BANSALCLASSESTARGET IIT JEE 2007 MATHEMATICS STERLING QUESTION BANK ON AREA UNDER THE CURVE & DIFFERENTIAL EQUATION Time Limit : 3 Sitting Each of 85 Minutes duration approx. Select the correct alternative : (Only one is correct) Q.1 Area common to the curve y = 9 x 2 & x² + y² = 6 x is : 3 3 3 3 3 (A) (B) (C) 3 (D) 3 4 4 4 4 Q.2 Spherical rain drop evaporates at a rate proportional to its surface area. The differential equation corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is K > 0, is dr dr dr (A) K = 0 (B) K = 0 (C) Kr (D) none dt dt dt Q.3 If y = 2 sin x + sin 2 x for 0 x 2 , then the area enclosed by the curve and the x-axis is : (A) 9/2 (B) 8 (C) 9 (D) 4 Q.4 Number of values of m N for which y = emx is a solution of the differential equation D3y – 3D2y – 4Dy + 12y = 0, is (A) 0 (B) 1 (C) 2 (D) more than 2 Q.5 The area bounded by the curve y = x2 + 4x + 5, the axes of co-ordinates & the minimum ordinate is: 2 2 2 (A) 3 (B) 4 (C) 5 (D) none 3 3 3 Q.6 The general solution of the differential equation, y + y (x) (x) . (x) = 0 where (x) is a known function is : (A) y = ce(x) + (x) 1 (B) y = ce+(x) + (x) 1 (C) y = ce(x) (x) + 1 (D) y = ce(x) + (x) + 1 where c is an arbitrary constant . Q.7 The area bounded by the curve y = x2 1 & the straight line x + y = 3 is : 9 7 17 17 17 (A) (B) 4 (C) (D) 2 2 6 Q.8 Orthogonal trajectories of family of the curve x 2 3 y 2 3 a 2 3 , where 'a' is any arbitrary constant, is (A) x 2 3 y 2 3 c (B) x 4 3 y 4 3 c (C) x 4 3 y 4 3 c (D) x1 3 y1 3 c Q.9 The area enclosed by the curve y2 + x4 = x2 is : 2 4 8 10 (A) (B) (C) (D) 3 3 3 3 Q.10 Equation of a curve passing through the origin if the slope of the tangent drawn at any of its point (x, y) is cos(x + y) + sin(x + y), is (A) y = 2 tan–1(ex – 1) + x (B) y = 2 tan–1(ex – 1) – x –1 x (C) y = 2 tan (e ) – x (D) y = 2 tan–1(ex) + x Bansal Classes Q. B. on AUC and Differential Equation [2] Q.11 The area enclosed between the curves y = sin x , y = cos x & the x-axis if 0 x is : 2 (A) 2 1 (B) 2 2 (C) 2 (D) 2 2 1 Q.12 The differential equation of all parabolas having their axis of symmetry coinciding with the axis of x has its order and degree respectively: (A) (2, 1) (B) (2, 2) (C) (1, 2) (D) (1, 1) Q.13 The area bounded by the curve y = x² + 1 & the tangents to it drawn from the origin is 2 4 1 (A) (B) (C) (D) 1 3 3 3 Q.14 Which one of the following functions is not homogeneous? 1 2 xy x (A) f (x, y) = 2 (B) f (x, y) = x ·y 3 3 tan 1 x y2 y 2x 2 y 2 x 2y 2 (C) f (x, y) = x (ln x y 2 – ln y)+yex/y (D) f(x,y)=x ln x ln ( x y) +y2tan 3x y Q.15 The area enclosed by the curve y = x & x = – y , the circle x2 + y2 = 2 above the x-axis, is 3 (A) (B) (C) (D) 4 2 2 Q.16 Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth y, where the constant of proportionality k > 0 depends on the acceleration due to gravity and the geometry of the 1 hole. If t is measured in minutes and k = then the time to drain the tank if the water is 4 meter deep 15 to start with is (A) 30 min (B) 45 min (C) 60 min (D) 80 min x Q.17 The area bounded by x² + y² 2 x = 0 & y = sin in the upper half of the circle is : 2 4 2 8 2 (A) (B) (C) (D) 2 4 2 Q.18 The solution to the differential equation y lny + xy' = 0, where y (1) = e, is x2 (A) x (ln y) = 1 (B) xy (ln y) = 1 (C) (ln y)2 = 2 (D) ln y + y = 1 2 Q.19 The ratio in which the x-axis divides the area of the region bounded by the curves y = x2 4 x & y = 2 x x2 is : 4 4 4 (A) (B) (C) (D) none 23 27 19 Bansal Classes Q. B. on AUC and Differential Equation [3] y y Q.20 A curve passes through the point 1 , & its slope at any point is given by cos2 . Then the 4 x x curve has the equation e 1 e (A) y=x tan–1(ln ) (B) y=x tan–1(ln + 2) (C) y = tan–1(ln ) (D) none x x x Q.21 The area enclosed by the curve y = (x 1) (x 2) (x 3) between the co-ordinate axes and the ordinate at x = 3 is : 9 11 11 9 (A) (B) (C) (D) 2 3 4 4 Q.22 The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point (1, 1) is x x y y y y x x (A) y e e (B) x e e (C) xe e (D) ye e 3 Q.23 The line y = mx bisects the area enclosed by the curve y = 1 + 4x x2 & the lines x = 0, x = & 2 y = 0. Then the value of m is: 13 6 3 (A) (B) (C) (D) 4 6 13 2 Q.24 The differential equation of all parabolas each of which has a latus rectum '4a' & whose axes are parallel to x-axis is : (A) of order 1 & degree 2 (B) of order 2 & degree 3 (C) of order 2 and degree 1 (D) of order 2 and degree 2 Q.25 The area bounded by the curve y = f (x), the x-axis & the ordinates x =1 & x = b is (b 1)sin(3b + 4). Then f (x) is: (A) (x 1) cos (3x + 4) (B) sin (3x + 4) (C) sin (3x + 4) + 3 (x 1) . cos (3x + 4) (D) none Q.26 The foci of the curve which satisfies the differential equation (1 + y2) dx xy dy = 0 and passes through the point (1 , 0) are : (A) 2 , 0 (B) 0, 2 (C) (0, ± 1) (D) (± 2, 0) Q.27 The area of the region for which 0 < y < 3 2x x2 & x > 0 is : 3 3 (A) 3 2 x x 2 dx 1 (B) 3 2 x x 2 dx 0 1 3 (C) 3 2 x x 2 dx 0 (D) 3 2 x x 2 dx 1 Bansal Classes Q. B. on AUC and Differential Equation [4] Q.28 A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x 0. 2 If I = f (x ) dx then 0 2 2 (A) < I < (B) < I < (C) 1 < I < (D) 0 < I < 1 2 4 4 2 2 Q.29 The area bounded by the curve y = f(x) , the co-ordinate axes & the line x = x1 is given by x1 . e x1 . Therefore f(x) equals: (A) ex (B) x ex (C) xex ex (D) x ex + ex Q.30 A curve is such that the area of the region bounded by the co-ordinate axes, the curve & the ordinate of any point on it is equal to the cube of that ordinate. The curve represents (A) a pair of straight lines (B) a circle (C) a parabola (D) an ellipse Q.31 The limit of the area under the curve y = ex from x = 0 to x = h as h is : 1 (A) 2 (B) e (C) (D) 1 e Q.32 Degree of the differential equation y = a 1 e x a , a being the parameter is (A) 1 (B) 2 (C) 3 (D) not applicables Q.33 The slope of the tangent to a curve y = f (x) at (x , f (x)) is 2x + 1 . If the curve passes through the point (1 , 2) then the area of the region bounded by the curve , the x-axis and the line x = 1 is : 5 6 1 (A) (B) (C) (D) 1 6 5 6 dy Q.34 A curve satisfying the initial condition, y(1) = 0, satisfies the differential equation, x = y – x2. The area dx bounded by the curve and the x-axis is 1 1 1 1 (A) (B) (C) (D) 2 3 4 6 1 1 Q.35 The graphs of f (x) = x2 & g(x) = cx3 (c > 0) intersect at the points (0, 0) & , 2 . If the region which c c lies between these graphs & over the interval [0, 1/c] has the area equal to 2/3 then the value of c is (A) 1 (B) 1/3 (C) 1/2 (D) 2 2 dy dy Q.36 Number of straight lines which satisfy the differential equation + x y = 0 is: dx dx (A) 1 (B) 2 (C) 3 (D) 4 Bansal Classes Q. B. on AUC and Differential Equation [5] Q.37 The area bounded by the curves y = x and x = y where x, y 0 (A) cannot be determined (B) is 1/3 (C) is 2/3 (D) is same as that of the figure bounded by the curves y = x ; x 0 and x = y ; y 0 dy Q.38 The solution of the differential equation, (x + 2y3) = y is : dx x x x2 y (A) = y + c (B) = y2 + c (C) = y2 + c (D) = x2 + c y2 y y x Q.39 The area bounded by the curves y = x (1 ln x) ; x = e1 and positive X-axis between x = e1 and x = e is : e 2 4 e 2 e 2 5 e 2 4 e 2 e 2 5 e 2 e 2 (A) (B) (C) (D) 5 4 5 4 Q.40 The real value of m for which the substitution, y = um will transform the differential equation, dy 2x4y + y4 = 4x6 into a homogeneous equation is : dx (A) m = 0 (B) m = 1 (C) m = 3/2 (D) no value of m Q.41 The area bounded by the curves y = x (x 3)2 and y = x is (in sq. units) : (A) 28 (B) 32 (C) 4 (D) 8 dy 1 1 Q.42 The solution of the differential equation, x2 .cos y sin = 1, where y 1 as x is dx x x 1 1 x1 (A) y = sin – cos (B) y = x x x sin x1 1 1 x1 (C) y = cos sin (D) y = x x x cos x1 Q.43 The positive values of the parameter 'a' for which the area of the figure bounded by the curve 5 y = cos ax, y = 0, x = , x = is greater than 3 are : 6a 6a (A) (B) (0, 1/3) (C) (3, ) (D) none of these Q.44 The equation of a curve passing through (1, 0) for which the product of the abscissa of a point P & the intercept made by a normal at P on the x-axis equals twice the square of the radius vector of the point P, is (A) x2 + y2 = x4 (B) x2 + y2 = 2 x4 (C) x2 + y2 = 4 x4 (D) none Q.45 The curvilinear trapezoid is bounded by the curve y = x2 + 1 and the straight lines x=1 and x=2. The co-ordinates of the point ( on the given curve) with abscissa x [1,2] where tangent drawn cut off from the curvilinear trapezoid an ordinary trapezium of the greatest area, is 3 13 (A) (1,2) (B) (2,5) (C) , (D) none 2 4 Bansal Classes Q. B. on AUC and Differential Equation [6] Q.46 The latus rectum of the conic passing through the origin and having the property that normal at each point (x, y) intersects the x axis at ((x + 1), 0) is : (A) 1 (B) 2 (C) 4 (D) none x 1 Q.47 The value of 'a' (a>0) for which the area bounded by the curves y = , y = 0, x = a and 6 x2 x = 2a has the least value, is (A) 2 (B) 2 (C) 21/ 3 (D) 1 dy Q.48 The solution of the differential equation, 2 x2y = tan (x2y2) 2xy2 given y(1) = is dx 2 (A) sinx2y2 = ex–1 (B) sin(x2y2) = x (C) cosx2y2 + x = 0 (D) sin(x2y2) = e.ex Q.49 Area of the region enclosed between the curves x = y2 – 1 and x = |y| 1 y 2 is (A) 1 (B) 4/3 (C) 2/3 (D) 2 dy 1 2 y 4 x Q.50 Solution of the differential equation, = is dx 1 y 2 x (A) 4x2 + 4xy + y2 2x 2y + c = 0 (B) 4x2 – 4xy – y2 2x 2y + c = 0 2 2 (C) 4x + 4xy + y 2x 2y + c = 0 (D) 4x2 + 4xy – y2 2x 2y + c = 0 Q.51 Let y = g (x) be the inverse of a bijective mapping f : R R f (x) = 3x3 + 2x. The area bounded by the graph of g (x), the x-axis and the ordinate at x = 5 is : 5 7 9 13 (A) (B) (C) (D) 4 4 4 4 dy yx Q.52 The solution of the differential equation, = , given y( 5) = 5 represents dx y x 1 (A) a pair of straight lines (B) a circle (C) parabola (D) hyperbola Q.53 Area enclosed by the curves y = lnx ; y = ln | x | ; y = | ln x | and y = | ln | x | | is equal to (A) 2 (B) 4 (C) 8 (D) cannot be determined x Q.54 If y = ln | c x | (where c is an arbitrary constant) is the general solution of the differential equation dy y x x = + then the function is : dx x y y x2 x2 y2 y2 (A) (B) – (C) (D) – y2 y2 x2 x2 Q.55 If the tangent to the curve y = 1 – x2 at x = , where 0 < < 1, meets the axes at P and Q. Also varies, the minimum value of the area of the triangle OPQ is k times the area bounded by the axes and the part of the curve for which 0 < x < 1 , then k is equal to 2 75 25 2 (A) (B) (C) (D) 3 16 18 3 Bansal Classes Q. B. on AUC and Differential Equation [7] d3y dy 3 13 Q.56 If the function y = e4x + 2e–x is a solution of the differential equation dx dx K then the value of y K is (A) 4 (B) 6 (C) 9 (D) 12 Q.57 If (a, 0); a > 0 is the point where the curve y = sin2x – 3 sinx cuts the x-axis first, A is the area bounded by this part of the curve , the origin and the positive x-axis, then (A) 4A + 8 cosa = 7 (B) 4A + 8 sina = 7 (C) 4A – 8 sina = 7 (D) 4A – 8 cosa = 7 2 ex Q.58 A function y = f (x) satisfies (x + 1) . f (x) – 2 (x2 + x) f (x) = , x 1 ( x 1) If f (0) = 5 , then f (x) is 3x 5 x 2 6x 5 x 2 6x 5 x 2 5 6x x2 (A) .e (B) .e (C) .e 2 (D) .e x 1 x 1 ( x 1) x 1 Q.59 The curve y = ax2 + bx + c passes through the point (1, 2) and its tangent at origin is the line y = x. The area bounded by the curve, the ordinate of the curve at minima and the tangent line is 1 1 1 1 (A) (B) (C) (D) 24 12 8 6 Q.60 The differential equation whose general solution is given by, y = c1 cos( x c 2 ) (c 3e ( x c4 ) ) (c5 sin x ) , where c1, c2, c3, c4, c5 are arbitrary constants, is d4 y d2 y d 3 y d 2 y dy (A) y0 (B) y0 dx 4 dx 2 dx 3 dx 2 dx d5 y d 3 y d 2 y dy (C) 5 y 0 (D) 3 2 y0 dx dx dx dx dy Q.61 A function y = f (x) satisfies the differential equation – y = cos x – sin x with initial condition that y is dx bounded when x . The area enclosed by y = f (x), y = cos x and the y-axis is 1 (A) 2 1 (B) 2 (C) 1 (D) 2 Q.62 The curve, with the property that the projection of the ordinate on the normal is constant and has a length equal to 'a', is 2 2 (A) x a ln y a y c (B) x a 2 y 2 c 2 (C) (y – a) = cx (D) ay = tan–1 (x + c) Bansal Classes Q. B. on AUC and Differential Equation [8] 5 Q.63 Area bounded by the curve y = min {sin2x, cos2x}and x-axis between the ordinates x = 0 and x = is 4 5 5( 2) (A) square units (B) square units 2 4 5( 2) 1 (C) square units (D) square units 8 8 2 Q.64 The equation to the orthogonal trajectories of the system of parabolas y = ax2 is x2 y2 x2 y2 (A) y 2 = c 2 (B) x = c (C) y 2 = c 2 (D) x = c 2 2 2 2 x Q.65 If t y( t )dt = x2 + y (x) then y as a function of x is a x 2 a 2 x 2 a 2 (A) y = 2 – (2 + a2) e 2 (B) y = 1 – (2 + a2) e 2 x 2 a 2 (C) y = 2 – (1 + a2) e 2 (D) none x2 1 dy Q.66 A curve y = f (x) passing through the point 1, satisfies the differential equation + x e 2 =0. e dx Then which of the following does not hold good? (A) f (x) is differentiable at x = 0. (B) f (x) is symmetric w.r.t. the origin. (C) f (x) is increasing for x < 0 and decreasing for x > 0. (D) f (x) has two inflection points. Q.67 The substitution y = z transforms the differential equation (x2y2 – 1)dy + 2xy3dx = 0 into a homogeneous differential equation for (A) = – 1 (B) 0 (C) = 1 (D) no value of . x Q.68 A curve passing through (2, 3) and satisfying the differential equation t y( t ) dt = x2y (x), (x >0) is 0 9 x 2 y2 (A) x2 + y2 = 13 (B) y2 = x (C) 1 (D) xy = 6 2 8 18 Q.69 Which one of the following curves represents the solution of the initial value problem Dy = 100 – y, where y (0) = 50 (A) (B) (C) (D) Q.70 Solution of the differential equation e x 2 e y 2 y dy e x 2 ( xy 2 x ) + = 0, is dx 2 2 2 2 (A) e x (y2 – 1) + e y = C (B) e y (x2 – 1) + e x = C 2 2 2 2 (C) e y (y2 – 1) + e x = C (D) e x (y – 1) + e y = C Bansal Classes Q. B. on AUC and Differential Equation [9] Direction for Q.71 to Q.73 (3 question together) Consider the function f (x) = x3 – 8x2 + 20x – 13 Q.71 Number of positive integers x for which f (x) is a prime number, is (A) 1 (B) 2 (C) 3 (D) 4 Q.72 The function f (x) defined for R R (A) is one one onto (B) is many one onto (C) has 3 real roots (D) is such that f (x1) · f(x2) < 0 where x1 and x2 are the roots of f ' (x) = 0 Q.73 Area enclosed by y = f (x) and the co-ordinate axes is 65 13 71 (A) (B) (C) (D) none 12 12 12 3 Q.74 The area enclosed by the curves y = cos x, y = 1 + sin 2x and x = equals 2 3 3 3 3 (A) – 2 (B) (C) 2 + (D) 1 + 2 2 2 2 Q.75 The area of the region under the graph of y = xe–ax as x varies from 0 to , where 'a' is a positive constant, is 1 1 1 1 1 1 (A) (B) 2 (C) (D) a a a a a2 a2 Q.76 The polynomial f (x) satisfies the condition f (x + 1) = x2 + 4x. The area enclosed by y = f (x – 1) and the curve x2 + y = 0, is 16 2 16 8 2 (A) (B) (C) (D) none 3 3 3 Select the correct alternatives : (More than one are correct) Q.77 Family of curves whose tangent at a point with its intersection with the curve xy = c2 form an angle of 4 is (A) y2 2xy x2 = k (B) y2 + 2xy x2 = k x cx (C) y = x - 2 c tan1 + k (D) y = c ln x + k c cx where k is an arbitrary constant . dy y Q.78 The general solution of the differential equation, x = y. ln is : dx x (A) y = xe1 cx (B) y = xe1 + cx (C) y = ex . ecx (D) y = xecx where c is an arbitrary constant. Bansal Classes Q. B. on AUC and Differential Equation [10] Q.79 Which of the following equation(s) is/are linear. dy y dy d 2y (A) + = ln x (B) y + 4x = 0 (C) dx + dy = 0 (D) = cos x dx x dx dx 2 Q.80 The function f(x) satisfying the equation, f2(x) + 4 f (x) . f(x) + [f (x)]2 = 0 . (A) f(x) = c . e 2- 3 x (B) f(x) = c . e 2+ 3 x (C) f(x) = c . e 32 x (D) f(x) = c . e 2+ 3 x where c is an arbitrary constant. Q.81 The equation of the curve passing through (3 , 4) & satisfying the differential equation, 2 dy dy y + (x y) – x = 0 can be dx dx (A) x y + 1 = 0 (B) x2 + y2 = 25 (C) x2 + y2 5x 10 = 0 (D) x + y 7 = 0 Q.82 The area bounded by a curve, the axis of co-ordinates & the ordinate of some point of the curve is equal to the length of the corresponding arc of the curve. If the curve passes through the point P(0, 1) then the equation of this curve can be 1 1 (A) y = (ex e – x + 2) (B) y = (ex + ex) 2 2 2 (C) y = 1 (D) y = e ex x Q.83 Identify the statement(s) which is/are True. y (A) f(x , y) = ey/x + tan is homogeneous of degree zero x y y2 y (B) x . ln dx + sin1 dy = 0 is homogeneous of degree one x x x (C) f(x , y) = x2 + sin x . cos y is not homogeneous (D) (x2 + y2) dx - (xy2 y3) dy = 0 is a homegeneous differential equation . Q.84 The graph of the function y = f (x) passing through the point (0 , 1) and satisfying the differential equation dy + y cos x = cos x is such that dx (A) it is a constant function (B) it is periodic (C) it is neither an even nor an odd function (D) it is continuous & differentiable for all x . dy sin 2 x Q.85 A function y = f (x) satisfying the differential equation ·sin x – y cos x + 2 = 0 is such that, dx x y 0 as x then the statement which is correct is / 2 (A) Lim f(x) = 1 (B) f(x) dx is less than 2 x 0 0 / 2 (C) f(x) dx is greater than unity (D) f(x) is an odd function 0 Bansal Classes Q. B. on AUC and Differential Equation [11] [12] Q. B. on AUC and Differential Equation Bansal Classes Select the correct alternative : (Only one is correct) Q.1 D Q.2 A Q.3 B Q.4 C Q.5 B Q.6 A Q.7 D Q.8 B Q.9 B Q.10 B Q.11 B Q.12 A Q.13 A Q.14 D Q.15 D Q.16 C Q.17 A Q.18 A Q.19 A Q.20 A Q.21 C Q.22 A Q.23 A Q.24 C Q.25 C Q.26 A Q.27 C Q.28 A Q.29 D Q.30 C Q.31 D Q.32 D Q.33 A Q.34 D Q.35 C Q.36 B Q.37 B Q.38 B Q.39 B Q.40 C Q.41 D Q.42 A Q.43 B Q.44 A Q.45 C Q.46 B Q.47 D Q.48 A Q.49 D Q.50 A Q.51 D Q.52 C Q.53 B Q.54 D Q.55 A Q.56 D Q.57 A Q.58 B Q.59 A Q.60 B Q.61 A Q.62 A Q.63 C Q.64 A Q.65 A Q.66 B Q.67 A Q.68 D Q.69 B Q.70 A Q.71 C Q.72 B Q.73 A Q.74 C Q.75 D Q.76 A Select the correct alternatives : (More than one are correct) Q.77 A,B,C,D Q.78 A,B,C Q.79 A,C,D Q.80 C,D Q.81 A,B Q.82 B,C Q.83 A,B,C Q.84 A,B,D Q.85 A,B,C ANSWER KEY
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