.7 A,B,C,D Q.8 B,C Q.9 A,B Q.0 A,B,C Q. A,B Q. A,C,D Q. A,B,C,D Q. B,D Q.5 A,B Q.6 B,C,D Q.7 B,C Q.8 A,B,D Q.9 A,C Q.0 C,D Q. A,B,D Q. A,B