List of top Mathematics Questions

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$ . If $P(T)$ denotes the probability of occurrence of the event $T$, then

The circle passing through the point (-1,0) and touching the Y-axis at (0, 2) also passes through the point

Let $f : R \rightarrow R$ be a function such that $f (x + y) = f (x) + f (y), \forall x, y \in R$. If $f (x )$ is differentiable at $x = 0$, then

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4y^2=4.$ If the hyperbola passes through a focus of the ellipse, then

Let $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}+\hat{k} $ and $\overrightarrow{c}=\overrightarrow{i}-\overrightarrow{j}-\overrightarrow{k} $ be three vectors. A vector $\hat{v}$ in the plane of $\overrightarrow{a}$ and $ \overrightarrow{b},$ whose projection on $\overrightarrow{c} $ is $\frac{1}{\sqrt{3}},$ is given by

The vector(s) which is/are coplanar with vectors $\widehat{i} + \widehat{j}+2\widehat{k}$ and $\widehat{i} + 2\widehat{j}+\widehat{k},$ are perpendicular to the vector $\widehat{i} + \widehat{j}+\widehat{k}$ is / are

Let P(6,3) be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ If the norm al at the point P intersects the X -axis at (9, 0), then the eccentricity of the hyperbola is

If $ H_n = 1+\frac{1}{2} + ..... + \frac{1}{n}, $ then the value of $ S_n = 1+\frac{3}{2} + \frac{5}{3} + .....+\frac{2n-1}{n} is $

If the normal at the point $ P(\theta) $ to the ellipse $ \frac{x^2}{14} + \frac {y^2}{5} = 1 $ intersects it again at the point $ Q(2\theta), $ then $ cos \theta $ equals to

The eccentricity of the hyperbola with latus rectum $ 12 $ and semi-conjugate axis $ 2\sqrt{3} $ , is

If the parabolas $ y^2 = 4x $ and $ x^2 = 32y $ intersect at $ (16, 8) $ at an angle $ \theta $ , then $ \theta $ equals to

If $ asin^{-1}x - bcos^{-1} x = c $ , $ a sin^{-1}x + b cos^{-1} x $ is equal to

If algebraic sum of distances of a variable line from points $ (2,0) $ , $ (0,2) $ and $ (-2- 2) $ is zero, then the line passes through the fixed point

The length of the perpendicular distance of the point $ (-1,\,4,\,0) $ from the line $ \frac{x}{1}=\frac{y}{3}=\frac{z}{1} $ is equal to

The number of lines making equal angles with the coordinate axes in three dimensional geometry is equal to

The projection of a line segment $OP$ through origin $O$, on the coordinate axes are $8, 5, 6$. Then, the length of the line segment $OP$ is equal to

The domain of the function \( f(x) = \frac{\sqrt{9 - x^2}}{\sin^{-1}(3 - x)} \) is ________.

The term independent of \( x \) in the expansion of \( \left[ \sqrt{\frac{x}{3}} + \sqrt{\frac{3}{2}} x^2 \right]^{10} \) is ________.

In a \( \Delta ABC \), \( \angle B = 90^\circ \), then \( \tan^2\left(\frac{A}{2}\right) \) is

In a trapezoid of the vector \( \vec{BC} = \lambda \vec{AD} \). We will, then find that \( \vec{P} = \vec{AC} + \vec{BD} \) is collinear with \( \vec{AD} \). If \( \vec{P} = \mu \vec{AD} \), then

If \( P(\vec{p}) \), \( Q(\vec{q}) \), \( R(\vec{r}) \), and \( S(\vec{s}) \) are four points such that \( 3\vec{p} + 8\vec{q} = 6\vec{r} + 5\vec{s} \), then the lines PQ and RS are

Let \( \vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} + \hat{j} \), if \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}| \), \( |\vec{c} - \vec{a}| = 2\sqrt{2} \), and the angle between \( \vec{a} \times \vec{b} \) and \( \vec{c} \) is \( 30^\circ \), then \( |(\vec{a} \times \vec{b}) \times \vec{c}| \) is equal to

Consider a tetrahedron with faces \( F_1, F_2, F_3, F_4 \). Let \( \vec{v_1}, \vec{v_2}, \vec{v_3}, \vec{v_4} \) be area vectors perpendicular to these faces in the outward direction, then \( |\vec{v_1} + \vec{v_2} + \vec{v_3} + \vec{v_4}| \) equals

If \( V \) is the volume of the parallelepiped with edges \( \vec{a}, \vec{b}, \vec{c} \), then the volume of the parallelepiped with edges \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) (defined by dot products) is

Define the length of \( a\hat{i} + b\hat{j} + c\hat{k} \) as \( |a| + |b| + |c| \). This definition coincides with the usual definition if and only if