Transcript
Question Bank Vector
3–D Geometry
Area Under Curve
Differential Equation
394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93141-87482 , 93527-21564 IVRS No. 0744-2439051, 0744-2439052, 0744-2439053 www.motioniitjee.com,
[email protected]
Page # 2
Vector, 3–D, AUC, Diff. Eq.
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
Vector, 3–D, AUC, Diff. Eq.
Page # 3
SINGLE CORRECT CHOICE TYPE 1.
Spherical rain drop evaporates at a rate proportional to its surface area. The differentia l equation corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is K > 0, is (A)
2.
dr + K = 0 dt
(B)
dr – K = 0 dt
(C)
dr = Kr dt
(D) none
The area bounded in the first quadrant by the normal at (1, 2) on the curve y2 = 4x, x-axis & the curve is given by (A)
10 3
(B)
7 3
(C)
4 3
(D)
9 2
3.
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.
A function y = f(x) satisfies the diff erential equation f(x) . sin 2x – cos x + (1 + sin 2x) f ’ (x) = 0 with initial condition y(0) = 0. The value of f(/6) is equal to (A) 1/5 (B) 3/5 (C) 4/5 (D) 2/5
5.
Suppose y = f(x) and y = g(x) are two functions whose graphs intersect at the three points (0, 4), (2, 2) and 4
(4, 0) with f(x) > g(x) for 0 < x < 2 and f(x) < g(x) for 2 < x < 4. If [f ( x) g( x )] dx 10 and 0
then area between two curves for 0 < x < 2 is (A) 5 (B) 10 6.
(C) 15
4
[g(x) f ( x)] dx 5, 2
(D) 20
Consider the lines L1 : x = 3 – t, y = 2 + t, z = 5t, intersecting the plane x – y + 2z = 9 at the point A L2 : x = 1 + 2t, y = 4t, z = 2 – 3t, intersecting the plane x + 2y – z + 1 = 0 at the point B and L3 : x = y – 1 = 2z, intersecting the plane 4x – y + 3z = 8 at the point C. The points A, B, C (A) constitute a right triangle. (C) constitute an obtuse triangle.
7.
The general solution of the differential equation
(B) constitute an acute triangle (D) do not form a triangle. 1 x dy = is a family of curves which looks most like which y dx
of the following ?
(A)
8.
(B)
(C)
(D)
Let ‘a’ be a positive constant number. Consider two curves C1 : y = ex, C2 : y = ea – x. Let S be the area of the S part surrounding by C1, C2 and the y-axis, then aLim 0 a 2 equals (A) 4
(B) 1/2
(C) 0
394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
(D) 1/4
Vector, 3–D, AUC, Diff. Eq.
Page # 4 9.
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 hole. 1 then the time to drain the tank if the water is 4 meter deep to start with is 15 (B) 45 min (C) 60 min (D) 80 min
If t is measured in minutes and k = (A) 30 min 10.
If the differential equation of the fam ily of curve given by y = Ax + Be2x where A and B are arbitrary constant is of the form (1 – 2x)
d dx
(A) (2, –2)
d dy y + k y = 0 then the ordered pair (k, l ) is dx dx (B) (–2, 2)
(C) (2, 2)
(D) (–2, –2)
11.
3 points O(0, 0), P(a, a2), Q(–b, b2) (a > 0, b > 0) are on the parabola y = x2. Let S1 be the area bounded by the line PQ and the parabola and let S 2 be the area of the triangle OPQ, the minimum value of S1/S2 is (A) 4/3 (B) 5/3 (C) 2 (D) 7/3
12.
A curve passes through the point 1,
y & its slope at any point is given by y – cos2 . Then the curve 4 x x
has the equation (A) y = x tan –1 (n
13.
e ) x
(B) y = x tan –1 (n + 2)
(C) y =
1 e tan –1(n ) x x
(D) none
Given the position vectors of the vertices of a triangle ABC, A ( a ) ; B ( b ) ; C ( c ). A vector r is parallel to the altitude drawn from the vertex A, making an obtuse angle with the positive Y-axis.
If | r | = 2 34 ; a 2ˆi jˆ 3kˆ ; b ˆi 2 jˆ 4kˆ ; c 3ˆi jˆ 2kˆ, then r is
(A) 6ˆi 8 jˆ 6kˆ
14.
(B) 2/3
(D) 6ˆi 8 jˆ 6kˆ
| x | is
(C) 1/6
(D) 1
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 (A) y e
16.
(C) 6ˆi 8 jˆ 6kˆ
The area of the region(s) enclosed by the curves y = x2 and y = (A) 1/3
15.
(B) 6ˆi 8 jˆ 6kˆ
x y
e
(B) x e
x y
e
(C)
y xex
e
(D)
y yex
e
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 f ( x) dx then If I =
0
(A)
17.
2
< I <
2 4
(B)
4
< I <
2 2
(C) 1 < I <
2
(D) 0 < I < 1
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 394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
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Vector, 3–D, AUC, Diff. Eq.
Page # 5
18.
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
19.
Area enclosed by the graph of the function y = n2x – 1 lying in the 4th quadrant is (A)
2 e
(B)
4 e
(C) 2 e
1 e
(D) 4 e
20.
Locus of the point P, for which OP represents a vector with direction cosine cos =
1 e
1 (‘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
21.
dy dy – y = 0 is Number of straight lines which satisfy the differential equation + x dx dx (A) 1
22.
(B) 2
(C) 3
(D) 4
The values of for which the following equations sinx – cosy + ( + 1)z = 0; cosx + siny – z = 0; x + ( + 1)y + cos z = 0, have non trivial solution, is (A) = n, R – {0} (B) = 2n, is any rational number (C) = (2n + 1),
R+, n I
(D) = (2n + 1)
2
, R, n I
23.
The value of the constant ‘m’ and ‘c’ for which y = mx + c is a solution of the differential equation D2y – 3Dy – 4y = – 4x. (A) is m = – 1; c = 3/4 (B) is m = 1; c = – 3/4 (C) no such real m, c (D) is m = 1 ; c = 3/4
24.
The area bounded by y = 2 – |2 – x| & y =
(A)
25.
4 3 n 3 2
(B)
3 is |x|
4 3 n 3 2
(C)
3 + n 3 2
(D)
1 + n 3 2
Four vectors a, b, c and x satisfy the relation (a . x ) b c x where b . a 1 . The value of x in terms of
a , b and c is equal to
(a . c ) b c(a .b 1) (A) (a .b 1)
26.
(B)
c
a.b 1
2 (a . c ) b c (C) a .b 1
The real value of m for which the substitution, y = um will transform the differential equation, 2x4y into a homogeneous equation is (A) m = 0 (B) m = 1
(C) m = 3/2
394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
2 (a . c ) c c (D) a .b 1 dy + y4 = 4x6 dx
(D) no value of m
Vector, 3–D, AUC, Diff. Eq.
Page # 6 27.
x The area bounded by the curve y = f(x), the co-ordinate axes & the line x = x1 is given by x1 . e 1 , Therefore f(x) equals (A) ex (B) x ex (C) xex – ex (D) x ex + ex
28.
The distance of the plane passing through the point P(1, 1, 1) and perpendicular to the line from the origin is (A) 3/4
29.
30.
32.
(C) 7/5
(D) 1
Consider the two statements Statement-1 : y = sin kt satisfies the differential equation y” + 9y = 0. 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
The solution of the differential equation, x2
(A) y = sin
31.
(B) 4/3
1 1 – cos x x
x sin
(D) 3
dy 1 1 . cos – ysin = – 1, where y –1 as x is dx x x
x 1
(B) y =
x 1 y 1 z 1 = = 3 0 4
(C) y = cos
1 x
1 1 + sin x x
(D) y =
x 1 x cos 1x
Suppose g(x) = 2x + 1 and h (x) = 4x2 + 4x + 5 and h (x) = (fog)(x). The area enclosed by the graph of the function y = f(x) and the pair of tangents drawn to it from the origin, is (A) 8/3 (B) 16/3 (C) 32/3 (D) none
The value of the triple product a × b × b × c
3
(B) a b c
b × c × c × a c × a × a × b is equal to
2
(A) a b c
4
(C) a b c
(D) a b c
33.
Where a , b , c , are non-zero non-coplanar vectors. Number of t riplets of a, b & c for which the system of equations, ax – by = 2a – b and (c + 1)x + cy = 10 – a + 3 b has infinitely many solutions and x = 1, y = 3 is one of the solutions is (A) exactly one (B) exactly two (C) exactly three (D) infinitely many
34.
A wet porous substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet hung in the wind loses half its moisture during the first hour, then the time when it would have lost 99.9% 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.
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 (A)
36.
5 6
(B)
6 5
(C)
1 6
(D) 1
If P(x, y, z) is a point on the line segment joining Q(2, 3, 4) and R(3, 5, 6) such that the projections of the
vector OP on the axes are (A) 2 : 3
13 21 26 , , respectively. The P divides QR in the ratio 5 5 5
(B) 3 : 1
(C) 1 : 3
(D) 3 : 2
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
Vector, 3–D, AUC, Diff. Eq.
Page # 7
37.
A curve C passes through origin and has the property that at each point (x, y) on it the normal line at that point passes through (1, 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
38.
If y =
x (where c is an arbitrary constant) is the general solution of the differential equation n | cx |
dy y = + dx x
(A)
39.
x
x x y then the function y is
2
(B) –
y2
x
2
(C)
y2
x and x = –
The area bounded by the curves y = – (A) cannot be determined
(B) is 1/3
y2
(D) –
x2
y2 x2
y where x, y 0 (C) is 2/3
x ; x 0 and x = y ; y 0
(D) is same as that of the figure bounded by the curves y =
2
40.
ex A function y = f(x) satisfies (x + 1) . f ’ (x) – 2(x + x) f(x) = , ( x 1) 2
3x 5 x 2 .e x 1
6 x 5 x 2 . e x 1
(A)
41.
42.
(B)
(C)
a2 y c
(D) 8
(B) x a2 y 2 c
(C) (y – a)2 = cx
(D) ay = tan –1 (x + c)
Given | p | = 2; | q | = 3 and p . q = 0. 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
The differential equation corresponding to the family of curves y = ex (ax + b) is d2y
dy (A) 2 + 2 dx – y = 0 dx
45.
5 6x x 2 .e x 1
(D)
The curve, with the property that the projection of the ordinate on the normal is constant and has a length equal to ‘a’, is
44.
6 x 5 x 2 ( x 1)2 . e
The area bounded by the curves y = x (x – 3) 2 and y = x is (in sq. units) (A) 28 (B) 32 (C) 4
(A) x a n y 2
43.
x > –1. If f(0) = 5, then f(x) is
d2y
dy (B) 2 – 2 dx + y = 0 dx
d2 y
dy (C) 2 + 2 dx + y = 0 dx
d2 y dy (D) 2 – 2 dx – y = 0 dx
y = f(x) is a function which satisfies (i) f(0) = 0 ; (ii) f ’’ (0) = f ’ (x) and (iii) f ’ (0) = 1 then the area bounded by the graph of y = f(x), the lines x = 0, x – 1 = 0 and y + 1 = 0, is (A) e (B) e – 2 (C) e – 1 (D) e + 1 394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
Vector, 3–D, AUC, Diff. Eq.
Page # 8 46.
The equation to the orthogonal trajectories of the system of parabolas y = ax2 is (A)
47.
x2 2
+ y2 = c
(B) x2 +
2
= c
(C)
x2 2
– y2 = c
Area of the region enclosed between the curves x = y2 – 1 and x = |y| (A) 1
48.
y2
(B) 4/3
Consider the differential equation
(D) x2 –
y2 2
= c
1 y 2 is
(C) 2/3
(D) 2
dy + y tan x = x tan x + 1. Then dx
(A) The integral curves satisfying the differential equation and given by y = x + c sin x. (B) The angle at which the integral curves cut the y-axis is
2
.
(C) Tangents to all the integral curves at their point of intersection with y-axis are parallel. (D) none of these x
49.
If
t y(t) dt = x + y(x) then y as a function of x is 2
a
(A) y
x 2 a2 2 2 (2 a )e 2
(B) y
x 2 a2 1 (2 a2 ) e 2
(C) y
2
2 (1 a
x 2 a2 )e 2
(D) none
50.
The area bounded by the curve y = x e –x ; 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.
Which one of the following lines is parallel to the line L : (x, y, z) = (1, 0, –2) + t(–1, 3, 0), t (A)
x 1 z 3 = ,y=3 3 2
(B) 1 – x =
y = z + 2 3
(C) 1 – x =
y ,z=5 3
R
(D) x + 1 =
y ,z=2 3
1
52.
A function f(x) satisfying
f (tx) dt = n f(x), where x > 0, is 0
(A) f ( x )
1 n c.x n
(B) f ( x ) c . x
n n 1
(C) f ( x )
1 c.x n
(D) f ( x ) c . x(1 n)
53.
Area enclosed by the curves y = nx ; y = n | x | ; y = | n x | and y = | n | x | | is equal to (A) 2 (B) 4 (C) 8 (D) cannot be determined
54.
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.
If (a, 0); a > 0 is the point where the curve y = sin2x –
3 sin x 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
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
Vector, 3–D, AUC, Diff. Eq.
Page # 9
x
56.
A curve passing through (2, 3) and satisfying the differential equation
t y(t ) dt = x y(x), (x > 0) is 2
0
(A) x2 + y2 = 13
57.
9 x 2
(B) y2 =
(C)
x2 8
+
y2 18
= 1
(D) xy = 6
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 (A)
58.
1 24
(B)
1 12
(C)
1 8
(D)
1 6
Which one of the following curves represents the solution of the initial value problem Dy = 100 – y, where y(0) = 50
100
100
(A)
(B)
50 O
59.
y
y
100
100
y
y
x
O
x
A function y = f(x) satisfies the differential equation
50
(C)
50
O
x
50
(D)
O
x
dy – 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 in the 1st quadrant (A)
60.
2 1
(B)
2
(C) 1
(D)
1 2
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, 3) and one in (–1, 0)
(C) two values one in (3, 4) and one in (–2, –1)
(D) none of these
COMPREHENSION TYPE Paragraph for Question Nos. 61 to 62 Let 61.
Distance between these lines is (A)
62.
ˆ and L : ˆi jˆ kˆ (2ˆi 3 jˆ 6kˆ ) L1 : 2iˆ 3ˆj 4kˆ (2iˆ 3 ˆj 6k) 2
10 7
(B)
10
(C)
10 7
(D) 7 10
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
394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
(D) none
Vector, 3–D, AUC, Diff. Eq.
Page # 10
63.
Paragraph for Question Nos. 63 to 65 Consider the function, f(x) = x3 – 8x2 + 20x – 13 Number of positive integers x for which f(x) is a prime number, is (A) 1 (B) 2 (C) 3
64.
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
65.
Area enclosed by y = f(x) and the co-ordiante axes is (A)
65 12
(B)
13 12
(C)
(D) 4
71 12
(D) none
Paragraph for Question Nos. 66 to 68 Consider a tetrahedron D-ABC with position vectors of its angular points as A(1, 1, 1) ; B(1, 2, 3) and C(1, 1, 2)
3 ˆ 3 ˆ ˆ i j 2k 2 4
Also the position vector of the centre ‘G’ of the tetrahedron are 66.
The volume of the tetrahedron D-ABC is (A)
67.
(B)
1 3
(C) 1
(D) 2
(C) 1
(D)
Shortest distance between the skew lines AB and CD is (A)
68.
2 3
0.9
(B)
0.8
1 3
If N is the foot of the perpendicular from the point D on the face ABC then the position vector of N are
1 (B) 1 2
1 (A) 1 2
1 (C) 1 2
(D) None of these
Paragraph for Question Nos. 69 to 71 x
–x
x
Let y = f(x) satisfies the equation f(x) = (e + e ) cos x – 2x –
(x t) f ' (t) dt . 0
69.
70.
71.
y satisfies the differential equation (A)
dy + y = ex(cos x – sin x) – e –x(cos x + sin x) dx
(B)
dy – y = ex(cos x – sin x) + e –x (cos x + sin x) dx
(C)
dy + y = ex(cos x + sin x) – e –x (cos x – sin x) dx
(D)
dy – y = ex(cos x – sin x) + e –x(cos x – sin x) dx
The value of f ’ (0) + f ” (0) equals (A) – 1 (B) 2
(C) 1
(D) 0
f(x) as a function of x equals 2 ex (A) e (cos x – sin x) + (3 cos x + sin x) + e –x 5 5 –x
(C) e –x(cos x – sin x) +
ex 5
(3 cos x – sin x) +
2 –x e 5
2 ex (B) e (cos x + sin x) + (3 cos x – sin x) – e –x 5 5 –x
(D) e –x(cos x + sin x) +
ex 5
(3 cos x – sin x) –
2 –x e 5
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
Vector, 3–D, AUC, Diff. Eq.
Page # 11
Paragraph for Question Nos. 72 to 74 A curve y = f(x) satisfies the differential equation (1 + x2) 72.
The function y = f(x) (A) is strictly increasing x R (C) is such that it has a maxima but no minima.
73.
75.
76.
(B)
4 n 2 3
(C)
2 n 2 3
(D)
1 n 2 3
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) f(x) is a transcedental function (D) y = f(x) is a bijective mapping. Paragraph for Question Nos. 75 to 77 Let P denotes the plane consisting of all points that are equidistant from the points A(–4, 2, 1) and B(2, –4, 3) and Q be the plane, x – y + cz = 1 where c R. 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 If the angle between the planes P and Q is 45º then the product of all possible values of ‘c’ is (A) – 17
77.
(B) is such that it has a minima but no maxima. (D) has no inflection point.
The area enclosed by y = f –1(x), the x-axis and the ordinate at x = 2/3 is (A) 2 n 2
74.
dy + 2yx = 4x2 and passes through the origin. dx
If the line L with equation
(B) – 2
(C) 17
(D)
24 17
x 1 y 2 z7 = = intersects the plane P at the point R(x0, y0, z0) then the 1 1 3
sum (x0 + y0 + z0) has the value equal to (A) 12 (B) – 15
(C) 13
(D) – 11
REASONING TYPE
78.
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-1 : Differential equation satisfying the curve is linear. Statement-2 : Degree of differential equation is one (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.
79.
Statement-1 : Differential equation corresponding to all lines, ax + by + c = 0 has the order 3. Statement-2 : General solution of a differential equation of nth order contains n independent arbitrary constants. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. 394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
Vector, 3–D, AUC, Diff. Eq.
Page # 12
80.
Statement-1 : Integral curves denoted by the first order linear differential equation
dy dx
–
1 x
y = –x are family
of parabolas passing through the origin. Statement-2 : Every differential equation geometrically represents a family of curve having some common property. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.
81.
x x = ny2 dx is sin = cenx y y
Statement-1 : The solution of (y dx – x dy) cot
Statement-2 : Such type of differential equations can only be solved by the substitution x = vy. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.
MULTIPLE CORRECT CHOICE TYPE
82.
Let T be the triangle with vertices (0, 0), (0, c2) and (c, c2) and let R be the region between y = cx and y = x2 where c > 0 then c3 (A) Area (R) = 6
c3 (B) Area of R = 3
Lim Area (T ) (C) c 0 =3 Area (R)
3 Lim Area (T ) (D) c 0 = Area (R) 2
83.
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 f rom the x-axis. Then the differential equation of the curve (A) is homogeneous. (B) can be converted into linear differential equation with some suitable substitution. (C) is the family of circles touching the x-axis at the origin. (D) the family of circles touching the y-axis at the origin.
84.
If a, b, c & d are the pv’s of the points A, B, C & D respect ively in three dimensional space & satisfy the
relation 3 a – 2 b + c – 2 d = 0, then (A) A, B, 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, b, c & d are linearly dependent.
85.
The differential equation, x
3 dy + dy = y2 dx dx
(A) is of order 1
(B) is of degree 2
(C) is linear
(D) is non linear
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
Vector, 3–D, AUC, Diff. Eq.
86.
Page # 13
Suppose f is defined from R [–1, 1] as f(x) =
x2 1 x2 1
where R is the set of real number. Then the statement
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. units 87.
The function f(x) satisfying the equation, f 2(x) + 4 f ’ (x) . f(x) + [f ’ (x)]2 = 0 (2 (A) f ( x ) c . e
(B) f ( x) c . e( 2 3 ) x
3)x
(C) f ( x ) c . e( 3 2) x
(D) f ( x ) c . e (2 3 ) x
(where c is an arbitrary constant)
88.
6 The vectors u 3 ; 2
v
2 6 ; 3
3 2 6
w
(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.
89.
If a, b, c are non-zero, non-collinear vectors such that a vector p = a b cos 2 a ^ b c and a
a ^ c b then
vector q = a c cos
p + q is
(A) parallel to a
(B) perpendicular to a
(C) coplanar with b & c
(D) coplanar with a and c 2
90.
A function y = f(x) satisfying the differential equation as x
sin x dy . sin x – y cos x + = 0 is such that, y dx x2
0
then the statement which is correct is /2 f ( x ) dx is less than (B)
Limit (A) x 0 f(x) = 1
0
/2 f ( x ) dx is greater than unity (C)
2
(D) f(x) is an odd function
0
91.
Which of the following statement(s) hold good ?
(A) if a . b a . c
(C) if a . b a . c and a b a c
b c ( a 0)
1
b c ( a 0)
(D) if v1, v 2,v 3 are non coplanar vectors and k1
b c ( a 0)
then k1.(k 2 k 3 )
(B) if a b a c
v2 v3 ; k 2 v1 . ( v 2 v 3 )
v 3 v1 and k 3 v1 . ( v 2 v 3 )
v1 . ( v 2 v 3 )
394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
v1 v 2 v1 . ( v 2 v 3 )
Vector, 3–D, AUC, Diff. Eq.
Page # 14
92.
0x cos x 2 2 Consider f(x) = such that f is periodic with period , then x x 2 2 2 4
(A) The range of f is 0,
(B) f is continuous for all real x, but not differentiable for some real x (C) f is continuous for all real x
(D) The area bounded by y = f(x) and the X-axis from x = – n to x = n is 2n 1
93.
3 for a given n N 24
If a line has a vector equation, r 2ˆi 6 jˆ ( ˆi 3 jˆ) then which of the following statem ents holds good ? (A) the line is parallel to 2ˆi 6 jˆ
(B) the line passes through the point 3ˆi 3 jˆ
(C) the line passes through the point ˆi 9 jˆ
(D) the line is parallel to xy plane
x
94.
A differentiable function satisfies f(x) = ( f ( t ) cos t cos( t x )) dt. Which of the following hold good ? 0
(A) f(x) has a minimum value 1 – e.
(C) f ” = e 2
95.
(B) f(x) has a maximum value 1 – e –1. (D) f ’ (0) = 1
If the line r 2ˆi jˆ 3kˆ ( ˆi jˆ 2 kˆ ) makes angle
with xy, 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.
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 , 1 27 5 23 (B) The coordinates of the turning point of the graph of y = f(x) occur at (1, 1) and , 3 27 23 , 1 . 27
(C) The value of p for which the equation f(x) = p has 3 distinct solutions lies in interval
(D) The area enclosed by y = f(x), the lines x = 0 and y = 1 as x varies from 0 to 1 is
7 . 12
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
Vector, 3–D, AUC, Diff. Eq.
97.
Page # 15
Which of the following pair(s) is/are orthogonal ? (A) 16x2 + y2 = c and y16 = kx (C) y = cx2 and x2 + 2y2 = k (where c and k are arbitrary constant)
(B) y = x + ce –x and x + 2 = y + ke –y (D) x2 – y2 = c and xy = k
98.
Consider the system of equations (4 – p 2)x + 2y = 0 and 2x + (7 – p 2)y = 0. 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.
The volume of a right triangular prism ABCA 1B1C1 is equal to 3. If the position vectors of t he vertices of the base ABC are A(1, 0, 1) ; B(2, 0, 0) and C(0, 1, 0) the position vectors of the vertex A1 can be (A) (2, 2, 2) (B) (0, 2, 0) (C) (0, –2, 2) (D) (0, –2, 0)
100.
Let
dy + y = f (x) where y is a continuous function of x with y(0) = 1 and f (x) = dx
Which of the following hold(s) good ? (A) y(1) = 2e –1 (B) y’ (1) = – e –1
x
e 2 e
if 0 x 2 if x
2
.
(D) y ’ (3) = – 2e –3
(C) y(3) = – 2e –3
101.
Consider the functions f (x) and g (x), both defined f rom R R and are defined as f (x) = 2x – x2 and g(x) = xn where n N. 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.
If a, b, c are different real numbers and a ˆi b jˆ ckˆ ; b ˆi c jˆ akˆ & c ˆi a jˆ bkˆ are position vectors of three non-collinear points A, B & C then (A) centroid of triangle ABC is
abc 3
ˆi jˆ kˆ
(B) ˆi jˆ kˆ 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.
SUBJECTIVE TYPE
103.
Let a , b and c be three non zero non coplanar vectors and p, q and r be three vectors defined as
p a b 2c ; q 3a 2b c and r a 4b 2c
If the volume of the parallelopiped determined by a , 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.
In a regular tetrahedron, the centres of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is
m , where m and n are relatively prime positive n
integers. Find the value of (m + n). 105.
Let u, v, w be the vectors such that u, v, w 0 , if | u | = 3, | v | = 4 & | w | = 5 then find the value of
| u . v v . w w .u | .
394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motionii tjee@gmail. com
Vector, 3–D, AUC, Diff. Eq.
Page # 16
ANSWER KEY 1.
A
2.
A
3.
C
4.
D
5.
C
6.
8.
D
9.
C
10.
A
11.
A
12.
A
15. A
16. A
17.
C
18.
C
19.
22.
7.
B
13. A
14.
B
B
20. B
21.
B
23.
B
24.
B
25.
A
26.
C
27. D
28.
C
29. A
30. A
31.
B
32.
C
33.
B
34. C
35. A
36.
D
37. A
38.
D
39.
B
40.
B
41. D
42. A
43.
D
44.
B
45.
C
46.
A
47.
D
48. C
49. A
50. A
51.
C
52.
A
53.
B
54.
A
55. A
56.
D
57. A
58.
B
59.
A
60.
C
61.
B
62. A
63.
C
64.
B
65. A
66.
B
67.
D
68.
C
69. A
70.
D
71.
C
72. A
73.
C
74.
C
75.
C
76. B
77. A
78.
B
79.
80.
D
81.
C
82. A,C
83. A,B,D
84. A,C,D
91. C,D
D
D
B
85. A,B,D
86. A,C,D
87. C,D
88. A,C,D
89. B,C
90. A,B,C
92. A,D
93. B,C,D
94. A,B,C
95. A,B,D
96. B,C,D
97. A,B,C,D 98. B,D
99. A,D
100. A,B,D
101. B,C,D
102. A,B,C,D
103.
104. 28
15
105. 25
394 - Rajeev G andhi Nagar Kota, P h. No. 0744-2209671, 93141-87482, 93527-21564 IVRS No. 0744-243905 1, 0744-24390 52, 0744-2439 053 www.motioni itjee.com, email-hr.motioniit
[email protected]
