Common to all branches
B.E., Mechanical, Civil, ECE, EEE, CSE, IT, Aeronautical, Petroleum etc
UNIT I DIFFERENTIAL CALCULUS
[ANS ARE HIGHLIGHTED]
1.Find domain and range of the function f(x) = √9-x^2
a) 0 ≤ x ≤ 3 ;0 ≤ y ≤ 3
b) -3 ≤ y ≤ 3 ;0 ≤ x ≤ 3
c) -3 ≤ x ≤ 3 ;-3 ≤ y ≤ 3
d) -3 ≤ x ≤ 3 ;0 ≤ y ≤ 3
2.Test the function f(x) = 3x^2+2x-1 is continuous at------------------.
a) x = 0 b) x =1 c) x= 2 d) x = 3
3.Find the critical point of f(x) = x^3+x^2-x
a) x = -1 , 1/3
b) x = -1 , -1/3
c) x = -1 , 1/2
d) x = 1 , 1/3
4.Find the critical value of f(x) = 3x^4-16x^3+18x^2
a) x = 0,1,2
b) x = 0,-1,2
c) x = 0,1,4
d) x = 0,1,3
5.What is the degree of the differential equation 4x^3-6x^2y^3+2y=0 ?
a) 3 b) 5 c) 1 d) 8
6.What is the order of the differential equation, y”+y’-x3y=sinx?
a) 2 b) 1 c) 0 d) 3
7.Which of the following correctly defines ordinary differential equations?
a) A differential equation in which a dependent variable (say ‘y’) depends on only one
independent variable (say ’x’)
b) A differential equation in which an independent variable (say ‘y’) depends on only one
dependent variable (say ’x’)
c) A differential equation in which a dependent variable (say ‘y’) depends on one or more
independent variables (say ’x’, ’t’ etc.)
d) A differential equation in which an independent variable (say ‘y’) depends on one or more
dependent variables (say ’x’, ’t’ etc.)
8.How many real solution to this function x^3+3x+1 = 0
a) 0 b) 1 c) 2 d) 3
9.Find the slope of the tangent line to the parabola y = 4x-x^2 at (1,3)
a) 3x^2+1 b) 3x^2-3 c) x^2-1 d) 3x^2-1
10.Find the equation of the tangent line to the parabola y = x^2-8x+9 at the point (3,-6)
a) y = 2x b) y = -2x c) y = -x d) y = x
11.Find the critical points to this function y = x^4-4x^3
a) 0,1 b) 0,2 c) 0,3 d) 0,4
UNIT II FUNCTIONS OF SEVERAL VARIABLES
12. Find the stationary values of 3x^2-y^2+x^3
a) (0,0) & (-3,0)
b) (0,0) & (-2,0)
c) (0,0) & (0,-2)
d) (-2,0)
13.Stationary point is a point where function f(x,y) have?
a) fx = 0 b) fy = 0 c) fx = 0 & fy = 0 d) none
14.Expand f(x) = 1⁄x about x = 1.
a) 1 – (x-1) + (x-1)2 – (x-1)3 + ….
b) 1 + (x-1) + (x-1)2 + (x-1)3 + ….
c) 1 + (x-1) – (x-1)2 + (x-1)3 + ….
d) 1 – (x+1) + (x+1)2 – (x+1)3 + ….
15.Find the maximum and minimum of x^2+y^2+6x+12
a) the point (-3,0) is a minimum point
b) the point (-2,0) is a minimum point
c) the point (-3,0) is a maximum point
d) the point (-2,0) is a maximum point
16.The point (0,0) in the domain of f(x, y) = sin(xy) is a point of ___________
a) saddle b) minima c) maxima d) constant
17.Find the minimum value of the function f(x, y) = x2 + y2 +199 over the real domain
a) 12 b) 13 c) 0 d) 199
18.What is the maximum value of the function f(x, y) = 3xy + 4x2y2 in the region?
x=0; y=0; 2x + y = 2
a) 1 b) 0 c) 100 d) 10
19.The jacobian of p,q,r w.r.t x,y,z given p=x+y+z, q=y+z, r=z is ________
a) 0 b) 1 c) 2 d) -1
UNIT III INTEGRAL CALCULUS
20.Evaluate à´½e^-x dx, the upper limit of integration is infinite
a) converges to 1
b) converges to 0
c) divergent
d) none
21.Find the value of ∫x3 Sin(x)dx.
a) x3 Cos(x) + 3x2 Sin(x) + 6xCos(x) – 6Sin(x)
b) – x3 Cos(x) + 3x2 Sin(x) – 6Sin(x)
c) – x3 Cos(x) – 3x2 Sin(x) + 6xCos(x) – 6Sin(x)
d) – x3 Cos(x) + 3x2 Sin(x) + 6xCos(x) – 6Sin(x)
22.Integration of (Sin(x) + Cos(x))ex is______________
a) ex Cos(x)
b) ex Sin(x)
c) ex Tan(x)
d) ex (Sin(x)+Cos(x))
23.Find the value of ∫∫xyex + y dxdy.
a) yey (xex-ex) b) (yey-ey)(xex-ex) c) (yey-ey)xex d) (yey-ey)(xex+ex)
UNIT IV MULTIPLE INTEGRALS
24.Evaluate ഽഽxy dx dy taken over the positive quadrant of the circle x^2+y^2 = a^2
a) a4/8 sq.units
b) a4/4 sq.units
c) a2/8 sq.units
d) a2/2 sq.units
UNIT V DIFFERENTIAL EQUATIONS
25.Solve (D^2+2D+1)y = 0
a) y = (Ax+B)e^x
b) y = (Ax+B)e^-x
c) y = (x+B)e^-x
d) y = A cos x + B sin x
26.Solve (D^3+D^2+4D+4)y = 0
a) y = Ae^-x + (Bcosx + Csin x )
b) y = ( A cos2x + B sin 2x )
c) y = Ae^-x + (Bcos2x+Csin2x)
d) y = Ae^-x - (Bcos2x+ Csin2x)
27.Solve (D^4-2D^2+1)y = 0
a) y = (Ax+B)e^-x
b) y = (Ax+B) + (Cx+D)e^x
c) y = (Ax+B)e^-x - (Cx+D)e^x
d) y = (Ax+B)e^-x + (Cx+D)e^x
28.Find the particular integral of (D^2+2D+1)y = e^-x x^2
a) P.I = 1/4x^4e^-y
b) P.I = x^4e^-y
c) P.I = 1/24x^4e^y
d) P.I = 1/24x^4 e^-y
29.Convert into differential form of (x^2D^2-xD+4)y = x^2sin(logx)
a) (D^2 - 2D + 4 )y = e^2z
b) (D^2 + 2D + 4 )y = e^2z sinz
c) (D^2 - 2D + 4 )y = e^2zsinz
d) (D^2 - 2D + 4 )y = sin z
30.Find Wronskian from the equation ( D^2 - D)y = e^xcosx
a) W = -e^x
b) W = e^-x
c) W = e^x
d) W = 2e^x
ALL THE BEST ! STUDY WELL
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