List of top Mathematics Questions

If the lines given by \( \vec{r} = 2\hat{i} + \lambda(\hat{i} + 2\hat{j} + m\hat{k}) \) and \( \vec{r} = \hat{i} + \mu(2\hat{i} + \hat{j} + 6\hat{k}) \) are perpendicular, then the value of \( m \) is

If \( A(-1,2,3) \), \( B(3,-2,1) \), \( C(2,1,3) \) and \( D(-1,-2,4) \) are the vertices of a tetrahedron, then its volume is

A metal wire \(108\) meters long is bent to form a rectangle. If the area of the rectangle is maximum, then its dimensions are

If \( y\sqrt{1-x^2} + x\sqrt{1-y^2} = 1 \), then \( \dfrac{dy}{dx} = \)

\( y = c^2 + \dfrac{c}{x} \) is the solution of the differential equation

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

If the vectors \( (2\mathbf{i}-q\mathbf{j}+3\mathbf{k}) \) and \( (4\mathbf{i}-5\mathbf{j}+6\mathbf{k}) \) are collinear, then the value of \( q \) is

If the vectors \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, then \( \dfrac{\left| \vec{a} + 2\vec{b} \;\; \vec{b} + 2\vec{c} \;\; \vec{c} + 2\vec{a} \right|} {\left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right|} \) is

Simplify \( \cos\left(\dfrac{3\pi}{4}+x\right) - \sin\left(\dfrac{\pi}{4}-x\right) \)

Evaluate the integral \( \displaystyle \int \frac{e^x}{\sqrt{x}}(1+2x)\,dx \)

Evaluate \( \displaystyle \int_{-\pi}^{\pi} \frac{2x}{1+\cos^2 x}\,dx \)

If \( f(x) = [x]^2 - 5[x] + 6 = 0 \), where \( [x] \) denotes the greatest integer function, then \( x \in \)

If \( \int_{0}^{1} (5x^2 - 3x + k) \, dx = 0 \), then \( k = \)

If \( \int x^x (1 + \log x) \, dx = k x^x + c \), then \( k = \)

A particle moves according to the law \( s = t^3 - 6t^2 + 9t + 25 \). The displacement of the particle at the time when its acceleration is zero is

For a sequence \( (t_n) \), if \( s_n = 7(3^n - 1) \), then \( t_n = \)

If the population grows at the rate of \( 8% \) per year, then the time taken for the population to be doubled is \([ \log 2 = 0.6912 ]\)

The measure of the acute angle between the lines given by the equation \( 3x^2 - 4\sqrt{3}xy + 3y^2 = 0 \) is

The area of the region bounded by the curve \( y = \log x \), x-axis and the lines \( x = 1 \), \( x = e \) is

The general solution of the differential equation \( (1 - x^2) \frac{dy{dx} + 2xy = x(1 - x^2)^{\frac{1}{2}} \) is}

Simplify the expression \( \sin \left( \frac{\pi}{3} + x \right) - \cos \left( \frac{\pi}{6} + x \right) \)

Evaluate the integral \( \int_0^1 \frac{x^2}{1 + x^2} dx \)

Evaluate the integral \( \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \log |9e^{2x} - 4| + c \), then (where \( c \) is the constant of integration)

If \( y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \), then \[ \frac{dy}{dx} = \]

The particular solution of the differential equation \( \cos \left( \frac{dy}{dx} \right) = a \), under the conditions \( a \in \mathbb{R} \) and \( y(0) = 2 \) is