GATE MATHEMATICS 2018
Previous Year Question Paper with Solution.
1. Consider the vector space V = {a0 + a1x + a2x2 : a1
R for i = 0, 1, 2} of polynomials of degree at most 2. Let f : V
R be a linear functional such that f(1 + x) = 0, f(1 – x2) = 0 and f(x2 – x) = 2. Then f(1 + x + x2) equals _________.
Ans. (a)
Sol. Let f : V R be a linear functional such that
f(1 + x) = 0, f(1 – x2) = 0
and f(x2 – x) = 2
We need to find the image of (1 + x + x2) under the linear function f.
Let there exist scalar and
such that
On comparing the constant term coefficient of x and x2, we get
Solving for and
, we get
Therefore, from Eq. (i), we get
Taking the image under linear function f, we get
Using the property of linearly, we get
Substituting the values of f(1 + x), f(1 – x2) and f(x2 – x), we get
2. Let A be a 7 × 7 matrix such that 2A2 – A4 = I, where I is the identity matrix. If A has two distinct eigenvalues and each eigenvalue has geometric multiplicity 3, then the total number of non-zero entries in Jordan canonical form of A equals _________.
Ans.
Sol. Let A = [aij]n × n be a square matrix of order n.
We know that,
Geometric multiplicity of
= Number of linearly independent eigenvector of matrix A.
=
According to question,
Total number of non-zero entries on the Jordan canonical form of A are 4.
(Since, if the number of the vectors are linearly independent, then the rank of the matrix of the given is equal to the number of vectors).
3. Let f(z) = (x2 + y2) + i2xy and g(z) = 2xy + i(y2 – x2) for z = x + iy C. Then, in the complex plane C
(a) f is analytic and g is not analytic
(b) f is not analytic and g is analytic
(c) Neither f nor g is analytic
(d) Both f and g are analytic
Ans. (b)
Sol. Given, f(z) = (x2 + y2) + i(2xy) ...(i)
and g(z) = 2xy + i(y2 – x2) ...(ii)
For z = x + iy C
From Eq. (i),
Here u(x, y) = x2 + y2 and v(x, y) = y
Since, u and v are polynomials in x and y, therefore they are continuous at each point.
f(z) is continuous everywhere,
u and v do not satisfy Cauchy-Riemann equations.
f(z) is not analytic.
From Eq. (ii),
Here, u(x, y) = 2xy and v(x, y) = y2 – x2
Since, u and v are polynomials in x and y, therefore they are continuous at each point.
g(z) is continuous.
u and v satisfy Cauchy-Riemann equations.
g(z) is analytic.
4. If is the Laurent series of the function f(z) =
, then a–2 equals _________.
Ans. 48
Sol.
5. Let fn : [0, 1] R be given by fn(x) =
, n = 1, 2, ... . Then, the sequence (fn)
(a) converges uniformly on [0, 1]
(b) does not converge uniformly on [0, 1] but has a subsequence that converges uniformly on [0, 1]
(c) does not converge pointwise on [0, 1]
(d) converges pointwise on [0, 1] but does not have a subsequence that converges uniformly on [0, 1]
Ans. (d)
Sol. We have, fx : [0, 1] R be given by
But depends on x, hence fn(x) is pointwise converse to f(x) but not uniformly.
6. Let C : x2 + y2 = 9 be the circle on R2 oriented positively.
Then, equals _________.
Ans. 36
Sol.
7. Consider the following statements:
P : There exists an unbounded subset of R whose Lebesgue measure is equal to 5.
Q : If f : R R is continuous and g : R
R is such that f = g almost everywhere on R, then g must be continuous almost everywhere on R.
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (d)
Sol. P : The set of rational numbers Q which is an unbounded subset of R with Lebesque outer measure is zero.
In addition, R is an unbounded subset of itself with Lebesque outer measure is +.
Also, we know that every set of finite measure is 'nearly bounded'.
Q : If f is continuous on [a, b], then function g defined by 222 22
g(x) =
is continuous on [a, b] and differentiable on (a, b) and
8. If x3y2 is an integerating factor of (6y2 + axy)dx + (6xy + bx2)dy = 0, where a, b R, then
(a) 3a – 5b = 0
(b) 2a – b = 0
(c) 3a + 5b = 0
(d) 2a + b = 0
Ans. (a)
Sol. Here, m = 6y2 + axy and n = 6xy + bx2
Multiplying both sides the above differential equation by, xαy β, so that it reduces in the form of
m1dx + n1dy = 0
From Eq. (i), we get
From Eq. (ii), we get
Therefore, integrating factor (IF)
Now, from question,
Given that, IF = x3y2
3a – 5b = 0.
9. If x(t) and y(t) are the solutions of the system = y and
= –x with the initial conditions x(0) = 1 and y(0) = 1, then
equals _________.
Ans. 0
Sol.
Similarly, y(t) = –c1 sin t + c2 cos t
10. If y = 3e2x + e–2x – ax is the solution of the initial value problem +
y = 4
x, y(0) = 4 and
(0) = 1, where
,
R, then
(a) = 3 and
= 4
(b) = 1 and
= 2
(c) = 3 and
= –4
(d) = 1 and
= –2
Ans. (c)
Sol. If G is a non-abelian p-group and |G| = pn
from question pn = 53
n = 3.
11. Let G be a non-abelian group of order 125. Then, the total number of elements in Z(G) = {x G : gx = xg for all g
G} equals _________.
Ans. 3
Sol. Given,
F1 consisting number of elements = 29
F2 consisting number of elements = 26
then total number of elements in F1
F2
= 2gcd (9, 6)
= 23 = 8.
12. Let F1 and F2 be subfields of a finite field F consisting of 29 and 26 elements, respectively. Then, the total number of elements in F1
F2 equals _________.
Ans. 8
Sol. Hahh-Banach extension theorem
Let Y be a subspace of X and let f : Y K be a bounded linear functional we wish to find a linear function g : X
k such that
(i) g(y) = f(y) for every y Y and (ii) ||g|| = ||f||
If this happen, then we say that g is a norm-preserving extension of f above implies that ||f|| ≤ ||g||.
According to question,
(i) ||g|| = ||f||
and also,
From Eqs. (i) and (ii), we get, = 0 and
= 0
13. Consider the normed linear space R2 equipped with the norm given by ||(x, y)|| = |x| + |y| and the subspace X = {(x, y) R2 : x = y}. Let f be the linear functional on X given by f(x, y) = 3x. If g(x, y) =
x +
y,
,
R, is a Hahn-Banach extension of f on R2, then
–
equals _________.
Ans. 0
Sol.
14. For n Z, define cn =
, where i2 = –1. Then
equals
(a)
(b)
(c)
(d)
Ans. (d)
Sol.
15. If the fourth order divided difference f(x) = ax4 + 5x3 + 3x + 2
R, at the points 0.1, 0.2, 0.3, 0.4, 0.5 is 5, then a equals _________.
Ans. 5
Sol. We have, f(x) = x4 + 5x3 + 3x + 2,
R
At the points 0.1, 0.2, 0.3, 0.4, 0.5, fourth order divided difference is 5.
Then, f(0. 1) = 0.001 + 2.305
f(0. 2) = 0.0016 + 2.64
f(0. 3) = 0.0081 + 3.035
f(0. 4) = 0.0256 + 3.52
f(0. 5) = 0.0625 + 4.125
16. If the quadrature rule where c1, c2
R, is exact for all polynomials of degree < 1, then c1 + 3c2 equals _________.
Ans. 1
Sol.
17. If u(x, y) = 1 + x + y + f(xy), where f : R2
R is differentiable function, then u satisfies
(a)
(b)
(c)
(d)
Ans. (c)
Sol.
18. The partial differential equation, = 0 is
(a) hyperbolic along the line x + y = 0
(b) elliptic along the line x – y = 0
(c) elliptic along the line x + y = 0
(d) parabolic along the line x + y = 0
Ans. (d)
Sol.
19. Let X and Y be topological space and let f : X Y a continuous subjective function. Which one of the following statement is true?
(a) If X is separable, then Y is separable
(b) If X is first countable, then Y is first countable
(c) If X is Hausdorff, then Y is Hausdorff
(d) If X is regular, then Y is regular
Ans. (b)
Sol. If f : X Y is a continuous surjective function on topological spaces X and Y, then if X is a first countable space, then Y is also first countable space.
Any function taking limits of sequence to limit of sequence is continuous. In particular, the converse holds, if X is a metric space. When using nets instead of sequence, this converse holds for a general topological space.
20. Consider the topology T = {U Z : Z/U is finite 0
U} on Z. Then, the topological space (Z, T) is
(a) Compact but not connected
(b) Connected but not compact
(c) Both compact and connected
(d) Neither compact nor connected
Ans. (a)
Sol.
21. Let F(x) be the distribution function of a random variable X. Consider the functions
G1(x) = {F(x)}3, x R,
G2(x) = 1 – {1 – F(x)}5, x R
Which of the above functions are distribution functions?
(a) Neither G1 nor G2
(b) Only G1
(c) Only G2
(d) Both G1 and G2
Ans. (d)
Sol. Since, f(x) be the distribution function of a random variable X.
G1(x) be the distribution function of a random variable X.
And also,
G2(x) be the distribution function of a random variable X.
22. Let X1, X2, ..., Xn (n > 2) be independent and identically distributed random variables with finite variance and let
Then, the convariance between
and X1 –
is
(a) 0
(b)
(c)
(d)
Ans. (a)
Sol. We know that, if X and Y are two random variables, then covariance between them in defined as cov(X, Y) = E(XY) – E(X)E(Y)
and if X and Y are independent, then cov (X, Y) = 0
Given that, X1, X2, ..., Xn (n > 2) be independent and identically distributed random variables with finite variance .
and X1 –
are independent, therefore
= 0.
23. Let X1, X2, ... Xn (n > 2) be a random sample from a N(µ, ) population, where
= 144. The smallest n such that the length of the shortest 95% confidence interval for µ will not exceed 10 is _________.
Ans. 23
Sol.
24. Consider the linear programming problem (LPPT)
Maximize 4x1 + 6x2
x1 > 6, x2 is unrestricted in sign.
Then, the LPP has
(a) no optimal solution
(b) only one basic feasible solution and that is optimal
(c) more than one basic feasible solution and a unique optimal solution
(d) infinitely many optimal solutions
Ans. (b)
Sol. Since, x2 is unrestricted so that, we definie
Now, using these new variables and changing inequality into equation, we get
max z = 4x1 + 6x2
25. For a linear programming problem (LPP) and its dual, which one of the following is not true?
(a) The dual of the dual is primal
(b) If the primal LPP has an unbounded objective function, then the dual LPP is infeasible
(c) If the primal LPP is infeasible, then the dual LPP must have unbounded objective function
(d) If the primal LPP has a finite optimal solution, then the dual LPP also has a finite optimal solution
Ans. (c)
Sol. LPP and its dual.
(a) Dual of the dual is the primal (which is correct)
(b) Unboundness property: If the primal (dual) problem has an unbounded solution, then the dual (primal) problem is infeasible.
(c) If the primal is infeasible and that the dual feasible region is exactly the primal feasible region.
(d) Strong Duality Property: If the primal (dual) problem has a finite optimal solution, then the dual LPP also has finite optimal solution.
26. If U and V are the null space of and
, respectively, then the dimension of the subspace U + V equals _________.
Ans. 2
Sol. The nullity of a matrix A is the dimension of its null space and is denoted by nullity (A).
If A is an m × n matrix, then rank(A) + nullity(A) = n
From question
dim(U + V) = dim(U) + dim(V) – dim(U V)
= (4 – 2) + (4 – 2) – 2
= 2 + 2 – 2 = 2.
27. Given two n × n matrices A and B with entries in C, consider the following statements:
P : If A and B have the same minimal polynomial then A is similar to B.
Q : If has n distinct eigenvalues, then there exists u Cn such that u, Au, ..., An – 1 u are linearly independent.
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (c)
Sol. P : If two matrices are similar, then their minimal polynomial are the same one.
Q : We know that, if be distinct eigenvalues of an n × n matrix, the corresponding eigencectors X1, X2, ..., Xn from a linear independent set.
28. Let A = (aij) be a 10 × 10 matrix such that aij = 1 for i j and aij =
+ 1, where
> 0. Let
and µ be the largest and the smallest eigenvalues of A, respectively. If
+ µ = 24, then
equals _________.
Ans. 7
Sol.
29. Let C be the simple, positively oriented circle of radius 2 centred at the origin in the complex plane. Then equals _________.
Ans. 3
Sol.
30. Let Re(z) and Im(z), respectively, denote the real part and the imaginary part of a complex number z. Let T : be the bilinear transformation such that T(6) = 0, T(3 – 3i) = i and T(0) =
. Then, the image of D = {z
C : |z – 3| < 3} under the mapping w = T(z) is
(a) {w C : Im(w) < 0}
(b) {w C : Re(w) < 0}
(c) {w C : Im(w) > 0}
(d) {w C : Re(w) > 0}
Ans. (d)
Sol.
31. Let (xn) and (yn) be two sequences in a complete metric space (X, d) such that d(xn, xn + 1) and d(yn, yn + 1)
for all n
N. Then
(a) Both (xn) and (yn) converge
(b) (xn) converges but (yn) need not converge
(c) (yn) converges but (xn) need not converge
(d) Neither (xn) nor (yn) converges
Ans. (b)
Sol. Let (X, d) be a matric space, let {xn} be a sequence in x and suppose that
For all n 1, the distance between two consecutives terms xm, xm + 1 halves every times in increases.
Then, {xn} is Cauchy. To show this, first note that by the assumption, we have
where, we used the formula for geometric series on the last line. This last quantity converse to 0 as m
. So, for
> 0.
Let N be large enough that < for m > N.
Then, d(xn, xm) < whenever n, m > N this prove that {xn} is Cauchy.
Also, we know that, a matric space (x, d) is said to be complete if every Cauchy sequence in x converges.
32. Let f : [0, 1] R be given by f(x) = 0 if x is rational, and if x is irrational, then f(x) = 9n, where n is the number of zeroes immediately after the decimal point in he decimal representation of x. Then the Lebesgue integral
equals _________.
Ans. 9
Sol.
33. Let f : R2
R be defined by
f(x, y) = .
Then, at (0, 0)
(a) f is continuous and the directional derivative of f does not exist in some direction
(b) f is not continuous and the directional derivatives of f exist in all directions
(c) f is not differentiable and the directional derivative of f exist in all directions
(d) f is differentiable
Ans. (c)
Sol. Firstly, we prove that
We have,
For any given > 0,
> 0 such that |f(x, y) – 0| <
, whenever |x| <
and |y| <
But it is given that f(0, 0) = 0
f(x, y) is continuous at (0, 0).
Now, we check differentiability at (0, 0).
We have,
So, that A = 0, B = 0, which does not depends on h and k and
Let us h = mk, we get
Which does not depends on m.
f is not differentiable at (0, 0).
Not define, if a1 = 0
Directional derivative of f does not exist in some direction.
34. Let D be the region in R2 bounded by the parabola y2 = 2x and the line y = x. Then, equals _________.
Ans. 2
Sol. If D be the region bounded by the parabola y2 = 2x and the line y = x, then intersection points are (0, 0) and (2, 2) Taking limit according to x-axis
35. Let y1(x) = x3 and y2(x) = x2 |x| for x R.
P : y1(x) and y2(x) are linearly independent solutions of + 6y = 0 on R.
Q : The Wronskian = 0 for all x
R.
Ans.
Sol.
36. Let and
with
>
be the roots of the indicial equation of
– y = 0 at x = –1. Then
– 4
equals _________.
Ans. 2
Sol.
37. Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. Then the total number of elements of S9 that commute with = (123)(4567) in S9 equals _________.
Ans. 24
Sol. We have, = (1 2 3) (4 5 6 7)
=
= (1 2 3) (4 5 6 7) (8) (9)
Hence,
So, number of permutation which commutes with 𝜏 are
= 12 × 31 × 41 × 2! × 1! × 1!
= 12 × 2! = 24.
38. Let Q[x] be the ring of polynomials over Q. Then, the total number of maximal ideals in the quotient ring equals _________.
Ans. 3
Sol.
39. Let {en : n N} be an orthonormal basis of a Hilbert space H. Let T : H
H be given by Tx =
. For each n
N, define Tn : H
H by Tnx =
. Then
(a) ||Tn – T|| 0 and n
(b) ||Tn – T|| 0 and n
but for each x
H, ||Tnx – Tx||
(c) For each x H, ||Tnx – Tx||
0 and n
∞ but the sequence (||T||) is unbounded
(d) There exist x, y H such that
Ans. (a)
Sol. Let B{e1, e2, ...} = {en: n N}
then B is orthonormal basis because ||en|| = 1 for each n and if n ≠ m, then < en, em > = 0
Also, if =
It is easy to see that < x, en > = xn.
The idea of an orthonormal basis is that we can express Tx
For each n N, define Tn : H
H
40. Consider the subspace V = {(xn)
l2 :
the Hilbert space l2 of all square summable sequences. For n
N, define Tn : V
R by Tn((xk)) =
.
P : {Tn : n N} is pointwise bounded on V
Q : {Tn : n N} is uniformly bounded {x
V : ||x||2 = 1}
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (b)
Sol. Consider the space of square summable sequences.
l2 =
It is a subspace with respect to pointwise operations.
We define, (x, y) =
This series converges absolutely as for every finite n N, the Cauchy-Schwarz inequality implies
and the right hand side is uniformly bounded. It can be shown that l2 is complete. Moreover, it is easy to show that the subset of rational-valued sequence.
41. Let p(x) be the polynomial of degree at most 2 that interpolates the data (–1, 2), (0, 1) and (1, 2). If q(x) is polynomial of degree at most 3 such that p(x) + q(x) interpolates the data (–1, 2), (0, 1), (1, 2) and (2, 11) then q(3) equals _________.
Ans. 24
Sol. The difference table for p(x) is obtained as.
The forward difference polynomial (h = 1) is
Since, p(x) be the polynomial of degree at most 2.
= 2 + (x + 1) (–1) + (x + 1)(x)
= 2 – x – 1 + x2 + x
and, the difference table for is obtained as
The forward difference polynomial (h = 1) is
Since, h(x) be the polynomial of degree at most 3.
p(x) + q(x) = 2 – (x + 1) + (x + 1) (x) + (x2 – 1) (x)
(x2 + 1) + q(x) = 2 – x – 1 + x2 + x + x3 – x
(x2 + 1) + q(x) = x3 + x2 – x + 1
q(x) = x3 + x2 – x + 1 – (x2 + 1)
q(x) = x3 – x
q(x) = (3)3 – 3 = 24.
42. Let J be the Jacobi iteration matrix of the linear system .
Consider the following statements:
P : One of the eigenvalues of J lies in the interval [2, 3].
Q : The Jacobi iteration converges for the system.
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans.
Sol.
43. Let u(x, y) be the solution of = 4u satisfying the condition u(x, y) = 1 on the circle x2 + y2 = 1. Then u(2, 2) equals _________.
Ans. 64
Sol.
Let parametric form for circle x2 + y2 = 1 be
So, we have
Squaring both sides, we get
From Eqs. (ii) and (iii),
From Eq. (i),
44. Let u(r, ) be the bounded solution of the following
Then
(a) 1
(b)
(c)
(d)
Ans. (c)
Sol.
45. Let Tu and Td denote the usual topology and the discrete topology on R, respectively. Consider the following three topologies
T1 = Usual topology on R2 = R × R,
T2 = Topology generated by the basis {U × V : U Td, V
Tu) on R × R
T3 = Dictionary order topology on R × R.
Then
(a)
(b)
(c)
(d)
Ans. (d)
Sol. Let X1 denotes the topological space R with discrets topology and let X2 be R with usual topology, then the product topology on R × R is nothing but the dictionary order topology on R2. Since the basis for the product topology on R × R is given by [{X1} × (a, b) : x1, a, b,
R] any open set in the dictionary order topology is union of open sets in the product topology.
We also note that the product topology is finer than the usual topology on R2. In fact, any basis element (a, b) × (c, d) of the usual topology can be expressed as the union Ua < x < b{x} × (c, d) of open sets {x} × {c, d} in the product topology
.
46. Let X be a random variable with probability mas function P(n) = for n = 1, 2, ...
Then, E(X – 3|X > 3) equals _________.
Ans. 4
Sol.
47. Let X and Y be independent and identically distributed variables with probability mass functions p(n) = 2–n, n = 1, 2, ... . Then P(X > 2Y) equals (rounded to 2 decimal places) _________.
Ans. 0.28
Sol.
48. Let X1, X2, ... be a sequence of independent and identically distributed Poisson random variables with mean 4. Then equals _________.
Ans. 0.69
Sol.
49. Let X and Y be identically distributed exponential random variables with probability densit function.
f(x) =
Then P(max(X, Y) < 2) equals (founded to 2 decimal places) _________.
Ans. 0.75
Sol.
50. Let E and F be any two events with P(E) = 0.4, P(F) = 0.33 and .
Then equals _________ (rounded to 2 decimal places).
Ans. 0.67
Sol.
51. Let X1, X2, ... Xm (m > 2) be a random sample from a binomial distribution with parameters n = 1 and p, p (0, 1) and let
. Then, a uniformly minimum variance unbaised estimator for p(1 – p) is
(a)
(b)
(c)
(d)
Ans. (a)
Sol.
52. Let X1, X2, ..., X9 be a random sample from a N(0, ) population. For testing H0 =
= 2 against H1 :
= 1, the most powerful test rejects H0 if
, where c is to be chosen such that the level of significance is 0.1. Then the power of this test equals _________.
Ans. 0.50
Sol.
53. Let X1, X2, ..., Xn (n > 2) be a random sample from a N(,
) population, where
> 0, and let W =
. Then the maximum likelihood estimator of
is
(a)
(b)
(c)
(d)
Ans. (d)
Sol.
54. Consider the following transporation problem. The entries inside the cells denote per unit cost of transportation from the origins to the destinations.
The optimal cost of transportation equals _________.
Ans. 590
Sol.
55. Consider the linear programming problem (LPP)
Maximize kx1 + 5x2
Subject to x1 + x2
1, 2x1 + 3x2
1, x1, x2
0
If x* = is an optimal solution of the above LPP with k = 2, then the largest value of k (rounded to 2 decimal places) for which x* remains optimal equals _________.
Ans. 3.33
Sol.