UG TRB BRTE 2023 / GAT 2023
QUESTION BOOKLET SERIES A
SUBJECT 003 MATHEMATICS
PART-B
31. A reciprocal equation of second type is:
(A) $6x^5+11x^4+33x^3- 33x^2-11x-6=0$
(B) $6x^{5}+11x^{4}-33x^{3}-33x^{2}+11x+6=0$
(C) $6x^5-11x4+33x333x²+11x-6=0$
(D)$ -6x^{3}-11x^{4}-33x^{3}+33x^{2}+11x+6=0$
ANSWER : B
32. The expansion of tan4θ is ---
(A) $${4tan\θ+4tan^{3}\θ} / {1-6tan^{2}\θ+tan^{4}\θ}$$
(B) $${4tan\θ+4tan^{3}\θ} / {1+6tan^{2}\θ+tan^{4}\θ}$$
(C) $${4tan\θ-4tan^{3}\θ} / {1-6tan^{2}\θ-tan^{4}\θ}$$
(D) $${4tan\θ-4tan^{3}\θ} / {1-6tan^{2}\θ+tan^{4}\θ}$$
ANSWER D
33. If $$tan ^(-1)(2-i)=x+iy $$ then:
(A) 4y=log2
(B) 4y= -log2
(C) y=4log2
(D) y=-log2
34. Which one of the following series is valid only in -1<x ≤1?
(A) logx
(B) log(1+x)
(C) log(1-x)
(D) $ {1}/{2}$$ log({1+x}/{1-x})$
ANSWER:B
35. The equation obtained by removing the second term in $x^{4}+20x³-143x²+430x+462=0$ is:
(A) $y^{4}+7y^{2}-12=0$
(B) $y^{4}-7y^{2}+12=0$
(C) $y^{4}+7y^{2}-180y-12=0$
(D) $y^{4}-7y^{2}+180y+12=0$
ANSWER:A
36. The radius of curvature at (3,4)on $x^{2}+y^{2}=25$ is
(A) ${1}/{5}$
(B) 5
(C) 25
(D) 625
ANSWER:B37. The slope of the tangent with the initial line for the cardioid r=a(1-cosθ) at θ= $π/6$ is:
(A) 1
(B)-1
(C) ∞
(D) ${1}/{2}$
ANSWER:A
38. The value of \int_{-\infty}^{\infty}∫cos(${π}/{2}$$x^{2}$)dx is
(A) 0
(B) 1
(C) ${π}/{2}$
(D) $$√{{π}/{2}}$$
ANSWER:B39. The points on the surface $z^{2}=xy+1 $ nearest to the origin are:
(A) (0, 0, 1) and (0, 0, -1)
(B) (1, 0, 0) and (-1, 0, 0)
(C) (0, 1, 0) and (0, -1, 1)
(D) (0, 1, 1) and (-1, 0, 1)
40 The area inside the circle r=3acosθ and outside the Cardioid r=a(1+cosθ)is :
(Α) 5πa²
(B) ${3πa²}/{2}$
(C) ${πa²}/{4}$
(D) πa²
A 41. If $√{3}-i7 $ is one of the roots of an equation, then the degree of that equation is
(A) 4
(B) 3
(C)12
(D) 1
ANSWER:A42. Which of the following is a root of $6x^{5}-x^{4}-43x^{3}+43x^{2}+x-6=0$
(A) ${1}/{2}$
(B) ${1}/{3}$
(C) ${1}/{4}$
(D) ${1}/{5}$
43. In which of the following, the equation x³-3x+1=0 has a root ?
(A) Between 1 and 2
(B) Between 2 and 3
(C) Between 3 and 4
(D) Between 4 and 5
44. Sum and product of the eigen values of the matrix
3 -1 1
-1 5 -1
1 -1 3
(A) 6, 36
(B) 11, 36
(C) 11, 18
(D) 36, 11
45. Sum of the divisors of 360 is:
(A) 1170
(B) 1710
(C) 1107
(D) 1071
46. The remainder when $2^{46}$ is divisible by 47 is:
(A) 1
(B) 2
(C) 3
(D) 4
ANSWER:A
47. The value of lim x → ${π}/{2}$
${sinx + cos2x}/{cos²x}$ is
(A) ${3}/{2}$
(B) ${5}/{2}$
(C) ${7}/{2}$
(D) ${1}/{2}$
ANSWER:A48. The value of Log(-i):
(A) i$$({2nπ}+ {π}/{4})$$
(B) i(2nπ-${π}/{2}$)
6) i(2nπ-${π}/{4}$)
(D) i(2nπ+${π}/{2}$)
ANSWER:B49. If ⍺+iß= $sinh^{-1}$(1+i), then :
(A) tan⍺=tanhẞ
(B) tanß= tanh⍺
(C) tanẞ=-tanh⍺
(D) tan⍺ = - tanhẞ
ANSWER:B50. If ⍺,ẞ,ɣ , 𝛅 be the roots of the biquadratic equation $x^{4}+px^{3}+qx^{2}+rx+s=0$ then the value of 𝚺⍺²β² is
(A) q²-2pr-14s
(B) q²-2pr+14s
(C) q²-2pr+2s
(D) q²+2pr+2s
ANSWER:C
51. The value of \int_{0} {2} ∫$x^{7}(2-x)^{8}dx$ is:
(A) ${2^{16}8! 9!}/{15!}$
(B) ${2^{14}7! 9!}/{15!}$
(C) ${2^{16}7! 8!}/{16!}$
(D) ${2^{14}8! 9!}/{16!}$
52. Using spherical polar coordinates, the volume common to the sphere x²+y²+z² =9 and one branch of the cone x²+y²=z² is:
(A) 9π(2+√2 ) cubic units
(B) ${π}/{3}$(2+√2 ) cubic units
(C) 9π(2 -√2 ) cubic units
(D) ${π}/{5}$(2 - √2 ) cubic units
ANSWER:C53. The cubic equation which has the same asymptotes as the cubic $x^{3}-6x^{2}y+11xy^{2}-6y^{3}+x+y+1=0$ and which touches the axis of y at the origin and goes through the point (3, 2) is:
(A) $x^{3}-6x^{2}y+11xy^{2}-6y^{3}=0$
(B) $x^{3}-6x^{2}y+11xy^{2}-6y^{3}=x$
(C) $x^{3}-6x^{2}y+11xy^{2}-6y^{3}=x^{2}$
(D) $x^{3}-6x^{2}y+11xy^{2}-6y^{3}=x+y$
ANSWER:B54 The area inside the circle r=3a cosθ and outside the cardioid r=a(1 +cosθ) is
(A) πa² sq. units
(B) πsq. units
(C) πa sq. units
(D) a² sq. units
ANSWER:A55. The radius of curvature at any point on the curve $r^{n}=a^{n} cosnθ is:
(A) ${a}/{n+1}$
(B) ${ar}/{n+1}$
(C) ${a^{n}}/{(n+1)r^{n-1}}$
(D) ${r}/{n+1}$
ANSWER:C56. The volume of the region above xy-plane bounded by the paraboloid z=x²+y² and the cylinder x²+y²=a² is=
(A) ${π}/{2}$
(B) ${π}/{2}$$a^{4}$
(C) a²
(D) πa²
ANSWER:B
57. The minimum value of total surface area of a rectangular box, open at the top, having a volume of 32 cubic feet is:
(A) 48 sq. feet
(B) 24 sq. feet
(C) 12 sq. feet
(D) 18 sq. feet
58. The value of ∫int 0 →${π}/{2}$∫${dθ}/{sinθ}$x ∫int 0 →${π}/{2}$ ∫ √sinθ dθ is
(A) ${π}/{4}$
(B) ${π}/{3}$
(C) ${π}/{2}$
(D) π
59. The radius of curvature for the curve y²=x³+8 at (-2, 0) is units.
(A) ${1}/{6}$
(B) 6
(C) 0(D) 8
ANSWER:B
60. Let S be the focus of the parabola y²=4ax and p is the radius of curvature at any point P on it. Then ⍴² varies depends on which values given below ?
(A) SP
(B) (SP)²
(C) (SP)³
(D $(SP)^{4}$
ANSWER:C61. Let G be the group of positive real numbers under multiplication and $\overline{G}$ be the group of all real numbers under addition. Define $Φ:G →\overline{G}$ by Φ(x)=log10 x then:
(A) Φ is 1-1, onto and a homomorphism
(B) Φ is not onto
(C) Φ is a homomorphism
(D) Φ is not 1-1
ANSWER:C62. If U is an ideal of the ring R, there is a homomorphism Φ:R → R/U given by Φ(a)=a+U a∈ R then R/U is:
(A)a ring and is a homomorphic image of R.
(B) not a ring.
(C) a ring and is not a homomorphic image of R
(D) a ring and is a isomorphic image of R.
ANSWER:A63. Let R be a ring and I an ideal of R then:
(A) R is commutative → R/I is commutative
(B) R/I is commutative →R is commutative
(C) R/I is a ring with identity → R is a ring with identity
(D) R/1 is an integral domain → R is an integral domain
ANSWER:A64. Choose the correct statement:
(A) Every homomorphism is an isomorphism.
(B) A homomorphism is 1-1 iff its kernel is (0).
(C) A homomorphic image of an integral domain is an integral domain.
(D) Homomorphic image of a field is a field.
65. Choose the correct statement:
(A) Every Integral domain is a field.
(B) Z is an Integral domain but not a field.
(C) The ring of integers is a field.
(D) Z is an Integral domain and a field.
66. If An is an alternating group of n symbols and Sn is a symmetric group of n symbols, then An contains elements.
(A) ${n}/{2}$
(B) ${n!}/{2}$
(C) ${(n-1)!}/{2}$
(D) ${n-1}/{2}$
67. In V3(R), the vectors (1, 4, 2), (a, 1, 3) and (-4, 11, 5) are linearly dependent. Then the value of a is:
(A) -2
(B) 2
(C) -1
(D) 1
ANSWER: A68. If the three vectors (a, 2, 3), (2, 5, 1) and (-1, 1, 4) form a basis for V3(R), then the value of a is:
(A) 0
(B) 1
(C) 2
(D) 3
ANSWER: B (QUESTION DOUBT)
69. The set of all positive rational number forms an abelian group under the operation. Define a is: a*b= ${ab}/{2}$. The inverse of an element
(A) ${1}/{a}$
(B) ${2}/{a}$
(C) a
(D) ${4}/{3}$
ANSWER: B70. The order of an element -1 in (z, +) is:
(A1
(B) 2
(C) 5
(D) Infinite
ANSWER:D71. If a²=a ∀a∈ R then the ring R is called a
(A) Euclidean ring
(B) Boolean ring
(C) Integral domain
(D) Division ring
72. If dim F V = m then dim Hom (V, F) is:
(A) m²
(B) m
(C) m-1
(D) m+1
ANSWER: A
73. Let G be the group of all 2 x 2 matrices (a b c d ) where ad-bc ≠ 0 and a, b, c, d are integer modula 3 relative to matrix multiplication. then O(G) is:
(A) 36
(B) 48
(C) 32
(D) 26
74. From the following statement which one represent vector space ?
(A) R+ over R
(B) Z over Q
(C) Q(x) over R
(D) Z over Z5
75. Let W₁ and W₂ be sub spaces of a finite dimensional inner product space then:
(A) (W₁ + W₂) 丄 = W₁丄 ⋂ W₂ 丄
(B) (W₁+W₂)丄 = W₁丄 U W₂丄
(C) (W₁丄⋂ W₂) 丄=W₂ 丄U W₂丄
(D) (W₁ 丄 ⋂ W₂丄) =W₂ 丄 ⋂ W₂丄
76. If f(x)=x² (0 ≤ x ≤1) and for each n∊z+, σn= {0,${1}/{n}$,${2}/{n}$ ,..........${n}/{n}$} [0, 1], then the value of lim n→∞ L[f;σn] is:
(A) 0
(B) 1
(C) ${1}/{3}$
(D) ${2}/{3}$
77. If gn(x)= ${x}/{1+nx}$ (0 ≤ x < ∞), then the sequence {gn(x)} {n=1}→∞ is
(A) Converges uniformly to 0 on [0, ∞)
(B) Converges uniformly to 1 on [0, ∞)
(C) Converges uniformly to ${1}/{2}$ on [0, ∞)
(D) Diverges
78. If B={${1}/{2}$, ${3}/{4}$,...,${2^{n}-1}/{2^{n}$},...}. Then the values of g.l.b. and Lu.b. of B are:
(A) ${1}/{2}$ and 1
(B) 1 and ${1}/{2}$
(C)0 and ${1}/{2}$
(D) 0 and 1
79 The value of lim n→∞ ${3n²-b}/{5n²+4}$ is:
(A) ${3}/{2}$
(B) 0
(C) ∞
(D) ${3}/{5}$
80. Which of the following sequence is an element of l²?
(A) ${{1}/√{n}}$ _{n=1}→∞
(B) √{n} _{n=1}→∞
(C) $(-1)^{n}$ _{n=1}→∞
(D) ${{1}/{n}}$ _{n=1}→∞
81. If the roots of the auxiliary equation for a second order ordinary differential equation are ɑ 士β then the complementary function (C.F.) is
(A) c1 $(e)^{ax}$ cos(√βx+c2)
(B) c1 $(e)^{ax}$ cosh(√βx+c2)
(C) $(e)^{ax}$ (c1cos√βx+c2 sin√βx)
(D) $(e)^{ax}$ (c1cos√βx- c2 sin√βx)
82. The Laplace transform of the differential equation y" +2y'-3y=sint is
(A) L(y(t))= ${1}/{(s-1)(s+3)(s^{2}+1)}$
(B) L(y(t))=${1}/{(s^{2}-1)(s+3)(s^{2}+1)}$
(C) L(y(t))=${1}/{(s-1)(s+3)(s+1)}$
(D) L(y(t))=${1}/{(s^{2}-1)(s+3)(s-1)}$
83. A differential equation satisfied by the family of concentric circles whose centre at (0, 0) is
(A) x+ ${dy}/{dx}$ =0
(B) y+x ${dy}/{dx}$ =0
(C) x+y ${dy}/{dx}$ =0
(D) x+y+y ${dy}/{dx}$ =0
ANSWER: C84. The singular solution of a partial differential equation represents the of the family of surfaces represented by the complete solution of that partial differential equation.
(A) Parabola
(B) Involute
(C) Evolute
(D) Envelope
85. The equation of surface satisfying 4yzp+q+2y=0 and passing through y²+z²=1, x+y=2 is
(A) y²+z²+x+z+3=0
(B) y²+z²+x²+y+z=0
(C) y²+z²+x+z-3=0
(D) x²+y²+z²-z+y=0
86. The value of \int_{0}^{\infty}\frac{sin^{2}tx}{x^{2}}dxis
(A) \pi/3~t
(B) 1\pi/4~t
(C) \pi/2~t
(D) \pi/5~t
87. The Partial Differential Equations of all surfaces of revolution having z-axis as the axis of rotation.
(A) y\frac{\partial z}{\partial y}=x\cdot\frac{\partial z}{\partial x}
(B) x\frac{\partial z}{\partial p}=y\cdot\frac{\partial z}{\partial q}
(C) y\frac{\partial z}{\partial x}=xy\frac{\partial z}{\partial y},\_
(D) y\cdot\frac{\partial z}{\partial y}=x\cdot\frac{\partial z}{\partial y}
88. The equation (\alpha xy^{3}+y~cos~x)dx+(x^{2}y^{2}+\beta~sin~x)dy=0 is exact if
(A) \alpha=\sqrt[3]{2},\beta=1
(B) \alpha=1,\beta=\frac{3}{2}
(C) \alpha=\frac{2}{3} \beta=1
(D) \alpha=1 \beta=\frac{2}{3}
ANSWER: C89. The partial differential equation obtained after eliminating arbitrary functions from z=f(x+t)+g(x-t) is:
(A) r+t=0
(B) p=q
(Chr r-t=0
(D) p+q=0
90. The function which is not piecewise continuous among the following is
(A) f(t)=t sin, t\ne0 and f(0)=1 over (-\infty,\infty)
f(t)=\frac{1}{t}sin\frac{1}{t}, t\ne0 and f(0)=1 over (-\infty,\infty)
(C) f(t)=e^{-t}over(-\infty,\infty)
(D) f(t)=\frac{sint}{t} (-\infty,\infty) t\ne0 and f(0)=1 on
91. The differential equation \frac{dy}{dx}+Py=Qy^{n} , where P and Q are functions of x only known as
(A) Legendre's equation
(B) Bessel's equation
(C) Lagrange's equation
(D) Bernoulli's equation
ANSWER: D92. The value of \int_{0}^{\infty}te^{-3} cost dt is
.te^{-3t},p~int
(A) \frac{3}{13}
2
(B)7 \frac{8}{25}
\frac{1}{25} (0)
(D) \frac{2}{25}
ANSWER: D93. Which of the following is not an inverse Laplace transform of \frac{1}{s^{2}} ?
(A) f(t)=\begin{cases}0,t=2\\ t,t\ne2\end{cases}
(B) f(t)=\begin{cases}5,&t=1\\ 2,&t=6\\ t,&t\ne1,6\end{cases}
\bigvee f(t)=\begin{cases}t&t\ne6\\ 0,&t=6\end{cases}
(D) f(t)=\begin{cases}e^{t},&t\ne5,8\\ 6,&t=5\\ 0,&t=8\end{cases}
ANSWER: D
94. The equation which is not an exact equation among the following is
(A) (cosx cosy-cott) dx-sinx siny dy=0
(B) 3x(xy-2)dx+(x^{3}+2y)dy=0
(C) ydx + (x + xy²)dy=0
(D) 2xydx+(x^{2}+3y^{2})dy=0
ANSWER: C95. The value of \int_{\Omega}(sin~3t)\delta(t-\frac{\pi}{2})dt\cdot is the integral
(A) 00-0
(B) 1
-1
(D) π
ANSWER: C96. If~F(s)=\sqrt{\frac{2}{\pi}}\frac{sinas}{s}and f(x)=\begin{cases}1&for|x|<a\\ 0&for|x|>a>0\end{cases} then the value of the integral equal to: \int_{0}^{\infty}(\frac{sint}{t})^{2}d is
(A) \frac{\pi}{4}
(B) \frac{\pi}{3}
\frac{\pi}{2}
(D) π
ANSWER: C
97. The value of \iint_{S}^{\rightarrow}F\cdot\vec{n}ds sphere x^{2}+y^{2}+(z-1)^{2}=1 \vec{F}=x\vec{i}-y\vec{j}+2z\vec{k} is equal to: over the where
(A) \frac{6\pi}{3}
(B) \frac{7\pi}{3}
(C \frac{8\pi}{3}
D \frac{9\pi}{3}
ANSWER:98. If F[f(x)]=F(s) then the value of F[f'(x)] is equal to:
(A) is F(s)
(B)-is F(s)
(C) -\frac{1}{s}F(s)
(D) -\frac{i}{s}F(s)
ANSWER: B99. The value of the integral is equal to: 0
(A) (e+1)\vec{i}+\frac{1}{2}(e^{-2}+1)\vec{j}-\frac{1}{2}\vec{k}
(e-1)\vec{i}+\frac{1}{2}(e^{-2}+1)\vec{j}+\frac{1}{2}\vec{k}
(C) (e-1)\vec{i}-\frac{1}{2}(e^{-2}-1)\vec{j}+\frac{1}{2}\vec{k}
(D) (e+1)\vec{i}-\frac{1}{2}(e^{-2}+1)\vec{j}+\frac{1}{2}\vec{k}
ANSWER: C100. Let f(x)=1² be the function defined on the interval (0, 2). Then the Fourier Series constant a for n=0 is equal to:
(A) \frac{4}{=n^{2}}
\frac{4}{n^{2}}
\frac{4}{nn}
\frac{-4}{nm^{2}}
ANSWER: B101. The value of \oint\frac{dz}{z(z^{2}+4)}.C:|z|=1is:
(A)
(B) \frac{\pi i}{2}
\sim^{2\pi}
(D) \frac{-i}{4}
ANSWER: B102. The angle of rotation at the point z=\frac{1+i}{2} under the mapping w=z^{2} is
\frac{\pi}{2} (A)
\frac{\pi}{3} (B)
\frac{\pi}{4}
(D)
ANSWER: A
103. The function f(z)=\frac{e^{z}}{z}
(A) only one singular point z=0
(B) two singular points 2=0 and z=\infty
two singular points z=0 and :z=-\infty
(D) three singular points z=0, z= and
ANSWER: D104. The radius of convergence of the power series -\frac{n!}{n^{n}}z^{n}/i
(A) e
(B) \frac{1}{e}
(C)
(D) 1
ANSWER: A105. The value of Z which satisfies |z|-z=1+2i is:
(A) \frac{3}{2}+2i
(B) 1+i
41 1\pm i
(D \frac{3}{2}i
ANSWER: A
106. If a power series in Z is convergent at Z=Z then in the circular open disc |Z|<|Z_{1}| it is
(A) convergent
(B) not convergent
(C) absolutely converge
(D) uniformly convergent
ANSWER:C107. If u=e^{-y_{sin}x} is harmonic, the analytic function having u as its real part is:
(A) e^{-y}(sin~x-i~cos~x)
( e^{-x}(sin~y-cos~y)
(C) e^{-y}(sin~x+1~cos~x)
(D) e^{-y}(cos~x-sin~x)
ANSWER:A108. A mapping which is isogonal but not conformal is:
(A) w=z+a
(B) \overline{z}
\frac{1}{z}
(D) kz
ANSWER: B109. The residue of z^{2}e^{\frac{1}{z}} at the point z=0~i :
\frac{2\pi i}{3}
(B) {\pi i/}_{3}
(C) \frac{1}{6}
) D -\frac{1}{6}
ANSWER: C110. The centre and radius of |z-2-i|=3 are
(A) 2+i~3
(B) 2-1, 3
1097-2+1,3
(D) -2-i,-3
ANSWER:A111. The order of the pole z=0 of the function 1-sinz f(2)= is:
(A) 5
(B) 0
(G\cap1
(D) 3
ANSWER:A112. By Cauchy's residue theorem, \int_{C}f(z); dz is equal to:
\pi i\sum_{j=1}^{n}Res\{f(z);z_{j}\} (A)
2\pi i\sum_{j=1}^{\infty}Res\{f(z);z_{j}\}\cap (B)
(C) 2\pi i\sum_{j=1}^{n}Res\{f(z);z_{j}\}
(D)\pi i\sum_{j=1}^{\infty}Res\{f(z);z_{j}\}
ANSWER: C113. The value of the constant 'a' so that n(x,y)=ax^{2}-y^{2}+xy is harmonic:
(A) 1
(B) 2
(C) 3
(D)의
ANSWER: A114. The expansion of \frac{1}{3^{2}} as 1-32-n+3(z-1)^{2}-4(2-1)^{3}+ is valid for all z in
(A) |z|=1
(B) 1
|z-1|<1
(D) 1
ANSWER:C115 The zeros of f(z)=\frac{z^{3}-1}{z^{3}+1} .
1,\frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}
(B) 1,0,-1
\frac{1-i\sqrt{3}}{2},\frac{1+i\sqrt{3}}{2}
(D) 1,2i-2i
ANSWER:A116. A cone whose height is equal to four times the radius of its base is hung from a point in the circumference of its base. Then the position of equilibrium:
(A) \frac{\pi}{2}
(B) \frac{-5}{3}
19 \frac{\pi}{4}
(D)
ANSWER:A117. A point is moving with a velocity of 10 m/sec and at a subsequent instant it is moving at the same rate in a direction inclined at 30° to the former direction. Then the change of velocity is m/sec.
(A) 3.176
(B) 5.176
(C) 4.176
(D) 6.176
ANSWER:118. A man can swim directly across a river of width 'a' metres in t₁ seconds (no current) and when there is current in to seconds. Then the speed of the current is
att2
m/sec
(A) √2-22
at m/sec
(B) 12/2-122
(C) t1t2 m/sec
(D) √2-122 m/sec att2
ANSWER: C119. Four forces equal to P, 2P, 3P and 4P are respectively, acting along the four sides of square ABCD taken in order then the magnitude of the resultant force:
(A) 2√2P
(B) 4/2P
(C) √2P
(D) 6/2P
ANSWER: A120. A steel ball is let fall through a height of 0.64 m on a plate of steel. The height through which it rebounds is 0.36 m. Calculate the coefficient of restitution.
(A) 1
(B) 0.75
(C) 0.50
(D) 0.25
ANSWER:C121. If \int\vec{r}\times\frac{d^{2}\vec{r}}{dt^{2}}dt=\vec{r}\times\frac{d\vec{r}}{dt}+C value of\vec{r}\times\frac{d\vec{x}}{dt} 15 then the (Where 7=517+- and C is a constant)
(A) 2t^{3}\vec{1}+25t^{4}\vec{j}+5t^{2}\vec{k}
(B) 2t^{3}\vec{1}+25t^{4}j-5t^{2}\vec{k}
( -2t^{3}\vec{i}+5t^{4}\vec{j}-5t^{2}\vec{k}
(D) -2t^{3}\vec{1}-25t^{4}\vec{j}-5t^{2}\vec{k}
ANSWER: B122. If F =(2xy+z^{3})\vec{1}+x^{2}\vec{j}+3xz^{2}\vec{k} is a conservative vector field, then the value of is equal to: 2^{3}+3^{3}+x^{2}+1xy^{2}
(A) x²y+xz+C
(B) xy^{2}+x^{3}z+C
(C) x²y +xz³+C
(D) xy^{2}+xz^{3}+C
ANSWER: C123. Necessary and sufficient condition for the vector V(t) to have a constant direction is:
(A ) ${V↖{→}⋅{dV↖{→}}/{dt}} = 0$
(B) ${V↖{→} X {dV↖{→}}/{dt}} = 0$
(C) ${V↖{→} = 0$
(D) ${dV↖{→}}/{dt} = 0$
ANSWER:B124. If C is a closed curve, then to: \oint_{r} dr is equal
\[∮_\{r↖{→}⋅{dr↖{→}\]
\[∫_\Cd\bo ω =∫_{∂\Δ}d\bo ω\]
(A) г
(B) r²
(0) ${1}/{r}$
(D) 0
125. If ${F↖{→}} = 2xy{i↖{→}}+yz²{j↖{→}}+xz{k↖{→}}$ and V is bounded by the planes x=0,y=0,z =0, x=2, y=1, z=3 then the value of ∭Δ FdV is equal to
(A) 29
(B) 30
(C) 31
(D) 32
126. if(${π-x}/{2})²$= ${π²}/{12}$+\sum_{n=1}^{\infty}\frac{1}{n^{2}} cost in the interval 1(0, 2π), then the value of \sum_{n=1}^{\infty}\${1}/{n²}}is
equal to:
(A) S{π^²}/{12}$
(B) S{π^²}/{8}$
(C) S{π^²}/{6}$
D) S{π^²}/{4}$
127. If f(x)=x sinx is defined in the interval 0<x<2π then the value of the Fourier constant a0 is equal to:
(A)-2
(B) 2
(C) 1
(D-1
128. The Fourier transform of f(x) for f(x)=\begin{cases}1,|x|<a\\ 0,|x|>a>0\end{cases}isequal~tr :
(A) \sqrt{\frac{2}{\pi}}\frac{sin~as}{s}
(B) \sqrt{\frac{\pi}{2}}\frac{sin~as}{s},
(C) \sqrt{\frac{2}{\pi}}\frac{sin~s}{s}
(D) \sqrt{\frac{\pi}{2}}\frac{sin~s}{s}.
129. If A and $\vec{B}$ are irrotational then $ \vec{A}\ X \vec{B}$ :
(A) Solenoidal
(B) Irrotational
(C) Neither solenoidal nor irrotational
(D) Both solenoidal and irrotational
130. If C is the triangle bounded by the lines y=0, x=1, y=x then the value of the integral ∫(xy-x²)dx+x²y dy is equal to
(A) ${1}/{12}$
(B) ${-1}/{12}$
(C) 12
(D) -12
131. f(x)= ${sinx}/{x}$,x ∈ ${R}^{'}$ , x≠0 is:
(A) Continuous at x=0
(B) Not continuous at x=0
(C) Differentiable at x=0
(D) Continuous everywhere
132. Which of the following is true?
(A) [0, x), 0<x<1, is not open in [0, 1]
(B) (0, 1), 0 < x < 1, is open in [0, 1]
(C) [0, x), 0<x<1, is open in R
(D) (x, 1], 0<x<1, is open in R
133. The value of c of Rolle's theorem for f(x)=sin in[0,π]
(A) ${π}/{6}$
(B) ${π}/{4}$
(C) ${π}/{3}$
(D) ${π}/{2}$
134. Which of the following is not a convergent sequence?
(A) \{\frac{sin~n}{n}\}_{n=1}^{\infty}
(B) \{\frac{n^{3}}{2^{n}}\}_{n=1}^{\infty}
\{\frac{3^{n}}{n!}\}_{n=1}^{\infty}
(D) \{(-1)^{n}\}_{n=1}^{\infty}
135 f(x)= {$ {x}^{P}$ sin(${1}/{x}$),&x ≠ 0,
x= 0, is differentiable
at x=0 if:
(A) p<1
() B p=1
(C) p>1
(D) p=0
ANSWER: C
136. Which of the following is an open set in R² ?
(A) (x,y) ∈ R²,ax+by<c}
(B) {(x,y) ∈ R²,ax+by ≥ c}
(C) \{(x,y)∈ R²,a ≤ x ≤ b}
(D) \{(x,y) ∈ R²,ax+by ≤ c}
137. Which of the following function is not uniformly continuous ?
(A) f(x)=x² 0 ≤ x ≤1
(B) f(x)= ${1}/{x}$ 0<x<1
(C) f(x)=x,x ∈ R
(D) f(x)=x³, 0 ≤ x ≤ 3
138. If G1 and G2 are open subsets of the metric space M, then G1⋂G2 is :
(A) Open
(B) Closed
(C) Neither open nor closed
(D) Both open and closed
139. The function
f(x)= {x~sin(${1}/{x}$) if x≠0
0, if x=0 is
(A) Continuous at x=0 but not differentiable at x=0
(B) Continuous at x=0 and differentiable at x=0
(C) Not continuous at x=0 and not differentiable at x=0
(D) Not continuous at x=0 but differentiable at x=0
140. 1: Every cauchy sequence contains convergent subsequence
II Every monotone sequence contains convergent subsequence
III: Every bounded sequence contains convergent subsequence
(A) I and II alone is true
(B) I and III alone is true
(C) All are true
(D) III alone is true
141. A ball of mass 'm' impinges on another of mass '2m' which is moving in the same direction as the first but with one-seventh of its velocity e=${3}/{4}$, first ball after impact? what is the velocity of
(A V1=3
(B) V1=2
(C) V₁=1
(D V1=0
142. Identify the components of velocity of a particle in the radial and transverse direction.
(A) \dot{r}-r\dot{\theta}^{2}
(B) \dot{r}-\dot{r}\dot{\theta}^{2}
(C) \dot{r}-\dot{r}^{2}\dot{\theta}
(D) \dot{r}-\dot{r}^{2}\theta
143. If the maximum velocity of a particle moves exactly in sample harmonic motion is 1 m/sec and its period ${1}/{5}$ of the second, then the Amplitude is:
(A) ${10}/{π}$ metre
(B) ${1}/{10π}$ metre
(C) ${1}/{π}$ metre
(D) ${1}/{5π}$ metre
144. The magnitude of the resultant of two equal forces inclined at an angle ꭤ is
(A) 2P cosꭤ
(B) P²cos²ꭤ
(C) 2Pcos ${α}/{2}$
(D) Pcos ${α}/{2}$
145. A particle is projected with velocity'u' making an angle 'a' with the horizontal, then the time of flight is
(A) ${u² sin²α}/{2g}$
(B) ${u sinα}/{g}$
(C) ${2u sinα}/{g}$
(D) ${u² sin2α}/{2}$
ANSWER: C
146. One ship is sailing due east at the rate of 12 km per hour, and another ship is sailing due north at the rate of 16 km per hour, then the relative velocity of first ship with respect to second is
(A) 28 km per hour
(B) 4 km per hour
(C) 25 km per hour
(D) 20 km per hour
147. Two forces of magnitudes 100 N and 150 N are acting simultaneously at a point. If the angle between the forces is 45°, then the resultant force is
(A) 232 N
(B) 230 N
(C) 180 N
(D) 200 N
148. The moment of inertia of a solid right circular cone of height 'h' base radius 'r' and semi-vertical angle 'a' about its axis is
(A) ${Mr} /{10}$
(B) ${3Mr} /{10}$
(C) ${3M} /{10}$
(D) ${3Mr²} /{10}$
149. The equation of a central orbit in polar co-ordinates is
(A) ${d²u}/{dy²}$ + u = ${ф(r)}/{h}$
(B) ${d²u}/{dy²}$ + u² = - ${ф(r)}/{hu}$
(C) ${d²u}/{dy²}$ - u = ${ф(r)}/{u²h}$
(D) ${d²u}/{dy²}$ + u = - ${ф(r)}/{h²u²}$
150. A couple consists of which of the following?
(A) Two like parallel forces of same magnitude
(B) Two like parallel forces of different magnitude
(C) Two unlike parallel forces of same magnitude
(D) Two unlike parallel forces of different magnitude
151. In (M/M/I): (/FIFO) model the effective arrival rate of customers is:
(A) μ(1-P0)
(B) ${ג}/{μ}$
(C) ${1}/{μ}$
(D) ג(1-P0)
152. Critical path of the following network problem is
(A) (1)→ (2)→(6)
(B)(1)→( 2)→(4)→(5)→ (6)
(C) ( 1)→(4)→(5)→(6)
(D) (1)→(3) →(6)
ANSWER: B
153. An item is produced at the rate of 50 items per day. The demand occurs at the rate of 25 items per day. If the set up cost is Rs. 100,00 and holding cost is Rs. 0.01 per unit of item per day, then what is the economic lot size for one run, if no shortage is allowed ?
(A) 600 units
(B) 800 units
(C) 1000 units
(D) 1200 units
ANSWER: C
154. If the following pay off matrix is strictly determinable,
Player B
B1 B2 B3
A1 λ 6 2
A2 -1 λ -7
A3 -2 4 λ
Player A
then the value of λ lies in the interval:
(A) [0, 4]
(B) [1, 3]
(C) [ -1, 2]
(D) [0,3]
155. A television repair man finds that the time spent on his jobs has an exponential distribution with mean 30 units. If he repairs sets in the order in which they came in, and if the arrival of sets is approximately Poisson Distribution with an average rate of 10 per 8 hour day, the repair man's expected idle time in each day (in hours) is:
(A) 1
(B) 2
C3
(D) 4
ANSWER: DOUBT
156. Customers arrive at a clinic at the rate of 8/hour, and the doctor can serve at the rate of 9/hour, arrivals follow Poisson distribution and service time follow exponential distribution. Then the probability that a customer does not join the queue and walks into the doctor's room is:
(A) 0
(B) ${1}/{9}$
(C) ${2}/{9}$
(D) ${2}/{7}$
157. The optimal solution of the following linear programming problem
minimize z=20x1 + 10x2 subject to the constraints
x1+2x2≤40
3x1+x2≥30
4x1+3x2≥60x1,x2≥0
(A) x1=6, x2=12, z=240
(B) X1=10, x2=4, z=240
(C) x1=5, x2=14 , z=240
(D) Infeasible
ANSWER: A
158. The optimum strategy of player A for the following
game A= $[\table \0, \1; \2, \1]$
(A) (${1}/{2}$,$ {1}/{2}$)
(B) (${1}/{3}$,$ {1}/{3}$)
(C) (0, 1)
(D) (1, 0)
159. The Assignment Cost (in rupees) for the following Assignment problem.
Persons
$[\table \JOBS, \I, \II, \III, \IV, \V; \A, \10, \5, \13, \15, \16; \B, \3, \9, \18, \13, \6; \C, \10, \7, \2, \2, \2; \D, \7, \11, \9, \7, \2; \E, \7, \9, \10, \4, \12]$
(A) Rs. 33
(B) Rs. 22
(C) Rs. 30
(D) Rs. 23
160. The following linear programming problem Maximize z=5x1+3x2 subject to the constraints 2x1+x2≤1
x1 +4x2≥6
x1,x2≥0 has:
(A) unique solution
(B) unbounded solution
(C) no feasible solution
(D) more than one solution
ANSWER: C
161. There are 64 beds in a garden and 3 seeds of a particular type of flower are sown in each bed. The probability of a flower being white is ${1}/{4}. Find the number of beds with 3 white flowers
(A) 9
(B) 3
(C) 27
(D) 1
162. In a series of houses actually invaded by small pox, 70% of the inhabitants are attacked and 85% have been vaccinated. What is the lowest percentage of vaccinated that must have been attacked?
(A) 64.7%
(B) 35.3%
(C) 74.7%
(D) 25.3%
163. In a certain distribution the following result were obtained X=45, median=48, coefficient of skewness-0.4. Find standard deviation.
(A) 21.5
(B) 22.5
(C) 23.5
(D) 24.5
ANSWER:B
164. Fisher's ideal index is
(A) Arithmetic mean of Laspeyre's and Paasche's Index
(B) Median of Laspeyre's and Paasche's Index
(C) Geometric mean of Laspeyre's and Paasche's index
(D) Harmonic mean of Laspeyre's and Paasche's index
165. Sixty percent of the employees of the XYZ corporation are college graduates. Of these 10 percent are in sales. Of the employees who did not graduate from college, eighty percent are in sales. What is the probability that an employee selected at random is in sales ?
(A) 0.41
(B) 0.08
(C) 0.38
(D) 0.21
166. Suppose two Events A and B are independent, then whether the following is/ are true
(1) The events A and BC are independent
(1) The events AC and B are independent
(iii) The events AC and BC are independent
(A) (1) alone true
(B) (ii) alone true
(C) (iii) alone true
(D) (1), (ii), (iii) true
ANSWER: D
167. A letter is known to have come either from TATANAGAR or from CALCUTTA. On the envelope just two consecutive letters TA are visible. What is the probability that the letter came from CALCUTTA ?
(A) ${1 }/{2}$
(B) ${4}/{11}&
(C) ${2}/{8}$
(D)$ {1}/{7}$
168. Ten coins are tossed simultaneously. The probability of getting atleast seven heads
(A) ${716 }/{1024}$
(B) ${617}/{1024}&
(C) ${176}/{1024}$
(D)$ {671}/{1024}$
169. The quartiles of Normal distribution are 8 and 14 respectively, then the mean and standard deviation respectively are
(A) 12, 4.5
(B) 9, 4.2
(C) 10, 4.3
(D) 11, 4.4
ANSWER: A
170. Given σ² =${11}/{25}$;μ3 = ${32}/{875}$;μ4=${3693}/{8750}. find the coefficient of skewness and coefficient of kurtosis.
(A) -0.1253,-2.172
(B) 0.1253, -2.172
(C) 0.1253, 2.172
(D) -0.1253, 2.172
171. The range of values of p and q that will render the entry (2, 2) a saddle point for the game:
Player B
B_{1} B_{2} B_{3}
A_{1} 2 4 5
A_{2} 10 7 q
A_{3} 4 P 6
Player A
(A) p>7 q>7
(B) p ≤7 q>7
(C) p<6 , q>6
(D) p≤5 , q≥7
ANSWER: B
172. Two players X and Y play a game of matching two coins. X wins one unit of value when both are head and he gets nothing when both 1 are tail. He loses ${1}/{2}$ unit of value when the coins do not match. Then the value of the game is:
(A)- ${1}/{2}$
(B) ${1}/{2}$
ANSWER: B
173. The initial basic feasible solution by using Vogel's approximation for the following transportation problem is:
To
D E F G Availability
From
A 11 13 17 14 250
B 16 18 14 10 300
C 21 24 13 10 400
Demand 200 225 275 250
(A) 12000
(B) 12075
(C) 13000
(D) 12025
ANSWER: B
174. In an assignment problem, if there are n jobs and n workers, then the total number of possible assignments is
(A) n! solutions
(B) (n-1)! solutions *
(C) n solutions
(D) $(n!)^n $ solutions
175. In a service counter, the arrival of customers follow poisson distribution with the arrival rate of 12 customers per hour and the service time follows exponential distribution with the service rate of 15 customers per hour. The probability of having 5 customers in the entire system is:
(A) $(0.8)^5$
(B) $(0.2) (0.8)^5$
(C) $(0.2)^5$
(D) $(0.16)^5$
ANSWER: B
176. Find the median of continuous random variable X, given that its density function is $f(x)=$ {1}/{π(1+x^{2})},- ∞ <x<∞
(A) 0
(B) 1
(C) 3
(D) 2
177. 'A' speaks truth 4 out of 5 times. A die is tossed. He reports that there is a six. What is the chance that actually there was six ?
(A) $5/9$
(B) $4/9$
(C) $7/9$
(D) $6/9$
ANSWER: B
178. The arithmetic mean of a series of 20 items were calculated by a student as 20 cm but while calculating an item 13 was misread as 30. Find the correct arithmetic mean.
(A) 19.10
(B) 19.15
(C) 19.20
(D) 19.25
ANSWER: B179. If Cov(X, Y)=10, Var(X)=16 and Var(Y) = 9, find the coefficient of correlation.
(A) 0.333
(B) 0.833
(C) 0,245
(D) 0.924
ANSWER:B
180. For a moderately skew data, the arithmetic mean is 200, the coefficient of variance is 8 and coefficient of skewness is 0.3. Find the mode.
(A) 198.4
(B) 195.2
(C) 197.3
(D) 193.1
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