List of top Mathematics Questions

If the equation of the plane passing through the point $A(-2, 1, 3)$ and perpendicular to the vector $3\vec{i} + \vec{j} + 5\vec{k}$ is $ax + by + cz + d = 0$, then $\dfrac{a + b}{c + d} =$
Three vectors of magnitudes $ a, 2a, 3a $ are along the diagonals of three adjacent faces of a cube that meet at a point. Find the magnitude of the sum of these vectors.
Let $ \vec{a}, \vec{b}, \vec{c} $ be unit vectors such that: - $ \vec{a} \perp \text{plane of } \vec{b}, \vec{c} $ - Angle between $ \vec{b} $ and $ \vec{c} $ is $ \frac{\pi}{3} $ Find: $$ |\vec{a} + \vec{b} + \vec{c}| $$
The vector \( \vec{x} \) is perpendicular to the vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 18\hat{i} - 22\hat{j} - 5\hat{k} \) and makes an obtuse angle with \( \hat{j} \). If \( |\vec{x}|=14 \), then \( \vec{x} = \)
The set of real values of \( \lambda \) for which the vectors \( \lambda\hat{i} - 3\hat{j} + 5\hat{k} \) and \( 2\lambda\hat{i} - \lambda\hat{j} + \hat{k} \) are perpendicular to each other is
Three vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfy: \[ \vec{a} + \vec{b} + \vec{c} = \vec{0},\quad |\vec{a}| = 1,\quad |\vec{b}| = 3,\quad |\vec{c}| = 4 \] Then the value of: \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = ? \]
The point of intersection of the lines \[ \vec{r} = 2\vec{b} + t(6\vec{c} - \vec{a}) \quad \text{and} \quad \vec{r} = \vec{a} + s(\vec{b} - 3\vec{c}) \text{ is:} \]
PQRS is a quadrilateral and \( \vec{PQ} = \vec{a} \), \( \vec{QR} = \vec{b} \), \( \vec{SP} = \vec{a} - \vec{b} \).
\( M \) is the midpoint of \( QR \), and \( X \) lies on \( \vec{SM} \) such that \( \vec{SX} = \frac{4}{5} \vec{SM} \).
If \( \vec{SM} = m(4\vec{a} - \vec{b}) \), and \( \vec{SX} = n(4\vec{a} - \vec{b}) \), then \( m + n = \):
If \( C \) is the midpoint of line segment \( \overline{AB} \), and \( P \) is any point outside the line \( AB \), then:
$OABC$ is a tetrahedron. If $D, E$ are the midpoints of $OA$ and $BC$ respectively, then $\overrightarrow{DE} =$
If $\vec{a} + \vec{b} + \vec{c} = 0$, and $|\vec{a}| = 7$, $|\vec{b}| = 5$, $|\vec{c}| = 3$, then the angle between $\vec{b}$ and $\vec{c}$ is
If $ \vec{a} $ is collinear with $ \vec{b} = 3\hat{i} + 6\hat{j} + 6\hat{k} $ and $ \vec{a} \cdot \vec{b} = 27 $, then find $ |\vec{a}| $.
Given: $$ \vec{F} = 2\hat{i} + 2\hat{j} + 5\hat{k},\quad A = (1, 2, 5),\quad B = (-1, -2, -3) $$ and: $$ \overrightarrow{BA} \times \vec{F} = 4\hat{i} + 6\hat{j} + 2\lambda \hat{k} $$ Find the value of $ \lambda $.
\( \vec{a}, \vec{b} \) are two vectors such that \( |\vec{a}|=\sqrt{3}, |\vec{b}|=\sqrt{2} \). If \( \vec{x} \) is a unit vector satisfying \( \vec{x} \times \vec{a} = \vec{b} \) then \( \vec{x} = \)
In triangle $ ABC $, given: $$ a = 6,\quad b = 5,\quad c = 4,\quad \text{find } \cos 2A $$
Which of the following trigonometric values are negative?
  • I) $ \sin(-292^\circ) $
  • II) $ \tan(-193^\circ) $
  • III) $ \cos(-207^\circ) $
  • IV) $ \cot(-222^\circ) $
The sides of a triangle \( ABC \) are given by \( a = 3 \), \( b = 5 \), and \( c = 3 \). Then \( \cos A = \)
In a triangle $ABC$, $\dfrac{a}{b} = 2 + \sqrt{3}$ and $\angle C = 60^\circ$. Then the measure of $\angle A$ is
In a triangle ABC, if a = 4, b = 5 and c = 7, then \( \sin(A/2) = \)
In a triangle ABC, $a = 2,\ b = 3,\ c = 4$, then $\tan\left(\dfrac{A}{2}\right) = $ ?
Let ABC be an acute-angled triangle with area R. Then \( \sqrt{a^2b^2-4R^2} + \sqrt{b^2c^2-4R^2} + \sqrt{c^2a^2-4R^2} = \)
If $(1 + \tan1^\circ)(1 + \tan2^\circ)\dots(1 + \tan45^\circ) = 2^n$, then $n =$ ?
If \( \sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = 1 \), then the value of \( x \) in the interval \( [0, \pi] \) is:
\( \cos\theta (\csc\theta - \sec\theta) - \cot\theta = \)
If the identity $\cos^4\theta = a\cos4\theta + b\cos2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$, then $(a, b, c) =$ ?