Tribhuvan University
Institute of Science and Technology
TU Exam Year: 2080
Candidates are required to give their answers in their own words as far as practicable.
SECTION A
Attempt all questions.
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1.
Given \( \vec{a} = (4,0,3) \) and \( \vec{b} = (-2,1,5) \), find \( |\vec{a}| \), \( 3\vec{b} \), \( \vec{a} + \vec{b} \), and \( 2\vec{a} + 5\vec{b} \).
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2.
Estimate the value of \( \lim_{x \rightarrow 0} \frac{\sqrt{x^{2}+9} - 3}{x^{2}} \).
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3.
The area of the parabola \( y = x^{2} \) from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface.
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4.
Find the solution of the equation \( y^{2}dy = x^{2}dx \) that satisfies the initial condition \( y(0) = 2 \).
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5.
As dry air moves upward, it expands and cools. If the ground temperature is \( 20^{\circ}C \) and the temperature at a height of 1 km is \( 10^{\circ}C \), express the temperature \( T(\text{in } ^{\circ}C) \) as a function of height (in kilometers), assuming that a linear model is appropriate.
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6.
Draw a graph of the function in part (a). What does the slope represent?
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7.
What is the temperature at a height of 2.5 km?
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8.
Integrate \( \int_{0}^{1} x^{2} \sqrt{x^{3} + 1} \, dx \).
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9.
Find the Maclaurin series expansion of \( f(x) = e^{x} \) at \( x = 0 \).
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10.
Find where the function \( f(x) = 3x^{4} - 4x^{3} - 12x^{2} + 5 \) is increasing and where it is decreasing.
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11.
Find \( y^{\prime} \) if \( x^{3} + y^{3} = 6xy \).
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12.
Show that \( y = x - \frac{1}{x} \) is a solution of the differential equation \( xy^{\prime} + y = 2x \).