Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1
The ratio of boys to girls in a class is 7 to 3.Among the options below, an acceptable value for the total number of students in the class is:
(A)
21
(B)
37
(C)
50
(D)
73
Q.2
A polygon is convex if, for every pair of points, P and Q belonging to the polygon, the line segment PQ lies completely inside or on the polygon.Which one of the following is NOT a convex polygon?.
Q.3
Consider the following sentences:Everybody in the class is prepared for the exam.Babu invited Danish to his home because he enjoys playing chess.Which of the following is the CORRECT observation about the above two sentences?
(A)
(i) is grammatically correct and (ii) is unambiguous
(B)
(i) is grammatically incorrect and (ii) is unambiguous
(C)
(i) is grammatically correct and (ii) is ambiguous
(D)
(i) is grammatically incorrect and (ii) is ambiguous
Q.4
A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like.
Q.5
_______is to surgery as writer is to____Which one of the following options maintains a similar logical relation in the above sentence?
(A)
Plan, outline
(B)
Hospital, library
(C)
Doctor, book
(D)
Medicine, grammar
Q. 6 – Q.10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).
Q.6
We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm x 1 cm each. Sheet M is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal square patches and assembled to form the largest possible closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that of the cube is_______
(A)
𝜋 / 2
(B)
3/𝜋
(C)
9 / 𝜋
(D)
3𝜋
Q.7
Details of prices of two items P and Q are presented in the above table. The ratio of cost of item P to cost of item Q is 3:4. Discount is calculated as the difference between the marked price and the selling price. The profit percentage is calculated as the ratio of the difference between selling price and cost, to the cost
(𝐏𝐫𝐨𝐟𝐢𝐭 % = × 𝟏𝟎𝟎).
The discount on item Q, as a percentage of its marked price, is________
(A)
25
(B)
12.5
(C)
10
(D)
5
Q.8
There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.The probability that at least two chocolates are identical is
(A)
0.3024
(B)
0.4235
(C)
0.6976
(D)
0.8125
Q.9
Given below are two statements 1 and 2, and two conclusions I and II.
Statement 1: All bacteria are microorganisms.
Statement 2:All pathogens are microorganisms.
Conclusion I: Some pathogens are bacteria.
Conclusion II: All pathogens are not bacteria.
Based on the above statements and conclusions, which one of the following options is logically CORRECT?
(A)
Only conclusion I is correct
(B)
Only conclusion II is correct
(C)
Either conclusion I or II is correct.
(D)
Neither conclusion I nor II is correct.
Q.10
Some people suggest anti-obesity measures (AOM) such as displaying calorie information in restaurant menus. Such measures sidestep addressing the core problems that cause obesity: poverty and income inequality.Which one of the following statements summarizes the passage?
(A)
The proposed AOM addresses the core problems that cause obesity.
(B)
If obesity reduces, poverty will naturally reduce, since obesity causes poverty.
(C)
AOM are addressing the core problems and are likely to succeed.
(D)
AOM are addressing the problem superficially.
Mathematics (MA)
– Q.14 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1
Let 𝑨 be a 𝟑 × 𝟒 matrix and 𝑩 be a 𝟒 × 𝟑 matrix with real entries such that 𝑨𝑩 is non-singular. Consider the following statements: P: Nullity of 𝑨 is 𝟎.Q: 𝑩𝑨 is a non-singular matrix.Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.2
Let (𝒛) = 𝒖(𝒙, 𝒚) + 𝒊 𝒗(𝒙, 𝒚) for 𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ, where 𝒙 and 𝒚 are real numbers, be a non-constant analytic function on the complex plane ℂ. Let 𝒖𝒙,𝒗𝒙and 𝒖𝒚, 𝒗𝒚denote the first order partial derivatives of 𝒖(𝒙, 𝒚) = 𝑹𝒆(𝒇(𝒛))and 𝒗(𝒙, 𝒚) = 𝑰𝒎(𝒇(𝒛)) with respect to real variables 𝒙 and 𝒚, respectively. Consider the following two functions defined on ℂ: 𝒈𝟏(𝒛) = 𝒖𝒙(𝒙, 𝒚) − 𝒊 𝒖𝒚 (𝒙, 𝒚) 𝐟𝐨𝐫 𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ,𝒈𝟐(𝒛) = 𝒗𝒙(𝒙, 𝒚) + 𝒊 𝒗𝒚(𝒙, 𝒚) 𝐟𝐨𝐫 𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ. Then
(A)
both 𝑔1(𝑧) and 𝑔2(𝑧) are analytic in ℂ
(B)
𝑔1(𝑧) is analytic in ℂ and 𝑔2(𝑧) is NOT analytic in ℂ
(C)
𝑔1(𝑧) is NOT analytic in ℂ and 𝑔2(𝑧) is analytic in ℂ
(D)
neither 𝑔1(𝑧) nor 𝑔2(𝑧) is analytic in ℂ
Q.3
Let 𝑻(𝒛) =, 𝒂𝒅 − 𝒃𝒄 ≠ 𝟎, be the Möbius transformation which maps the points 𝒛𝟏 𝟎, 𝒛𝟐 = −𝒊, 𝒛𝟑 = ∞ in the 𝒛-plane onto the points 𝒘𝟏 = 𝟏𝟎,𝒘𝟐 = 𝟓 − 𝟓𝒊, 𝒘𝟑 = 𝟓 + 𝟓𝒊 in the 𝒘-plane, respectively. Then the image of the set 𝑺 = {𝒛 ∈ ℂ ∶ 𝑹𝒆(𝒛) < 𝟎} under the map 𝒘 = 𝑻(𝒛) is
(A)
{𝑤 ∈ ℂ ∶ |𝑤| < 5}
(B)
{𝑤 ∈ ℂ ∶ |𝑤| > 5}
(C)
{𝑤 ∈ ℂ ∶ |𝑤 − 5| < 5}
(D)
{𝑤 ∈ ℂ ∶ |𝑤 − 5| > 5}
Q.4
Let 𝑹 be the row reduced echelon form of a 𝟒 × 𝟒 real matrix 𝑨 and let the third column of 𝑹 be . Consider the following statements:P: If is a solution of 𝑨𝐱 = 𝟎, then = 𝟎.Q: For all 𝐛 ∈ ℝ𝟒, 𝒓𝒂𝒏[𝑨| 𝐛] = 𝒓𝒂𝒏𝒌[𝑹| 𝐛]. Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.5
The eigenvalues of the boundary value problem
𝒙 ∈ (𝟎, 𝝅), 𝝀 > 𝟎, 𝒚(𝟎) = 𝟎, 𝒚(𝝅) − (𝝅) = 𝟎,
are given by
(A)
𝜆 = (𝑛𝜋)2, 𝑛 = 1,2,3, …
(B)
𝜆 = 𝑛2, 𝑛 = 1,2,3, …
(C)
𝜆 = 𝑘𝑛2, where 𝑘𝑛 , 𝑛 = 1,2,3, … are the roots of 𝑘 − tan(𝑘𝜋) = 0
(D)
𝜆 = 𝑘𝑛2, where 𝑘𝑛 , 𝑛 = 1,2,3, … are the roots of 𝑘 +tan(𝑘𝜋) = 0
Q.8
Consider the fixed-point iterationwith,and the initial approximation x0=3.25Then, the order of convergence of the fixed-point iteration method is
(A)
1
(B)
2
(C)
3
(D)
4
Q.9
Let {𝒆𝒏 ∶ 𝒏 = 𝟏, 𝟐, 𝟑, … } be an orthonormal basis of a complex Hilbert space𝑯. Consider the following statements:P: There exists a bounded linear functional 𝒇: 𝑯 → ℂ such that 𝒇(𝒆𝒏 ) = for 𝒏 = 𝟏, 𝟐, 𝟑, … .Q: There exists a bounded linear functional 𝒈: 𝑯 → ℂ such that 𝒈(𝒆𝒏 ) =for 𝒏 = 𝟏, 𝟐, 𝟑, … .Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.10
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.11
Consider the following statements:
Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.12
Let 𝒇: ℝ𝟑 → ℝ be a twice continuously differentiable scalar field such that 𝒅𝒊𝒗(𝛁𝒇) = 𝟔. Let 𝑺 be the surface 𝒙𝟐 + 𝟐 + 𝒛𝟐 = 𝟏 and 𝒏̂ be unit outward normal to 𝑺. Then the value of ∬𝑺 (𝛁𝒇 ⋅ 𝒏̂) 𝒅𝑺 is
(A)
2 𝜋
(B)
4 𝜋
(C)
6 𝜋
(D)
8 𝜋
Q.13
Consider the following statements:P: Every compact metrizable topological space is separable. Q: Every Hausdorff topology on a finite set is metrizable.Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.14
Consider the following topologies on the set ℝ of all real numbers:Τ𝟏 = {𝑼 ⊂ ℝ ∶ 𝟎 ∉ 𝑼 𝐨𝐫 𝑼 = ℝ},Τ𝟐 = {𝑼 ⊂ ℝ ∶ 𝟎 ∈ 𝑼 𝐨𝐫 𝑼 = ∅} ,Τ𝟑 = Τ𝟏 ∩ Τ𝟐.Then the closure of the set {𝟏} in (ℝ, Τ𝟑) is
(A)
{1}
(B)
{0,1}
(C)
ℝ
(D)
ℝ\{0}
Q.15 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).
Q.15
Q.16
Let ᴦ denote the boundary of the square region 𝑹 with vertices (𝟎, 𝟎), (𝟐, 𝟎), (𝟐, 𝟐) and (𝟎, 𝟐) oriented in the counter- clockwise direction. Then∮ᴦ(𝟏 − 𝒚𝟐) 𝒅𝒙 + 𝒙 𝒅𝒚 = .
Q.17
The number of 𝟓-Sylow subgroups in the symmetric group 𝑺𝟓of degree 𝟓 is__________.
Q.18
Let 𝑰 be the ideal generated by 𝒙𝟐 + 𝒙 + 𝟏 in the polynomial ring 𝑹 = ℤ[𝒙], where ℤ𝟑denotes the ring of integers modulo 𝟑. Then the number of units in the quotient ring 𝑹/𝑰 is.
Q.19
Let 𝑻: ℝ𝟑 → ℝ𝟑be a linear transformation such thatThen the rank of 𝑻 is________.
Q.20
Let 𝒚(𝒙) be the solution of the following initial value problemThen (𝟒) = .
Q.21
Let𝒇(𝒙) = 𝒙𝟒 + 𝟐 𝒙𝟑 − 𝟏𝟏 𝒙𝟐 − 𝟏𝟐 𝒙 + 𝟑𝟔 𝐟𝐨𝐫 𝒙 ∈ ℝ.The order of convergence of the Newton-Raphson methodwith 𝒙𝟎 = 𝟐. 𝟏, for finding the root 𝑎 = 𝟐 of the equation 𝒇(𝒙) = 𝟎 is______.
Q.23
Consider the Linear Programming Problem 𝑷:
Maximize 𝟐𝒙𝟏 + 𝟑𝒙𝟐 subject to
𝟐𝒙𝟏 + 𝒙𝟐 ≤ 𝟔,
−𝒙𝟏 + 𝒙𝟐 ≤ 𝟏,
𝒙𝟏 + 𝒙𝟐 ≤ 𝟑,
𝒙𝟏 ≥ 𝟎 and 𝒙𝟐 ≥ 𝟎.
Then the optimal value of the dual of 𝑷 is equal to.
Q.24
Consider the Linear Programming Problem 𝑷:
Minimize 𝟐𝒙𝟏 − 𝟓𝒙𝟐subject to
𝟐𝒙𝟏 + 𝟑𝒙𝟐 + 𝒔𝟏 = 𝟏𝟐,
−𝒙𝟏 + 𝒙𝟐 + 𝒔𝟐 = 𝟏,
−𝒙𝟏 + 𝟐𝒙𝟐 + 𝒔𝟑 = 𝟑,
𝒙𝟏 ≥ 𝟎, 𝒙𝟐 ≥ 𝟎, 𝒔𝟏 ≥ 𝟎, 𝒔𝟐 ≥ 𝟎, and 𝒔𝟑 ≥ 𝟎.
If is a basic feasible solution of 𝑷, then 𝒙𝟏 + 𝒔𝟏 + 𝒔𝟐 + 𝒔𝟑 = .
Q.25
Let 𝑯 be a complex Hilbert space. Let 𝒖, 𝒗 ∈ 𝑯 be such that 〈𝒖, 𝒗〉 = 𝟐. Then
.26 – Q.43 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).
Q.26
Let ℤ denote the ring of integers. Consider the subring𝑹 = {𝒂 + 𝒃 √−𝟏𝟕 ∶𝒂, 𝒃 ∈ ℤ} of the field ℂ of complex numbers. Consider the following statements:
P: 𝟐 + √−𝟏𝟕 is an irreducible element.
Q: 𝟐 + √−𝟏𝟕 is a prime element.Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.27
Consider the second-order partial differential equation (PDE)
Consider the following statements:
P: The PDE is parabolic on the ellipse + 𝒚𝟐 = 𝟏.
Q: The PDE is hyperbolic inside the ellipse + 𝒚𝟐 = 𝟏.Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.28
If 𝒖(𝒙, 𝒚) is the solution of the Cauchy problem 𝒖(𝒙, 𝟎) = −𝒙𝟐, 𝒙 > 𝟎,then 𝒖(𝟐, 𝟏) is equal to
(A)
1 − 2 𝑒−2
(B)
1 + 4 𝑒−2
(C)
1 − 4 𝑒−2
(D)
1 + 2 𝑒−2
Q.30
The critical point of the differential equation
+β𝟐𝒚 = 𝟎, 𝑎 > β > 𝟎,is a
(A)
node and is asymptotically stable
(B)
spiral point and is asymptotically stable
(C)
node and is unstable
(D)
saddle point and is unstable
Q.31
The initial value problem𝒕 > 𝟎, 𝒚(𝟎) = 𝟏,where 𝒇(𝒕, 𝒚) = 𝟏𝟎 𝒚, is solved by the following Euler method𝒚𝒏+𝟏 = 𝒚𝒏 + 𝒉 𝒇(𝒕𝒏, 𝒚𝒏), 𝒏 ≥ 𝟎,with step-size h. Then 𝒚𝒏 → 𝟎 as 𝒏 → ∞, provided
(A)
0 < ℎ < 0.2
(B)
0.3 < ℎ < 0.4
(C)
0.4 < ℎ < 0.5
(D)
0.5 < ℎ < 0.55
Q.32
Consider the Linear Programming Problem 𝑷: Maximize 𝒄𝟏𝒙𝟏 + 𝒄𝟐𝒙𝟐subject to 𝒂𝟏𝟏𝒙𝟏 + 𝒂𝟏𝟐𝒙𝟐 ≤ 𝒃𝟏, 𝒂𝟐𝟏𝒙𝟏 + 𝒂𝟐𝟐𝒙𝟐 ≤ 𝒃𝟐, 𝒂𝟑𝟏𝒙𝟏 + 𝒂𝟑𝟐𝒙𝟐 ≤ 𝒃𝟑, 𝒙𝟏 ≥ 𝟎 and 𝒙𝟐 ≥ 𝟎, where 𝒂𝒊𝒋, 𝒃𝒊 and 𝒄𝒋 are real numbers (𝒊 = 𝟏, 𝟐, 𝟑; 𝒋 =𝟏, 𝟐).Let be a feasible solution of 𝑷 such that 𝒑𝒄𝟏 + 𝒒𝒄𝟐= 𝟔 and let all feasible solutionsof 𝑷 satisfy −𝟓 ≤ 𝒄𝟏𝒙+𝒄𝟐𝒙𝟐 ≤ 𝟏𝟐.Then, which one of the following statements is NOT true?
(A)
𝑃 has an optimal solution
(B)
The feasible region of 𝑃 is a bounded set
(C)
If is a feasible solution of the dual of 𝑃, then 𝑏1𝑦1 + 𝑏2𝑦2 + 𝑏3𝑦3 ≥ 6
(D)
The dual of 𝑃 has at least one feasible solution
Q.33
Let 𝑳𝟐[−𝟏, 𝟏] be the Hilbert space of real valued square integrable functions on [−𝟏, 𝟏] equipped with the norm ‖𝒇‖ = (∫−𝟏𝟏 |𝒇(𝒙)|𝟐 𝒅𝒙)1/2.Consider the subspace 𝑴 = {𝒇 ∈ 𝑳2[−𝟏, 𝟏] ∶ ∫−𝟏𝟏 𝒇(𝒙)𝒅𝒙 = 𝟎}.For (𝒙) = 𝒙𝟐, define 𝒅 = 𝐢𝐧𝐟 {‖𝒇 − 𝒈‖ ∶ 𝒈 ∈ 𝑴 }. Then
(A)
(B)
(C)
(D)
Q.34
Let 𝑪[𝟎, 𝟏] be the Banach space of real valued continuous functions on [𝟎, 𝟏] equipped with the supremum norm. Define 𝑻: 𝑪[𝟎, 𝟏] → 𝑪[𝟎, 𝟏] byLet (𝑻) denote the range space of 𝑻. Consider the following statements:P: 𝑻 is a bounded linear operator.Q: 𝑻−𝟏: (𝑻) → 𝑪[𝟎, 𝟏] exists and is bounded. Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.35
Let 𝑃𝟏 = {𝒙 = (𝒙(𝟏), 𝒙(𝟐), … , 𝒙(𝒏), … ) | ∑𝒏=𝟏∞ |𝒙(𝒏)| < ∞} be the sequence space equipped with the norm ‖𝒙‖ = ∑𝒏=𝟏∞|𝒙(𝒏)|. Consider the subspaceand the linear transformation 𝑻: 𝑿 → 𝑃𝟏given by(𝑻𝒙)(𝒏) = 𝒏 (𝒏) for 𝒏 = 𝟏, 𝟐, 𝟑, … .Then
(A)
𝑇 is closed but NOT bounded
(B)
𝑇 is bounded
(C)
𝑇 is neither closed nor bounded
(D)
𝑇−1 exists and is an open map
Q.36
Let 𝒇𝒏: [𝟎, 𝟏𝟎] → ℝ be given by 𝒇n(𝒙) = 𝒏 𝒙𝟑 𝒆−𝒏𝒙for 𝒏 = 𝟏, 𝟐, 𝟑, … . Consider the following statements:P: (𝒇𝒏) is equicontinuous on [𝟎, 𝟏𝟎].Q: ∑𝒏=𝟏∞𝒇𝒏does NOT converge uniformly on [𝟎, 𝟏𝟎].Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.37
Let 𝒇: ℝ𝟐 → ℝ be given by
Consider the following statements:P: 𝒇 is continuous at (𝟎, 𝟎) but 𝒇 is NOT differentiable at (𝟎, 𝟎).Q: The directional derivative 𝑫𝒖(𝟎, 𝟎) of 𝒇 at (𝟎, 𝟎) exists in the direction of every unit vector 𝒖 ∈ ℝ𝟐.Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.38
Let 𝑽 be the solid region in ℝ𝟑bounded by the paraboloid 𝒚 = (𝒙𝟐 + 𝒛𝟐) and the plane 𝒚 = 𝟒. Then the value of ∭𝑽𝟏𝟓 √𝒙𝟐 + 𝒛𝟐 𝒅𝑽 is
(A)
128 𝜋
(B)
64 𝜋
(C)
28 𝜋
(D)
256 𝜋
Q.39
Let 𝒇: ℝ𝟐 → ℝ be given by (𝒙, 𝒚) = 𝟒𝒙𝒚 − 𝟐 𝒙𝟐 − 𝒚𝟒. Then 𝒇 has
(A)
a point of local maximum and a saddle point
(B)
a point of local minimum and a saddle point
(C)
a point of local maximum and a point of local minimum
(D)
two saddle points
Q.40
The equation 𝒙𝒚 − 𝒛 𝐥𝐨𝐠 𝒚 + 𝒆𝒙𝒛 = 𝟏 can be solved in a neighborhood of the point (𝟎, 𝟏, 𝟏) as 𝒚 = (𝒙, 𝒛) for some continuously differentiable function 𝒇. Then
(A)
∇𝑓(0, 1) = (2, 0)
(B)
∇𝑓(0, 1) = (0, 2)
(C)
∇𝑓(0, 1) = (0, 1)
(D)
∇𝑓(0, 1) = (1, 0)
Q.41
Consider the following topologies on the set ℝ of all real numbers.Τ𝟏is the upper limit topology having all sets (𝒂, 𝒃] as basis.Τ𝟐 = {𝑼 ⊂ ℝ ∶ ℝ\𝑼 𝐢𝐬 𝐟𝐢𝐧𝐢𝐭𝐞} 𝖴 {∅}.Τ𝟑is the standard topology having all sets (𝒂, 𝒃) as basis. Then
(A)
Τ2 ⊂ Τ3 ⊂ Τ1
(B)
Τ1 ⊂ Τ2 ⊂ Τ3
(C)
Τ3 ⊂ Τ2 ⊂ Τ1
(D)
Τ2 ⊂ Τ1 ⊂ Τ3
Q.42
Let ℝ denote the set of all real numbers. Consider the following topological spaces.𝑿𝟏 = (ℝ, Τ𝟏), where Τ𝟏 is the upper limit topology having all sets (𝒂, 𝒃] as basis.𝑿𝟐 = (ℝ, Τ𝟐), where Τ𝟐 = {𝑼 ⊂ ℝ ∶ ℝ\𝑼 𝐢𝐬 𝐟𝐢𝐧𝐢𝐭𝐞} 𝖴 {∅}. Then
(A)
both 𝑋1 and 𝑋2 are connected
(B)
𝑋1 is connected and 𝑋2 is NOT connected
(C)
𝑋1 is NOT connected and 𝑋2 is connected
(D)
neither 𝑋1 nor 𝑋2 is connected
Q.43
Let 〈∙, ∙〉: ℝ𝒏 × ℝ𝒏 → ℝ be an inner product on the vector space ℝ𝒏over ℝ. Consider the following statements:P: |〈𝒖, 𝒗〉| ≤ (〈𝒖, 𝒖〉 + 〈𝒗, 𝒗〉) for all 𝒖, 𝒗 ∈ ℝ𝒏.Q: If 〈𝒖, 𝒗〉 = 〈𝟐𝒖, −𝒗〉 for all 𝒗 ∈ ℝ𝒏, then 𝒖 = 𝟎.Then
(A)
both P and Q are TRUE
(B)
P is TRUE and Q is FALSE
(C)
P is FALSE and Q is TRUE
(D)
both P and Q are FALSE
Q.44 -Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).
Q.44
Let 𝑮 be a group of order 𝟓𝟒with center having 𝟓𝟐elements. Then the number of conjugacy classes in 𝑮 is____.
Q.45
Let 𝑭 be a finite field and 𝑭×be the group of all nonzero elements of 𝑭 under multiplication. If 𝑭×has a subgroup of order 𝟏𝟕, then the smallest possible order of the field 𝑭 is______.
Q.46
Let 𝑹 = {𝒛 = 𝒙 + 𝒊𝒚 ∈ ℂ ∶ 𝟎 < 𝒙 < 𝟏 a𝐧𝐝 − 𝟏𝟏 𝝅 < 𝒚 < 𝟏𝟏 𝝅} and ᴦ be the positively oriented boundary of 𝑹. Then the value of the integralis ___________.
Q.47
Let 𝑫 = {𝒛 ∈ ℂ ∶ |𝒛| < 𝟐𝝅} and 𝒇: 𝑫 → ℂ be the function defined by
If (𝒛) = ∑∞𝒏=𝟎𝒂𝒏 𝒛𝒏for 𝒛 ∈ 𝑫, then 𝟔𝒂𝟐=____________ .
Q.48
The number of zeroes (counting multiplicity) of 𝑷(𝒛) = 𝟑𝒛𝟓 + 𝟐𝒊 𝒛𝟐 + 𝟕𝒊 𝒛 +𝟏 in the annular region {𝒛 ∈ ℂ ∶ 𝟏 < |𝒛| < 𝟕} is____________.
Q.49
Let 𝑨 be a square matrix such that 𝐝𝐞(𝒙𝑰 − 𝑨) = 𝒙𝟒(𝒙 − 𝟏)𝟐(𝒙 − 𝟐)𝟑, where 𝐝𝐞𝐭(𝑴) denotes the determinant of a square matrix 𝑴.If 𝐫𝐚𝐧(𝑨𝟐) < 𝐫𝐚𝐧𝐤(𝑨𝟑) = 𝐫𝐚𝐧𝐤(𝑨𝟒), then the geometric multiplicity of the eigenvalue 𝟎 of 𝑨 is.
Q.50
If 𝒚 = ∑∞𝒌=𝟎𝒂𝒌𝒙𝒌, (𝒂𝟎 ≠ 𝟎) is the power series solution of the differentialequation− 𝟐𝟒 𝒙𝟐𝒚 = 𝟎, then =_________.
Q.51
If 𝒖(𝒙, 𝒕) = 𝑨 𝒆−𝒕 𝐬𝐢𝐧 𝒙 solves the following initial boundary value problem
then 𝝅 𝑨 = ______.
Q.52
Let 𝑽 = {𝒑 ∶ (𝒙) = 𝒂𝟎 + 𝒂𝟏𝒙 + 𝒂𝟐𝒙𝟐, 𝒂𝟎, 𝒂𝟏, 𝒂𝟐 ∈ ℝ } be the vector space of all polynomials of degree at most 𝟐 over the real field ℝ. Let 𝑻: 𝑽 →𝑽 be the linear operator given by 𝑻(𝒑) = (𝒑(𝟎) − 𝒑(𝟏)) + (𝒑(𝟎) + 𝒑(𝟏)) 𝒙 + 𝒑(𝟎) 𝒙𝟐.Then the sum of the eigenvalues of 𝑻 is _____ .
Q.53
The quadrature formula𝟐∫ 𝒙 𝒇(𝒙) 𝒅𝒙 ≈ 𝑎 𝒇(𝟎) + 𝖰 𝒇(𝟏) + 𝒇(𝟐)𝟎is exact for all polynomials of degree ≤ 𝟐. Then 𝟐β−🇾= .
Q.54
For each 𝒙 ∈ (𝟎, 𝟏], consider the decimal representation 𝒙 = ∙ 𝒅𝟏𝒅𝟐𝒅𝟑 ⋯ 𝒅𝒏 ⋯. Define 𝒇: [𝟎, 𝟏] → ℝ by 𝒇(𝒙) = 𝟎 if 𝒙 is rational and 𝒇(𝒙) = 𝟏𝟖 𝒏 if 𝒙 is irrational, where 𝒏 is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of 𝒙. Then the Lebesgue integral ∫𝟏𝟎 (𝒙) 𝒅𝒙 = _______.
Q.55
Let𝒙̃=be an optimal solution of the following Linear Programming Problem 𝑷:
Maximize 𝟒𝒙𝟏 + 𝒙𝟐 − 𝟑𝒙𝟑subject to
𝟐𝒙𝟏 + 𝟒𝒙𝟐 + 𝒂𝒙𝟑 ≤ 𝟏𝟎,
𝒙𝟏 − 𝒙𝟐 + 𝒃𝒙𝟑 ≤ 𝟑,
𝟐𝒙𝟏 + 𝟑𝒙𝟐 + 𝟓𝒙𝟑 ≤ 𝟏𝟏,
𝒙𝟏 ≥ 𝟎, 𝒙𝟐 ≥ 𝟎 and 𝒙𝟑 ≥ 𝟎,
where 𝒂, 𝒃 are real numbers.If 𝒚̃ =is an optimal solution of the dual of 𝑷, then 𝒑 + 𝒒 + 𝒓 =_________(Round off to two decimal places).
Q.1
× 𝟏𝟎𝟎).
, 𝒂𝒅 − 𝒃𝒄 ≠ 𝟎, be the Möbius transformation which maps the points 𝒛𝟏 𝟎, 𝒛𝟐 = −𝒊, 𝒛𝟑 = ∞ in the 𝒛-plane onto the points 𝒘𝟏 = 𝟏𝟎,𝒘𝟐 = 𝟓 − 𝟓𝒊, 𝒘𝟑 = 𝟓 + 𝟓𝒊 in the 𝒘-plane, respectively. Then the image of the set 𝑺 = {𝒛 ∈ ℂ ∶ 𝑹𝒆(𝒛) < 𝟎} under the map 𝒘 = 𝑻(𝒛) is
𝒙 ∈ (𝟎, 𝝅), 𝝀 > 𝟎, 𝒚(𝟎) = 𝟎, 𝒚(𝝅) − 
(𝝅) = 𝟎, 
with ,and the initial approximation x0=3.25Then, the order of convergence of the fixed-point iteration method is
for 𝒏 = 𝟏, 𝟐, 𝟑, … .Q: There exists a bounded linear functional 𝒈: 𝑯 → ℂ such that 𝒈(𝒆𝒏 ) =
for 𝒏 = 𝟏, 𝟐, 𝟑, … .Then
Then the rank of 𝑻 is________ .
Then (𝟒) = .
with 𝒙𝟎 = 𝟐. 𝟏, for finding the root 𝑎 = 𝟐 of the equation 𝒇(𝒙) = 𝟎 is______.
+ 𝒚𝟐 = 𝟏.
𝒖(𝒙, 𝟎) = −𝒙𝟐, 𝒙 > 𝟎,then 𝒖(𝟐, 𝟏) is equal to
+β𝟐𝒚 = 𝟎, 𝑎 > β > 𝟎,is a
𝒕 > 𝟎, 𝒚(𝟎) = 𝟏,where 𝒇(𝒕, 𝒚) = 𝟏𝟎 𝒚, is solved by the following Euler method𝒚𝒏+𝟏 = 𝒚𝒏 + 𝒉 𝒇(𝒕𝒏, 𝒚𝒏), 𝒏 ≥ 𝟎,with step-size h. Then 𝒚𝒏 → 𝟎 as 𝒏 → ∞, provided
is a feasible solution of the dual of 𝑃, then 𝑏1𝑦1 + 𝑏2𝑦2 + 𝑏3𝑦3 ≥ 6
Let (𝑻) denote the range space of 𝑻. Consider the following statements:P: 𝑻 is a bounded linear operator.Q: 𝑻−𝟏: (𝑻) → 𝑪[𝟎, 𝟏] exists and is bounded. Then
and the linear transformation 𝑻: 𝑿 → 𝑃𝟏 given by(𝑻𝒙)(𝒏) = 𝒏 (𝒏) for 𝒏 = 𝟏, 𝟐, 𝟑, … . Then
(〈𝒖, 𝒖〉 + 〈𝒗, 𝒗〉) for all 𝒖, 𝒗 ∈ ℝ𝒏.Q: If 〈𝒖, 𝒗〉 = 〈𝟐𝒖, −𝒗〉 for all 𝒗 ∈ ℝ𝒏, then 𝒖 = 𝟎.Then
is ___________.
be an optimal solution of the following Linear Programming Problem 𝑷: 
is an optimal solution of the dual of 𝑷, then 𝒑 + 𝒒 + 𝒓 =_________(Round off to two decimal places).