GATE MATHEMATICS 2016
Previous Year Question Paper with Solution.
1. Let {X, Y, Z} be a basis of R3. Consider the following statements P and Q.
P : {X + Y, Y + Z, X – Z} is a basis of R3.
Q : {X + Y + Z, X + 2Y + Z, X – 3Z} is a basis of R3.
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (c)
Sol.
But (X, Y, Z) is basis of R3.
Hence, {X + Y, Y + Z, X – Z} is not a basis of R3.
So, statement P is false.
But (X, Y, Z) is basis or R3.
So, {X + Y + Z, X + 2Y + Z, X – 3Z} is a linearly independent set of dimension 3.
Hence, {X + Y + Z, X + 2Y + Z, X – 3Z} is a basis is R3.
2. Consider the following statements P and Q.
P : If M = , then M is singular.
Q : Let S be a diagonalisable matrix. If T is a matrix such that S + 5T = Id, then T is diagonalisable.
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (c)
Sol. P: Here, det(M) = 1(2 × 9 – 3 × 4) – 1 (1 × 9 – 1 × 4) + 1(1 × 3 – 2 × 1) = 2
M is non-singular matrix.
Thus, Statement P is false.
Point (1): Difference of two diagonalisable matrices is diagonalisable.
Point (2): Scalar multiplication of a diagonalisable matrix is diagonalisable.
Hence, T must be diagonalisable.
Thus, Statement Q is true.
3. Consider the following statements P and Q.
P : If M is an n × n complex matrix, then R(M) =
Q : There exists a unitary matrix with an eigen value such that
< 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. P: A transformation f : M M, defined by f(A) =
does not change the dimension of the nullity.
Also, we know that Rank(A) = Rank Thus, R(M) =
. Hence, statement P is true.
Q: U is a complex matrix, where UU* = I
Where, U* denotes the cojugate transpose of U and I denotes the identity matrix.
Take Ux = x as some eigen equation.
Then, by properties of conjugate transpose,
x*U* = * x*
Multiplying these two equations together gives us
4. Consider a real vector space V of dimension n and a non-zero linear transformation T : V V. If dimension T(V) < n and T2 =
T, for some
R – {0}, then which of the following statements is true?
(a) determinant (T) =
(b) There exists a non-trival subspace V1 of V such that T(X) = 0 for all X V1
(c) T is invertible
(d) is the only eigen value of T
Ans. (b)
Sol. Here, dim V = n
R(T) < n
Now, using rank-nullity theorem
R(T) + N(T) = dim V
There exists a non-trival subspace V1 of V such that
5. Let S = [0, 1] [2, 3] and f : S
R be a strictly increasing function such that f(s) in connected.
Which of the following statements is true?
(a) f has exactly one discontinuity
(b) f has exactly two discontinuities
(c) f has infinitely many discontinuities
(d) f is continuous
Ans. (d)
Sol. Consider the function given in the following graph.
Here, f is strictly increasing function on S = [0, 1] [2, 3] and f(S) = [0, 2] is connected.
Now, function f(x) has no discontinuity.
Option (a), (b) and (c) are incorrect.
6. Let a1 = 1 and an = an – 1 + 4, n 2. Then,
is equal to.
Ans. 0.25
Sol.
On adding, we have
7. Maximum {x + y : (x, y) clouser B(0, 1)} is euqal to _________.
Ans. 1.414
Sol.
8. Let a, b, c, d R such that c2 + d2
0. Then, the cauchy problem
aux + buy = ex + y, x, y R,
u(x, y) = 0 on cx + dy = 0
has a unique solution, if
(a) ac + bd 0
(b) ad – bc 0
(c) ac – bd 0
(d) ad + bc 0
Ans. (c)
Sol. Result: (Existence and Uniqueness Theorem)
Consider the first order quasi-linear PDE in the domain R3
where a, b and c are of class C1
is an initial smooth curve in
and
b(x0(s), y0(s),u0(s))
a(x0(s), y0(s), u0(s) ≠ 0,
Then, there exists one and only one solution u = u(x, y) defined in a neighbourhood N of the initial curve T, which satisfies the equation a(x, y, u)ux + b(x, y, u)uy = c(x, y, u) and the initial condition, u0(s) = u(x0(s), y0(s), .
Now, solution is here cx + dy = 0
9. Let u(x, t) be the Alembert's solution of the initial value problem for the wave equation.
utt – c2 uxx = 0
u(x, 0) = f(x), ut(x, 0) = g(x),
where c is a positive real number and f(x), g(x) are smooth odd functions. Then, u(0, 1) is equal to _________ .
Ans. 0
Sol.
But f(x) and g(x) are odd function
Note g(x) is odd function,
Hence, u(0, 1) = 0.
10. Let the probability density function of a random variable X be
f(x) =
Then, the value of c is equal to _________.
Ans. 5.25
Sol.
11. Let V be the set of all solutions of the equation + by = 0 satisfying y(0) = y(1), where a, b are positive real numbers. Then, dimension (V) is equal to _________.
Ans. 1
Sol. Dimension (V) = Number of linearly independent solutions – Number of initial conditions= 2 – 1 = 1.
12. Let where p(x) and q(x) are continuous functions.
If y1(x) = sin(x) – 2 cos(x) and y2(x) = 2 sin(x) + cos(x) are two linearly independent solutions of the above equation, then |4p(0) + 2q(1)| is equal to _________.
Ans. 2
Sol.
Thus, we have p(x) y2 + q(x)y1 = y1 – p(x)y1 + q(x)y2 = y2
On solving, we get
q(x) = 1 for all x
p(x) y1(x) y2(x) = 0 for all x
Now, at x = 0, p(0)y1 (0)y2 (0) = 0
Thus, |4p(0) + 2q(1)| = 4 × 0 + 2 × 1 = 2
13. Let Pn(x) be the Legendre polynomial of degree n and I = where k is a non-negative integer. Consider the following statements P and Q.
P : I = 0, if k < n.
Q : I = 0 if n – k is an odd integer.
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (a)
Sol.
But in both statement k n.
Hence, both the statement are true.
14. Consider the following statements P and Q.
P : has two linearly independent Frobenius series solutions near x = 0.
Q : has two linearly independent Frobenius series solutions near x = 0.
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.
Which is Bessell's equation.
Here, x = which is not integer.
Hence, it has two linearly independent Frobenius series solutions near x = 0.
Hence, Statement P is correct but Statement Q is not correct.
15. Let the polynomial x4 be approximated by a polynomial of degree 2, which interpolates x4 atx = –1, 0 and 1. Then, the maximum absolute interpolation error over the interval [–1, 1] is equal to _________.
Ans. 0.093
Sol. Here x0 = –1, x1 = 0, x2 = 1
y = f(x) = x4
y0 = 1, y1 = 0, y2 = 1
Using Lagrange's formula
16. Let (zn) be a sequence of distinct points in D(0, 1) = {z C : |z| < 1} with
Consider the following statements P and Q.
P : There exists a unique analytic function f on D(0, 1) such that f(zn) = sin(zn) for all n.
Q : There exists an analytic function f on D(0, 1) such that f(zn) = 0, if n is even and f(zn) = 1, if n is odd.
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (b)
Sol.
Hence, Statement P is correct.
whereas, Statement Q is not correct.
17. Let (R, ) be a topological space with the cofinite topology. Every infinite subset of R is
(a) compact but not connected
(b) both compact and connected
(c) not compact but connected
(d) neither compact nor connected
Ans. (b)
Sol. |Ac is finite} is a cofinite topology.
Let A be infinite subset of R.
Suppose that A is disconnected.
Then, there exist open sets B and C such that
A = B C
where, B C = ϕ
But (B C)c = R
Bc
Cc = R
But Bc and Cc both are finite, whereas Bc
Cc = R which is infinite. This is a contradiction.
Hence, A is connected. Now, every infinite subset of cofinite topology is compact. Thus, every infinite subset at cofinite topology is both compact and connected.
18. Let c0 = {(xn): xn
R, xn
0} and M = {(xn)
c0 : x1 + x2 + ... + x10 = 0}.
Then, dimension (c0/M) is equal to _________.
Ans. 1
Sol. dim(c0/M) = dim c0 – dim M = 1.
19. Consider where
= Max {|x|, |y|}. Let f : R
R be defined by f(x, y) =
and
the norm preserving linear extension of f to
Then, f(1, 1, 1) is equal to _________.
Ans. 1
Sol. Here, f(1, 1, 1) . Hence, f(1, 1, 1) = 1.
20. f:[0, 1] [0, 1] is called a shrinking map, if |f(x) – f(y)| < |x – y| for all x, y
[0, 1] and a contraction if there exists and
< 1 such that |f(x) – f(y)|
|x – y| for all x, y
[0, 1].
Which of the following statements is true for the function f(x) =?
(a) f is both a shrinking map and a contraction
(b) f is a shrinking map but not a contraction
(c) f is not a shrinking map but a contraction
(d) f is neither a shrinking map nor a contraction
Ans. (b)
Sol. Here, |f(x) – f(y)| |x – y|(1–|x + y|), where x, y
[0, 1]
Put = 1 – |x + y|. Now, take x = y = 0, then
= 1. which shows that f is not a contraction map.
Again, take x = y = , then
=
, which shows that f not shrinking map.
21. Let M be the set of all n × n real matrices with the usual norm topology.
Consider the following statements P and Q.
P : The set of all symmetric positive definite matrices in M is connected.
Q : The set of all invertible matrices in M is compact.
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. P: The set of all symmetric positive definite matrices in M is connected statement P is true.
Q: Consider the sequence of invertible matrices in M as An =
which is converging to A = but det(A) = 0. Hence, A is not ivertible.
So, the set of all invertible matrices in M is not closed. Hence, not compact. So, statement Q is not correct.
22. Let X1, X2, X3, ..., Xn be a random sample from the following probability density function for 0 < µ < < 1
Here, and µ are unknown parameters. Which of the following statements is true?
(a) Maximum likelihood estimator of only µ exists
(b) Maximum likelihood estimator of only exists
(c) Maximum likelihood estimators of both µ and exist
(d) Maximum likelihood estimator of neither µ nor exists.
Ans. (d)
Sol.
23. Suppose X and Y are two random variables such that aX + bY is a normal random variable for all a, b R.
Consider the following statements P, Q, R and S.
P : X is a standard normal random variable.
Q : The conditional distribution of X given Y is normal.
R : The conditional distribution of X given X + Y is normal
S : X – Y has mean 0.
Which of the above statements always hold true?
(a) Both P and Q
(b) Both Q and R
(c) Both Q and S
(d) Both P and S
Ans. (b)
Sol. aX + bY is a normal random variable for all a, b R.
Put a = 1 and b = 0, then X is a normal random variable.
Put a = 0 and b = 1, then Y is a normal random variable.
Hence X/Y is a normal random variable. So, statement Q is correct. Now, put a = 1 and b = 1, then X + Y is a normal random variable. Hence, X/(X + Y) is a normal random variable. So, statement R is correct.
24. Consider the following statements P and Q.
P : If H is a normal subgroup of order 4 of the symmetric group S4, then S4/H is abelian.
Q : If Q = is the quaternion group, then Q8/{–1, 1} is abelian.
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (c)
Sol. Here, = {{1, – 1}, {i, – i}, {j, – j}, {k, – k}} is a group of order 4. Every gorup of order 4 is abelian.
Hence, statement Q is true.
25. Let F be a field of order 32. Then, the number of non-zero solution (a, b) F × F of the equation x2 + xy + y2 = 0 is equal to _________.
Ans. 0
Sol. If y = 0, clearly also x = 0. So, suppose y 0, multiply by y2, and set t = xy1 to obtain t2 + t + 1 = 0.
Now, you should know that the solutions of the latter equations are the two elements different from 0, 1 in the field of order 22 = 4.
Since, 32 = 25, the field of order 4 is not a subfield of F.
Hence, answer is 0.
26. Let be oriented in the counter-clockwise directions. Let
Then, the value of I is equal to _________.
Ans. 0.041
Sol. Here, z = 0 is the only singularly which is isolated essential singularly.
27. Let u(x, y) = x3 + ax2y + bxy2 + 2y3 be a harmonic function and v(x, y) its harmonic conjugate. If v(0, 0) = 1, then |a + b + v(1, 1)| is equal to _________.
Ans. 11
Sol.
28. Let be the triangular path connecting the points (0, 0), (2, 2) and (0, 2) in the counter clockwise direction in R2. Then
I =
is equal to _________.
Ans. 16
Sol.
Integral on OA
Here, y = x and x = 0 to x = 2.
Integral on AB
Here, y = 2 and x = 2 to x = 0.
Integral on BO
Here, x = 0
29. Let y be the solution of + y = |x|, x
R, y(–1) = 0. Then, y(1) is equal to
(a)
(b)
(c)
(d) 2 – 2e
Ans. (a)
Sol.
30. Let X be a random variable with the following cumulative distribution function
F(x) =
Then, is equal to _________.
Ans. 0.68
Sol.
31. Let y be the curve which passes through (0, 1) and intersects each curve of the family y = cx2 orthogonally. Then, y also passes through the point
(a)
(b)
(c) (1, 1)
(d) (–1, 1)
Ans. (b)
Sol.
Perpendicular to the family of curve y = cx2
y-axis passes through .
32. Let S(x) = a0 + be the fourter series to the
periodic function defined by f(x) = x2 + 4 sin(x) cos(x),
Then
is equal to _________.
Ans. 17.35
Sol.
33. Let y(t) be a continuous function on If y(t) = t(1 – 4
then
is equal to _________.
Ans. 0.5
Sol. y(t) = t(1 – 4C1 + 4C2)
34. Let Sn = Then, S10 + I10 is equal to
(a) ln 10 + 1
(b) ln 10 – 1
(c)
(d)
Ans. (a)
Sol.
Thus, S10 + I10 = ln 10 + ln 10 – ln 10 + ln 10 + 1 = ln 10 + 1.
35. For any (x, y) let f(x, y) distance ((x, y,)
= infimum
Then,
is equal to _________.
Ans. 1
Sol.
36. Let f(x) = Then,
is equal to _________.
Ans. 1.137
Sol.
37. Let M = be a real matrix with eigen values 1, 0 and 3. If the eigen vectors corresponding to 1 and 0 are (1, 1, 1)T and (1, –1, 0)T respectively, then the value of 3f is equal to _________.
Ans. 7
Sol.
On solving above equations, we get f = 7
38. Let M = and eM = Id + M +
M3 + ... If eM = [bij], then
is equal to _________.
Ans. 5.5
Sol.
39. Let the integral I = where
f(x) =
Consider the following statements P and Q.
P : If I is the value of the integral obtained by the composite trapezoidal rule with two equal subintervals, then I is exact.
Q : If I the value of the integral obtained by the composite trapezoidal rule with three equal subintervals, then I is exact.
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. P: For two subinterval
x0 = 0, y0 = 0, x1 = 2, y1 = 2 and x2 = 4, y2 = 4
Trapezoidal rule with two equal subintervals
Q: For three subinterval
Trapezoidal rule with two equals subintervals
Thus, statement P is true.
40. The difference between the least two eigen values of the boundary value problem
is equal to _________.
Ans. 2
Sol. Case-I: Let = 0. Then, solutions is y(x) = Ax + B.
Using given conditions, we get y(x) = 0. So, there is no eigen function corresponding to = 0.
Case-II: Let = – µ 2, where μ
0.
Then, solution is y(x) = Aeµx + Be–µx.
Using given conditions, we get y(x) = 0. So, there is no eigen function corresponding to = – µ 2.
Case-III: Let = µ 2.Then, solution is
y(x) = Acos(µx) + Bsin(µx)
Using given conditions, we get
A = 0 and Bcos = 0
Now, consider Bcos = 0. If B = 0, then with A = 0, we get y(x) = 0, which is not an eigen function. So, B
0 for the existence of the eigen functions.
Since B 0, we have
Hence, the values are given by
Thus, the least given values are
Hence, the difference between the least two eigen values is 2.
41. The number of roots of the equation x2 – cos(x) = 0 in the interval is equal to _________.
Ans. 2
Sol.
From graph, it is clear that x2 – cos x = 0 has two solutions in .
42. For the fixed point iteration xk + 1 = g(xk), k = 0, 1, 2 ... consider the following statements P and Q.
P : If g(x) = then the fixed iteration converges to 2 for all x
[1, 100].
Q : If g(x) = then the fixed point iteration converges to 2 for all x
[0, 100].
Which of the above statements hold true?
(a) Both P and Q
(b) Only P
(c) Only Q
(d) Neither P nor Q
Ans. (a)
Sol. P : Here > 1 for x = 1. Hence, statement P is correct.
Q : Here, < 1, for all x
[0, 100]. Hence, xk + 1 =
converges a root of x2 – x – 2 = 0 that is 2 for all x
[0, 100]. Thus, statement Q is correct.
43. Let T : l2
l2 be defined by
T(x1, x2, ..., xn, ...) = (x2 – x1, x3 – x2, ..., xn + 1 – xn, ...)
Then,
(a) ||T|| = 1
(b) ||T|| > 2 but bounded
(c) 1 < ||T|| 2
(d) ||T|| is bounded
Ans. (c)
Sol.
44. Minimise w = x + 2y subject to 2x + y 3, x + y
2, x
0, y
0
Then, the minimum value of w is equal to _________.
Ans. 2
Sol. The points of intersection of line 2x + y = 3 on the coordinates axis are and (0, 3). The points of intersection of line x + y = 2 on the coordinates axis are (2, 0) and (0, 2).
The graph is as follows.
The shaded region is the feasible region and the corner points are the solution points,
Now, w(1, 1) = 1 + 2 × 1 = 3
w(0, 3) = 0 + 2 × 3 = 6
w(2, 0) = 2 + 2 × 0 = 2.
Hence, minimum value of w = 2.
45. Maximise w = 11 x – z subject to 10x + y – z 1, 2x – 2y + z
2, x, y, z
0
Then, the maximum value of w is equal to _________.
Ans. 1.25
Sol. Standard form of LPP is
Maximise = 11x – z + 0s1 + 0s2
Subject to 10x + y – z + s1 + 0s2 = 1
2x – 2y + z + 0s1 + s2 = 2
x 0, y
0, z
0, s1
0, s2
0.
The simplex table is as follows:
Which is an optimal table.
46. Let X1, X2, X3, ... be a sequence of i.i.d random variables with mean 1. If N is a geometric random variable with the probability mass function P(N = K) = k = 1, 2, 3... and it is independent of the
s then E(X1 + X2 +...+XN) is equal to _________.
Ans. 2
Sol. Here, X1, X2, X3, ... be a sequence of i.i.d random variable with
E(Xi) = 1 = E(X)
On subtraction, we get
47. Let X1 be an exponential random variable with mean 1 and X2 a gamma random variable with mean 2 distributed, then P(X1 < X2) is equal to _________.
Ans. 0.75
Sol.
48. Let X1, X2, X3, ... be a sequence of i.i.d. uniform (0, 1) random variables. Then, the value of P(–In(1 – X1) – ... – ln (1 – Xn)
n) is equal to _________.
Ans. 0.5
Sol.
49. Let X be a standard normal random variable. Then, P(x < 0| |[x] = 0) is equal to
(a)
(b)
(c)
(d)
Ans. (a)
Sol.
50. Let X1, X2, X3, ..., Xn be a random sample from the probability density function
f(x) =
where are parameters. Consider the following testing problem:
Which of the following statements is true?
(a) Uniformly Most Powerful test doest not exist
(b) Uniformly Most Powerful test does not exist for some 0 < c <
(c) Uniformly Most Powerful test is of the form for some 0 < c <
(d) Uniformly Most Powerful test is of the form c1 for some 0 < c1 < c2 <
Ans. (c)
Sol.
51. Let X1, X2, X3,... be a sequence of i.i.d. N(µ, 1) random variables. Then, is equal to _________.
Ans. 0.71
Sol.
52. Let X1, X2, X3,...Xn be a random sample from uniform [1, ], for some
> 1.
If X(n) = Maximum (X1, X2, X3, ..., Xn), then the UMVUE of is
(a)
(b)
(c)
(d)
Ans. (b)
Sol.
53. Let x1 = x2 = x3 = 1, x4 = x5 = x6 = 2 be a random sample from a Poisson random variable with mean , where
{1, 2}. Then, the maximum likelihood estimator of
is equal to _________.
Ans. 2
Sol.
54. The remainder when 98! is divided by 101 is equal to _________.
Ans. 50
Sol. Here, (98!) (99) (100) = 100!
Using Wilson's Theorem,
(98!) (–2) (–1)
–1(mod 101)
Let x = 98!. Then, 2x = –1 (mod 101).
If follows that 2x 100(mod 101) and therefore x
50 (mod 101).
Hence, the required remainder is 50.
55. Let G be a group whose presentation is
G = {x, y| x5 = y2 = e, x2y = yx}
Then, G is isomorphic to
(a) Z5
(b) Z10
(c) Z2
(d) Z30
Ans. (c)
Sol. Consider Z2 = {0, 1}
If x = 0, y = 1
then 05 = 0, 12 = 1 + 1 = 0, 02 = 0 – 1.0
Z2
G
Option (c) is correct.
Consider Z5 = {0, 1, 2, 4}.
If n = y = 3
then 15 = 1 + 1 + 1 + 1 + 1 = 0
y2 = 32 = 3 + 3 = 1 ≠ 0
Z5
G
Consider Z10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
If x = 1, y = 3
then 15 = 1 + 1 + 1 + 1 + 1 = 5 0
32 = 3 + 3 = 9 0
Z10
G
Similarly, we can see that Z30
G.