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MATHEMATICS- PAGE 2

1:-The area of a triangle is equal to that of a square whose side measures 60 m. The side of the triangle whose corresponding altitude is 90 m is
 A:-60 m
 B:-40 m
 C:-80 m
 D:-90 m
Ans: C

2:-The height of an arc of a circle is 10 cm and its diameter is 12.5 cm. The chord of the arc is of length 
 A:-10 cm
 B:-12 cm
 C:-8 cm
 D:-11 cm
Ans: A

3:-`A` is an `nxxn` matrix, `K` is a scalar and |`KA`|= `r` |`A`|, then the value of `r` is
 A:-`Kn`
 B:-`n^K`
 C:-`K^n`
 D:-`(K)/(n)`
Ans: C

4:-For the function `f(x) = x^2 e^-x`, the maximum occurs when `x` is equal to
 A:-2
 B:-1
 C:-0
 D:-`(1)/(2)`
Ans: A

5:-The value of `int_0^{pi/4} sec^4(x)dx` is
 A:-`(3)/(4)`
 B:-`(4)/(3)`
 C:-`(pi)/(3)`
 D:-`(pi)/(4)`
Ans: B

6:-For the differential equation `(dy)/(dx)+5y = 0` ; with `y(0) = 1`, the general solution is
 A:-`e^5^x`
 B:-`5e^-5^x`
 C:-`-5e^-5^x`
 D:-`e^-5^x`
Ans: D

7:-The area bounded by the curve `y = log x`, the `x`-axis and the ordinates at `x=1` and `x = 2` is
 A:-`log 4 - 1`
 B:-`log 4-3`
 C:-`log 2- 2`
 D:-`2 log 2 + 1`
Ans: A

8:-The coefficients of three consecutive terms in the expansion of `(1 + x)^n` are in the ratio 1 : 3 : 5, then the value of `n` is
 A:-3
 B:-5
 C:-7
 D:-9
Ans: C

9:-Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points `(k, 24)` and (8, 12) ; then the value of `k` is
 A:--2
 B:-4
 C:-8
 D:-24
Ans: B

10:-The value of `(sin x - sin y)/(cos x + cos y)` is
 A:-`tan ((x-y)/(2))`
 B:-`tan ((x+y)/(2))`
 C:-`cot ((x-y)/(2))`
 D:-`cot ((x+y)/(2))`
Ans: A

11:-If the line `y = 3x + c` is a tangent to the curve `y = 2x^2` , then the value of `c` is
 A:-`(3)/(4)`
 B:-`(9)/(8)`
 C:-`(-3)/(4)`
 D:-`(-9)/(8)`
Ans: D

12:-The solution of the following simultaneous equations `5x + 3y + 3z = 48;` `2x + 6y - 3z = 18` ; `8x - 3y + 2z = 21 ` is
 A:-`x = 3, y = 6, z= 5`
 B:-`x = 3, y = 5, z = 6`
 C:-`x = 6, y = 3, z = 5`
 D:-`x = 5, y = 6, z = 3`
Ans: B

13:-A sector with radius 18 cm and centre angle 60°. While folding to make a cone, what will be the radius of that cone?
 A:-3 cm
 B:-9 cm
 C:-30 cm
 D:-36 cm
Ans: A

14:-If `quad x_n = (n+1)^2 - (n-1)^2` , what is its `quad x_2` ?
 A:-10
 B:-12
 C:-20
 D:-8
Ans: D

15:-Find the half of `2^(14)`.
 A:-`2^(7)`
 B:-`1^(7)`
 C:-`7^(2)`
 D:-`2^(13)`
Ans: D

16:-The sides of a triangle are 10 cm, 13 cm, 13 cm. What is its area?
 A:-`60 text{cm}^(2)`
 B:-`36 text{cm}^(2)`
 C:-`130 text{cm}^(2)`
 D:-`13 text{cm}^(2)`
Ans: A

17:-Sum of first '`quad n`' terms of an arithmetic sequence is `quad 2 n^2 + 3n` , what is its second term?
 A:-9
 B:-8
 C:-10
 D:-4
Ans: A

18:-`quad x^2 - x - 6 = 0` , what are the two factors of this equation?
 A:-`quad (x+2) (x+ 3)`
 B:-`quad (x-3) (x+2)`
 C:-`quad (x-2) (x-3)`
 D:-`quad (x+3) (x-2)`
Ans: B

19:-An equilateral triangle with one side 8 cm. What is the radius of its incircle?
 A:-4
 B:-6
 C:-`4/sqrt(3)`
 D:-2
Ans: C

20:-In a right angled triangle, small sides are 3.5 cm; 2.5 cm; what is its hypotenuse?
 A:-6
 B:-4.3
 C:-3.5
 D:-5.5
Ans: B
21:-Two lines are intersect in a point. How many angles are formed?
 A:-2
 B:-4
 C:-8
 D:-10
Ans: B

22:-How many edges are in a square pyramid?
 A:-4
 B:-12
 C:-8
 D:-5
Ans: C

23:-A solid body with length, breadth and height are equal, what is its name?
 A:-Rectangle
 B:-Circle
 C:-Cube
 D:-Parallelogram
Ans: C

24:-6, 10, 12, 15, 16, 18, 21. What is its Arithmetic Mean?
 A:-12
 B:-14
 C:-15
 D:-18
Ans: B

25:-What is the total surface area of a sphere with its radius 2.5 cm?
 A:-`6.25 Pi text{cm}^(2)`
 B:-`2.5 Pi text{cm}^2`
 C:-`25 Pi text{cm}^2`
 D:-`3.14 text{cm}^(2)`
Ans: C

26:-What is sin 45°?
 A:-`sqrt(2)`
 B:-1
 C:-`sqrt(3)`
 D:-`1/sqrt(2)`
Ans: D

27:-In a parallelogram one side is 15 cm and distance from this side to its opposite side is 9 cm, what is its area?
 A:-`135 text{cm}^(2)`
 B:-`67.5 text{cm}^(2)`
 C:-`80 text{cm}^(2)`
 D:-`24 text{cm}^(2)`
Ans: A

28:-Find the value of `quad 0.05 + 1 1/2+2.45`
 A:-4
 B:-5
 C:-4.05
 D:-3.45
Ans: A

29:-Find the value of `quad 2 8/11 + 3 4/12`
 A:-`quad 5 12/23`
 B:-`200/33`
 C:-`100/33`
 D:-`90/43`
Ans: B

30:-The trigonometric term 'Tangent' is originate from
 A:-French
 B:-Greek
 C:-Latin
 D:-Arabic
Ans: C

31:-If `quad 5^2 xx 5^4 xx 5^6 xx ... xx 5^{2n} = (0.04)^{-28}, text {what is the value of } 'n'?`
 A:-8
 B:-7
 C:-14
 D:-12
Ans: B

32:-30°, 60°, 90° are the angles of a triangle; opposite side of 30° is two metre, what is the length of side opposite to 60°?
 A:-`sqrt(3)`
 B:-`2/sqrt(3)`
 C:-`2 sqrt(3)`
 D:-`1/sqrt(2)`
Ans: C
33:-The inverse of `[[2,5], [1,3]]` is
 A:-`[[3,-1],[-5,2]]`
 B:-`[[3,-5],[-1,2]]`
 C:-`[[-3,1],[5,-2]]`
 D:-`[[3,1],[5,2]]`
Ans: B

34:-If `A=[[12,17,23],[3,6,9],[12,24,36]]` , then `det A` is
 A:-74
 B:--74
 C:-`(1)/(74)`
 D:-0
Ans: D

35:-`sum_(r=0)^n nC_r =......`
 A:-1
 B:-0
 C:-`2^(n)`
 D:-`n^(2)`
Ans: C

36:-sin`(180 + theta) = ......`
 A:-sin`theta`
 B:--sin`theta`
 C:-cos`theta`
 D:--cos`theta`
Ans: B

37:-The equation of any line perpendicular to `Ax + By + C = 0` is
 A:-`Bx - Ay + K = 0`
 B:-`Ax - By + K = 0`
 C:-`Bx + Ay + K = 0`
 D:-`Ax + By + K = 0`
Ans: A

38:-`Lt_(x->(pi)/(4)) (log tan x)/(((pi)/(4) - x)) = ......`
 A:-0
 B:-`(1)/(2)`
 C:--2
 D:-`-(1)/(2)`
Ans: C

39:-Find `(dy)/(dx)` if `y = sqrt(f (x) + sqrt(f (x) + sqrt(f (x) + ......))) `
 A:-`(f' (x))/(2y-1)`
 B:-`(2y-1)/(f'(x))`
 C:-`(f'(x))/(2y+1)`
 D:-`(2y+1)/(f'(x))`
Ans: A

40:-`int_-pi^Pi x^(3) cos xdx = ......`
 A:-`2pi`
 B:-`-(pi)/(2)`
 C:-`+(pi)/(2)`
 D:-0
Ans: D

41:-`int (dx)/(sqrt(a^2-x^2)) = ...... `
 A:-`sin^-1 ((x)/(a)) + C`
 B:-`cos^-1 ((x)/(a)) + C`
 C:-`sin^-1 ((a)/(x)) + C`
 D:-`cos^-1 ((a)/(x)) + C `
Ans: A
​
42:-Integrating factor for the differential equation `(dy)/(dx) + (3y)/(x) = (1)/(x^2)` is
 A:-`y^3`
 B:-`e^{y^ 3}`
 C:-`x^3`
 D:-`e^{x^3}`
Ans: C
43:-If the sum of the distances of a point in the plane from two fixed perpendicular lines is l, then the locus of the point is :
 A:-Ellipse
 B:-Hyperbola
 C:-Square
 D:-Parabola
Ans: C

44:-The quadratic equation whose roots are the squares of the roots of the equation `2x^2-2x+5=0` is :
 A:-`4x^2+4x+25=0`
 B:-`4x^2+16x+25=0`
 C:-`4x^2-4x-25=0`
 D:-`16x^2-4x-25=0`
Ans: B

45:-The product of the eigen values of the matrix `A = [[1, 2],[-1, 3]]` is :
 A:-5
 B:-1
 C:-4
 D:-None of these
Ans: A

46:-If `f (x) = (ln (sin 2x))^2` then f' (x) is :
 A:-4 cot 2x
 B:-2 tan 2x
 C:-2 cot 2x. ln(sin 2x)
 D:-None of these
Ans: D

47:-The ratio of the diameter of a sphere to the height of the right circular cone having the greatest volume which can be inscribed in the sphere is :
 A:-3 : 2
 B:-`3 : 2 sqrt(2)`
 C:-4 : 3
 D:-2 : 1
Ans: A

48:-The dimension of the dual space of the vector space of 2`xx`2 matrices with complex entries, over the field of real numbers is :
 A:-3
 B:-8
 C:-0
 D:-`-1`
Ans: B

49:-If f is a non-zero linear functional defined on a three dimensional vector space X, then the dimension of the null space of f is :
 A:-3
 B:-2
 C:-1
 D:-None of these
Ans: B

50:-Which of the following is not a Hilbert space ?
 A:-C [0, 1]
 B:-`L^2 [0, 1]`
 C:-`l^2`
 D:-None of these
Ans: A

51:-Let `{x_n}` be a convergent sequence in R, the set of real numbers. Then the number of limit points of `{x_n}` is :
 A:-Exactly one
 B:-Exactly one if R is under usual topology
 C:-There is no limit for limit points
 D:-Cannot say
Ans: B

52:-For what values of 'a' the function :
`f (x)=x^2+a, x < 2`
`= 2a - x^2, x>=2` is continuous at every x.
 A:-a = 0
 B:-a = 2
 C:-a = 8
 D:-a = 4
Ans: C

53:-If a real valued function f is defined and continuous on a closed and bounded set F of real numbers, then :
 A:-f is a constant
 B:-f is not differentiable on F
 C:-f is measurable
 D:-f is uniformly continuous on F
Ans: D

54:-Let f be a family of measurable real functions. Let G be the set of limit points of sequences of real functions in f. Then G is :
 A:-measurable
 B:-non-measurable
 C:-countable
 D:-none of the above
Ans: A

55:-Let f : R`->`R is differentiable for every x `epsi` R, such that `lim_(x->oo)` f'(x) = . Then `lim_(x->oo)` `f(x)/x` is :
 A:-0
 B:-
 C:-
 D:-`oo`
Ans: C

56:-Let t be a positive integer and define a sequence `{x_n}` by `x_(n+1)=t+` for all n`>=`0 with `x_0=0`. If the sequence is convergent, then :
 A:-`t = 1/4`
 B:-`t > 1/4`
 C:-`t < 1/4`
 D:-`t < (-1)/(4)`
Ans: A

57:-Find the number of continuous onto maps from [0, 1] to (0, 1).
 A:-Countably many
 B:-None
 C:-Exactly one
 D:-Uncountably many
Ans: B

58:-Let `f (x) = a_1` sin `x + a_2` sin `2x +...+ a_n` sin nx such that |f (x)|`<=`| sin x| `AA`x`epsi`R. Then `|a_1+2a_2+...+ na_n|` is :
 A:-cannot find
 B:-0
 C:-greater than `(n (n+1))/2` ``
 D:-less than or equal to 1
Ans: D

59:-Let `{S_n}` be a sequence of real numbers defined by `S_1=sqrt(2)` and `S_(n+1)=sqrt(2+S_n),` then `{S_n}` is :
 A:-Monotonically increasing for all n
 B:-Monotonically decreasing for all n
 C:-Monotonically increasing for even n and decreasing for odd n
 D:-Neither increasing nor-decreasing
Ans: A

60:-Let f be a function on real line such that |f| is measurable. Then :
 A:-f is measurable if it is monotonically increasing
 B:-f is always measurable
 C:-f need not be measurable
 D:-none of the above
Ans: C

61:-The value of `int_C^``1/z^2 dz,` where C is the curve `|z - i| = 1/2,` is :
 A:-`pi`i
 B:-0
 C:-2`pi`i
 D:-`- 1/pi^3`
Ans: B

62:-Which of the following is a bounded complex valued function :
 A:-f (z) = sin z, in the complex plane
 B:-`f (z) = e^z/z,` in the complex plane with |z| > 0
 C:-f (z) = cosh z, in the complex plane
 D:-None of the above (A), (B) and (C)
Ans: D

63:-The function `g(z) = (z)/(e^z-1), z!= 0,` has :
 A:-Essential singularity at z = 1
 B:-Pole at the origin
 C:-Essential singularity at the origin
 D:-Removable singularity at the origin
Ans: D

64:-Which of the following is the value of :
 A:-1
 B:-2`Pi`i
 C:-8
 D:-10
Ans: C

65:-The number of roots of `z^4-6z+3=0,` with absolute value less than 1 is :
 A:-4
 B:-1
 C:-0
 D:-3
Ans: B

66:-If R is the set of real numbers, then which of the following is a metric on R`xx`R.
For `X = (x_1, y_1) and Y = (x_2, y_2)` in R`xx`R.
 A:-d (X, Y) = Max `{ |x_1-y_1|, |x_2-y_2|}`
 B:-d (X, Y) = `|x_1-x_2|`
 C:-d (X, Y) = `sqrt ((x_1-y_1)^2+(x_2-y_2)^2)`
 D:-d (X, Y) = `(sqrt((x_1-x_2)^2+(y_1-y_2)^2))/(1+sqrt((x_1-x_2)^2+(y_1-y_2)^2))`
Ans: D

67:-If (X, d) is an arbitrary metric space and E is a subset of X, choose the false statement from the following.
 A:-Every interior point of E is a limit point of E
 B:-The largest open set contained in E can be a closed set
 C:-The closure of E can be an open set
 D:-Every limit point of E is the limit of a sequence in E
Ans: A

68:-Let R be the set of real numbers and be the semi open interval topology on R. Then, which of the following is true for (R, ).
 A:-(R, ) is a second countable space.
 B:-(R, ) is a metrizable space.
 C:-(R, ) is a separable space.
 D:-(R, ) is a compact space.
Ans: C

69:-Choose the correct statement from the following :
 A:-[0, 1) is homeomorphic to the unit circle `S^1,` both have subspace topologies of the usual topologies of R and R `xx`R respectively.
 B:-{[a, b] : a `epsi` R, b `epsi` R, a < b} is a base for a topology on R.
 C:-The projection maps from a product space to the coordinate spaces are always closed.
 D:-The torus surface is a quotient space of the unit square.
Ans: D

70:-Let X be an uncountable set and be the co-finite topology on X. Then, choose the correct one from the following :
 A:-(X, ) is a connected, `T_1` - space which is metrizable.
 B:-(X, ) is a second countable `T_2` - space which is compact
 C:-(X, ) is a separable compact space which is not metrizable
 D:-(X, ) is a second countable `T_1` - space which is metrizable
Ans: C

71:-From the following given operations on the set of positive integers, choose the one which is a binary operation :
 A:-a*b = `(ab)/(2)`
 B:-a*b=ab` - `a
 C:-a*b=ab+b
 D:-a*b=a`-`b
Ans: C

72:-Let G = (V, E) be a simple graph. Choose the wrong statement from below :
 A:-G is bipartite implies G has a perfect matching
 B:-Any plane Graph has a dual graph
 C:-Two isomorphic graphs are having the same degree sequence
 D:-Any Hamiltonian graph is 2-connected
Ans: A

73:-Let (X, d) be a metric space and `(x_n)` be a sequence in X. Choose the correct statement from the following :
 A:-If `(x_n)` is a Cauchy sequence, then it converges.
 B:-If `(x_n)` converges in X, any subsequence converges to the same limit.
 C:-If `(x_n)` is bounded, then it has a convergent subsequence.
 D:-If `(x_n)` is unbounded, then it cannot have a convergent subsequence.
Ans: B

74:-Let `X = R^2`. From the following maps from X to X, choose the one which is a linear homeomorphism :
 A:-`F (x_1, x_2) = (x_1 - x_2, x_2 - x_1)`
 B:-`F (x_1, x_2) = (2x_1 - x_2, 0)`
 C:-`F (x_1, x_2) = (3x_1, 4x_1)`
 D:-`F (x_1, x_2) = (x_1-x_2, x_1+x_2)`
Ans: D

75:-Let E = { x `epsi` R : x is rational, 0 `<=` x `<=` 2}. The Lebesgue measure of E is :
 A:-2
 B:-1
 C:-not Lebesgue measurable
 D:-0
Ans: D

76:-Among the following equations, choose the exact differential equation :
 A:-`12x^2ydx+4x^3dy=0`
 B:-`(x+y)dx+3x^2ydy=0`
 C:-`(dx)/(x) + (dy)/(xy^2) = 0`
 D:-`(4x^2+3y)dx+6x^2y^2dy=0`
Ans: A

77:-Let X = (`C^2 `, `||.||_1`). Let A`in`B L (X) be defined by A`(x_1, x_2)=(2x_2, x_1).` Then ||A|| is :
 A:-3
 B:-`3/2`
 C:-2
 D:-1
Ans: C

78:-Let `mu` denote the Mobius function then `mu`(75) is :
 A:-1
 B:-0
 C:-15
 D:-`-1`
Ans: B

79:-The point on the plane curve `f (x) = - ` `x^3+3x+9x - 12` where the slope of the tangent is maximum is :
 A:-`(2,-12)`
 B:-(1, 1)
 C:-`(1, -1)`
 D:-None of these
Ans: D

80:-Let `X = R^2,` Y = {(x, 0) : x `epsi` R} where, `||.||_1` is defined on X. Let g : Y `->` R be defined by g((x,0))=x. From the following choose a Hahn-Banach extension of g :
 A:-`f (x_1, x_2) = x_1 + 2x_2`
 B:-`f (x_1, x_2) = (x_1 + x_2)/(2)`
 C:-`f (x_1, x_2) = x_1 - x_2`
 D:-`f (x_1, x_2) = 3x_1 + 4x_2`
Ans: C

81:-Which of the following is not a solution of the two-dimensional Laplace equation :
 A:-`u=e^xsiny`
 B:-`u=x^2-y^2`
 C:-`u=x^2+y^2`
 D:-`u = log (x^2+y^2)`
Ans: C

82:-A partial differential equation of the form `=F(x, y, u, u_x, u_y)` is said to be elliptic if :
 A:-`ac-b^2=0`
 B:-`ac-b^2 > 0`
 C:-`ac-b^2 < 0`
 D:-a+c=2b
Ans: B

83:-General Integral of `yzz_x+xzz_y=xy` is :
 A:-`F (x^2-y^2, z^2-y^2) = 0`
 B:-
 C:-F `(yz, xz)=0`
 D:-
Ans: A

84:-Which of the following function is nowhere analytic :
 A:-`f(z)=3x+y+i` `(3y-x)`
 B:-`f(z)=2xy+i` `(x^2-y^2)` ``
 C:-`f(z)=x^2-y^2+i` `2xy`
 D:-`f(z)=cosh` `x` `cosy+i` `sinh` `x` `siny`
Ans: B

85:-If atleast two elements a, b other than identity element are such that 0(a) = finite and 0(b) = `oo`, then the group is called :
 A:-Torsion free group
 B:-Torsion group
 C:-Mixed group
 D:-P-group
Ans: C

86:-Which of the following pair of groups are isomorphic ?
 A:-`Z_(24) and Z_8 o+ Z_(3)`
 B:-`Z_(25) and Z_5 o+ Z_5`
 C:-`Z_(24) and Z_5 o+ Z_5`
 D:-`Z_(20) and Z_2 o+ Z_10`
Ans: A

87:-Let `phi`(x)=x be an iteration function for solving the equation f(x)=0 by fixed point iteration method. The iterations converges to a root, if :
 A:-|`phi'``(x_k)`|>1, for k=0, 1, 2, . . .
 B:-|`phi'``(x_k)` |<1, for k=0, 1, 2, . . .
 C:-|`phi` `(x_(k+1))` |`<=`|`phi'` `(x_k)`|, for k=0, 1, 2, . . .
 D:-|`phi''` `(x_k)`|`<=``1/2` (|`phi'` `(x_k)`| + | `phi` `(x_k)`|), for k=0, 1, 2, . . .
Ans: B

88:-Let `x_0` be an initial approximation to the root of an algebraic equation f(x)=0 and let `x_k` be the `k^(th)` iterate obtained in Newton-Rapson method. Then for k=1, 2, 3, . . .
 A:-`x_(k+1) = x_k` + `f (x_k)/(f' (x_k)` `(x_k-x_0)`
 B:-`x_(k+1) = x_k` + `(f' (x_k))/(f (x_k))` `(x_k-x_0)`
 C:-`x_(k+1) = x_k` - `(f' (x_k))/(f (x_k))`
 D:-`x_(k+1) = x_k` - `(f (x_k))/(f' (x_k))`
Ans: D

89:-What is the largest possible value for the rank of a 2`xx`5 matrix ?
 A:-10
 B:-5
 C:-3
 D:-None of these
Ans: D

90:-Approximate value of f (5) by Lagrange's interpolation polynomial using the following data :
 A:-18
 B:-23.5
 C:-4.5
 D:-47
Ans: A

91:-A Linear transformation T on `l^2 ` `(Z_N)` = {z(1), z(2),..... z(N)/z(i) ε C, 1 ≤ i ≤ N} is a Fourier multiplier operator iff the matrix of T in the Fourier basis is __________.
 A:-Hermitian
 B:-Skew Hermitian
 C:-Orthogonal
 D:-Diagonal
Ans: D

92:-Which of the following is an elliptic function ?
 A:-Weirstrass ρ function
 B:-Weirstrass `sigma` function
 C:-Weirstrass Zeta function
 D:-`e^z`
Ans: A

93:-Domain of convergence of Riemann Zeta function is __________.
 A:-Re z > 0
 B:-Re z > 1
 C:-|z| > 1
 D:-z > 1
Ans: B

94:-The relative cardinality of the Fuzzy set A={(a, 0.1), (b, 0.5), (c, 0.9)} is __________.
 A:-1.5
 B:-0.5
 C:-1
 D:-0
Ans: B

95:-Tychnoff's theorem states that :
 A:-Arbitrary product of compact spaces is compact in the product topology
 B:-Arbitrary product of hausdorff spaces is hausdorff in the product topology
 C:-Arbitrary product of regular spaces is regular in the product topology
 D:-Arbitrary product of normal spaces need not be normal in the product topology
Ans: A

96:-The language generated by the Grammar G = ({S}, Σ ={a, b} S, {S `->` aSb |λ}) is __________.
 A:-`{a^nb^n, n>= 1}`
 B:-`{b^na^n, n >= 1}`
 C:-`{a^nb^n, n>= 0}`
 D:-`{b^na^n, n>= 0}`
Ans: C

97:-Which of the following is true for the following differential equation ?
`(1 - x^2) y'' - 2xy' + 2y = 0`
 A:-x = `oo` is an ordinary point
 B:-x = `oo` is a regular singular point
 C:-x = `oo` is an irregular singular point
 D:-none of these
Ans: B

98:-The integral curve through (1, - 1) of the gradient field of f (x, y) = `(x^2+y^2)/2` is __________.
 A:-`(e^t, e^t)`
 B:-`(- e^t, e^t)`
 C:-`(- e^t, - e^t)`
 D:-`(e^t, - e^t)`
Ans: D

99:-The integral equation corresponding to the initial value problem `y'' + xy' + (1 - x^2) y=f(x),` y(0)=0, y'(0)=0 is a __________.
 A:-Volterra equation of first kind
 B:-Fredholm equation of first kind
 C:-Volterra equation of second kind
 D:-Fredholm equation of second kind
Ans: C

100:-If a source of strength m is at the origin, the corresponding stream function is ψ=
 A:-- mθ
 B:-mθ
 C:-θ/m
 D:-- θ/m
Ans: A
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