MATHEMATICS- PAGE 1
1. Average of first 20 natural numbers is
(a). 190
(b). 200
(c). 210
(d). 220
Answer: Option-C
2. What is the perimeter of a rectangle whose area is 500 sq.cm and its breadth is 10 cm ?
(a). 110 cm
(b). 120 cm
(c). 126 cm
(d). 130 cm
Answer: Option-B
3. Two numbers are in the ratio 3:4. If sum of the numbers is 140, then numbers are
(a). 50 and 90
(b). 65 and 75
(c). 60 and 80
(d). 55 and 85
Answer: Option-C
4. A man completes 3 /8 of a job in 6 days. At this rate, how many more days will it take him to finish the job ?
(a). 7
(b). 8
(c). 9
(d). 10
Answer: Option-D
5. What percent of 1/ 9 is 1/ 6 ?
(a). 140 %
(b). 150 %
(c). 160 %
(d). 170 %
Answer: Option-B
6. By selling an article for Rs.100, a person gets Rs.10. Then his gain in percent is
(a). 10 %
(b). 11 %
(c). 111 9 %
(d). 112 9 %
Answer: Option-C
7. At what rate per annum will a sum of money double in 5 years ?
(a). 15 %
(b). 20 %
(c). 22.5 %
(d). 25 %
Answer: Option-B
8. Babu can cover a certain distance in 1hr 30 min by covering half of the distance at 3 kmh and the rest at 6 kmh. What is the total distance ?
(a). 6 km
(b). 7 km
(c). 7.5 km
(d). 8 km
Answer: Option-A
9. A man completes 3 /8 of a job in 6 days. At this rate, how many more days will it take him to finish the job ?
(a). 7
(b). 8
(c). 9
(d). 10
Answer: Option-D
10. Average of first 20 natural numbers is
(a). 190
(b). 200
(c). 210
(d). 220
Answer: Option-C
11. What is the perimeter of a rectangle whose area is 500 sq.cm and its breadth is 10 cm ?
(a). 110 cm
(b). 120 cm
(c). 126 cm
(d). 130 cm
Answer: Option-B
12. Two numbers are in the ratio 3:4. If sum of the numbers is 140, then numbers are
(a). 50 and 90
(b). 65 and 75
(c). 60 and 80
(d). 55 and 85
Answer: Option-C
13. If 40% of 3 /8 of a number 12, then the number is
(a). 80
(b). 70
(c). 60
(d). 50
Answer: Option-A
14. What is the difference between the biggest and smallest fractions among 1/2, 2/3, 7/12, 5/6?
(a). 1 /6
(b). 1/ 5
(c). 1/ 4
(d). 1/3
Answer: Option-D
15. LCM of three different numbers is 120. Which of the following cannot be their HCF?
(a) 8
(b) 24
(c) 12
(d) 35
Answer: 35
16. Arithmetic mean of first ten odd numbers will be
1) 10
2) 15
3) 20
4) 5
Answer: 1
17. Probability of throwing an even number with a die will be
1) 1/6
2) 1/2
3) 1/3
4) 1/4
Answer: 2
18. On the basis of number of variables how many types of correlations are recognised?
1) 1
2) 2
3) 3
4) 4
Answer: 4
19. John Napier is an inventor of :
1) Algorithm
2) Logarithm
3) Algebra
4) Trignometry
Ans: 2
20. In which of the following numbers all zeros are significant ?
1) 0.0005
2) 0.0500
3) 50.000
4) 0.0050
Ans: 3
(a). 190
(b). 200
(c). 210
(d). 220
Answer: Option-C
2. What is the perimeter of a rectangle whose area is 500 sq.cm and its breadth is 10 cm ?
(a). 110 cm
(b). 120 cm
(c). 126 cm
(d). 130 cm
Answer: Option-B
3. Two numbers are in the ratio 3:4. If sum of the numbers is 140, then numbers are
(a). 50 and 90
(b). 65 and 75
(c). 60 and 80
(d). 55 and 85
Answer: Option-C
4. A man completes 3 /8 of a job in 6 days. At this rate, how many more days will it take him to finish the job ?
(a). 7
(b). 8
(c). 9
(d). 10
Answer: Option-D
5. What percent of 1/ 9 is 1/ 6 ?
(a). 140 %
(b). 150 %
(c). 160 %
(d). 170 %
Answer: Option-B
6. By selling an article for Rs.100, a person gets Rs.10. Then his gain in percent is
(a). 10 %
(b). 11 %
(c). 111 9 %
(d). 112 9 %
Answer: Option-C
7. At what rate per annum will a sum of money double in 5 years ?
(a). 15 %
(b). 20 %
(c). 22.5 %
(d). 25 %
Answer: Option-B
8. Babu can cover a certain distance in 1hr 30 min by covering half of the distance at 3 kmh and the rest at 6 kmh. What is the total distance ?
(a). 6 km
(b). 7 km
(c). 7.5 km
(d). 8 km
Answer: Option-A
9. A man completes 3 /8 of a job in 6 days. At this rate, how many more days will it take him to finish the job ?
(a). 7
(b). 8
(c). 9
(d). 10
Answer: Option-D
10. Average of first 20 natural numbers is
(a). 190
(b). 200
(c). 210
(d). 220
Answer: Option-C
11. What is the perimeter of a rectangle whose area is 500 sq.cm and its breadth is 10 cm ?
(a). 110 cm
(b). 120 cm
(c). 126 cm
(d). 130 cm
Answer: Option-B
12. Two numbers are in the ratio 3:4. If sum of the numbers is 140, then numbers are
(a). 50 and 90
(b). 65 and 75
(c). 60 and 80
(d). 55 and 85
Answer: Option-C
13. If 40% of 3 /8 of a number 12, then the number is
(a). 80
(b). 70
(c). 60
(d). 50
Answer: Option-A
14. What is the difference between the biggest and smallest fractions among 1/2, 2/3, 7/12, 5/6?
(a). 1 /6
(b). 1/ 5
(c). 1/ 4
(d). 1/3
Answer: Option-D
15. LCM of three different numbers is 120. Which of the following cannot be their HCF?
(a) 8
(b) 24
(c) 12
(d) 35
Answer: 35
16. Arithmetic mean of first ten odd numbers will be
1) 10
2) 15
3) 20
4) 5
Answer: 1
17. Probability of throwing an even number with a die will be
1) 1/6
2) 1/2
3) 1/3
4) 1/4
Answer: 2
18. On the basis of number of variables how many types of correlations are recognised?
1) 1
2) 2
3) 3
4) 4
Answer: 4
19. John Napier is an inventor of :
1) Algorithm
2) Logarithm
3) Algebra
4) Trignometry
Ans: 2
20. In which of the following numbers all zeros are significant ?
1) 0.0005
2) 0.0500
3) 50.000
4) 0.0050
Ans: 3
21. The chord of a circle is ___________.
A:-A straight line joining the end of an arc
B:-A part of diameter
C:-Circle contained by two radius
D:-A straight line touches a circle in one point
Correct Answer:- Option-A
22. A four side figure in which all the sides equal but the angles are not the right angles
A:-Square
B:-Trapezium
C:-Rectangle
D:-Rhombus
Correct Answer:- Option-D
A:-A straight line joining the end of an arc
B:-A part of diameter
C:-Circle contained by two radius
D:-A straight line touches a circle in one point
Correct Answer:- Option-A
22. A four side figure in which all the sides equal but the angles are not the right angles
A:-Square
B:-Trapezium
C:-Rectangle
D:-Rhombus
Correct Answer:- Option-D
23. A unit vector perpendicular to the surface x 2 +y2 +z2 =3 at the point(1,1,1) is
A:-(i+j+k)/√3
B:-(i+j+k)/2
C:-(i+2j+3k)/√3
D:-(i+j+k)/√2
Correct Answer:- Option-A
24. If B = Curl A, the value of ∫B.ds over a closed surface S is
A:-4Π
B:-A
C:-B
D:-0
Correct Answer:- Option-D
25. Number of independent components of an antisymmetric tensor of rank 2 in 4-dimension is
A:-16
B:-6
C:-4
D:-8
Correct Answer:- Option-B
26. If square of a Hermitian matrix is a unit matrix,then its eigen values are
A:-(0,1)
B:-(2,-2)
C:-(1,-1)
D:-(0,2)
Correct Answer:- Option-C
27. The trace of metric tensor for Minkowski space is
A:-2
B:-1
C:--1
D:-0
Correct Answer:- Option-A
27. Gauss's theorem for a vector function A is
A:-∫A.ds = ∫CurlA dV
B:-∫A.ds = ∫DivA dV
C:-∫CurlA.ds = ∮ A . dl
D:-∫A.ds = ∮ A . dl
Correct Answer:- Option-B
A:-(i+j+k)/√3
B:-(i+j+k)/2
C:-(i+2j+3k)/√3
D:-(i+j+k)/√2
Correct Answer:- Option-A
24. If B = Curl A, the value of ∫B.ds over a closed surface S is
A:-4Π
B:-A
C:-B
D:-0
Correct Answer:- Option-D
25. Number of independent components of an antisymmetric tensor of rank 2 in 4-dimension is
A:-16
B:-6
C:-4
D:-8
Correct Answer:- Option-B
26. If square of a Hermitian matrix is a unit matrix,then its eigen values are
A:-(0,1)
B:-(2,-2)
C:-(1,-1)
D:-(0,2)
Correct Answer:- Option-C
27. The trace of metric tensor for Minkowski space is
A:-2
B:-1
C:--1
D:-0
Correct Answer:- Option-A
27. Gauss's theorem for a vector function A is
A:-∫A.ds = ∫CurlA dV
B:-∫A.ds = ∫DivA dV
C:-∫CurlA.ds = ∮ A . dl
D:-∫A.ds = ∮ A . dl
Correct Answer:- Option-B
28. The residue of `z/((z-a)(z-b))` at infinity is
A:-a/b
B:--b/a
C:-1
D:--1
Correct Answer:- Option-D
29. Which one of the following is a tensor of order zero, if A and B are vectors?
A:-A + B
B:-A - B
C:-A . B
D:-A x B
Correct Answer:- Option-C
30. Aij and Bij represent symmetric and anti symmetric real valued tensor respectively in three dimension. The number of independent components of Aij and Bij are
A:-3 and 6
B:-6 and 3
C:-6 and6
D:-9 and 6
Correct Answer:- Option-B
31. If F(s) is the Laplace transform of F (t) the Laplace transform of F (at) is
A:-`1/a` F(s)
B:-`1/a`F(s/a)
C:-F(s)
D:-F(s/a)
Correct Answer:- Option-B
32. The matrix `[[0,-1,0],[1,0,0],[0,0,1]]` is
A:-orthogonal
B:-hermitian
C:-anti symmetric
D:-None of the above
Correct Answer:- Option-A
A:-a/b
B:--b/a
C:-1
D:--1
Correct Answer:- Option-D
29. Which one of the following is a tensor of order zero, if A and B are vectors?
A:-A + B
B:-A - B
C:-A . B
D:-A x B
Correct Answer:- Option-C
30. Aij and Bij represent symmetric and anti symmetric real valued tensor respectively in three dimension. The number of independent components of Aij and Bij are
A:-3 and 6
B:-6 and 3
C:-6 and6
D:-9 and 6
Correct Answer:- Option-B
31. If F(s) is the Laplace transform of F (t) the Laplace transform of F (at) is
A:-`1/a` F(s)
B:-`1/a`F(s/a)
C:-F(s)
D:-F(s/a)
Correct Answer:- Option-B
32. The matrix `[[0,-1,0],[1,0,0],[0,0,1]]` is
A:-orthogonal
B:-hermitian
C:-anti symmetric
D:-None of the above
Correct Answer:- Option-A
33:-A sphere of radius 4 cm is carved from a homogeneous sphere of radius 8 cm and mass 160 g. The mass of the smaller sphere is
A:-80 g
B:-60 g
C:-40 g
D:-20 g
Ans: D
34:-A pendulum swings through an angle of 30° and describes an arc 8.8 cm in length. The length of the pendulum is (Use `Pi=` `(22)/(7))`
A:-8.8 cm
B:-16.8 cm
C:-12.4 cm
D:-10. 2 cm
Ans: B
35:-A solid cube is cut into two cuboids of equal volumes. The ratio of the total surface area of the given cube to that of one of the cuboids is
A:-2 : 1
B:-3 : 2
C:-4 : 1
D:-4 : 3
Ans: B
36:-What is the value of `(1)/(5+ (1)/(5 + (1)/(5+...)))?`
A:-`(-5 + sqrt(29))/(2)`
B:-`(-5 - sqrt(29))/(2)`
C:-`(-5+- sqrt(29))/(2)`
D:-7
Ans: A
37:-`2^1000000` mod 7 is
A:-5
B:-3
C:-2
D:-4
Ans: C
38:-When `x^5 + x^4 + 5x^2 -3` is divided by `x+2,` the remainder is
A:-0
B:-1
C:-2
D:-3
Ans: B
39:-A tree with 7 vertices has __________ edges.
A:-8
B:-7
C:-5
D:-6
Ans: D
40:-The number of distinct spanning trees of `K_4` is
A:-16
B:-12
C:-32
D:-8
Ans: A
41:-If the identity element `e in S` exists in a semigroup (S, `*` ), then it is a
A:-Group
B:-Groupoid
C:-Monoid
D:-None of the above
Ans: C
42:-The number of generators of `(Z_24, +)` is
A:-2
B:-6
C:-8
D:-10
Ans: C
43:-A Sylow 3-subgroup of a group of order 12 has order
A:-2
B:-3
C:-1
D:-12
Ans: B
44:-Consider `Z_5` and `Z_20` as rings modulo 5 and 20 respectively. Then the number of homomorphism φ
`:Z_5 -> Z_20` is
A:-1
B:-4
C:-5
D:-2
Ans: D
45:-Let `Q` be the field of rational numbers and `Z_2` is a field modulo 2. Then the polynomial `f (x) = x^3 -9x^2 + 9x + 3` is
A:-irreducible over `Q` but reducible over `Z_2`
B:-irreducible over both `Q` and `Z_2`
C:-reducible over `Q` but irreducible over `Z_2`
D:-reducible over both ` Q and` `Z_2`
Ans: A
46:-Let `A =` `[[3,1,-1],[2,2,-1],[2,2,0]]`. The characteristic polynomial of `A` is
A:-`x^3 + 5x^2+8x+4`
B:-`x^2+5x`
C:-`x^3-5x^2+8x-4`
D:-`x^3+8x+4`
Ans: C
47:-The eigen values of the matrix `[[4,-2],[-2,1]]` are
A:-1, 4
B:--1, 2
C:-0, 5
D:-Cannot be determined
Ans: C
48:-Let `V` be a finite dimensional vector space, `I` be the identity transformation on `V` , then the null space
of `I` is
A:-`{0}`
B:-`phi`
C:-`V`
D:-None of the above
Ans: A
49:-If `V` is a vector space with dim `V=n` , then the dimension of the hyperspace of `V` is
A:-`n`
B:-`n-1`
C:-`n+1`
D:-0
Ans: B
50:-Let `V` be a vector space of all 2 × 2 matrices over `R` . Let `T` be the linear mapping `T : V -> V` such that
`T (A)= AB-BA` where `B = [[2,1],[0,3]]` . Then the nullity of `T` is
A:-1
B:-2
C:-3
D:-4
Ans: A
51:-Banach space is a
A:-Complete normed vector space
B:-Normed vector space
C:-Complete vector space
D:-None of the above
Ans: A
52:-Which of the following is true?
A:-All normed spaces are inner product spaces
B:-All inner product spaces are normed spaces
C:-All inner product spaces are Banach spaces
D:-All inner product spaces are Hilbert spaces
Ans: B
53:-Banach space is a Hilbert space if
A:-Pythagorean theorem holds
B:-Projection theorem holds
C:-Parallelogram law holds
D:-None of the above
Ans: C
54:-If `T` is a bounded linear operator on a Hilbert space `H```, which of the following is not true?
A:-`T` is normal if `T` is self-adjoint
B:-`T` is normal if `T` is unitary
C:-`T` is self-adjoint if `T` is normal
D:-None of the above
Ans: C
55:-The equation of the normal at the point `(a sec Theta, b tanTheta)` on the hyperbola `(x^2)/(a^2)- (y^2)/(b^2)= 1` is
A:-` (x)/(a) sec Theta - (y)/(b) tan Theta = 1`
B:-`(x)/(a) sec Theta + (y)/(b) tan Theta = 1`
C:-`(ax)/(sec Theta) - (by)/(tan Theta) = a^2 + b^2`
D:-`(ax)/(sec Theta) + (by)/(tan Theta) = a^2 + b^2`
Ans: D
56:-`lim_(x->oo) (log x)/(x^n)` is
A:-`oo`
B:-`-oo`
C:-1
D:-0
Ans: D
57:-`(x * y) + (x' + y')` is equal to
A:-`x * y`
B:-`x' + y'`
C:-0
D:-1
Ans: D
58:-Let `a` be any element in a Boolean algebra `B` . If `a+x=1` and `ax=0` , then
A:-`x=1`
B:-`x=0`
C:-`x=a`
D:-`x=a'`
Ans: D
59:-Which of the following is reflexive?
A:-`l^2`
B:-`l^1`
C:-`L^1 [a,b]`
D:-`l^oo`
Ans: A
60:-If `1 < p < oo` and `q` is conjugate of `p` , then
A:-`l^{p'} = l^q`
B:-`l^{p'} = l^p`
C:-`l^{p'} < l^q`
D:-`l^{p' }> l^q`
Ans: A
61:-If`S` is a non-empty set of real numbers, then
A:-Inf `S` = Sup `S`
B:-Inf `S` = -Sup `(-S)`
C:-Inf `S` = Sup `(-S)`
D:-Inf `S` = -Sup `S`
Ans: B
62:-Every infinite set has
A:-an uncountable subset
B:-a countable subset
C:-both countable and uncountable subsets
D:-none of the above
Ans: B
63:-A real valued function `f` has discontinuity of the second kind at `x=a` if
A:-`f (a+)` exist only
B:-`f (a-)` exist only
C:-Neither `f (a+)` nor `f (a-)` exist
D:-Both `f(a+)` and `f (a-)` exist
Ans: C
64:-For the sequence `{x_n}` ``, where `x_n= (-1)^n n` , the `ullim x_n` is
A:-1
B:-0
C:-`+oo`
D:-`-oo`
Ans: D
65:-Every open set of real numbers is the union of
A:-countable collection of disjoint closed intervals
B:-uncountable collection of disjoint closed intervals
C:-countable collection of disjoint open intervals
D:-uncountable collection of disjoint open intervals
Ans: C
66:-A set `E` is nowhere dense if
A:-closure of `E` contains non-empty open sets
B:-closure of `E` contains no non-empty open sets
C:-closure of `E` contains empty open set
D:-none of the above
Ans: B
67:-If `f_1` and `f_2` are two real-valued bounded functions defined on `[a,b]` then for every partition `P`
on `[a,b]`
A:-`U (P, f_1+f_2) = U (P, f_1) + U (P, f_2)`
B:-`U (P, f_1+f_2)<= U (P, f_1) + U (P, f_2)`
C:-`U (P, f_1+f_2)>= U (P, f_1) + U (P, f_2)`
D:-None of the above
Ans: B
68:-If `f : [a,b] -> R` is continuous and monotonic function then
A:-`f` is Riemann integrable on `[a,b]`
B:-`f` is not Riemann integrable on `[a,b]`
C:-`f` is Riemann integrable on `R` ``
D:-None of the above
Ans: A
69:-Which of the following is true?
A:-The set `[0,1]` is not countable
B:-If `E_1` and `E_2` are Lebesgue measurable, then `E_1 uu E_2` is Lebesgue measurable
C:-The family `M` of Lebesgue measurable sets is an algebra of sets
D:-All of the above
Ans: D
70:-Given `int_0^1 (sin {1/(x)})/(sqrt(x))dx` , then
A:-Integral is divergent
B:-Integral is absolutely convergent
C:-Integral is not absolutely convergent
D:-None of the above
Ans: B
71:-If `f` satisfies the conditions of Lagrange's mean value theorem and if `f' (x) = 0 AA x in [a,b]` , then which of the following is true?
A:-`f` is constant on `[a,b]`
B:-`f` is strictly increasing in `[a,b]`
C:-`f` is strictly decreasing in `[a,b]`
D:-None of the above
Ans: A
72:-`lim_(z->0) (barz)/(z)` is
A:-0
B:-1
C:-`(1)/(2)`
D:-Does not exist
Ans: D
73:-The radius of convergence of the power series `sum_(n=0)^oo (2n!)/((n!)^2) (2-3i)^n ` is
A:-1
B:-0
C:-`(1)/(2)`
D:-`(1)/(4)`
Ans: D
74:-A function is said to be harmonic if
A:-`(del^2u)/(delx^2) + (del^2v)/(delx^2) = 0`
B:-`(del^2u)/(delx^2) + (del^2u)/(dely^2) = 0`
C:-`(delu)/(delx) + (delu)/(dely) = 0`
D:-`(delv)/(delx) + (delv)/(dely) = 0`
Ans: B
75:-The value of `int_c log z dz` where `c` is the unit circle is
A:-`Pi i`
B:-`2Pi i`
C:-`4Pi i`
D:-0
Ans: B
76:-The image of the unit circle `|z| = 1` under the transformation `w=2z+z^2` is
A:-Circle
B:-Straight line
C:-Parabola
D:-Cardioid
Ans: D
77:-If `X` is any set, `T` is a collection of all subsets of `X` then `(X, T)` is
A:-Discrete topology
B:-Indiscrete topology
C:-Trivial topology
D:-None of the above
Ans: A
78:-Let `X` and `Y` are topological spaces. The function `f` is a homeomorphism if
A:-`f : X -> Y` is a bijective function
B:-`f` is continuous
C:-`f^{-1} : Y ->X` is continuous
D:-All of the above
Ans: D
79:-Every compact subset of a Hausdorff space is
A:-Closed set
B:-Open set
C:-Null set
D:-None of the above
Ans: A
80:-The order and degree of the differential equation `(d)/(dx) ((d^2y)/(dx^2))^4 =0` is
A:-1, 4
B:-2, 4
C:-3, 1
D:-3, 4
Ans: C
81:-The value of Wronskian `W (x, x^2, x^3)` is
A:-`2x^2`
B:-`2x^4`
C:-`2x^3`
D:-`x^2`
Ans: C
82:-The general solution of `(del^2u)/(delx^2) + (del^2u)/(dely^2) = 0` is of the form
A:-`u= f(x + iy) - g (x - iy)`
B:-`u = f(x - iy) - g (x - iy)`
C:-`u = f(x + iy) + g (x - iy)`
D:-`u = f(x - iy) + g (x + iy)`
Ans: C
83:-The partial differential equation formed by eliminating the arbitrary function from `z = f ((y)/(x))` is
A:-`x(delz)/(delx) +(delz)/(dely) = 0`
B:-`(delz)/(delx) +(delz)/(dely) = 0`
C:-`(delz)/(delx) + y (delz)/(dely) = 0`
D:-`x (delz)/(delx) + y (delz)/(dely) = 0`
Ans: D
84:-The orthogonal trajectory of the family of curves `x^2-y^2 = k` is given by
A:-`x^2+y^2 =c`
B:-`xy=c`
C:-`y=c`
D:-`x=0`
Ans: B
85:-The general solution of the wave equation `(del^2y)/(delt^2) = c^2 (del^2y)/(delx^2)` is
A:-`y (x, t) = Phi (x +ct) + psi (x - ct)`
B:-`y (x, t) = f (x +ct)`
C:-`y (x, t) = f (x-ct)`
D:-No general solution exists
Ans: A
86:-Stirling's formula is the ___________ of Gauss' forward and backward formulae.
A:-Arithmetic mean
B:-Geometric mean
C:-Harmonic mean
D:-None of the above
Ans: A
87:-The interpolating polynomial of the highest degree which corresponds the functional values `f (-1) = 9, f(0)=5,f (2) = 3, f (5) = 15` is
A:-`x^3+x^2+2x+5`
B:-`x^2-3x+5`
C:-`x^4+4x^3 +5x^2+5`
D:-`x+5`
Ans: B
88:-The solution of the integral equation `Phi (x) = x+ int_0^x (Xi -x) Phi (Xi) dXi` is
A:-`cos x`
B:-`tan x`
C:-`sin x`
D:-`sec x`
Ans: C
89:-The minimizing curve must satisfy a differential equation called
A:-Lagrange's equation
B:-Euler-Lagrange equation
C:-Gauss equation
D:-None of the above
Ans: B
90:-A solid figure of revolution, for a given surface area, has maximum volume is in the case of
A:-a circle
B:-a sphere
C:-an ellipse
D:-a parabola
Ans: B
91:-A rigid body moving in space with one point fixed has degree of freedom
A:-3
B:-1
C:-6
D:-9
Ans: A
92:-A particle of unit mass is moving under gravitational field, along the cycloid `x = phi - sin phi, y =1 + cos phi` .
Then the Lagrangian for motion is
A:-`phi^2 (1+cos phi) - g (1- cos phi)`
B:-`phi^2 (1-cos phi) + g (1+ cos phi)`
C:-`phi^2 (1-cos phi) - g (1+ cos phi)`
D:-`2phi^2 (1-cos phi) - g (1+ cos phi)`
Ans: C
93:-`L^-1 [(1)/(s (s^2+a^2))]` is
A:-`(1)/(a^2) (1- cos at)`
B:-`(2 sin h t)/(t)`
C:-`(1)/(a^2) (e^{at} -1)`
D:-`(1)/(a^2) sin h at`
Ans: A
94:-`int_0^oo e^{-x^2}dx` is
A:-` (1)/(2)`
B:-`(pi)/(2)`
C:-`(sqrt(pi))/(2)`
D:-`-sqrt(pi)`
Ans: C
95:-Using Fourier series, representing `x` in the interval `[-pi, pi]`, the sum of the series `1-(1)/(3) + (1)/(5) -(1)/(7) + ...` is
A:-0
B:-1
C:-`(pi)/(2)`
D:-`(pi)/(4)`
Ans: D
96:-The only idempotent t-conorm is
A:-algebraic sum
B:-drastic union
C:-standard fuzzy union
D:-bounded sum
Ans: C
97:-Using fuzzy arithmetic operations on intervals [4,10]/[1,2] is
A:-[4,5]
B:-[2,10]
C:-[2,8]
D:-[4,20]
Ans: B
98:-The language generated by the grammar `G = ({S}, {a,b}, S, P)` where `P` is given by is `S -> aSb, S->lambda` is
A:-` {a^n b^n : n>=0}`
B:-`{a^n b^{n+1} : n>=0}`
C:-` {a^{n+1} b^n : n >= 0}`
D:-`{a^{n+2} b^n : n >= 1}`
Ans: A
99:-Which of the following is not true in the derivative of a smooth vector field `X`?
A:-` grad_v (X+Y) = grad_v X + grad_v Y`
B:-`grad_v (fX) = (grad_v f) X (p) + f(p) (grad_v X)`
C:-`grad_v (X * Y) = (grad_v X) * Y (p) + X (p) * (grad_v Y)`
D:-`grad_v (fX) = f(grad_vX)`
Ans: D
100:-Let `X` be a non-empty compact Hausdorff space. If every point of `X` is a limit point of `X`, then
A:-`X` is disjoint
B:- `X` is countable
C:-`X` is uncountable
D:-None of the above
Ans: C
A:-80 g
B:-60 g
C:-40 g
D:-20 g
Ans: D
34:-A pendulum swings through an angle of 30° and describes an arc 8.8 cm in length. The length of the pendulum is (Use `Pi=` `(22)/(7))`
A:-8.8 cm
B:-16.8 cm
C:-12.4 cm
D:-10. 2 cm
Ans: B
35:-A solid cube is cut into two cuboids of equal volumes. The ratio of the total surface area of the given cube to that of one of the cuboids is
A:-2 : 1
B:-3 : 2
C:-4 : 1
D:-4 : 3
Ans: B
36:-What is the value of `(1)/(5+ (1)/(5 + (1)/(5+...)))?`
A:-`(-5 + sqrt(29))/(2)`
B:-`(-5 - sqrt(29))/(2)`
C:-`(-5+- sqrt(29))/(2)`
D:-7
Ans: A
37:-`2^1000000` mod 7 is
A:-5
B:-3
C:-2
D:-4
Ans: C
38:-When `x^5 + x^4 + 5x^2 -3` is divided by `x+2,` the remainder is
A:-0
B:-1
C:-2
D:-3
Ans: B
39:-A tree with 7 vertices has __________ edges.
A:-8
B:-7
C:-5
D:-6
Ans: D
40:-The number of distinct spanning trees of `K_4` is
A:-16
B:-12
C:-32
D:-8
Ans: A
41:-If the identity element `e in S` exists in a semigroup (S, `*` ), then it is a
A:-Group
B:-Groupoid
C:-Monoid
D:-None of the above
Ans: C
42:-The number of generators of `(Z_24, +)` is
A:-2
B:-6
C:-8
D:-10
Ans: C
43:-A Sylow 3-subgroup of a group of order 12 has order
A:-2
B:-3
C:-1
D:-12
Ans: B
44:-Consider `Z_5` and `Z_20` as rings modulo 5 and 20 respectively. Then the number of homomorphism φ
`:Z_5 -> Z_20` is
A:-1
B:-4
C:-5
D:-2
Ans: D
45:-Let `Q` be the field of rational numbers and `Z_2` is a field modulo 2. Then the polynomial `f (x) = x^3 -9x^2 + 9x + 3` is
A:-irreducible over `Q` but reducible over `Z_2`
B:-irreducible over both `Q` and `Z_2`
C:-reducible over `Q` but irreducible over `Z_2`
D:-reducible over both ` Q and` `Z_2`
Ans: A
46:-Let `A =` `[[3,1,-1],[2,2,-1],[2,2,0]]`. The characteristic polynomial of `A` is
A:-`x^3 + 5x^2+8x+4`
B:-`x^2+5x`
C:-`x^3-5x^2+8x-4`
D:-`x^3+8x+4`
Ans: C
47:-The eigen values of the matrix `[[4,-2],[-2,1]]` are
A:-1, 4
B:--1, 2
C:-0, 5
D:-Cannot be determined
Ans: C
48:-Let `V` be a finite dimensional vector space, `I` be the identity transformation on `V` , then the null space
of `I` is
A:-`{0}`
B:-`phi`
C:-`V`
D:-None of the above
Ans: A
49:-If `V` is a vector space with dim `V=n` , then the dimension of the hyperspace of `V` is
A:-`n`
B:-`n-1`
C:-`n+1`
D:-0
Ans: B
50:-Let `V` be a vector space of all 2 × 2 matrices over `R` . Let `T` be the linear mapping `T : V -> V` such that
`T (A)= AB-BA` where `B = [[2,1],[0,3]]` . Then the nullity of `T` is
A:-1
B:-2
C:-3
D:-4
Ans: A
51:-Banach space is a
A:-Complete normed vector space
B:-Normed vector space
C:-Complete vector space
D:-None of the above
Ans: A
52:-Which of the following is true?
A:-All normed spaces are inner product spaces
B:-All inner product spaces are normed spaces
C:-All inner product spaces are Banach spaces
D:-All inner product spaces are Hilbert spaces
Ans: B
53:-Banach space is a Hilbert space if
A:-Pythagorean theorem holds
B:-Projection theorem holds
C:-Parallelogram law holds
D:-None of the above
Ans: C
54:-If `T` is a bounded linear operator on a Hilbert space `H```, which of the following is not true?
A:-`T` is normal if `T` is self-adjoint
B:-`T` is normal if `T` is unitary
C:-`T` is self-adjoint if `T` is normal
D:-None of the above
Ans: C
55:-The equation of the normal at the point `(a sec Theta, b tanTheta)` on the hyperbola `(x^2)/(a^2)- (y^2)/(b^2)= 1` is
A:-` (x)/(a) sec Theta - (y)/(b) tan Theta = 1`
B:-`(x)/(a) sec Theta + (y)/(b) tan Theta = 1`
C:-`(ax)/(sec Theta) - (by)/(tan Theta) = a^2 + b^2`
D:-`(ax)/(sec Theta) + (by)/(tan Theta) = a^2 + b^2`
Ans: D
56:-`lim_(x->oo) (log x)/(x^n)` is
A:-`oo`
B:-`-oo`
C:-1
D:-0
Ans: D
57:-`(x * y) + (x' + y')` is equal to
A:-`x * y`
B:-`x' + y'`
C:-0
D:-1
Ans: D
58:-Let `a` be any element in a Boolean algebra `B` . If `a+x=1` and `ax=0` , then
A:-`x=1`
B:-`x=0`
C:-`x=a`
D:-`x=a'`
Ans: D
59:-Which of the following is reflexive?
A:-`l^2`
B:-`l^1`
C:-`L^1 [a,b]`
D:-`l^oo`
Ans: A
60:-If `1 < p < oo` and `q` is conjugate of `p` , then
A:-`l^{p'} = l^q`
B:-`l^{p'} = l^p`
C:-`l^{p'} < l^q`
D:-`l^{p' }> l^q`
Ans: A
61:-If`S` is a non-empty set of real numbers, then
A:-Inf `S` = Sup `S`
B:-Inf `S` = -Sup `(-S)`
C:-Inf `S` = Sup `(-S)`
D:-Inf `S` = -Sup `S`
Ans: B
62:-Every infinite set has
A:-an uncountable subset
B:-a countable subset
C:-both countable and uncountable subsets
D:-none of the above
Ans: B
63:-A real valued function `f` has discontinuity of the second kind at `x=a` if
A:-`f (a+)` exist only
B:-`f (a-)` exist only
C:-Neither `f (a+)` nor `f (a-)` exist
D:-Both `f(a+)` and `f (a-)` exist
Ans: C
64:-For the sequence `{x_n}` ``, where `x_n= (-1)^n n` , the `ullim x_n` is
A:-1
B:-0
C:-`+oo`
D:-`-oo`
Ans: D
65:-Every open set of real numbers is the union of
A:-countable collection of disjoint closed intervals
B:-uncountable collection of disjoint closed intervals
C:-countable collection of disjoint open intervals
D:-uncountable collection of disjoint open intervals
Ans: C
66:-A set `E` is nowhere dense if
A:-closure of `E` contains non-empty open sets
B:-closure of `E` contains no non-empty open sets
C:-closure of `E` contains empty open set
D:-none of the above
Ans: B
67:-If `f_1` and `f_2` are two real-valued bounded functions defined on `[a,b]` then for every partition `P`
on `[a,b]`
A:-`U (P, f_1+f_2) = U (P, f_1) + U (P, f_2)`
B:-`U (P, f_1+f_2)<= U (P, f_1) + U (P, f_2)`
C:-`U (P, f_1+f_2)>= U (P, f_1) + U (P, f_2)`
D:-None of the above
Ans: B
68:-If `f : [a,b] -> R` is continuous and monotonic function then
A:-`f` is Riemann integrable on `[a,b]`
B:-`f` is not Riemann integrable on `[a,b]`
C:-`f` is Riemann integrable on `R` ``
D:-None of the above
Ans: A
69:-Which of the following is true?
A:-The set `[0,1]` is not countable
B:-If `E_1` and `E_2` are Lebesgue measurable, then `E_1 uu E_2` is Lebesgue measurable
C:-The family `M` of Lebesgue measurable sets is an algebra of sets
D:-All of the above
Ans: D
70:-Given `int_0^1 (sin {1/(x)})/(sqrt(x))dx` , then
A:-Integral is divergent
B:-Integral is absolutely convergent
C:-Integral is not absolutely convergent
D:-None of the above
Ans: B
71:-If `f` satisfies the conditions of Lagrange's mean value theorem and if `f' (x) = 0 AA x in [a,b]` , then which of the following is true?
A:-`f` is constant on `[a,b]`
B:-`f` is strictly increasing in `[a,b]`
C:-`f` is strictly decreasing in `[a,b]`
D:-None of the above
Ans: A
72:-`lim_(z->0) (barz)/(z)` is
A:-0
B:-1
C:-`(1)/(2)`
D:-Does not exist
Ans: D
73:-The radius of convergence of the power series `sum_(n=0)^oo (2n!)/((n!)^2) (2-3i)^n ` is
A:-1
B:-0
C:-`(1)/(2)`
D:-`(1)/(4)`
Ans: D
74:-A function is said to be harmonic if
A:-`(del^2u)/(delx^2) + (del^2v)/(delx^2) = 0`
B:-`(del^2u)/(delx^2) + (del^2u)/(dely^2) = 0`
C:-`(delu)/(delx) + (delu)/(dely) = 0`
D:-`(delv)/(delx) + (delv)/(dely) = 0`
Ans: B
75:-The value of `int_c log z dz` where `c` is the unit circle is
A:-`Pi i`
B:-`2Pi i`
C:-`4Pi i`
D:-0
Ans: B
76:-The image of the unit circle `|z| = 1` under the transformation `w=2z+z^2` is
A:-Circle
B:-Straight line
C:-Parabola
D:-Cardioid
Ans: D
77:-If `X` is any set, `T` is a collection of all subsets of `X` then `(X, T)` is
A:-Discrete topology
B:-Indiscrete topology
C:-Trivial topology
D:-None of the above
Ans: A
78:-Let `X` and `Y` are topological spaces. The function `f` is a homeomorphism if
A:-`f : X -> Y` is a bijective function
B:-`f` is continuous
C:-`f^{-1} : Y ->X` is continuous
D:-All of the above
Ans: D
79:-Every compact subset of a Hausdorff space is
A:-Closed set
B:-Open set
C:-Null set
D:-None of the above
Ans: A
80:-The order and degree of the differential equation `(d)/(dx) ((d^2y)/(dx^2))^4 =0` is
A:-1, 4
B:-2, 4
C:-3, 1
D:-3, 4
Ans: C
81:-The value of Wronskian `W (x, x^2, x^3)` is
A:-`2x^2`
B:-`2x^4`
C:-`2x^3`
D:-`x^2`
Ans: C
82:-The general solution of `(del^2u)/(delx^2) + (del^2u)/(dely^2) = 0` is of the form
A:-`u= f(x + iy) - g (x - iy)`
B:-`u = f(x - iy) - g (x - iy)`
C:-`u = f(x + iy) + g (x - iy)`
D:-`u = f(x - iy) + g (x + iy)`
Ans: C
83:-The partial differential equation formed by eliminating the arbitrary function from `z = f ((y)/(x))` is
A:-`x(delz)/(delx) +(delz)/(dely) = 0`
B:-`(delz)/(delx) +(delz)/(dely) = 0`
C:-`(delz)/(delx) + y (delz)/(dely) = 0`
D:-`x (delz)/(delx) + y (delz)/(dely) = 0`
Ans: D
84:-The orthogonal trajectory of the family of curves `x^2-y^2 = k` is given by
A:-`x^2+y^2 =c`
B:-`xy=c`
C:-`y=c`
D:-`x=0`
Ans: B
85:-The general solution of the wave equation `(del^2y)/(delt^2) = c^2 (del^2y)/(delx^2)` is
A:-`y (x, t) = Phi (x +ct) + psi (x - ct)`
B:-`y (x, t) = f (x +ct)`
C:-`y (x, t) = f (x-ct)`
D:-No general solution exists
Ans: A
86:-Stirling's formula is the ___________ of Gauss' forward and backward formulae.
A:-Arithmetic mean
B:-Geometric mean
C:-Harmonic mean
D:-None of the above
Ans: A
87:-The interpolating polynomial of the highest degree which corresponds the functional values `f (-1) = 9, f(0)=5,f (2) = 3, f (5) = 15` is
A:-`x^3+x^2+2x+5`
B:-`x^2-3x+5`
C:-`x^4+4x^3 +5x^2+5`
D:-`x+5`
Ans: B
88:-The solution of the integral equation `Phi (x) = x+ int_0^x (Xi -x) Phi (Xi) dXi` is
A:-`cos x`
B:-`tan x`
C:-`sin x`
D:-`sec x`
Ans: C
89:-The minimizing curve must satisfy a differential equation called
A:-Lagrange's equation
B:-Euler-Lagrange equation
C:-Gauss equation
D:-None of the above
Ans: B
90:-A solid figure of revolution, for a given surface area, has maximum volume is in the case of
A:-a circle
B:-a sphere
C:-an ellipse
D:-a parabola
Ans: B
91:-A rigid body moving in space with one point fixed has degree of freedom
A:-3
B:-1
C:-6
D:-9
Ans: A
92:-A particle of unit mass is moving under gravitational field, along the cycloid `x = phi - sin phi, y =1 + cos phi` .
Then the Lagrangian for motion is
A:-`phi^2 (1+cos phi) - g (1- cos phi)`
B:-`phi^2 (1-cos phi) + g (1+ cos phi)`
C:-`phi^2 (1-cos phi) - g (1+ cos phi)`
D:-`2phi^2 (1-cos phi) - g (1+ cos phi)`
Ans: C
93:-`L^-1 [(1)/(s (s^2+a^2))]` is
A:-`(1)/(a^2) (1- cos at)`
B:-`(2 sin h t)/(t)`
C:-`(1)/(a^2) (e^{at} -1)`
D:-`(1)/(a^2) sin h at`
Ans: A
94:-`int_0^oo e^{-x^2}dx` is
A:-` (1)/(2)`
B:-`(pi)/(2)`
C:-`(sqrt(pi))/(2)`
D:-`-sqrt(pi)`
Ans: C
95:-Using Fourier series, representing `x` in the interval `[-pi, pi]`, the sum of the series `1-(1)/(3) + (1)/(5) -(1)/(7) + ...` is
A:-0
B:-1
C:-`(pi)/(2)`
D:-`(pi)/(4)`
Ans: D
96:-The only idempotent t-conorm is
A:-algebraic sum
B:-drastic union
C:-standard fuzzy union
D:-bounded sum
Ans: C
97:-Using fuzzy arithmetic operations on intervals [4,10]/[1,2] is
A:-[4,5]
B:-[2,10]
C:-[2,8]
D:-[4,20]
Ans: B
98:-The language generated by the grammar `G = ({S}, {a,b}, S, P)` where `P` is given by is `S -> aSb, S->lambda` is
A:-` {a^n b^n : n>=0}`
B:-`{a^n b^{n+1} : n>=0}`
C:-` {a^{n+1} b^n : n >= 0}`
D:-`{a^{n+2} b^n : n >= 1}`
Ans: A
99:-Which of the following is not true in the derivative of a smooth vector field `X`?
A:-` grad_v (X+Y) = grad_v X + grad_v Y`
B:-`grad_v (fX) = (grad_v f) X (p) + f(p) (grad_v X)`
C:-`grad_v (X * Y) = (grad_v X) * Y (p) + X (p) * (grad_v Y)`
D:-`grad_v (fX) = f(grad_vX)`
Ans: D
100:-Let `X` be a non-empty compact Hausdorff space. If every point of `X` is a limit point of `X`, then
A:-`X` is disjoint
B:- `X` is countable
C:-`X` is uncountable
D:-None of the above
Ans: C