List of top Mathematics & Statistics Questions

Write the unit vector in the opposite direction to \( \vec{u} = 8\hat{i} + 3\hat{j} - \hat{k} \).
Using truth table, prove that the statement patterns \( p \leftrightarrow q \) and \( (p \land q) \lor (\sim p \land \sim q) \) are logically equivalent.
The vector equation of the line passing through the point having position vector \( 4\hat{i} - \hat{j} + 2\hat{k} \) and parallel to vector \( -2\hat{i} - \hat{j} + \hat{k} \) is given by .........
Select and write the correct answer of the following multiple choice type questions:
(i) If \( A = \{1, 2, 3, 4, 5\} \), then which of the following is not true?
Show that: \( \int_0^\pi \log (1 + \tan x) \, dx = \frac{\pi}{8} \log 2 \)
Find the area of region between parabolas \( y^2 = 4ax \) and \( x^2 = 4ay \).
Solve the L.P.P. by graphical method. Minimize \( z = 8x + 10y \) subject to \( 2x + y \geq 7 \), \( 2x + 3y \geq 15 \), \( y \geq 0 \), \( x \geq 0 \).
Using vectors prove that the altitudes of a triangle are concurrent.
Solve the following system of equations by the method of inversion:
\( x - y + z = 4 \)
\( 2x + y - 3z = 0 \)
\( x + y + z = 2 \)
A wire of length 36 meters is bent to form a rectangle. Find its dimensions if the area of the rectangle is maximum.
Solve \( x + y \frac{dy}{dx} = \sec (x^2 + y^2) \)
If \( y = e^{m \tan^{-1} x} \), then show that \( (1 + x^2) \left( \frac{dy}{dx} \right)^2 + (2x - 2m) \frac{dy}{dx} = 0 \).
Find the vector equation of the plane passing through the point \( A(-1, 2, -5) \) and parallel to the vectors \( 4\hat{i} - \hat{j} + 3\hat{k} \) and \( \hat{i} + \hat{j} - \hat{k} \).
If \( A(\vec{a}) \) and \( B(\vec{b}) \) be any two points in space and \( R(\vec{r}) \) be a point on the line segment \( AB \) dividing it internally in the ratio \( m:n \), then prove that \( \vec{r} = \frac{m \vec{b} + n \vec{a}}{m + n} \).
If \( A(5, 1, p) \), \( B(1, q, p) \), and \( C(1, -2, 3) \) are vertices of a triangle and \( G \left( -\frac{4}{3}, \frac{1}{3}, -\frac{4}{3} \right) \) is its centroid, then find the values of \( p \), \( q \) by vector method.
If angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) is equal to the angle between the lines represented by \( 2x^2 - 5xy + 3y^2 = 0 \), then show that \( 100 (h^2 - ab) = (a + b)^2 \).
If \( 2 \tan^{-1} (\cos x) = \tan^{-1} (2 \csc x) \), then find the value of \( x \).
If \( X \sim B(n, p) \) and \( E(X) = 6 \) and \( \text{Var}(X) = 4.2 \), then find \( n \) and \( p \).
Form the differential equation of all lines which make intercept 3 on x-axis.
Solve the differential equation \( y \frac{dy}{dx} + x = 0 \)
Evaluate: \( \int \sec^3 x \, dx \)
Find the equation of tangent to the curve \( y = 2x^3 + x^2 + 2 \) at \( \left( \frac{1}{2}, 2 \right) \).
Find the cartesian equation of the plane passing through \( A(1, 2, 3) \) and the direction ratios of whose normal are 3, 2, 5.
Find the value of \( k \), if \( 2x + y = 0 \) is one of the lines represented by \( 3x^2 + kxy + 2y^2 = 0 \)
Find the principal solutions of \( \cot \theta = 0 \)