CSIT 1st Semester
Math I Board Question Paper 2080

(a) If \(\vec{a} = (4, 0, 3)\) and \(\vec{b} = (-2, 1, 5)\), find \(|\vec{a}|, 3\vec{b}, \vec{a} + \vec{b}\) and \(2\vec{a} + 5\vec{b}\). [1+1+1+2]
(b) Estimate the value of \(\lim_{x \to 0} \frac{\sqrt{x^2 + 9} - 3}{x^2}\)

(a) 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.
(b) Find the solution of the equation \(y^2dy = x^2dx\) that satisfies the initial condition \(y(0) = 2\).

As dry air moves upward, it expands and cools. If the ground temperature is \(20°C\) and the temperature at height of 1 km is \(10°C\), express the temperature \(T\) (in \(°C\)) as a function of the height \(h\) (in kilometer), assuming that linear model is appropriate.
(a) Draw a graph of the function in part (a). What does the slope represent?
(c) What is the temperature at a height of \(2.5 km\)?

Integrate \(\int_0^1 x^2 \sqrt{x^3 + 1} \, dx\).

Find the Maclaurin series expansion of \(f(x) = e^x\) at \(x = 0\).

Find where the function \(f(x) = 3x^4 - 4x^3 - 12x^2 + 5\) is increasing and where it is decreasing.

Show that \(y = x - \frac{1}{x}\) is a solution of the differential equation \(xy' + y = 2x\).

Sketch the graph and find the domain and range of the function \(f(x) = 2x - 1\).

Determine whether the series \(\sum_{n=1}^{\infty} \frac{n^2}{5n^2 + 4}\) converges or diverges.

If \(f(x,y) = x^3 + x^2y^3 - 2y^2\), find \(f_x(2,1)\) and \(f_y(2,1)\).

Show that the function \(f(x) = x^2 + \sqrt{7 - x}\) is continuous at \(x = 4\).