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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