List of top Mathematics Questions

In a triangle ABC, if \(r_1 = 3, r_2 = 4, r_3 = 6\), then \(b =\)
Let the position vectors of the vertices of triangle ABC be \(\vec{a}, \vec{b}, \vec{c}\). If a point \(P\) on the plane of triangle has a position vector \(\vec{r}\) such that \(\vec{r} - \vec{b} = \vec{a} - \vec{c}\) and \(\vec{r} - \vec{c} = \vec{a} - \vec{b}\), then \(P\) is the
If the variance of the first \(n\) natural numbers is 10 and the variance of the first \(m\) even natural numbers is 16, then \(n : m =\)
The point of intersection of the lines represented by \(\vec{r} = (\hat{i} - 6\hat{j} + 2\hat{k}) + t(\hat{i} + 2\hat{j} + \hat{k})\) and \(\vec{r} = (4\hat{j} + \hat{k}) + s(2\hat{i} + \hat{j} + 2\hat{k})\) is
If the points A, B, C, D with position vectors \(\vec{i} + \vec{j} - \vec{k}, -\vec{i} + 2\vec{k}, \vec{i} - 2\vec{j} + \vec{k}, 2\vec{i} + \vec{j} + \vec{k}\) form a tetrahedron, then angle between faces ABC and ABD is
If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and \(\vec{a} \perp \vec{b}\), and \((\vec{a} - \vec{c}) \cdot (\vec{b} + \vec{c}) = 0\), and \(\vec{c} = l\vec{a} + m\vec{b} + n(\vec{a} \times \vec{b})\), then \(n^2 =\)
In \(\triangle ABC\), if \(a + c = 5b\), then \(\cot\dfrac{A}{2} \cdot \cot\dfrac{C}{2} =\)
Evaluate the expression:
\[ \cos^3 \left( \frac{3\pi}{8} \right) \cos \left( \frac{3\pi}{8} \right) + \sin^3 \left( \frac{3\pi}{8} \right) \sin \left( \frac{3\pi}{8} \right) \]
If \( A + B + C = \frac{\pi}{4} \), then evaluate the expression:
\[ \sin 4A + \sin 4B + \sin 4C \]
In a triangle ABC, if \(\sin\frac{A}{2} = \dfrac{1}{4}\sqrt{\dfrac{5}{\sqrt{5}}}, a = 2, c = 5\), and \(b\) is an integer, then the area (in sq. units) of triangle ABC is
If \[ \binom{p}{q} = \binom{p}{q} \quad \text{and} \quad \sum_{i=0}^m \binom{10}{i} \binom{20}{m-i} \text{ is maximum, then find } m. \]
If \( ax^2 + bx + e>0 \) for all \( x \in \mathbb{R} \) and the expressions \( cx^2 + ax + b \) and \( ax^2 + bx + c \) have their extreme values at the same point \( x \), then for the expression \( cx^2 + ax + b \), find the correct statement regarding its extreme value.
If \( x^2 - 5x + 6 \) is a factor of \( f(x) = x^4 - 17x^3 + kx^2 - 247x + 210 \), find the other quadratic factor of \( f(x) \).
If $a = \ln \left( \frac{1}{z^2} \right)$ and $z$ is any non-zero complex number such that $|z| = 1$, then which of the following is the correct expression for $a$?
The general solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{y}{x} e^x$ is
Identify the correct option from the following:
$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos x - \sin x}{\sin 2x} \, dx =$
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$\int_0^{\frac{\pi}{2}} \frac{\sin x}{\cos x + \sin x} \, dx =$
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$\lim_{n \to \infty} \left[ \frac{n+1}{n^2+1^2} + \frac{n+2}{n^2+2^2} + \frac{n+3}{n^2+3^2} + \dots + \frac{n+2n}{n^2+(2n)^2} \right] =$
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If $\int \frac{dx}{\sin^3 x + \cos^3 x} = A \log \left| \sqrt{2} + t \right| + B \tan^{-1} (t) + c$, then $\left( \frac{B}{A}, t \right) =$
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$\int (x^3 y^3) \, dx =$
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The value of $c$ of Rolle's theorem for the function $f(x) = 2 \sin x + \sin 2x$ in the interval $[0, \pi]$ is
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$\int \frac{e^{\sin x} (\sin 2x - 8 \cos x)}{2 (\sin x - 3)^2} \, dx =$
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Consider the following functions
I) $f(x) = \left| \frac{1}{2 - x}, x<\frac{1}{2} \right|$
II) $f(x) = \left| \frac{1}{(2 - x)^2}, x \neq 2 \right|$
III) $f(x) = |x|$
IV) $f(x) = |x|$
Then on $[0, 1]$, Lagrange's mean value theorem is applicable to the functions Identify the correct option from the following:
The area (in square units) of the triangle formed by the X-axis, the tangent and the normal drawn at (1, 1) to the curve $x^3 + y^3 = 2xy$ is
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$\lim_{x \to 0} \frac{(\sqrt{2})^{-\sqrt{1 + \cos x}}}{15 + \cos 2x - 4} =$
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