Engineering Mathematics GATE-2006

Q 1: The ordinary differential equation \frac{dY}{dt}=f(Y) is solved using the approximation Y(t+\triangle t)=Y(t)+f\lbrack Y(t)\rbrack\triangle t . The numerical error introduced by the approximation at each step is

A) proportional to ∆t
B) proportional to (∆t)²
C) independent of ∆t
D) proportional to (1/∆t)

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Q 2: The trapezoidal rule of integration, when applied to \int_a^bf(x)\operatorname dx will give the exact value of the integral

A) if f (x) is a linear function of x
B) if f (x) is a quadratic function of x
C) for any f (x)
D) for no f (x)

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Q 3:  The value of α for which the following three vectors are coplanar is

a=i+2j+k
b=3j+k
c=2i+\alpha j

A) 4
B) 0
C) -2
D) -10

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Q 4: The derivative of \vert x\vert with respect to x when x\neq0 is

A) \frac{\vert x\vert}x
B) -1
C) 1
D) undefined

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Q 5: If the following represents the equation of a line \begin{vmatrix}x&2&4\\y&8&0\\1&1&1\end{vmatrix}=0 , then the line passes through the point

A) (0, 0)
B) (3, 4)
C) (4, 3)
D) (4, 4)

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Q 6: If A=\begin{bmatrix}2&1\\2&3\end{bmatrix} , then the eigen values of A³ are

A) 27 and 8
B) 64 and 1
C) 12 and 3
D) 4 and 1

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Q 7:  With y=e^ax , if the sum S=\frac{dy}{dx}+\frac{d^2y}{dx^2}+\cdots+\frac{d^ny}{dx^n} approaches 2y as n\rightarrow\infty , then the value of a is

A) 1/3
B) 1/2
C) 2/3
D) 2

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Q 8: Determine the following integral I=\int_s\overrightarrow r.d\overrightarrow s where, \overrightarrow r is the position vector field \overrightarrow r=\widehat ix+\widehat jy+\widehat kz and S is the surface of a sphere of radius R.

A) 4\pi R^2
B) 3/4\pi R^3
C) \pi R^3
D) 4\pi R^3

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Q 9: The solution to the following equation x^2\frac{d^3y}{dx^3}+2x\frac{d^2y}{dx^2}-2\frac{dy}{dx}=0 is given by

A) y=C_1x+C_2x^{-2}+C_3
B) y=C_1x^2+C_2x^{-2}+C_3
C) y=C_1x^2+C_2x^{-1}+C_3
D) y=C_1x+C_2x^{-1}+C_3

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Q 10: The value of the contour integral  I=\frac1{2\mathrm{πi}}\oint_C\frac{e^z}{\left(z+1\right)\left(z+3\right)}dz where C is the circle \vert z\vert=2 is

A) \frac1{2e}
B) \frac12\left(\frac1e-\frac1{e^3}\right)
C) zero
D) \frac1{\left(2\mathrm{πi}\right)e^3}

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Q 11: The Newton Raphson method is used to solve the equation, (x-1)^2+x-3=0 . The method will fail in the very first iteration, if the initial guess is

A) 0
B) 0.5
C) 1
D) 3

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Q 12: A pair of fair dice is rolled three times. What is the probability that 10 (the sum of the numbers on the two faces) will show up exactly once?

A) 121/1728
B) 363/1728
C) 121/576
D) 363/576

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Q 13: A company purchased components from three firms P, Q, and R as shown in the table below.

Firm	Total number of components purchased	Number of components likely to be defective
P	1000	5
Q	2500	5
R	500	2

The components are stored together. One of the components is selected at random and found to be defective. What is the probability that it was supplied by firm R?

A) 1/250
B) 1/12
C) 1/8
D) 1/6

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