List of practice Questions

Solve the differential equation: \[ \cos{x} \cos{y} \, dy - \sin{x} \sin{y} \, dx = 0 \]

Find the mean of numbers randomly selected from 1 to 15.

Find the area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \).

Find the general solution of \( \sin \theta + \sin 3\theta + \sin 5\theta = 0 \)

If \( -1 \leq x \leq 1 \), prove that \( \sin^{-1}{x} + \cos^{-1}{x} = \frac{\pi}{2} \)

If \( \theta \) is the acute angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \), then prove that \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are \( -2, 1, -1 \) and \( -3, -4, 1 \).

Find the shortest distance between lines: \[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}, \quad \frac{x-2}{4} = \frac{y-4}{5} = \frac{z-5}{6} \]

Lines \[ \overrightarrow{r} = (i + j - k) + \lambda(2i - 2j + k), \quad \overrightarrow{r} = (4i - 3j + 2k) + \mu(i - 2j + 2k) \] are coplanar. Find the equation of the plane determined by them.

Write the inverse and contrapositive of the following statement: If \( x<y \) then \( x^2<y^2 \).

Find the Cartesian co-ordinates of the point whose polar co-ordinates are \( \left( \sqrt{2}, \frac{\pi}{4} \right) \).

If \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of lines and \( h^2 = ab \neq 0 \), then find the ratio of their slopes.

If \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) are the position vectors of the points A, B, C respectively and \( 5\overrightarrow{a} + 3\overrightarrow{b} - 8\overrightarrow{c} = 0 \), then find the ratio in which the point C divides the line segment AB.

Solve the following inequalities graphically and write the corner points of the feasible region: \[ 2x + 3y \leq 6, \quad x + y \geq 2, \quad x \geq 0, \quad y \geq 0 \]

Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.

Evaluate: \[ \int_0^{\frac{\pi}{2}} \sqrt{1 - \cos{4x}} \, dx \]

Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).

The solution of the differential equation \( \frac{dx}{dt} = x \log \frac{x}{t} \) is

i. Write the joint equation of coordinate axes.

ii. Find the values of \( c \) which satisfy \( |c\vec{u}| = 3 \) where \( \vec{u} = \hat{i} + 2\hat{j} + 3\hat{k} \).

Write \( \int \cot{x} \, dx \).

Write the degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} = x \).

The area of the triangle with vertices \( (1, 2, 0), (1, 0, 2) \), and \( (0, 3, 1) \) in square units is

If the corner points of the feasible solution are \( (0, 10), (2, 2) \), and \( (4, 0) \), then the point of minimum \( z = 3x + 2y \) is

If \( y \) is a function of \( x \) and \( \log(x + y) = 2xy \), then the value of \( y'(0) \) is