Engineering Mathematics GATE-2006

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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





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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





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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





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





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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





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





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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





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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.





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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





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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





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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





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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?





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

FirmTotal number of components purchasedNumber of components likely to be defective
P10005
Q25005
R5002

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?





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