List of top Mathematics Questions

f and g are continuous functions on interval \( [0, a] \). Given that \( f(a - x) = f(x) \) and \( g(x) + g(a - x) = a \), show that \( \int_{0}^{a} f(x) g(x) dx = \frac{a}{2} \int_{0}^{a} f(x) dx \).
The feasible region along with corner points for a linear programming problem are shown in the graph. Write all the constraints for the given linear programming problem.
feasible region along with corner points
Evaluate: \( \int_{0}^{\pi} \frac{\sin 2px}{\sin x} dx \), \( p \in N \).
Prove that \( f : N \rightarrow N \) defined as \( f(x) = ax + b \) (\( a, b \in N \)) is one-one but not onto.
The integral \( \int \frac{dx}{\sin^2 x \cos^2 x} \) is equal to
If \( \vec{PQ} \times \vec{PR} = 4 \hat{i} + 8 \hat{j} - 8 \hat{k} \), then the area \( (\triangle PQR) \) is
In the following probability distribution, the value of p is:
probability distribution
The principal value of \( \sin^{-1} \left( \cos \frac{43\pi}{5} \right) \) is
The values of \(x\) for which the angle between the vectors \( \vec{a} = 2x^2 \hat{i} + 4x \hat{j} + \hat{k} \) and \( \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k} \) is obtuse, is:

If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \] has infinitely many solutions, then \( \lambda + \mu \) is equal to:}

The equation of the circle with extremities \( (1, 3) \) and \( (5, 7) \) of the diameter is:
The centre of the circle of curvature for the curve \( y = e^x \) at \( (0, 1) \) is:
The coefficient of \( (y-2) \) in the Taylor's series expansion of \( f(x, y) = x^2 + xy + y^2 \) in powers of \( (x-1) \) and \( (y-2) \) is:
The integral value of \( \int \frac{\cos 2x}{\sin^2 x \cos^2 x} \, dx = \):
The curvature of the straight line \( y = 2x + 3 \) at \( (1, 5) \) is:
491400 =
Evaluate the following statement: \[ f(x) = \sin(|x|) - |x| \text{ is not differentiable at } x = \underline{\hspace{2cm}} \]
Evaluate \( (\sec A + \tan A)(1 - \sin A) = \ ? \)
Let {an}n=0∞ be a sequence such that a0=a1=0 and an+2=3an+1−2an+1,∀ n≥0. Then a25a23−2a25a22−2a23a24+4a22a24  is equal to
Which of the following can be both a symmetric and skew-symmetric matrix?
A die with numbers 1 to 6 is biased such that \( P(2) = \frac{3}{10} \) and the probability of other numbers is equal. Find the mean of the number of times number 2 appears on the die, if the die is thrown twice.
Let \( R \) be a relation on the set of real numbers \( \mathbb{R} \) defined as \[ R = \{(x, y) : x - y + \sqrt{3} \text{ is an irrational number}, x, y \in \mathbb{R} \}. \] Verify \( R \) for reflexivity, symmetry, and transitivity.
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \), for \( -1<x<1, x \neq y \), then prove that \[ \frac{dy}{dx} = \frac{-1}{(1 + x)^2}. \]
If \( |\mathbf{a}| = 2 \), \( |\mathbf{b}| = 3 \) and \( \mathbf{a} \cdot \mathbf{b} = 4 \), then evaluate \( |\mathbf{a} + 2\mathbf{b}| \).

If \( f(x) = \begin{cases} 2x - 3, & -3 \leq x \leq -2 \\x + 1, & -2<x \leq 0 \end{cases} \), check the differentiability of \( f(x) \) at \( x = -2 \).