BANSALCLASSES TARGET 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
(A)
Q.2
9 x 2 & x² + y² = 6 x is :
Area Area common common to the curve curve y =
3
(B)
4
3
(C) 3
4
3
4
3 3
4
(D) 3
Spherical Spherical rain rain drop drop evaporates evaporates at a rate proportion proportional al to its surface surface area. The differential differential equation equation corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is K > 0, is (A)
dr dt
K= 0
(B)
dr dt
K = 0
(C )
dr
Kr
dt
(D) none
Q.3 Q.3
If y = 2 sin sin x + sin 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
Numb Number er of valu values es of m N for which y = emx is a solution of the differential equation D3y – 3D 2y – 4Dy + 12y = 0, is (A) 0 (B) 1 (C ) 2 (D) more than 2
Q.5
The area bounded bounded by the curve curve y = x 2 + 4x + 5 , the axes of co-ordinates & the minimum ordinate is: (A) 3
2
(B) 4
3
2
(C) 5
3
2
(D) none
3
Q.6
The general general solution solution of the differe differentia ntiall equatio equation, n, 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 bounde bounded d by the curve curve y = x 2 1 & the straight line x + y = 3 is : (A)
Q.8
9 2
7 17
3
c
(B) x 4 3 y 4
3
c
(D)
2
17 17 6
arbitra ry constant, is a 2 3 , where 'a' is any arbitrary
(C) x 4 3 y 4
3
c
(D) x1 3 y1 3 c
The area enclos enclosed ed by the curve curve y2 + x4 = x2 is : (A)
Q.10
(C )
Orthogo Orthogonal nal trajectori trajectories es of family family of the curve curve x 2 3 y 2 3 (A) x 2 3 y 2
Q.9
(B) 4
2 3
(B)
4 3
(C)
8 3
(D)
10 3
Equation of a curve curve passing through the origin origin if the slope slope of the tangent drawn drawn at any any of its point (x, (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 (C) y = 2 tan –1(ex) – x (D) y = 2 tan –1(ex) + x
Bansal C lasses
Q. B. on AUC and Differential Equation
[2]
Q.11 Q.11
The area enclosed between between the curves curves y = sin x , y = cos x & the x-axis if 0 x (A)
2
1
(B) 2 2
(C)
2
2
is :
2 1
(D) 2
Q.12 Q.1 2
The differential differential equation of all parabolas parabolas having having their their axis of symmetry 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 t he tangents to it drawn from the origin is (A)
Q.14 Q.1 4
2
3
(B)
4
3
1
xy 2
x
(B) f (x, y) = x 3 ·y
y2
(C) f (x, y) = x (l ( l n x 2 y 2 – l n y)+yex/y
The area enclosed enclosed by the curve y = (A)
Q.16
4
(B)
3
1 15
tan 1
x y
(D)
2
(B) 45 min
then the time to drain the tank if the water is 4 meter deep (C) 60 min
The area bounde bounded d by x² + y² 2 x = 0 & y = sin (A)
2
4
(B)
4
2
x 2
(D) 80 min
in the upper half of the circle is :
(C)
8
(D)
2
2
The solution to the differential equation y lny + xy' xy' = 0, where y (1) = e, is (A) x (l (l n y) = 1
Q.19
3
Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. 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
to start with is (A) 30 min
Q.18
2
y , the circle x2 + y2 = 2 above the x-axis, is
(C)
2
2x 2 y 2 x 2y l n l n ( x y ) +y2tan (D) f(x,y)=x x 3x y
x & x =–
hole. If t is measured in minutes and k =
Q.17
(D) 1
3
Which one of the following functions is not homogeneous? homogeneous? (A) f (x, y) =
Q.15
1
(C)
(B) xy ( l n y) = 1
2
(C) (l n y) = 2
x 2 y = 1 (D) l n y + 2
The ratio in which the x-axis divides divides the area of the region bounded bounded by the curves y = x 2 4 x & y = 2 x x2 is : (A)
4 23
Bansal C lasses
(B)
4 27
(C)
4 19
(D) none
Q. B. on AUC and Differential Equation
[3]
Q.20
y y & its slope at any point is given by cos2 . Then the x x 4
A curve passes through through the point 1 , curve has the equation (A) y=x tan –1(l n
Q.21
9
2
(B) (B) y=x tan tan –1(l n + 2)
(C) (C) y =
1 x
tan –1(l n
e x
) (D) none none
(B)
11
3
11
(C)
4
(D)
9 4
The x-intercept x-intercept of the tangent tangent to a curve curve is equal equal to the ordinate of of the point point of contact. contact. The equation of of the curve through the point (1, 1) is (A) y e
Q.23
x
)
The area enclosed enclosed by the curve y = (x 1) (x 2) (x 3) between between the co-ordinate axes and and the the ordinate at x = 3 is : (A)
Q.22
e
x y
x
e
(B) x e
y
y
e
(C) x e x
y
e
(D) y e x
e
The line y = mx bisects the area enclosed enclosed by the curve y = 1 + 4x x2 & the lines x = 0, x =
3 2
&
y = 0. Then the value of m is: (A)
13
6
(B)
6
13
(C)
3
(D) 4
2
Q.24 Q.2 4
The differential differential equation of all parabolas each of which which has a latus rectum '4a' & whose whose axes are parallel to x-axis 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 Q.25
The area bounded 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) (C) sin sin (3x (3x + 4) + 3 (x 1) . cos (3x +4) (D) none
Q.26
The foci of the the curve curve which satisfies the differen differential tial equatio equation n (1 + y2) dx xy dy = 0 and passes through the point (1 , 0) are :
(A) Q.27
(B) 0, 2
2, 0
(C) (0, ± 1)
(D) (± 2, 0)
The area of the region region for which 0 < y < 3 2x x2 & x > 0 is : 3
(A)
3 2 x x2 dx
3
(B)
1
3 2 x x dx 2
0
Bansal C lasses
2
0
1
(C)
3 2 x x dx 3
(D)
3 2 x x dx 2
1
Q. B. on AUC and Differential Equation
[4]
Q.28
A function function y = f (x) satisfies satisfies the condition condition f '(x) sin x + f (x) cos cos x = 1, f (x) being bounded bounded when x
If I =
0.
2
f (x) dx then 0
(A)
Q.29
2
< I <
2 4
(B)
4
< I <
2
(C) 1 < I <
2
(D) 0 < I < 1
2
The area bounded bounded by the curve curve y = f(x) , the co-ordina co-ordinate te axes axes & the line x = x 1 is given by x1 . e x1 . Therefore f(x) equals: (A) ex (B) (B) x ex
(C) xex ex
(D) (D) x ex + e x
Q.30
A curve is such that the area of the region region bounded bounded by the the co-ordinate co-ordinate axes, the curve & the ordinate of 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 under the curve y = e x from x = 0 to x = h as h is : 1 (A) 2 (B) e (C ) (D) 1 e
Q.32
Degree Degree of the differentia differentiall equati equation on 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 tangent to a curve curve y = f (x) at (x , f (x)) is 2x + 1 . If the curve curve passes passes through 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)
Q.34 Q.3 4
5 6
(B)
6 5
(C)
1
(D) 1
6
A curve curve satisfying satisfying the initial initial condition, condition, y(1) = 0, satisfies the differential differential equation, equation, x
dy dx
= y – x2. The area
bounded bounded by the curve curve and and the x-axis is (A)
Q.35
1 2
(B)
1 3
(C)
1
(D)
4
1 6
1 1 The graph graphss of f (x) = x 2 & 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 Q.36 Number of straight lines which satisfy the differential equation + x y = 0 is: dx dx dy
(A) 1
Bansal C lasses
(B) 2
(C ) 3
(D) 4
Q. B. on AUC and Differential Equation
[5]
Q.37
The area bounded 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 = Q.38
The solution solution of the different differential ial equation, equation, (x + 2y 2y3) (A)
Q.39
x y
2
=y+c
(B)
y
2
=y + c
= y is :
dx x
(C )
2
= y2 + c
y
(D)
y x
e 2 5 e 2 (B) 4
4 e 2 e 2 (C) 5
5 e 2 e 2 (D) 4
The real value value of m for which which the substituti substitution, on, y = u m will transform the differential equation, 2x4y
dy dx
+ y4 = 4x6 into a homogeneous homogeneous equation is :
(A) m = 0
(B) m = 1
(C) m = 3/2
(D) no value of m
Q.41
The area bounded bounded by the curves curves y = x (x 3)2 and y = x is (in sq. units) : (A) 28 (B) 32 (C ) 4 (D) 8
Q.42
The solution solution of the different differential ial equation equation,, x 2 (A) y = sin
(C) y = cos Q.43
= x2 + c
The area bounded bounded by the curves y = x (1 l n x) ; x = e1 and positive X-axis between x = e1 and x = e is :
e 2 4 e 2 (A) 5 Q.40
x
dy
x ; x 0 and x = y ; y 0
1 x 1 x
– cos
sin
1
dy dx
.cos
1 x
y sin
(B) y =
x 1
(D) y =
x
1 x
= 1, where y 1 as x is
x 1 x sin x1 x 1 x cos x1
The positive values of the parameter parameter 'a' for which the area of the figure bounded by the curve curve y = cos ax, y = 0, x = (A)
6a
, x =
5 6a
(B) (0, 1/3)
is greater than 3 are : (C) (3, )
(D) none of these
Q.44
The equation of a curve curve passing through (1, 0) for which the product of the abscissa of a point P & the intercept made ma de by a normal at P on the x-axis equals twice the square s quare of the radius vector of the point P, 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 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 (A) (1,2)
Bansal C lasses
(B) (2,5)
(C )
3 , 13 2 4
(D) none
Q. B. on AUC and Differential Equation
[6]
Q.46 Q.4 6
The latus rectum rectum of the the conic passing passing through through the origin origin and having having the the property property that normal at each point point (x, y) intersects the x axis at ((x + 1), 0) is : (A) 1 (B) 2 (C ) 4 (D) none
Q.47
The value of 'a' (a>0) for which the area bounded bounded by by the curves y =
x 6
1 x2
, y = 0, x = a and
x = 2a has the least value, is (A) 2 Q.48
(B)
(B) sin(x2y2) = x
(B) 4/3 dy dx
=
1 y 2x
is
(D) sin sin(x2y2) = e.ex
x = y2 – 1 and x = |y| 1 y 2
1 2 y 4x
2
is
(D) 2 is
(B) 4x2 – 4xy – y2 2x 2y + c = 0 (D) 4x2 + 4xy – y2 2x 2y + c = 0
Let y = g (x) be the inverse inverse of a bijectiv bijectivee mapping 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 : (A)
Q.52
dx
(C) 2/3
Solution of the differential differential equation, equation,
(D) 1
= tan (x2y2) 2xy2 given y(1) =
(C) cosx2y2 + x = 0
(A) 4x2 + 4xy + y2 2x 2y + c = 0 (C) 4x2 + 4xy + y2 2x 2y + c = 0 Q.51
dy
Area of the region region enclosed enclosed between between the curves (A) 1
Q.50
(C) 21/ 3
The solution solution of the differential differential equation, equation, 2 x 2y (A) sinx2y2 = ex–1
Q.49
2
5
(B)
4
7
(C)
4
The solution of the differential differential equation, equation,
dy dx
=
(A) a pair of straight lines (C) parabola
9 4
(D)
13 4
x , given y( 5) = 5 represents y x 1 y
(B) a circle (D) hyperbola
Q.53
Area enclosed enclosed by the curves curves y = l nx nx ; y = l n | x | ; y = | l n x | and y = | l n | x | | is equal to (A) 2 (B) 4 (C ) 8 (D) cannot be determined
Q.54 Q.54
If y = dy dx (A)
Q.55
x l n | c x |
y
=
x
x x then the function is : y y
+
x2 y
(where c is an arbitrary constant) is the general solution of the differential equation
2
(B) –
x2 y
2
(C)
y2 x2
(D) –
y2 x2
If the tangent tangent to the curve y = 1 – x 2 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 curve for which 0 < x < 1 , then then k is equal equal to (A)
2 3
Bansal C lasses
(B)
75 16
(C)
25 18
(D)
Q. B. on AUC and Differential Equation
2 3
[7]
d3y Q.56
If the functi function on y = e4x + 2e 2e –x is a solution of the differential equation dx K is (A) 4
Q.57
(B) 6
13 3
dy dx
y
(C ) 9
K then the value of
(D) 12
If (a, 0); a > 0 is the the point point where the curve y = sin2x sin2x –
3 sinx cuts the x-axis first, A is the area
bounded bounded by this part of the the curve curve , the the origin origin and the positive positive x-axis x-axis,, then then (A) 4A + 8 cosa = 7 (B) 4A + 8 sina = 7 (C) 4A – 8 sina = 7 (D) 4A – 8 cosa = 7 Q.58
A function function y = f (x) satisfies (x + 1) . f (x) – 2 (x2 + x) f (x) =
ex
2
( x 1)
, x
1
If f (0) = 5 , then f (x) is
3x 5 . e x 2 (A) x 1 Q.59
1
(B)
24
(A)
(C)
1 12
(C)
d4 y dx 4 d5y dx
5
d2 y
( x c 4 )
1
(D)
8
1 6
y0
dx 2
y0
) (c5 sin x ) , where c1, c2, c3, c4, c5 are arbitrary constants, is (B)
(D)
A functio function n y = f (x) (x) satisfies satisfies the differ differentia entiall equation equation
d3y dx 3 d3y dx
3
d2y dx 2 d2y dx
2
dy dx dy dx
y0 y0
dy
– y = cos x – sin x with initial condition that y is dx bounded when when x . The area enclosed by y = f (x), y = cos x and the y-axis y-axis is (A)
Q.62 Q.6 2
5 6 x . e x 2 (D) x 1
The differential equation whose general solution is given given by, by, y = c1 cos( x c 2 ) (c 3e
Q.61
(C)
6x 5 x 2 . e 2 ( x 1 )
The curve 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)
Q.60 Q.6 0
6x 5 . e x 2 (B) x 1
2 1
(B)
2
(C ) 1
(D)
1 2
The curve, curve, with the the property that the projection projection of the ordinate ordinate on the normal is constant and has a length length equal to 'a', is
(A) x a l n y
2
(C) (y – a) 2 = cx
Bansal C lasses
a 2 y c
(B) x
a2 y2 c
(D) ay = tan –1 (x + c)
Q. B. on AUC and Differential Equation
[8]
Q.63
Area bounded bounded by the curve y = min {sin 2x, cos2x}and x-axis between the ordinates x = 0 and x = (A) (C)
Q.64
5 2
square units
5( 2) 8
(B)
5( 2) 4
4
is
square units
1 8 2
(D)
square units
5
square units
The equation equation to the orthogona orthogonall trajectories trajectories of of the system system of parabolas parabolas y = ax 2 is (A)
x2 2
2
y =c
(B) x
2
y2
=c
2
(C )
x2 2
y
2
(D) x
=c
2
y2 2
=c
x
Q.65 Q.65
If
t y(t)dt = x + y (x) then y as a function of x is 2
a
Q.66
(A) y = 2 – (2 +
x 2 a 2 a 2) e 2
(B) y = 1 – (2 + a ) e
(C) y = 2 – (1 +
x 2 a 2 a 2) e 2
(D) none
a
x 2
A curve curve y = f (x) passing passing through through the point point 1,
2
1
satisfies the differential equation dy + x e e dx
x2 2
=0.
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 substitu substitution tion 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
t y(t ) dt = x y (x), (x >0) is 2
A curve curve passing passing through through (2, 3) and and satisfying satisfying the differential equation
0 2
2
2
(A) x + y = 13 Q.69 Q.6 9
(B) y =
2
x
(C )
x2 8
y2 18
1
(D) xy = 6
Which one one of the following following curves represents represents the solution of of the initial value problem Dy = 100 – y, y, where y (0) = 50
(A) Q.70
9
(B)
(C)
(D)
Solution Solution of the differen differential tial equatio equation n 2 2 dy e x e y y dx
x2
+ e ( xy
2
x ) = 0,
is
2
2
(B) e y (x2 – 1) + e x = C
2
2
(D) e x (y – 1) + e y = C
(A) e x (y2 – 1) + e y = C (C) e y (y2 – 1) + e x = C
Bansal C lasses
2
2
2
2
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 Number of positive positive integers x for which f (x) is a prime prime number number,, is (A) 1 (B) 2 (C ) 3
(D) 4
Q.72
The functio function n f (x) defined defined for R R (A) is one one onto (B) is many one onto (C) has 3 real roots (D) is such that f (x 1) · f(x2) < 0 where x 1 and x2 are the roots of f ' (x) = 0
Q.73
Area enclosed by y = f (x) and and the co-ordinate co-ordinate axes axes is (A)
Q.74
12
13
(C)
12
71
(D) none
12
3 2
–2
(B)
3
(C) 2 +
2
3 2
3
equals
(D) 1 +
2
3 2
The area area of of the region region under under the graph graph of y = xe –ax as x varies from 0 to , where 'a' is a positive constant, is (A)
Q.76
(B)
The area enclosed by the curves y = cos x, y = 1 + sin 2x and x = (A)
Q.75
65
1
(B)
a
1 a
1 a2
(C)
1 a
1 a
2
(D)
1 a2
The The poly polyno nomia miall f (x) satisfies the condition f condition f (x + 1) = x2 + 4x. The area enclosed by y = f = f (x – 1) and the 2 curve x + y = 0, is (A)
16 2
(B)
3
16 3
(C)
8 2
(D) none
3
Select the correct correct alternatives : (More than one are correct) correct)
Q.77 Q.7 7
Family of curves curves whose whose tangent at a point with with its intersection with the curve xy = c2 form an angle of is (A) y2 2xy x2 = k
4
(B) y2 + 2xy x2 = k
x c
(C) y = x - 2 c tan1 + k
(D) y = c l n
cx c
x
x + k
where k is an arbitrary constant .
Q.78
dy y = y. l n is : dx x
The general solution of the differential equation, x (A) y = xe1 cx
(B) y = xe1 + cx
(C) y = ex . ecx
(D) y = xecx
where c is an arbitrary constant.
Bansal C lasses
Q. B. on AUC and Differential Equation
[10]
Q.79
Which of the following following equation(s) equation(s) is/are linear. linear. (A)
Q.80
dy dx
+
y x
dy (B) y + 4x = 0 dx
= l n x
(D)
dx2
= cosx
The function function f(x) satisfying satisfying the equation, equation, f 2(x) + 4 f (x) . f(x) + [f (x)]2 = 0 . (A) f(x) = c . e
2- 3 x
(B) f(x) = c . e
3 2 x
(C) f(x) = c . e where c is an arbitrary constant. Q.81
(C) dx + dy = 0
d 2y
2+ 2+
(D) f(x) = c . e
3 x
2+
3 x
The equation equation of of the curve curve passing passing through through (3 (3 , 4) & satisfy satisfying ing the the differential differential equation, equation, 2
dy dy + (x y) – x = 0 can be y dx dx (A) x y + 1 = 0 (B) x 2 + y2 = 25 Q.82
(D) x + y 7 = 0
The area bounded bounded by a curve, the axis of co-ordinates co-ordinates & the the ordinate ordinate of some point point of the the curve 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 (A) y =
1 2
(ex e – x + 2)
1
(B) y =
(C ) y = 1 Q.83
(C) x2 + y2 5x 10 = 0
2
(ex + ex)
2
(D) y =
ex
e x
Identify Identify the statement(s) statement(s) which is/are True. True. (A) f(x , y) = e y/x + tan (B) x . l n
y x
dx +
y2 x
y
is homogeneo homogeneous us of degree zero
x
sin1
y x
dy = 0 is homogeneous of degree one
(C) f(x , y) = x 2 + sin x . cos y is not homogen homogeneous eous (D) (x2 + y2) dx - (xy2 y3) dy = 0 is a homegeneous differential equation . Q.84
The graph graph of the function 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 (C) it is neither an even nor an odd function Q.85
(B) it is periodic (D) it is continuous continuous & differentiabl differentiablee for all x .
A functi function on y = f (x) satisfying satisfying the differential equation
dy dx
·sin x – y cos x +
sin 2 x x2
= 0 is such that,
y 0 as x then the statement which is correct is /2
(A) Lim f(x) = 1
(B)
x 0
f(x) dx is less than
0
2
/2
(C)
f(x) dx is grea greate terr than than unit unity y f(x)
(D) (D) f(x) f(x) is an odd odd func functi tion on
0
Bansal C lasses
Q. B. on AUC and Differential Equation
[11]
ANSWER KEY C , B , A 5 . Q 8 D , B , A 4 8 . Q
C , B , A 3 8 . Q
C , B 2 8 . Q
B , A 1 8 . Q
D , C 0 8 . Q
D , C , A 9 7 . Q
C , B , A 8 7 . Q
, C , B , A 7 7 . Q D
t e r o M ( : s t t c e l e S c e r a e n o n a h r e t l a t c e r r o c e h ) t c e r r o e v i t a n 7 . Q A 6
7 . Q D 5
7 . Q C 4
7 . Q A 3
7 . Q B 2
7 . Q C 1
A 0 7 . Q
B 9 6 . Q
D 8 6 . Q
A 7 6 . Q
B 6 6 . Q
A 5 6 . Q
A 4 6 . Q
C 3 6 . Q
A 2 6 . Q
A 1 6 . Q
B 0 6 . Q
A 9 5 . Q
B 8 5 . Q
A 7 5 . Q
D 6 5 . Q
A 5 5 . Q
D 4 5 . Q
B 3 5 . Q
C 2 5 . Q
D 1 5 . Q
A 0 5 . Q
4 . Q D 9
4 . Q A 8
4 . Q D 7
4 . Q B 6
4 . Q C 5
4 . Q A 4
B 3 4 . Q
4 . Q A 2
4 . Q D 1
4 . Q C 0
3 . Q B 9
3 . Q B 8
3 . Q B 7
3 . Q B 6
C 5 3 . Q
D 4 3 . Q
A 3 3 . Q
D 2 3 . Q
D 1 3 . Q
C 0 3 . Q
D 9 2 . Q
A 8 2 . Q
C 7 2 . Q
A 6 2 . Q
C 5 2 . Q
C 4 2 . Q
A 3 2 . Q
A 2 2 . Q
C 1 2 . Q
A 0 2 . Q
A 9 1 . Q
A 8 1 . Q
A 7 1 . Q
C 6 1 . Q
D 5 1 . Q
1 . Q D 4
1 . Q A 3
1 . Q A 2
1 . Q B 1
1 . Q B 0
. Q B 9
. Q B 8
. Q D 7
A 6 . Q
. Q B 5
C 4 . Q
. Q B 3
A 2 . Q
. Q D 1
l n O ( : e t t c e l e S ) t c e r r o c s i e n o y r e t l a t c e r r o c e h v i t a n
Bansal C lasses
Q. B. on AUC and Differential Equation
[12]
