Engineering Mathematics GATE-2005

Q 1: Match the following, where x is the spatial coordinate and t is the time

Group I	Group II
P. Wave function	1. \frac{\partial C}{\partial t}=\alpha\;\frac{\partial C}{\partial x}
Q. Heat equation	2. \frac{\partial C}{\partial t}=\alpha^2\frac{\partial^2C}{\partial x^2} 3. \frac{\partial^2C}{\partial t^2}=\alpha^2\frac{\partial C}{\partial x} 4. \frac{\partial^2C}{\partial t^2}=\alpha^2\frac{\partial^2C}{\partial x^2}

A) P-4, Q-2
B) P-2, Q-4
C) P-3, Q-1
D) P-1, Q-3

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Q 2: Two bags contain ten coins each, and the coins in each bag are numbered from 1 to 10. One coin is drawn at random from each bag. The probability that one of the coins has a value of 1, 2, 3, or 4, while the other has a value of 7, 8, 9, or 10, is

A) 2/5
B) 4/25
C) 8/25
D) 16/25

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Q 3: Given i=\sqrt{-1} the ratio \frac{1+2i}{i-2}  is given by

A) 1
B) -1
C) i
D) -i

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Q 4: How many solutions does the following system of equations have?

4x+2y+z=7
x+3y+z=3
3x+4y+2z=2

A) 0
B) 1
C) 2
D) ∞

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Q 5: The matrix A is given by A=\begin{bmatrix}1&4\a&2\end{bmatrix} . The eigenvalues of the matrix A are real and non-negative for the condition.

A) -\frac1{16}\leq a\leq\frac1{16}
B) -\frac12\leq a\leq\frac12
C) -\frac12\leq a\leq\frac1{16}
D) -\frac1{16}\leq a\leq\frac12

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Q 6: The divergence of a vector field A is always equal to zero, if the vector field A can be expressed as

A) the gradient of any scalar filed ϕ
B) the divergence of any scalar field ϕ
C) the divergence of any vector field B
D) the curl of any vector filed B

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Q 7: In the limit x→0, what is the limiting value of the function F (x) given below

F(x)=\frac{1-\cos2x}{x^2}

A) 0
B) 1
C) 2
D) ∞

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Q 8: If Z=x=iy  is a complex number, where i=\sqrt{-1} , then which of the following is an analytic function of z?

A) x^2+y^2
B) 2ixy
C) x^2+y^2-2ixy
D) x^2-y^2+2ixy

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Q 9: What condition is to be satisfied so that the solution of the differential equation \frac{d^2y}{dx^2}+a\frac{dy}{dx}+by=0  is of the form y=(C_1+C_2x)e^{mx} , where C₁ and C₂ are constants of integration?

A) a² = b
B) b² = a
C) a² = 4b
D) b² = 4a

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Q 10: In the domain -\infty<x<\infty , the function y(x)=x^3e^{-x}  has

A) no maximum and no minimum
B) one maximum and no minimum
C) no maximum and one minimum
D) one maximum and one minimum

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Q 11: If f(x) is the solution of the equation \frac{dy}{dx}+2xy+2x=0  and g(x) is the solution of the equation \frac{dy}{dx}+2xy-2x=0 , and the constant of integration in f (x) is equal to that in g (x) then which of the following is true?

A) g(x)=f(x)+2
B) g(x)=f(x)+1
C) g(x)=f(x)-1
D) g(x)=f(x)-2

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Q 12: The function f (x) satisfies the equation f (x) = 0 at x = x_e. The Newton-Raphson iterative method converges to the solution in one step, regardless of the initial guess, if

A) f (x) is a linear function of x
B) f (x) is a quadratic function of x
C) f (x) is a cubic function of x
D) f (x) is an exponential function of x

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