Question BankVector 3–D Geometry Area Under Curve Differential Equation 394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-2439051, 0744-2439052, 0744-2439053 www.motioniitjee.com, [email protected] Rajeev Gandhi Nagar Kota. 0744-2209671.Page # 2 Vector. 394 .motioniitjee@gmail. 93141-87482. 93527-21564 IVRS No. Ph.com. 0744-2439051.com . AUC. Eq. 0744-2439052. 0744-2439053 www.motioniitjee. No. email-hr. Diff. 3–D. 93141-87482. The area bounded in the first quadrant by the normal at (1. (2. The differential equation corre- sponding to the rate of change of the radius of the rain drop if the constant of proportionality is K > 0. email-hr. 0744-2439051. 0744-2439052.motioniitjee. The value of f(/6) is equal to (A) 1/5 (B) 3/5 (C) 4/5 (D) 2/5 5. 2) on the curve y2 = 4x. 0744-2439053 www. AUC. y = 4t. y = 2 + t. 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 4. sin 2x – cos x + (1 + sin2x) f’(x) = 0 with initial condition y(0) = 0. Eq. Page # 3 SINGLE CORRECT CHOICE TYPE 1. Let ‘a’ be a positive constant number.Rajeev Gandhi Nagar Kota. Consider the lines L1 : x = 3 – t. Let S be the area of the S part surrounding by C1. (D) do not form a triangle. The points A. B. No. If [f ( x) g( x )] dx 10 and [g( x ) f ( x )] dx 5.motioniitjee@gmail. 0 2 then area between two curves for 0 < x < 2 is (A) 5 (B) 10 (C) 15 (D) 20 6. C (A) constitute a right triangle. dy 1 x 7. 93527-21564 IVRS No. Spherical rain drop evaporates at a rate proportional to its surface area. A function y = f(x) satisfies the differential equation f(x) .Vector. C2 and the y-axis. intersecting the plane 4x – y + 3z = 8 at the point C. (B) constitute an acute triangle (C) constitute an obtuse triangle. z = 2 – 3t. is dr dr dr (A) +K=0 (B) –K=0 (C) = Kr (D) none dt dt dt 2. Suppose y = f(x) and y = g(x) are two functions whose graphs intersect at the three points (0. 0744-2209671.com . 2) and 4 4 (4. intersecting the plane x – y + 2z = 9 at the point A L2 : x = 1 + 2t. 3–D. The general solution of the differential equation = is a family of curves which looks most like which dx y of the following ? (A) (B) (C) (D) 8. then aLim equals 0 a2 (A) 4 (B) 1/2 (C) 0 (D) 1/4 394 . C2 : y = ea – x. intersecting the plane x + 2y – z + 1 = 0 at the point B and L3 : x = y – 1 = 2z. 4). x-axis & the curve is given by 10 7 4 9 (A) (B) (C) (D) 3 3 3 2 3. 0) with f(x) > g(x) for 0 < x < 2 and f(x) < g(x) for 2 < x < 4. Diff. Consider two curves C1 : y = ex. Ph.com. z = 5t. The area of the region(s) enclosed by the curves y = x2 and y = | x | is (A) 1/3 (B) 2/3 (C) 1/6 (D) 1 15. Q(–b. Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. & its slope at any point is given by – cos2 . 0744-2209671.com . Diff. b > 0) are on the parabola y = x2.motioniitjee. b2) (a > 0. 93141-87482. making an obtuse angle with the positive Y-axis. P(a.com. b î 2 ĵ 4k̂ . Then the curve 4 x x has the equation e 1 e (A) y = x tan–1 (n ) (B) y = x tan–1 (n + 2) (C) y = tan–1(n ) (D) none x x x 13. Then f(x) is (A) (x – 1) cos (3x + 4) (B) sin(3x + 4) (C) sin(3x + 4) + 3(x – 1) . A vector r is parallel to the altitude drawn from the vertex A. 1 If t is measured in minutes and k = then the time to drain the tank if the water is 4 meter deep to start with is 15 (A) 30 min (B) 45 min (C) 60 min (D) 80 min 10. The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. c 3 î ĵ 2k̂. 9. C ( c ). If the differential equation of the family of curve given by y = Ax + Be2x where A and B are arbitrary constant is d d dy of the form (1 – 2x) y + k y = 0 then the ordered pair (k. A function y = f(x) satisfies the condition f ’(x) sin x + f(x) cos x = 1. A curve passes through the point 1. 3–D. 3 points O(0. The equation of the curve through the point (1. 2) (D) (–2. Page # 4 Vector. –2) (B) (–2. 93527-21564 IVRS No. l) is dx dx dx (A) (2. Let S1 be the area bounded by the line PQ and the parabola and let S2 be the area of the triangle OPQ. 1) is x x y y y y (A) y e e (B) x e e (C) x e x e (D) y e x e 16.Rajeev Gandhi Nagar Kota. the x-axis & the ordinates x = 1 & x = b is (b – 1) sin (3b + 4). 0). a 2 î ĵ 3k̂ . AUC. then r is (A) 6 î 8 ĵ 6k̂ (B) 6 î 8 ĵ 6k̂ (C) 6 î 8 ĵ 6k̂ (D) 6 î 8 ĵ 6k̂ 14. Given the position vectors of the vertices of a triangle ABC. /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 17. No. 0744-2439052. –2) 11. a2). B ( b ) . where the constant of proportionality k > 0 depends on the acceleration due to gravity and the geometry of the hole. It is known that the rate at which the water level drops is proportional to the square root of water depth y. 0744-2439053 www. A ( a ) .motioniitjee@gmail. The area bounded by the curve y = f(x). If | r | = 2 34 . the minimum value of S1/S2 is (A) 4/3 (B) 5/3 (C) 2 (D) 7/3 y y 12. Eq. 2) (C) (2. Ph. f(x) being bounded when x 0. 0744-2439051. cos (3x + 4) (D) none 394 . email-hr. AUC. R.motioniitjee@gmail. n I 2 23. 93141-87482. Page # 5 18.motioniitjee. is (A) = n. for which OP represents a vector with direction cosine cos = (‘O’ is the origin) is 2 (A) A circle parallel to y z plane with centre on the x-axis (B) a cone concentric with positive x-axis having vertex at the origin and the slant height equal to the magnitude of the vector (C) a ray emanating from the origin and making an angle of 60º with x-axis (D) a disc parallel to y z plane with centre on x-axis & radius equal to OP sin 60º 2 dy dy 21. R – {0} (B) = 2n.com. y = um will transform the differential equation. Area enclosed by the graph of the function y = n2x – 1 lying in the 4th quadrant is 2 4 1 1 (A) (B) (C) 2 e (D) 4 e e e e e 1 20. The value of the constant ‘m’ and ‘c’ for which y = mx + c is a solution of the differential equation D2y – 3Dy – 4y = – 4x. c (D) is m = 1 . c = 3/4 3 24.c)b c 2 (a . 0744-2439051. x ) b c x where b . Ph. Diff.Rajeev Gandhi Nagar Kota.com . 0744-2439052. is any rational number (C) = (2n + 1).b 1 a . R+.Vector.b 1 a . cosx + siny – z = 0. 2x4y + y4 = 4x6 dx into a homogeneous equation is (A) m = 0 (B) m = 1 (C) m = 3/2 (D) no value of m 394 .b 1) c 2 ( a . Four vectors a. c = – 3/4 (C) no such real m. have non trivial solution. The value of x in terms of a . Locus of the point P. No. n I (D) = (2n + 1) . Eq. (A) is m = – 1. 0744-2439053 www. A curve is such that the area of the region bounded by the co-ordinate axes. The real value of m for which the substitution. c ) c c (A) (B) (C) (D) (a .b 1 dy 26. x + ( + 1)y + cos z = 0. The curve represents (A) a pair of straight lines (B) a circle (C) a parabola (D) an ellipse 19. email-hr. c = 3/4 (B) is m = 1. The area bounded by y = 2 – |2 – x| & y = is |x| 4 3 n 3 4 3 n 3 3 1 (A) (B) (C) + n 3 (D) + n 3 2 2 2 2 25. b and c is equal to (a . c and x satisfy the relation (a . Number of straight lines which satisfy the differential equation + x – y = 0 is dx dx (A) 1 (B) 2 (C) 3 (D) 4 22. b.b 1) a. a 1 . 3–D. 93527-21564 IVRS No. the curve & the ordinate of any point on it is equal to the cube of that ordinate. 0744-2209671. c ) b c(a . The values of for which the following equations sinx – cosy + ( + 1)z = 0. 93527-21564 IVRS No. 27. Therefore f(x) equals (A) ex (B) x ex (C) xex – ex (D) x ex + ex x 1 y 1 z 1 28. ax – by = 2a – b and (c + 1)x + cy = 10 – a + 3 b has infinitely many solutions and x = 1. f(x)) is 2x + 1. 3. are non-zero non-coplanar vectors. 6) such that the projections of the 13 21 26 vector OP on the axes are . 4) and R(3. The area enclosed by the graph of the function y = f(x) and the pair of tangents drawn to it from the origin. b & c for which the system of equations. the x-axis and the line x = 1 is 5 6 1 (A) (B) (C) (D) 1 6 5 6 36. Number of triplets of a. If the curve passes through the point (1. If a sheet hung in the wind loses half its moisture during the first hour.Rajeev Gandhi Nagar Kota. 1) and perpendicular to the line = = 3 0 4 from the origin is (A) 3/4 (B) 4/3 (C) 7/5 (D) 1 29.9% of its moisture is (weather conditions remaining same) (A) more than 100 hours (B) more than 10 hours (C) approximately 10 hours (D) approximately 9 hours 35. 0744-2439053 www. is (A) 8/3 (B) 16/3 (C) 32/3 (D) none 32. z) is a point on the line segment joining Q(2. x2 . email-hr. A wet porous substance in the open air loses its moisture at a rate proportional to the moisture content. Page # 6 Vector. b .motioniitjee@gmail. respectively.motioniitjee. e x1 . 5. 0744-2439052. 1. AUC. 2) then the area of the region bounded by the curve.com. The value of the triple product a × b × b × c b × c × c × a c × a × a × b is equal to 2 (A) a b c 3 (B) a b c 4 (C) a b c (D) a b c 33. Where a . The distance of the plane passing through the point P(1. the co-ordinate axes & the line x = x1 is given by x1 . Consider the two statements Statement-1 : y = sin kt satisfies the differential equation y” + 9y = 0. 0744-2209671. The area bounded by the curve y = f(x). 93141-87482. If P(x. . Ph. where y –1 as x is dx x x 1 1 x 1 1 1 x 1 (A) y = sin – cos (B) y = (C) y = cos + sin (D) y = x x x sin 1x x x x cos 1x 31. No. Eq. 0744-2439051. Diff. then the time when it would have lost 99.com . Suppose g(x) = 2x + 1 and h (x) = 4x2 + 4x + 5 and h (x) = (fog)(x). y. 3–D. y = 3 is one of the solutions is (A) exactly one (B) exactly two (C) exactly three (D) infinitely many 34. Statement-2 : y = ekt satisfy the differential equation y” + y’ – 6y = 0 The value of k for which both the statements are correct is (A) – 3 (B) 0 (C) 2 (D) 3 dy 1 1 30. The slope of the tangent to a curve y = f(x) at (x. The solution of the differential equation. c . cos – ysin = – 1. The P divides QR in the ratio 5 5 5 (A) 2 : 3 (B) 3 : 1 (C) 1 : 3 (D) 3 : 2 394 . with the property that the projection of the ordinate on the normal is constant and has a length equal to ‘a’[email protected]. 0744-2439051.e (B) . Diff. The area bounded by the curves y = – x and x = – y where x. A curve C passes through origin and has the property that at each point (x. e 2 (D) . email-hr. (ii) f ’’(0) = f ’(x) and (iii) f ’(0) = 1 then the area bounded by the graph of y = f(x). 0744-2439053 www. q = 0. The equation of a common tangent to the curve C and the parabola y2 = 4x is (A) x = 0 (B) y = 0 (C) y = x + 1 (D) x + y + 1 = 0 x 38. If V = ( p × ( p × ( p × ( p × q )))) then the vector V is (A) collinear with p (B) V = 16 p (C) V = 48 q (D) V = 16 q 44. the lines x = 0. x 0 and x = y . Ph. y 0 2 ex 40.motioniitjee. Page # 7 37. 93527-21564 IVRS No. If y = (where c is an arbitrary constant) is the general solution of the differential equation n | cx | dy y x x = + y then the function y is dx x x2 x2 y2 y2 (A) (B) – (C) (D) – y2 y2 x2 x2 39. e (C) . 93141-87482. is (A) e (B) e – 2 (C) e – 1 (D) e + 1 394 . f ’(x) – 2(x2 + x) f(x) = . 0). The differential equation corresponding to the family of curves y = ex (ax + b) is d2 y dy d2 y dy d2 y dy d2 y dy (A) +2 –y=0 (B) –2 +y=0 (C) +2 +y=0 (D) –2 – y= 0 dx2 dx dx2 dx dx2 dx dx2 dx 45. x > –1. No.com. A function y = f(x) satisfies (x + 1) . y = f(x) is a function which satisfies (i) f(0) = 0 . The area bounded by the curves y = x (x – 3)2 and y = x is (in sq. | q | = 3 and p . AUC. If f(0) = 5. 0744-2439052. The curve.Rajeev Gandhi Nagar Kota. units) (A) 28 (B) 32 (C) 4 (D) 8 42. 3–D. 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 . Given | p | = 2. x – 1 = 0 and y + 1 = 0. y) on it the normal line at that point passes through (1. then f(x) is ( x 1) 3x 5 x 2 6x 5 x2 6x 5 x2 5 6x x 2 (A) . is (A) x a n y 2 a2 y c (B) x a2 y 2 c (C) (y – a)2 = cx (D) ay = tan–1 (x + c) 43.e x 1 x 1 ( x 1) x 1 41.com . 0744-2209671. Eq. Consider the differential equation + y tan x = x tan x + 1. x n (B) f ( x ) c . y = n | x | .Rajeev Gandhi Nagar Kota. x (1 n) 53. Area enclosed by the curves y = nx . A is the area bounded by this part of the curve. No. a > 0 is the point where the curve y = sin2x – 3 sin x cuts the x-axis first. t R x 1 z3 y y y (A) = . Ph. The equation to the orthogonal trajectories of the system of parabolas y = ax2 is x2 y2 x2 y2 (A) + y2 = c (B) x2 + =c (C) – y2 = c (D) x2 – =c 2 2 2 2 47. If t y(t) dt = x 2 + y(x) then y as a function of x is a x 2 a2 x 2 a2 x 2 a2 2 2 2 (A) y 2 (2 a )e 2 (B) y 1 (2 a )e 2 (C) y 2 (1 a )e 2 (D) none 50. y. is 0 1 n n 1 (A) f ( x ) c [email protected]=3 (B) 1 – x = =z+2 (C) 1 – x = .com.z=2 3 2 3 3 3 1 52. (D) none of these x 49. xy = 0 and x = c where c is the x-coordinate of the curve’s inflection point is (A) 1 – 3e–2 (B) 1 – 2e–2 (C) 1 – e–2 (D) 1 51. –2) + t(–1. where x > 0. x n (D) f ( x ) c . 0). Page # 8 Vector. 0). 0744-2439053 www.com . 0. 93527-21564 IVRS No. 0744-2209671. If (a. 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 48. Which one of the following lines is parallel to the line L : (x. Eq. 3. The area bounded by the curve y = x e–x . 0744-2439051. 0744-2439052. 3–D. 46. Then dx (A) The integral curves satisfying the differential equation and given by y = x + c sin x. The substitution y = z transforms the differential equation (x2y2 – 1)dy + 2xy3 dx = 0 into a homogeneous differential equation for (A) = – 1 (B) 0 (C) = 1 (D) no value of 55. Diff. (B) The angle at which the integral curves cut the y-axis is . 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 = 7 394 . AUC. 2 (C) Tangents to all the integral curves at their point of intersection with y-axis are parallel. z) = (1.z=5 (D) x + 1 = . x n 1 (C) f ( x ) c . y = | n x | and y = | n | x | | is equal to (A) 2 (B) 4 (C) 8 (D) cannot be determined 54. 93141-87482. email-hr.motioniitjee. A function f(x) satisfying f (tx) dt = n f(x). The area enclosed by y = f(x). Equation of the plane containing the lines L1 and L2 is (A) 3x – z – 2 = 0 (B) 2x + y – z = 0 (C) 3x – 2y – 3z = 0 (D) none 394 . 61 to 62 Let L1 : 2iˆ 3ˆj 4kˆ (2iˆ 3ˆj 6k) ˆ and L : î ĵ k̂ (2 î 3 ĵ 6k̂ ) 2 61. 3) and one in (–1.com . 4) and one in (–2. –1) (D) none of these COMPREHENSION TYPE Paragraph for Question Nos. If the area bounded between x-axis and the graph of y = 6x – 3x2 between the ordinates x = 1 and x = a is 19 square units then ‘a’ can take the value (A) 4 or – 2 (B) two values are in (2. Ph. Eq. Which one of the following curves represents the solution of the initial value problem Dy = 100 – y. 0744-2439052. The area bounded by the curve. AUC. where y(0) = 50 y y y y 100 100 100 100 (A) (B) 50 (C) 50 (D) 50 50 x x x x O O O O dy 59. 3–D. email-hr.Vector. The curve y = ax2 + bx + c passes through the point (1. A curve passing through (2. 3) and satisfying the differential equation t y(t ) dt = x y(x). Distance between these lines is 10 10 (A) (B) 10 (C) (D) 7 10 7 7 62.motioniitjee. No. 0744-2439051. 93141-87482. the ordinate of the curve at minima and the tangent line is 1 1 1 1 (A) (B) (C) (D) 24 12 8 6 58. 0744-2209671.motioniitjee@gmail. Diff. 0) (C) two values one in (3. 2) and its tangent at origin is the line y = x. 93527-21564 IVRS No. Page # 9 x 56. A function y = f(x) satisfies the differential equation – y = cos x – sin x. y = cos x and the y-axis in the 1st quadrant 1 (A) 2 1 (B) 2 (C) 1 (D) 2 60.Rajeev Gandhi Nagar Kota. with initial condition that y is dx bounded when x . 0744-2439053 www.com. (x > 0) is 2 0 9 x2 y2 (A) x2 + y2 = 13 (B) y2 = x (C) + =1 (D) xy = 6 2 8 18 57. 2) 3 3 Also the position vector of the centre ‘G’ of the tetrahedron are î ĵ 2k̂ 2 4 66. 1.com. 0 69. y satisfies the differential equation dy dy (A) + y = ex(cos x – sin x) – e–x(cos x + sin x) (B) – y = ex(cos x – sin x) + e–x (cos x + sin x) dx dx dy dy (C) + y = ex(cos x + sin x) – e–x (cos x – sin x) (D) – y = ex(cos x – sin x) + e–x(cos x – sin x) dx dx 70. 93141-87482. The volume of the tetrahedron D-ABC is 2 1 (A) (B) (C) 1 (D) 2 3 3 67. 63 to 65 Consider the function. f(x2) < 0 where x1 and x2 are the roots of f ’(x) = 0 65. 3) and C(1. 2. 69 to 71 x Let y = f(x) satisfies the equation f(x) = (e–x + ex) cos x – 2x – (x t) f ' (t) dt . 0744-2439053 www. Ph. 66 to 68 Consider a tetrahedron D-ABC with position vectors of its angular points as A(1. Number of positive integers x for which f(x) is a prime number. is (A) 1 (B) 2 (C) 3 (D) 4 64. No. email-hr. Eq. Paragraph for Question Nos.8 (C) 1 (D) 3 68.motioniitjee@gmail. 1. 3–D.9 (B) 0. 0744-2209671. B(1. Page # 10 Vector. 93527-21564 IVRS No. f(x) as a function of x equals ex 2 –x ex 2 –x (A) e–x(cos x – sin x) + (3 cos x + sin x) + e (B) e–x(cos x + sin x) + (3 cos x – sin x) – e 5 5 5 5 ex 2 ex 2 –x (C) e–x(cos x – sin x) + (3 cos x – sin x) + e–x (D) e–x(cos x + sin x) + (3 cos x – sin x) – e 5 5 5 5 394 . f(x) = x3 – 8x2 + 20x – 13 63.com .motioniitjee. Diff. 0744-2439052. If N is the foot of the perpendicular from the point D on the face ABC then the position vector of N are 1 1 1 (A) 1 (B) 1 (C) 1 (D) None of these 2 2 2 Paragraph for Question Nos. Shortest distance between the skew lines AB and CD is 1 (A) 0. 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) .Rajeev Gandhi Nagar Kota. 0744-2439051. AUC. 1) . Area enclosed by y = f(x) and the co-ordiante axes is 65 13 71 (A) (B) (C) (D) none 12 12 12 Paragraph for Question Nos. The value of f ’(0) + f ”(0) equals (A) – 1 (B) 2 (C) 1 (D) 0 71. 93527-21564 IVRS No. Page # 11 Paragraph for Question Nos. Statement-2 : General solution of a differential equation of nth order contains n independent arbitrary constants. 75. Eq. dx 72. statement-2 is true. Diff. 0744-2439051. statement-2 is true and statement-2 is NOT the correct explanation for statement-1. No. (B) Statement-1 is true. If the angle between the planes P and Q is 45º then the product of all possible values of ‘c’ is 24 (A) – 17 (B) – 2 (C) 17 (D) 17 x 1 y2 z7 77. The plane P is parallel to plane Q (A) for no value of c (B) if c = 3 (C) if c = 1/3 (D) if c = 1 76. 73. z0) then the 1 3 1 sum (x0 + y0 + z0) has the value equal to (A) 12 (B) – 15 (C) 13 (D) – 11 REASONING TYPE 78.com. 79. 75 to 77 Let P denotes the plane consisting of all points that are equidistant from the points A(–4. For the function y = f(x) which one of the following does not hold good ? (A) f(x) is a rational function (B) f(x) has the same domain and same range. (C) Statement-1 is true. email-hr. (C) is such that it has a maxima but no minima. 72 to 74 dy A curve y = f(x) satisfies the differential equation (1 + x2) + 2yx = 4x2 and passes through the origin. 0744-2209671. 93141-87482. the x-axis and the ordinate at x = 2/3 is 4 2 1 (A) 2 n 2 (B) n 2 (C) n 2 (D) n 2 3 3 3 74. (C) f(x) is a transcedental function (D) y = f(x) is a bijective mapping. (D) Statement-1 is false. Statement-1 : Differential equation corresponding to all lines. Paragraph for Question Nos. Statement-1 : Differential equation satisfying the curve is linear. statement-2 is false. statement-2 is true and statement-2 is correct explanation for statement-1. (D) has no inflection point.com . A curve C has the property that its initial ordinate of any tangent drawn is less than the abscissa of the point of tangency by unity. statement-2 is true.motioniitjee. statement-2 is false. –4. Ph. 1) and B(2. y0. AUC.motioniitjee@gmail. x – y + cz = 1 where c R. 0744-2439053 www. 394 . Statement-2 : Degree of differential equation is one (A) Statement-1 is true. (A) Statement-1 is true. statement-2 is true and statement-2 is NOT the correct explanation for statement-1. 0744-2439052. (B) Statement-1 is true. (D) Statement-1 is false. ax + by + c = 0 has the order 3.Vector. If the line L with equation = = intersects the plane P at the point R(x0. The area enclosed by y = f –1(x). The function y = f(x) (A) is strictly increasing x R (B) is such that it has a minima but no maxima. 3–D. statement-2 is true and statement-2 is correct explanation for statement-1. (C) Statement-1 is true.Rajeev Gandhi Nagar Kota. 2. 3) and Q be the plane. C & D respectively in three dimensional space & satisfy the relation 3 a – 2 b + c – 2 d = 0. (0. (B) can be converted into linear differential equation with some suitable substitution. c2) and let R be the region between y = cx and y = x2 where c > 0 then c3 c3 Lim Area (T ) Lim Area (T ) 3 (A) Area (R) = (B) Area of R = (C) c 0 =3 (D) c 0 = 6 3 Area (R) Area (R) 2 83. (C) Statement-1 is true. 84.motioniitjee@gmail. 93141-87482. statement-2 is true. statement-2 is true and statement-2 is NOT the correct explanation for statement-1. B. Statement-2 : Every differential equation geometrically represents a family of curve having some common property. x + dy = y2 dx dx (A) is of order 1 (B) is of degree 2 (C) is linear (D) is non linear 394 . statement-2 is true and statement-2 is correct explanation for statement-1. MULTIPLE CORRECT CHOICE TYPE 82. email-hr. c & d are the pv’s of the points A. dy 3 85. (C) Statement-1 is true. (A) Statement-1 is true. (B) Statement-1 is true. B. (D) Statement-1 is false.com . statement-2 is true and statement-2 is correct explanation for statement-1. statement-2 is false. Statement-1 : The solution of (y dx – x dy) cot y = ny2 dx is sin y = cenx Statement-2 : Such type of differential equations can only be solved by the substitution x = vy. Let T be the triangle with vertices (0. statement-2 is false. x x 81. 0744-2439053 www. (D) the family of circles touching the y-axis at the origin.com. Ph. No. statement-2 is true and statement-2 is NOT the correct explanation for statement-1.motioniitjee. b. c2) and (c. c & d are linearly dependent. (A) Statement-1 is true. 93527-21564 IVRS No. (D) Statement-1 is false. AUC. Then the differential equation of the curve (A) is homogeneous. Diff. 0744-2209671. A curve y = f(x) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of the point P from the x-axis. dy 1 80. Statement-1 : Integral curves denoted by the first order linear differential equation – y = –x are family dx x of parabolas passing through the origin. Page # 12 Vector. The differential equation. b. statement-2 is true.Rajeev Gandhi Nagar Kota. (C) is the family of circles touching the x-axis at the origin. 3–D. 0). Eq. If a. C & D are coplanar (B) the line joining the points B & D divides the line joining the point A & C in the ratio 2 : 1 (C) the line joining the points A & C divides the line joining the points B & D in the ratio 1 : 1 (D) the four vectors a. then (A) A. 0744-2439052. (B) Statement-1 is true. 0744-2439051. sin x – y cos x + = 0 is such that. 0744-2439053 www. Diff. email-hr. AUC.Vector. w 2 2 3 6 (A) form a left handed system (B) form a right handed system (C) are linearly independent (D) are such that each is perpendicular to the plane containing the other two. 93527-21564 IVRS No. Page # 13 x2 1 86. The function f(x) satisfying the equation. v 6 . A function y = f(x) satisfying the differential equation . c are non-zero.v 3 are non coplanar vectors and k1 . non-collinear vectors such that a vector p = a b cos 2 a ^ b c and a vector q = a c cos a ^ c b then p + q is (A) parallel to a (B) perpendicular to a (C) coplanar with b & c (D) coplanar with a and c dy sin2 x 90. b. k2 and k 3 1 2 v1 . b a . units 87. The vectors u 3 . ( v 2 v 3 ) 1 then k1. e( 2 3 ) x (C) f ( x ) c . f(x) + [f ’(x)]2 = 0 (A) f ( x ) c .motioniitjee@gmail. f2(x) + 4 f ’(x) .com . y 0 dx x2 as x then the statement which is correct is /2 (A) Limit x 0 f(x) = 1 (B) f (x) dx is less than 2 0 /2 (C) f ( x) dx is greater than unity (D) f(x) is an odd function 0 91. 0744-2209671. No. Ph. Suppose f is defined from R [–1. c b c ( a 0) (B) if a b a c b c ( a 0) (C) if a . 1] as f(x) = where R is the set of real number. v 2 .(k 2 k 3 ) . If a.com. Eq. b a . Which of the following statement(s) hold good ? (A) if a . v1 ( v 2 v 3 ) 394 . e(2 3)x (B) f ( x) c . 3–D. c and a b a c b c ( a 0) v2 v3 v 3 v1 v v (D) if v 1.motioniitjee. e ( 2 3 ) x (where c is an arbitrary constant) 6 2 3 88. 93141-87482. 89. ( v 2 v 3 ) v1 . 0744-2439052. 0744-2439051. ( v 2 v 3 ) v1 .Rajeev Gandhi Nagar Kota. Then the statement x2 1 which does not hold is (A) f is many one onto (B) f increases for x > 0 and decreases for x < 0 (C) minimum value is not attained even though f is bounded (D) the area included by the curve y = f(x) and the line y = 1 is sq. e( 3 2) x (D) f ( x ) c . motioniitjee. 12 394 . 27 7 (D) The area enclosed by y = f(x). 0744-2439053 www. 3 27 23 (C) The value of p for which the equation f(x) = p has 3 distinct solutions lies in interval .com. 93527-21564 IVRS No. 1) and . No. Which of the following hold good ? 0 (A) f(x) has a minimum value 1 – e. AUC. 0744-2439052. the lines x = 0 and y = 1 as x varies from 0 to 1 is . Ph. 1 27 5 23 (B) The coordinates of the turning point of the graph of y = f(x) occur at (1. r 2 î 6 ĵ ( î 3 ĵ ) then which of the following statements holds good ? (A) the line is parallel to 2 î 6 ĵ (B) the line passes through the point 3 î 3 ĵ (C) the line passes through the point î 9 ĵ (D) the line is parallel to xy plane x 94. Eq.motioniitjee@gmail. then x x 2 2 2 (A) The range of f is 0. 3–D. cos x 0x 2 92. A differentiable function satisfies f(x) = (f(t) cos t cos( t x)) dt. 0744-2209671. 93141-87482. (B) f(x) has a maximum value 1 – e–1. 0744-2439051. If the line r 2 î ĵ 3k̂ ( î ˆj 2 k̂ ) makes angle with xy. 1 . Which of the following statement(s) is/are True for the function f(x) = (x – 1)2(x – 2) + 1 defined on [0. 2] ? 23 (A) Range of f is . but not differentiable for some real x (C) f is continuous for all real x 3 (D) The area bounded by y = f(x) and the X-axis from x = – n to x = n is 2n 1 for a given n N 24 93. Page # 14 Vector. Consider f(x) = 2 such that f is periodic with period . Diff.Rajeev Gandhi Nagar Kota. email-hr. yz and zx planes respectively then which of the following are not possible ? (A) sin2 + sin2 + sin2 = 2 & cos2 + cos2 + cos2 = 1 (B) tan2 + tan2 + tan2 = 7 & cot2 + cot2 + cot2 = 5/3 (C) sin2 + sin2 + sin2 = 1 & cos2 + cos2 + cos2 = 2 (D) sec2 + sec2 + sec2 = 10 & cosec2 + cosec2 + cosec2 = 14/3 96.com . If a line has a vector equation. (C) f ” 2 = e (D) f ’(0) = 1 95. 4 (B) f is continuous for all real x. If the system has the solution other than x = y = 0 then the ratio x : y can be (A) – 1/2 (B) 1/2 (C) 2 (D) – 2 99.u | . No. B & C then abc (A) centroid of triangle ABC is 3 î ĵ k̂ (B) î ĵ k̂ is equally inclined to the three vectors (C) perpendicular from the origin to the plane of triangle ABC meet at centroid (D) triangle ABC is an equilateral triangle. q and r be three vectors defined as p a b 2c . Let a . 0) (C) (0. 1. v. v v . where m and n are relatively prime positive n integers. 105.com .Rajeev Gandhi Nagar Kota. 2. Find the value of (m + n). w w .motioniitjee. Page # 15 97. w be the vectors such that u. 0744-2439051. SUBJECTIVE TYPE 103. The ratio of the m volume of the smaller tetrahedron to that of the larger is . dx e if x 2 Which of the following hold(s) good ? (A) y(1) = 2e–1 (B) y’(1) = – e–1 (C) y(3) = – 2e–3 (D) y ’(3) = – 2e–3 101. 0. –2. Consider the system of equations (4 – p2)x + 2y = 0 and 2x + (7 – p2)y = 0. | v | = 4 & | w | = 5 then find the value of | u . 0) and C(0. –2. In a regular tetrahedron. Let + y = f(x) where y is a continuous function of x with y(0) = 1 and f(x) = e2 if 0 x 2 . 0744-2439053 www.motioniitjee@gmail. b and c be three non zero non coplanar vectors and p. If the area between f(x) and g(x) is 1/2 then n is a divisor of (A) 12 (B) 15 (C) 20 (D) 30 102. 0) the position vectors of the vertex A1 can be (A) (2. the centres of the four faces are the vertices of a smaller tetrahedron. if | u | = 3. If the position vectors of the vertices of the base ABC are A(1.Vector. 0744-2209671. c are different real numbers and a î b ĵ ck̂ . 2) (D) (0. w 0 . The volume of a right triangular prism ABCA1B 1C1 is equal to 3. If a. Ph. Diff. 2. AUC. v. b. 93527-21564 IVRS No. Eq. 3–D. 0) dy x 100. Let u. 0. Which of the following pair(s) is/are orthogonal ? (A) 16x2 + y2 = c and y16 = kx (B) y = x + ce–x and x + 2 = y + ke–y (C) y = cx2 and x2 + 2y2 = k (D) x2 – y2 = c and xy = k (where c and k are arbitrary constant) 98. 1) . b î c ĵ ak̂ & c î a ĵ bk̂ are position vectors of three non-collinear points A. 0744-2439052.com. both defined from R R and are defined as f(x) = 2x – x2 and g(x) = xn where n N. 394 . q 3a 2b c and r a 4b 2c If the volume of the parallelopiped determined by a . Consider the functions f (x) and g (x). B(2. b and c is V1 and that of the parallelopiped determined by p. q and r is V2 then V2 = KV1 implies that K is equal to 104. email-hr. 93141-87482. 2) (B) (0. C. 3–D.D 100.motioniitjee. C 10.com . A 13. 15 104. A.D 88. A 55. C 19.com. A 26. No. B. B 79.C. A. A 3. A.B. A 31. 0744-2439053 www. A.B. D 42. D 71. B 34. B 65.B. D 80. D 28.C 90. A. D 5. A 51. Eq. 0744-2439051. C 49. C 76. C. B. B. A 56. A 36.C. A. A 63. A 78. B 8. B 7. AUC. 0744-2209671.B. B 40. A.C 83. C 46. A 60. 93527-21564 IVRS No. D 57. D 39. B 32. C 35. B 62. ANSWER KEY 1. A. A 47. C 33. B. D 81.C. C 52.D 92.C 95.C 91. A 14. C 72. A. 25 394 . A 2. A 43. A 50. A 58. D 9.D 101. B 15. A 73. C 4.C. C 29. D 68. B 22. A 12.C.D 103. A 53. C 82. C 18.B. B 59. B 67.B. D 48.B. 0744-2439052.C.D 99.Rajeev Gandhi Nagar Kota. A 16. A 17.D 89. A. C 61.D 87. A. D 37. C 6.C.D 97.B. C 64. A. Ph.C. 28 105. C 69. B. C 74.D 86. B 20. A 66. A.D 93.motioniitjee@gmail. B 45. email-hr. B 41. A 30. B 54. A 11. Page # 16 Vector.D 96. C 27. D 44. B 24. A 38. C 75. A. A 70. B 25. B 77.D 102. D 23.D 85.D 94.D 84.D 98. B 21. 93141-87482. Diff.