List of top Mathematics Questions

\( (\vec{a}+2\vec{b}-\vec{c}) \cdot ((\vec{a}-\vec{b}) \times (\vec{a}-\vec{b}-\vec{c})) = \)
For a positive real number p, if the perpendicular distance from a point \( -\vec{i} + p\vec{j} - 3\vec{k} \) to the plane \( \vec{r} \cdot (2\vec{i} - 3\vec{j} + 6\vec{k}) = 7 \) is 6 units, then p =
An unbiased coin is tossed 8 times. The probability that head appears consecutively at least 5 times is
In \( \triangle ABC \), if \( r = 3 \) and \( R = 5 \) then \( \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \)
If the vector \( \vec{v} = \vec{i} - 7\vec{j} + 2\vec{k} \) is along the internal bisector of the angle between the vectors \( \vec{a} \) and \( \vec{b} = -2\vec{i} - \vec{j} + 2\vec{k} \) and the unit vector along \( \vec{a} \) is \( \hat{a} = x\vec{i} + y\vec{j} + z\vec{k} \) then \( x = \)
An aeroplane is flying at a constant speed, parallel to the horizontal ground at a height of 5 kms. A person on the ground observed that the angle of elevation of the plane is changed from \(15^\circ\) to \(30^\circ\) in the duration of 50 seconds, then the speed of the plane (in kmph) is
If \( \text{sech}^{-1}x = \log 2 \) and \( \text{cosech}^{-1}y = -\log 3 \), then \( (x+y) = \)
If the sides a,b,c of the triangle ABC are in harmonic progression, then \( \text{cosec}^2 A/2, \text{cosec}^2 B/2, \text{cosec}^2 C/2 \) are in
If \( \frac{ax+5}{(x^2+b)(x+3) = \frac{x+21}{12(x^2+b)} + \frac{c}{12(x+3)} \), then \( b^2 = \)}
If \( \alpha, \beta \) are the acute angles such that \( \frac{\sin \alpha}{\sin \beta} = \frac{6}{5} \) and \( \frac{\cos \alpha}{\cos \beta} = \frac{9}{5\sqrt{5}} \) then \( \sin \alpha = \)
If \( \left(\frac{\sin 3\theta}{\sin \theta}\right)^2 - \left(\frac{\cos 3\theta}{\cos \theta}\right)^2 = a \cos b\theta \), then \( a : b = \)
The terms containing \( x^r y^s \) (for certain r and s) are present in both the expansions of \( (x+y^2)^{13 \) and \( (x^2+y)^{14} \). If \( \alpha \) is the number of such terms, then the sum \( \sum_{r,s} \alpha (r+s) = \) (Note: The sum is over the common terms)}
The coefficient of \( x^3 \) in the power series expansion of \( \frac{1+4x-3x^2}{(1+3x)^3 \) is}
When the roots of \( x^3 + \alpha x^2 + \beta x + 6 = 0 \) are increased by 1, if one of the resultant values is the least root of \( x^4 - 6x^3 + 11x^2 - 6x = 0 \), then
If \( \alpha \neq 0 \) and zero are the roots of the equation \( x^2 - 5kx + (6k^2-2k) = 0 \), then \( \alpha = \)
The set of all real values of \(x\) for which \(f(x) = \sqrt{\frac{|x|-2}{|x|-3}}\) is a well defined function is
\( \sum_{k=1}^{n} k(k+1)(k+2)...(k+r-1) = \)
If the order and degree of the differential equation \(x \frac{d^2 y}{dx^2} = \left(1 + \left(\frac{d^2 y}{dx^2}\right)^2\right)^{-1/2}\) are \(k\) and \(l\) respectively, then \(k, l\) are the roots of
The equation of the curve passing through the point \( (0, \pi) \) and satisfying the differential equation \( ydx = (x + y^3 \cos y)dy \) is
Evaluate the integral \( \displaystyle \int_{1/5}^{1/2} \frac{\sqrt{x - x^2}}{x^3} \, dx \):
If \(\int \frac{1}{((x+4)^3 (x+1)^5)^{1/4}} \, dx = A \cdot \left(\frac{x+4}{x+1}\right)^n + c\), then
Evaluate: \[ \int \frac{dx}{(x+1)\sqrt{x^2+1}} \]
If \( m \) and \( M \) are the absolute minimum and absolute maximum values of the function \( f(x) = 2\sqrt{2 \sin x - \tan x} \) in the interval \( \left[0, \frac{\pi}{3} \right] \), then \( m + M = \)
The function \( f(x) = xe^{-x} \) for all \( x \in \mathbb{R} \) attains a maximum value at \( x = k \), then \( k = \)
The slope of a tangent drawn at the point \( P(\alpha, \beta) \) lying on the curve \( y = \frac{1}{2x - 5} \) is \( -2 \). If \( P \) lies in the fourth quadrant, then \( \alpha - \beta = \)