GATE MATHEMATICS 2014
Previous Year Question Paper with Solution.

1.    The function f(z) = |z|2 + is differentiable at

    (a) i

    (b) 1

    (c) –i

    (d) no point in

Ans.    (c)

Sol.    Given function is

        f(z) =    |z|2 +

    Let     z =    x + iy, then

        f(x, y) =    x2 + y2 + i(x – iy) + 1

        =    x2 + y2 + y + 1 + ix

        u(x, y) =    x2 + y2 + y + 1

        v(x, y) =    x

        ux(x, y) =    2x, uy(x, y) = 2y + 1

        vx(x, y) =    1, vy(x, y) = 0

    Cauchy Riemann equation becomes

        2x =    0 x = 0

        2y + 1 =    –1 y = –1

    Cauchy ux, uy, vx, vy are continuous at (0, – 1).

    Hence, f is differentiable at z = –i.

2.    The radius of convergence of the power series is _________.

Ans.

Sol.    We have,

    

    

3.    Let E1 and E2 be two non-empty subsets of a normed linear space X and let E1 + E2 = {x + y X : x E1 and y E2}.

    (a) If E1 and E2 are convex, then E1 + E2 is convex

    (b) If E1 or E2 is open, then E1 + E2 is open

    (c) E1 + E2 must be closed if E1 and E2 are closed

    (d) If E1 is closed and E2 is compact, then E1 + E2 is closed

Ans.    (c)

Sol.    Let E1 and E2 be two non-empty subsets of a normed linear space X and let E1 + E2 = {x + y X : x E1 and y E2}.

    Then, E1 + E2 must closed, if E1 and E2 are closed.

4.    Let y(x) be the solution to the initial value problem subject to y(1.2) = 2. Using the Euler method with the step size h = 0.05, the approximate value of y(1.3), correct to two decimal places, is _________.

Ans.

Sol.    Given, initial value problem is

    

    with     y(1.2)    = 2, h = 0.05

    Here,    f(x, y) =     x0 = 1.2, y0 = 2

    By euler method,

        y1 =    y(1.25) = y0 + hf(x0, y0)

        =    2 + 0.05

        y1 =    2.10

        y2 =    y(1.3) = y1 + hf(x1, y1)

        =    2.10 + 0.05

        y2 =    2.20.

5.    Let If is the the polynomial which interpolates the function f(x) = sin on [–1, 1] at all the zeroes of the polynomial 4x3 – 3x, then is _________.

Ans.

Sol.    We have,

        4x3 – 3x =    0

        x(4x2 – 3) =    0

    

    

    

    

    

6.    If u(x, t) is the D'Alembert's solution to the wave equation t > 0, with the condition u(x, 0) = 0 and = cos x, then is _________.

Ans.

Sol.    

    

    

    

7.    The solution to the integral equation is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

    

    

    

    From Eq. (ii), we get

    

8.    The general solution to the ordinary differential equation in terms of Bessel's functions, Jv(x), is

    (a) y(x) = C1J3/5(2x) + C2J–3/5(2x)

    (b) y(x) = C1J3/10(x) + C2J–3/10(x)

    (c) y(x) = C1J3/5(x) + C2J–3/5(x)

    (d) y(x) = C1J3/10(2x) + C2J–3/10(2x)

Ans.    (a)

Sol.    Given ordinary differential equation is

        

    Comparing with Bessel's equation

    

    Hence, required solution is

    

9.    The inverse Laplace transform of is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

    

10.    If X1, X2 are random sample of size 2 from an N(0, 1) population, then follows

    (a)

    (b) F2, 2

    (c) F2, 1

    (d) F1, 1

Ans.    (d)

Sol.    The above function of random variables follows a F distribution with dF 1, 1.

11.    Let Z ~ N(0, 1) be a random variable. Then, the value of E[max{Z, 0}] is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    So, after solving the expectation formula, we have the result.

12.    The number of non-isomorphic groups of order 10 is _________.

Ans.

Sol.    Non-isomorphic group of order 10 is two i.e.,

    (i) z10            (ii) z2 × z5.

13.    Let a, b, c and d be real numbers with a < c < d < b. Consider the ring C[a, b] with pointwise addition and multiplication.

    If S = {f C[a, b]: f(x) = 0 for all x [c, d]}, then

    (a) S is not an ideal of C[a, b]

    (b) S is an ideal of C[a, b] but not a prime ideal of C[a, b]

    (c) S is a prime ideal of C[a, b] but not a maximal ideal of C[a, b]

    (d) S is a maximal ideal of C[a, b]

Ans.    (b)

Sol.    Let a, b, c and d be real numbers with a < c < d < b. Consider the ring C[a, b] with pointwise addition and multiplication. If S = {f C[a, b] : f(x) = 0 for all x [c, d]}. Then, S is an ideal of C[a, b] but not a prime ideal of C[a, b].

14.    Let R is a ring. If R[x] is a principal ideal domain, then R is necessarly a

    (a) unique factorisation domain

    (b) principal ideal domain

    (c) euclidean domain

    (d) field

Ans.    (d)

Sol.    It is proposition.

    If R[x] is principal ideal domain, then R is necessarily a field.

15.    Consider the group homomorphism is given by = trace(A). The kernel of Ψ is isomorphic to which of the following groups?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

    

    

16.    Let X be a set with atleast two elements. Let be two topologies on X such that X}. Which of the following conditions is necessary for the identity function, id to be continuous?

    (a)

    (b)

    (c) no conditions or

    (d)

Ans.    (b)

Sol.    Let X be a set with atleast two elements and be two topologies on X such that

    Then, the identity function

    i.e.: to continuous, if

17.    Let A be such that det (A – I) = 0, where I denotes the 3 × 3 identity matrix. If the trace (A) = 13 and det (A) = 32, then the sum of squares of the eigenvalues of A is _________.

Ans.

Sol.    Given, |A – l| = 0, Tr A = 13 and |A| = 32

    So, possible 3 × 3 matrix is

    So, of eigenvalues = 8, 4, 1

    Sum of (eigenvalues)2 = 64 + 16 + 81.

18.    Let V denotes the vector space C5 [a, b] over and W = Then,

    (a) dim (V) = and dim (W) =

    (b) dim (V) = and dim (W) = 4

    (c) dim (V) = 6 and dim (W) = 5

    (d) dim (V) = 5 and dim (W) = 4

Ans.    (b)

Sol.    We have,

    

    Auxiliary equation is

    

        z2 + 2z – 1 = 0

    

    

    

    are linearly independent solution of given differential equation.

    

19.    Let V be a real inner product space of dimension 10. Let x, y V be non-zero vectors such that = 0. Then, the dimension of is _________.

Ans.

Sol.    Let V be a real inner product space of dimension 10.

    Let x, y V be non-zero vectors such that < x, y > = 0.

    Then, the dimension of is 8.

20.    Consider the following linear programming problem.

    Minimize x1 + x2

    Subject to    2x1 +    x2 8

        2x1 +    5x2 10

        x1, x2    0

    The optimal value to this problem is _________.

Ans.

Sol.    Given,     min z =    x1 + x2

    Subject to,     2x1 + x2    8        ...(i)

        2x1 + 5x2    10        ...(ii)

        x1, x2    0

    On solving Eqs. (i) and (ii), we get

        x1 =    

    At point A (8, 0), min z = 8

    At point C (0, 5), min z = 5

    At point

        =    4.25

    So, min z = 4.25.

21.    

    be a periodic function of period The coefficient of sin 3x in the Fourier series expansion of f(x) on the interval is _________.

Ans.

Sol.    

    

    

    

    

    

    

    

    

    So, the coefficient of sin 3x is 4.

22.    For the sequence of functions

        fn(x) =    

    Consider the following quantities expressed in terms of Lebesgue integrals

        

    Which of the following is true?

    (a) The limit in I does not exist

    (b) The integrand in II is not integrable on [1, )

    (c) Quantities I and II are well-defined but I II

    (d) Quantities I and II are well-defined and I = II

Ans.    (d)

Sol.    For the sequence of functions fn(x) = in terms of Lebesque integrals quantity.

        

    are well defined and I = II.

23.    Which of the following statements about the spaces lp and Lp [0, 1] is true?

    (a) l3l7 and L6[0, 1] L9[0, 1]

    (b) l3l7 and L9[0, 1] L6[0, 1]

    (c) l7l3 and L6[0, 1] L9[0, 1]

    (d) l7l3 and L9[0, 1] L6[0, 1]

Ans.    (b)

Sol.    l3l7 and L9[0, 1] L6[0, 1].

24.    The maximum modulus of on the set S = is

    (a) 2/e

    (b) e

    (c) e + 1

    (d) e2

Ans.    (b)

Sol.    The maximum modulus of is e.

25.    Let d1, d2 and d3 be metrics on a set X with atleast two elements. Which of the following is not a metric on X?

    (a) min{d1, 2}

    (b) max{d2, 2}

    (c)

    (d)

Ans.    (b)

Sol.    Let d1, d2 and d3 be metrices on a set with atleast two elements. Then,

    (a) min (d1, 2) is metric on X.

    (b) max (d2, 2) is not metric on X.

    (c) is metric on X.

    (d) is metric on X.

26.    Let = {z : Im(z) > 0} and let C be a smooth curve lying in with initial point –1 + 2i and final point 1 + 2i. The value of is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

    

    

        =    (1 + 4i)

        =    (1 + 4i) ln 2(1 + i) – (1 + 4i) ln 2i + 4 – (2 + 4i) [ln 2(1 + i) – ln 2i]

        =    ln2 (1 + i) + 4i ln2 (1 + i) – ln 2i – 4i ln 2i + 4 –2 ln 2(1 + i) + 2 ln 2i – 4i ln 2(1 + i) + 4i ln 2i

        =    ln 2i – ln 2(1 + i) + 4

    

    

27.    If a with |a| < 1, then the value of

    where is the sample closed curve |z| = 1 taken with the positive orientation, is _________.

Ans.

Sol.    2

28.    Consider C[–1, 1] equipped with the supremum norm given by = sup {|f(t)| : t [–1, 1]} for f C[–1, 1].

    Define a linear functional T on C[–1, 1] by T(f) = for all f C[–1, 1].

    Then, the value of is _________.

Ans.

Sol.    2

29.    Consider the vector space C[0, 1] over .

    Consider the following statements.

    P : If the set {t1f1, t2f2, t3f3} is linearly independent, then the set {f1, f2, f3} is linearly independent, where f1, f2, f3 C[0, 1] and tn represents the polynomial function t tn, n

    Q : If F : C[0, 1] is given by F(x) = for each x C[0, 1], then F is a linear map.

    Which of the above statements hold true?

    (a) Only P

    (b) Only Q

    (c) Both P and Q

    (d) Neither P nor Q

Ans.    (b)

Sol.    Clearly, statement P is incorrect. Let x, y [0, 1] and be scalars. Then,

    

    

    

    Thus, F is a linear map.

    So, the statement Q is correct.

30.    Using the Newton-Raphson method with the initial guess x(0) = 6, the approximate value of the real root of x log10 x = 4.77, after the second iteration, is _________.

Ans.

Sol.    f(x) = x log10 x – 4.77

        =    x[loge x . log10e] – 4.77

        =    x loge x log10 e – 4.77

    

        =    loge x log10 e + log10 e

        =    log10 e (1 + loge x)

    By Newton-Raphson method,

    

    Choose     x0 =    6

    

    

31.    Let the following discrete data be obtained from a curve y = y(x)

    

    Let S be the solid of revolution obtained by rotation the above curve about the x-axis betweenx = 0 and x = 1 and let V denotes its volume. The approximate value of V, obtained using Simpson's rule, is _________.

Ans.

Sol.    

    

    

    

32.    The integral surface of the first order partial differential equation

        

    passing through the curve x2 + y2 = 2x, z = 0 is

    (a) x2 + y2 – z2 – 2x + 4z = 0

    (b) x2 + y2 – z2 – 2x + 8z = 0

    (c) x2 + y2 + z2 – 2x + 16z = 0

    (d) x2 + y2 + z2 – 2x + 8z = 0

Ans.    (a)

Sol.    Given equation

    

    Let its answer be

        x2 + y2 + A z2 – 2x + Bz = 0    ...(ii)

    So, from Eq. (ii) differentiate w.r.t. x, we get

    

    Again, Eq. (ii) differentiate w.r.t. y,

    

    From Eqs. (i), (iii) and (iv), we get

    

      2y(z – 3)(2 – 2x) + (2x – z) (–2y) = y(2x – 3) (2Az + B)

      (2yz – 6y) (2 – 2x) – (2y) (2x – z) = (2xy – 3y) (2Az + B)

– 4yz – 12y – 4xyz + 12xy – 4xy + 2zy

        =    4A xyz – 6A zy + 2xy B – 3y B

      6zy – 4xyz + 8xy – 12y

        =    4A xyz – 6A zy + 2xyB – 3yB

    On comparing, we get A = – 1 and z

    So, required answer x2 + y2 – z2 – 2x + 4z = 0.

33.    The boundary value problem, and is converted into the integral equation = g(x) + where the kernel .

    Then, is _________.

Ans.

Sol.    –0.28

34.    If y1(x) = x is a solution to the differential equation (1 – x2) then its general solution is

    (a) y(x) = C1x + C2(x ln|1 + x2| – 1)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Given differential equation is

    

    Given that, y1(x) = x is solution of Eq. (i).

    Let y2(x) = xu(x) be another solution. Then, Eq. (i)

    Becomes

    

    Let v(x) = (x). Then, Eq. (ii) becomes

    

    Whose general solution is

    

    This gives ignoring C

    

    General solution of given differential equation is

        y(x) =    C1y1(x) + C2y2(x)

        

35.    The solution to the initial value problem sin t, y(0) = 0 and = 3, is

    (a) y(t) = et (sin t + sin 2t)

    (b) y(t) = e–t (sin t + sin 2t)

    (c) y(t) = 3et sin t

    (d) y(t) = 3e–t sin t

Ans.    (b)

Sol.    (D2 + 2D + 5) y =    3e–t sin t

        y(0) =    0, = 3

    Auxiliary equation m2 + 2m + 5 = 0

    

        CF = e–t [c1 sin 2t + c2 cos 2t]

    

    

    

    

    So,     y = e–t[sin 2t + sin t].

36.    The time to failure, in months, of light bulbs manufactured at two plants A and B obey the exponential distribution with means 6 and 2 months respectively. Plant B produces four times as many bulbs as plant A does. Bulbs from these plants plants are indistinguishable. They are mixed and sold together. Given that a bulb purchased at random is working after 12 months, the probability that it was manufactured at plant A is _________.

Ans.

Sol.    0.93

37.    Let X, Y be continuous random variables with joint density function

        fX, Y(x, y) =    

    The value of E[X + Y] is _________.

Ans.

Sol.    4

38.    Let X = [0, 1] (1, 2) be the subspace of , where is equipped with the usual topology.

    Which of the following is false?

    (a) There exists a non-constant continuous function f : X

    (b) X is homeomrphic to

    (c) There exists an onto continuous function f : [0, 1] where is the closure of X in

    (d) There exists an onto continuous function f : [0, 1] X

Ans.    (d)

Sol.

39.    Let X = A matrix P such that P–1 XP is diagonal matrix, is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    The characteristic equation is

    

    

    

    The eigenvector corresponding to = 2 is

        [A – 2l] X = [0]

    

        3x1 – 3x2 – 3x3 =    0

        –3x3 = 0 x3 =    0

    Thus, we have    x1 = x2 and x3 =    0

    

    The eigenvector corresponding to = –1 is

        [A + l] [X] =    [0]

    

        3x1 – 3x3 =    0

    Thus, we get x1 = x3 and x2 is arbitrary

    

    

    

    

    

40.    Using the Gauss-Seidel iteration method with initial guess {x1(0) = 3.5, x2(0) = 2.25, x3(0) = 1.625}, the second approximation {x1(2), x2(2), x3(2)} for the solution to the sytem of equations

        2x1 – x2 =    7

        –x1 + 2x2 – x3 =    1

        –x2 + 2x3 =    1

    is

    (a) x1(2) = 5.3125, x2(2) = 4.4491, x3(3) = 2.1563

    (b) x1(2) = 5.3125, x2(2) = 4.3125, x3(2) = 2.6563

    (c) x1(2) = 5.3125, x2(2) = 4.4491, x3(2) = 2.6563

    (d) x1(2) = 5.4991, x2(2) = 4.4491, x3(2) = 2.1563

Ans.    (b)

Sol.    

    

    

    

    

41.    The fourth order Runge-Kutta method given by uj + 1 = uj + [K1 + 2K2 + 2K3 + K4], j = 0, 1, 2, ..., is used to solve the initial value problem = u, u(0) =

    If u(1) = 1 is obtained by taking the step size h = 1, then the value of K4 is _________.

Ans.

Sol.    Given initial value problem is

    

    

    

    Also, given that u(1) = 1

    

    

    

        

42.    A particle P of mass m moves along the cycloid x = and y = Let g denotes the acceleration due to gravity. Neglecting the frictional force, the Lagrangian associated with the motion of the particle P is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.

43.    Suppose that X is a population random variables with probability density function

    

    where is a parameter. In order to test the null hypothesis H0 : = 2 against the alternative hypothesis H1 : = 3, the following test is used. Reject the null hypothesis if X1 1/2 and accept otherwise, where X1 is a random sample of size 1 drawn from the above population. Then, the power of the test is _________.

Ans.

Sol.    Power = 1 – P (type II error) = 1 – P (accept H0/H1)

    

44.    Suppose that X1, X2, ... , Xn is a random sample of size n drawn from a population with probability density function.

    

    where is a parameter such that > 0. The maximum likelihood estimator of is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    The solution comes out by minimizing the likelihood function L = .

45.    Let F be a vector field defined on \{(0, 0)} by F(x, y) = Let be defined by

    and          

    If then m is _________.

Ans.    7

Sol.

46.    

Then, is _________.

Ans.    75

Sol.

47.    Let T1, T2 : be linear transformations such that rank (T1) = 3 and nullity (T2) = 3.

    Let T3 : be a linear transformation such that T3 ° T1 = T2. Then, rank (T3) is _________.

Ans.

Sol.    Given that, T1, T2 : R5 R3 such that rank (T1) = 3 and nullity (T2) = 3.

      Nullity(T1) = 2 and Rank(T2) = 2

    Rank (T3 O T1) min[rank (T3), rank (T1)]

      2 min [Rank(T3), 3]

      Rank(T3) 2

    But Rank(T3 O T1) Rank(T3) + Rank(T1) – 5

      2 Rank(T3) + 3 – 5

      Rank(T3) 4

    Thus, 2 Rank(T3) 4.

48.    Let be the field of 3 elements and Let × be the vector space over . The number of distinct linearly dependent sets of the form {u, v}, where u, v × \{(0, 0)} and u v is _________.

Ans.    4

Sol.

49.    Let be the field of 125 elements. The number of non-zero elements such that is _________.

Ans.    4

Sol.

50.    The value of where R is the region in the first quadrant bounded by the curves y = x2, y + x = 2 and x = 0 is _________.

Ans.    0.97

Sol.

51.    Consider the heat equation t > 0, with the boundary conditions u(0, t) = 0, = 0 for t > 0 and the initial condition u(x, 0) = sin x. Then, is _________.

Ans.

Sol.    Given heat equation is t > 0 with the boundary condition.

    u(0, t) = 0, u(, t) = 0, for t > 0 and the initial condition u(x, 0) = sin x.

    Here, = 1, l = , f(x) = sin x

    

    Applying initial condition,

        u(x, 0) =    sin x, we get

        

    which is Fourier sine series.

    

    

52.    Consider the partial order in given by the relation (x1, y1) < (x2, y2) either if x1 < x2 or if x1 = x2 and y1 < y2. Then, in the order topology on defined by the above order

    (a) [0, 1] × [1] is compact but [0, 1] × [0, 2], is not compact

    (b) [0, 1] × [0, 1] is compact but [0, 1] × [1] is not compact

    (c) Both [0, 1] × [0, 1] and [0, 1] × [1] are compact

    (d) Both [0, 1] × [0, 1] and [0, 1] × [1] are not compact

Ans.    (d)

Sol.

53.    Consider the following linear programming problem.

    Minimize    x1 +    x2 + 2x3

    Subject to    x1 +    2x2 4

        x2 +    7x3 5

        x1 –    3x2 + 5x3 = 6

        x1, x2    0, x3 is unrestricted

    The dual to this problem

    Maximize    4y1 +    5y2 + 6y3

    Subject to    y1 +    y3 1

        2y1 +    y2 – 3y3 1

        7y2 +    5y3 = 2

    and further subject to

    (a) y1 0, y2 0 and y3 is unrestricted

    (b) y1 0, y2 0 and y3 is unrestricted

    (c) y1 0, y3 0 and y2 is unrestricted

    (d) y3 0, y20 and y1 is unrestricted

Ans.

Sol.    Given LPP is

    Minimize z = x1 + x2 + 2x3

    Subject to    x1 + 2x2    4

        x2 + 7x3    5

        x1 – 3x2 + 5x3 =    6

    x1, x2 0, x3 is unrestricted.

    

    Then, LPP becomes

    Minimize z = x1 + x2 + 2

    Subject to    –x1 + 2x2–4

    

    

    Let w1, w2, w3 and w4 be the dual constraints. Then,

    Maximize z = –4w1 + 5w2 + 6w3 – 6w4

    Subject to    – w1 + w3 – w4    1

        2w1 + w2 – 3w3 + 3w4    1

        7w2 + 5w3 – 5w4    2

        –7w2 – 5w3 + 5w4    2

        w1, w3, w4 0, w2    0

    Let    w1 = y1, w2 = y2 w3 =    w3 – w4

    Then, mazimize z = –4y1 + 5y2 + 6y3

    Subject to    –y1 + y3     1

        2y1 + y2 – 3y3    1

        7y2 + 5y3    2 and –7y2 – 5y3 2

    y1 0, y2 0, y3 – unrestricted.

54.    Let X = C1[0, 1]. For each f X, define

        p1(f) =    sup{|f(t)| : t [0, 1]

        p2(f) =    sup{|f' (t)| : t [0, 1]

        p3(f) =    p1(f) + p2(f).

    Which of the following statements is true?

    (a) (X, p1) is a Banach space

    (b) (X, p2) is a Banach space

    (c) (X, p3) is not a Banach space

    (d) (X, p3) does not have denumerable basis

Ans.    (d)

Sol.

55.    If the power series converges at 5i and diverges at – 3i, then the power series

    (a) converges at –2 + 5i and diverges at 2 – 3i

    (b) converges at 2 – 3i and diverges at –2 + 5i

    (c) converges at both 2 – 3i and –2 + 5i

    (d) diverges at both 2 – 3i and –2 + 5i

Ans.    (a)

Sol.    Given, elastic modulus of glass fibre = 75.59 Pa

    Elastic modulus of reinforced epoxy = 2.4 GPa

    Use,    YC =    WgfYgf + WreYre

        =    (1 – 0.6) × 72.5 + 0.6 × 2.4

      It is given that a glass fibre reinforced epoxy composite is made with 60 wt% unidirectional continuous glass fibre.

        Wre =    0.6 = 0.6

        Wgf =    1 – 0.6 = 0.4

        YC =    0.4 × 72.5 + 0.6 × 2.4 = 29 + 1.44

        YC =    30.4 GPa.