1. The pth derivative of a qth degree monic polynomial, where p, q are positive integers and 2p4 +
3pq3⁄2 = 3q3⁄2 + 2qp3 is given by?
a) Cannot be generally determined
b) (q – 1)!
c) (q)!
d) (q – 1)! * pq
Answer: c
Explanation: First consider the equation
2p4 + 3pq3⁄2 = 3q3⁄2 + 2qp3
After simplification, we get
(2p3 + 3q1⁄2) (p – q) = 0
This gives us two possibilities
2p3 = – 3q1⁄2
OR
p=q
The first possibility can’t be true as we are dealing with positive integers
Hence, we get
p=q
Thus the pth derivative of any monic polynomial of degree p(p = q) is
p! =q!
2. nth derivative of Sinh(x) is
a) 0.5(ex – e-x)
b) 0.5(e-x – ex)
c) 0.5(ex – (-1)n e-x)
d) 0.5((-1)-n e-x -ex)
Answer: c
Explanation: Y = Sinh(x)
Y = 0.5[ex – e-x].
y1 = 0.5 [ex – (-1)e-x].
y2 = 0.5 [ex – (-1)2 e-x].
Similarly,
yn = 0.5 [ex – (-1)n e-x].
3. If x = a(Cos(t) + t2) and y = a(Sin(t) + t2 + t3) then dy/dx equals to
a) (Cos(t) + 3t2 + 2t) / (-Sin(t) + 2t)
b) (Sin(t) + 3t2 + 2t) / (-Cos(t) + 2t)
c) (Sin(t) + 3t2 + 2t) / (Cos(t) + 2t)
d) (Cos(t) + 3t2 + 2t) / (Sin(t) + 2t)
Answer: a
Explanation: dx/dt = a(-Sin(t) + 2t)
dy/dt = a(Cos(t) + 2t + 3t2)
Then,
dy/dx = (Cos(t) + 3t2+2t)/(-Sin(t) + 2t).
4. For the function f(x) = x2 – 2x + 1. We have Rolles point at x = 1. The coordinate axes are then
rotated by 45 degrees in anticlockwise sense. What is the position of new Rolles’s point with
respect to the transformed coordinate axes?
a) 3⁄2
b) 1⁄2
c) 5⁄2
d) 1
Answer: a
Explanation:The coordinate axes are rotated by 45 degree then the problem transforms into
that of Lagrange mean value theorem where the point in some interval has the slope of
tan(45).
Hence differentiating the function and equating to tan(45).
We have
f'(x) = tan(45) = 2x – 2
2x – 2 = 1
x = 3⁄2.
5. What is the minimum angle by which the coordinate axes have to be rotated in anticlockwise
sense (in Degrees), such that the function f(x) = 3x3 + 5x + 1016 has at least one Rolles point
a) π⁄180 tan-1(5)
b) tan-1(5)
c) 180⁄π tan-1(5)
d) -tan-1(5)
Answer: c
Explanation: For the transformed function to have a Rolles point is equivalent to the existing
function having a Lagrange point somewhere in the real number domain, we are finding
the point in the domain of the original function where we have f'(x) = tan(α)
Let the angle to be rotated be α
We have
f'(x) = 9x2 + 5 = tan(α)
9x2 = tan(α) – 5
For the given function to have a Lagrange point we must have the right hand side be greater
than zero, so
tan(α) – 5 > 0
tan(α) > 5
α > tan-1(5)
In degrees we must have,
αdeg > 180⁄π tan-1(5).
6. For the function f(x) = x2 – 2x + 1. We have Rolles point at x = 1. The coordinate axes are then
rotated by 45 degrees in anticlockwise sense. What is the position of new Rolles point with
respect to the transformed coordinate axes?
a) 3⁄2
b) 1⁄2
c) 5⁄2
d) 1
Answer: a
Explanation:The coordinate axes are rotated by 45 degree then the problem transforms into
that of Lagrange mean value theorem where the point in some interval has the slope of
tan(45).
Hence differentiating the function and equating to tan(45).
We have
f'(x) = tan(45) = 2x – 2
2x – 2 = 1
x = 3⁄2.
7. Taylor’s theorem was stated by the mathematician _____________
a) Brook Taylor
b) Eva Germaine Rimington Taylor
c) Sir Geoffrey Ingram Taylor
d) Michael Eugene Taylor
8. What is the Taylor series expansion of f(x)= x2-x+1 about the point x=-1?
a) f(x) = -3-3(x+1)+(x+1)2
b) f(x) = -3-(x+1)+(x+1)2
c) f(x) = -3-3(x+1)+2(x+1)2
d) f(x) = -1-3(x+1)+2(x+1)2
Answer: a
Explanation: Given, f(x)= x2-x+1 , f(-1)=1+1+1=3
f'(x) = 2x-1,f'(-1)=2(-1)-1= -3
f”(x) = 2, f”(-1)=2
fn (x)=0, for n > 2
The Taylor series expansion of f(x) about x=a is,
f(x) = f(a)+(x-a) f'(a)+(x-a)2 f”2! (a)+⋯
Here a=-1,
f(x) = f(-1)+(x+1) f'(-1)+(x+1)2 f”2! (-1) since,for n > 2, fn (x) = 0.
f(x) = -3-3(x+1)+(x+1)2
9. What is the first term in the Taylor series expansion of f(x) = 8x5-3x2-5x about x=2?
a) 232
b) 244
c) 234
d) 222
Answer: c
Explanation: Given, f(x) = 8x5-3x2-5x
First term in the Taylor series for f(x) is f(a). Here, a=2. Therefore,
f(2) = 8(2)5-3(2)2-5(2)=8(32)-3(4)-10=234.
10. The initial condition for the recurrence relation of a factorial is ___________
a) 0! =0
b) 0! =1
c) 1! =1
d) 1! =0
Answer: b
Explanation: Factorial of a number is defined by the recurrence relation as, n! =n(n-1)! for n>0
and the initial condition is 0!=1.
11. Find lt(x,y)→(0,0)x3−y3(x−y)
a) -1⁄2
b) 0
c) ∞
d) -90
Answer: b
Explanation: Simplifying the expression we have
lt(x,y)→(0,0)(x−y)(x2+xy+y2)(x−y)
lt(x,y)→(0,0)(x2+xy+y2)1=(02+0.0+02)
= 0.
12. If F(x) = f(x)g(x)h(x) and F’(x) = 10F(x) and f’(x) = 10f(x) , g’(x) = 10g(x) and h’(x) = 10kh(x),
then find value of k.
a) 0
b) 1
c) -1
d) 2
Answer: c
Explanation: Given F(x) = f(x)g(x)h(x)
Differentiating,
F’(x) = f’(x)g(x)h(x) + f(x)g’(x)h(x) + f(x)g(x)h’(x)
Putting value of F’(x), f’(x), g’(x), h’(x)
We get
10 = 10 + 10 + 10k
K = -1.
13. Which of the following relations hold true?
a) i × i = j × j = k × k = 1
b) i × j = k, j × i = -k
c) i × i = j × j = k × k = -1
d) k × i = -j, i × k = j
Answer: b
Explanation: The properties of vector or cross product, for the orthogonal vectors, i, j, and k
are,
i × i = j × j = k × k = 0,
i × j = k, j × i = -k,
j × k = i, k × j = -i,
k × i = j, i × k = -j
14. What is the derivative of z=3x*logx+5x6 ex with respect to x?
a) 3+30x5 ex
b) 3+5x6 ex+30x5 ex
c) 3+5x6 ex
d) 3+3logx+5x6 ex+30x5 ex
Answer: d
Explanation: Given: z=3x*logx+5x6 ex
dzdx=3x(1x)+3logx+5x6 ex+30x5 ex
dzdx=3+3logx+5x6 ex+30x5 ex
15. f(x, y) = x2 + xyz + z Find fx at (1,1,1)
a) 0
b) 1
c) 3
d) -1
Answer: c
Explanation: fx = 2x + yz
Put (x,y,z) = (1,1,1)
fx = 2 + 1 = 3.
f(x, y) = sin(xy) + x2 ln(y) Find fyx at (0, π⁄2)
a) 33
b) 0
c) 3
d) 1
Answer: d
Explanation: fy = xcos(xy) + x2⁄y
fyx = cos(xy) – xysin(xy) + 2x⁄y
Put (x,y) = (0, π⁄2)
= 1.
16. f(x, y) = sin(xy + x3y) / x + x3 Find fxy at (0,1).
a) 2
b) 5
c) 1
d) undefined
Answer: c
Explanation: First find
fy = sin(xy + x3y)
Hence
fyx = fxy = (cos(xy + x3y)) . (y + 3x23y)
Now put (x,y) = (0,1)
= 1.
17. If z = xn f(y⁄x) then?
a) y ∂z⁄∂x + x ∂z⁄∂y = nz
b) 1/y ∂z⁄∂x + 1/x ∂z⁄∂y = nz
c) x ∂z⁄∂x + y ∂z⁄∂y = nz
d) 1/x ∂z⁄∂x + 1/y ∂z⁄∂y = nz
Answer: c
Explanation: Since the given function is homogeneous of order n , hence by euler’s theorem
x ∂z⁄∂x + y ∂z⁄∂y = nz.
18. For a homogeneous function if critical points exist the value at critical points is?
a) 1
b) equal to its degree
c) 0
d) -1
Answer: c
Explanation: Using Euler theorem we have
xfx + yfy = nf(x, y)
At critical points fx = fy = 0
f(a, b) = 0(a, b) → critical points.
19. If u = xx + yy + zz , find du⁄dx + du⁄dy + du⁄dz at x = y = z = 1.
a) 1
b) 0
c) 2u
d) u
Answer: d
Explanation: du⁄dx = xx (1+log⁡(x))
du⁄ = yy (1+log⁡(x))
dy
du⁄ = zz (1+log⁡(x))
dz
At x = y = z = 1,
du⁄ + du⁄ + du⁄ = u.
dx
dy
dz
20. At a distance of 120ft from the foot of a tower the elevation of its top is 60o. If the possible
error in measuring the distance and elevation are 1 inch and 1 minute, find approximate error
in calculated height.
a) 0.342
b) 0.284
c) 0.154
d) 0.986
Answer: b
Explanation: Tan(z) = h⁄x
h = x Tan(z)
Taking log and then differentiate we get,
∂h⁄ = ∂x⁄ + 1⁄
2
h
x
Tan(z) Sec (z)δz
o
Now h = 120 tan(60 ) = 120√3
Putting, δx = 1⁄12 ft, δz = π⁄(60*180)
Putting the values we get,
δh = 0.284.
21. Consider the f(x, y) = x2 + y2 – a. For what values of a do we have critical points for the
function.
a) independent of a
b) for any real number except zero
c) a ∊ (0, +∞)
d) a ∊ (-1, 1)
Answer: a
Explanation: Consider
fx = 2x
and
fy = 2y
There is no a here. Thus, independent of a.
22. What is the maximum value of the function f(x, y) = x2(1 + 3y) + x3 + y3 + y2(1 + 3x) + 2xy over
the region x=0; y=0; x + y=1.
a) 0
b) -1
c) Has no maximum value
d) 2
Answer: d
Explanation: Rewrite the function as
f(x, y) = (x + y)2 + (x + y)3
Put x + y = 1
= 2.
23. The maximum value of the function is?
f(x, y) = sin(x).cos(2y).cos(x + 2y) + sin(2y).cos(x + 2y).cos(x) in the region x=0; y=0; x+2y = 3
a) 90
b) cos(1)
c) sin(1).cos(1)
d) sin(3).cos(3)
Answer: d
Explanation: Rewrite the function as
f(x, y) = cos(x + 2y) * (sin(x).cos(2y) + cos(x).sin(2y))
f(x, y) = cos(x + 2y).sin(x + 2y)
Put x+2y = 3
= sin(3).cos(3).
24. 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
Answer: d
Explanation: Find
fx = 2x
fy = 2y
The critical point is
x=0
y=0
(0,0) is the critical point
Put it back into the function we get
z = 0 + 0 + 199 = 199 is the required minimum value.
25. For function f(x,y) to have minimum value at (a,b) value is?
a) rt – s2>0 and r<0
b) rt – s2>0 and r>0
c) rt – s2<0 and r<0
d) rt – s2>0 and r>0
Answer: b
Explanation: For the function f(x,y) to have minimum value at (a,b)
rt – s2>0 and r>0
where, r = ∂2f⁄∂x2, t=∂2f⁄∂y2, s=∂2f⁄∂x∂y, at (x,y) => (a,b).
26. For function f(x,y) to have maximum value at (a,b) is?
a) rt – s2>0 and r<0
b) rt – s2>0 and r>0
c) rt – s2<0 and r<0
d) rt – s2>0 and r>0
Answer: a
Explanation: For the function f(x,y) to have maximum value at (a,b)
rt – s2>0 and r<0
where, r = ∂2f⁄∂x2, t=∂2f⁄∂y2, s=(∂2f⁄∂x∂y, at (x,y) => (a,b).
27. For function f(x,y) to have no extremum value at (a,b) is?
a) rt – s2>0
b) rt – s2<0
c) rt – s2 = 0
d) rt – s2 ≠ 0
Answer: b
Explanation: For the function f(x,y) to have no extremum value at (a,b)
rt – s2 < 0 where, r = ∂2f⁄∂x2, t=∂2f⁄∂y2, s=∂2f⁄∂x∂y, at (x,y) => (a,b).
28. The drawback of Lagrange’s Method of Maxima and minima is?
a) Maxima or Minima is not fixed
b) Nature of stationary point is can not be known
c) Accuracy is not good
d) Nature of stationary point is known but can not give maxima or minima
Answer: b
Explanation: In lagrange’s theorem of maxima of minima one can not determine the nature of
stationary points.
29. What is the saddle point?
a) Point where function has maximum value
b) Point where function has minimum value
c) Point where function has zero value
d) Point where function neither have maximum value nor minimum value
Answer: d
Explanation: Saddle point is a point where function have neither maximum nor minimum value.
30. In a simple one-constraint Lagrange multiplier setup, the constraint has to be always one
dimension lesser than the objective function.
a) True
b) False
Answer: b
Explanation: This condition is not always necessary because the lesser dimension curve can
still be treated as a higher dimension curve.
31. Which of the following is the representation of a plane algebraic curve of nth degree?
a) f(x,y)=ayn+(bx+c) yn-1+(dx2+ex+f) yn-2+⋯+un (x)=0
b) f(x,y)=ayn-1+(bx+c) yn-2+(dx2+ex+f) yn-3+⋯+un (x)=0
c) f(x,y)=ayn+byn-1+cyn-2+⋯+un (x)=0
d) f(x,y)=ayn+(bx+c) yn+(dx2+ex+f) yn+⋯+un (x)=0
Answer: a
Explanation: Plane algebraic curve of nth degree is represented by,
f(x,y)=ayn+(bx+c) yn-1+(dx2+ex+f) yn-2+⋯+un (x)=0
Where a, b, c, d, e, f are all constants and un(x) is a polynomial in x of degree n.
32. What are the tangents to the curve x3+ y3=3axy at the origin?
a) x=0
b) x=0, y=0
c) y=0
d) x=y
Answer: b
Explanation: Given: x3+ y3=3axy
To find the tangent to the curve at the origin, we need to equate the lowest degree term to 0.
Therefore, 3axy=0, which gives x=0 and y=0 as two tangents to the curve at origin.
33. Which of the following represents the correct cartesian equation for Lemniscate of Bernoulli?
a) (x2+y2)2=2a2 (x2-y2)
b) (x2+y)2=2a2 (x2-y2)
c) (x2+y2)2=2(x2-y2)
d) (x2+y2)2=a2 (x2-y2)
Answer: a
Explanation: The Lemniscate of Bernoulli was first described as a modification of an ellipse in
1694 by Jakob Bernoulli. It is represented as,
Cartesian Equation: (x2+y2)2=2a2 (x2-y2)
Polar Equation: r2=2a2*cos2θ
34. What are the tangents to the curve y2=4ax at the origin?
a) x=0
b) x=0, y=0
c) y=0
d) x=y
Answer: a
Explanation: Given: y2=4ax
To find the tangent to the curve at the origin, we need to equate the lowest degree term to 0.
Therefore, 4ax=0, which gives x=0.
35. Which of the following is not correct regarding Folium of Descartes?
a) The equation is third degree in both x and y
b) Its asymptote is given by x+y+a=0
c) It is symmetrical about y = x
d) It forms a loop in the second quadrant
36. What is the order of the differential equation, y”+y’-x3y=sinx?
a) 2
b) 1
c) 0
d) 3
Answer: a
Explanation: Order of a differential equation is given by the highest order derivative appearing
in the differential equation. Hence for the given equation, y”+y’-x3y=sinx, the order is 2.
37. Which of the following is not a standard method for finding the solutions for differential
equations?
a) Variable Separable
b) Homogenous Equation
c) Bernoulli’s Equation
d) Orthogonal Method
38. With the help of _____________________ Mr.Melin gave inverse laplace transformation
formula.
a) Theory of calculus
b) Theory of probability
c) Theory of statistics
d) Theory of residues
39. Write the expression for spur of a matrix for a 3×3 matrix whose entries are in the form of a ij.
a) a11+ a12+a13
b) a11+ a21+a31
c) a12+ a22+a32
d) a11+ a22+a33
40. Which among the following is not a type of quadratic form?
a) Positive Semi-definite
b) Negative definite
c) Partial definite
d) Indefinite
41. Which of the following is true for matrices?
a) (AB)-1 = B-1A-1
b) (AT) = A
c) AB = BA
d) A*I = I
Answer: a
Explanation: The correct forms of the other options are:
(AT)T = A
AB ≠ BA
A*I = A
42. The determinant of the matrix whose eigen values are 7, 1, 9 is given by ________
a) 7
b) 63
c) 9
d) 17
Answer: b
Explanation: The product of the eigen values of a matrix gives the determinant of the matrix,
Therefore, ∆ = 63.
43. Del operator is also known as _________
a) Divergence operator
b) Gradient operator
c) Curl operator
d) Laplacian operator
44. If the function f(x) is even, then which of the following is zero?
a) an
b) bn
c) a0
d) nothing is zero
Answer: b
Explanation: Since bn includes sin(nx) term which is an odd function, odd times even function
is always odd. So, the integral gives zero as the result.
45. Which of the following is an example for first order linear partial differential equation?
a) Lagrange’s Partial Differential Equation
b) Clairaut’s Partial Differential Equation
c) One-dimensional Wave Equation
d) One-dimensional Heat Equation
Answer: a
Explanation: Equations of the form Pp + Qq = R , where P, Q and R are functions of x, y, z, are
known as Lagrange’s linear equation.
46. Which of the following is an example of non-linear differential equation?
a) y=mx+c
b) x+x’=0
c) x+x2=0
d) x”+2x=0
Answer: c
Explanation: For a differential equation to be linear the dependent variable should be of first
degree. Since in equation x+x2=0, x2 is not a first power, it is not an example of linear
differential equation.
47. Which of the following is the condition for a second order partial differential equation to be
hyperbolic?
a) b2-ac<0
b) b2-ac=0
c) b2-ac>0
d) b2-ac=<0
Answer: c
Explanation: For a second order partial differential equation to be hyperbolic, the equation
should satisfy the condition, b2-ac>0.
48. Which of the following is an iterative method?
a) Gauss Seidel
b) Gauss Jordan
c) Factorization
d) Gauss Elimination
49. Which of the methods is a direct method for solving simultaneous algebraic equations?
a) Relaxation method
b) Gauss seidel method
c) Jacobi’s method
d) Cramer’s rule
50. What is the other name for factorization method?
a) Muller’s Method
b) Decomposition Method
c) Lin Bairstow Method
d) Doolittle’s Method