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
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
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
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
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