List of top Mathematics Questions

The curve \( y = ax^3 + bx^2 + cx + 5 \) touches the x-axis at point \( P(-2, 0) \). Then the value of \( c \) is:
If \( ax + by = 1 \) is a normal to the parabola \( y^2 = 4px \), then the condition is
Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function satisfying $f(x) - x = \lambda$ (constant), $\forall x \in \mathbb{R}^+$ and $f(xf(y)) = f(y) + x$, $\forall x, y \in \mathbb{R}^+$. Then $\displaystyle \lim_{x \to 0} \dfrac{(f(x))^{\frac{5}{3}} - 1}{(f(x))^{\frac{2}{3}} - 1}$ is
A circle passes through \( (0, a) \) and \( (b, h) \), and its center is at \( (c, 0) \). Find \( c \).
Two brothers X and Y appeared for an exam. Let:
- $ A $: event that X passed
- $ B $: event that Y passed
Given: $$ P(A) = \frac{1}{7},\quad P(B) = \frac{2}{9} $$ Assuming independence, find the probability that both passed, i.e., $ P(A \cap B) $.
Let plane $ \pi $ pass through $ (1, 0, 1) $ and be perpendicular to both: - $ 2x + 3y - z = 2 $ - $ x - y + 2z = 1 $ Let another plane pass through point $ (11, 7, 5) $, be parallel to $ \pi $, and have the form: $$ ax + by - z + d = 0 $$ Then evaluate: $$ \frac{a}{b} + \frac{b}{d} $$
If $x \neq 0$ and $f(x)$ satisfies $8f(x) + 6f\left(\dfrac{1}{x}\right) = x + 5$, then $\dfrac{d}{dx} \left(x^2 f(x)\right)$ at $x = 1$ is
The total number of permutations of $n$ different things taken not more than $r$ at a time, when each thing may be repeated any number of times is:
\( \vec{a}, \vec{b} \) 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 \( \vec{x} = \)
If \( f(x) \) is the antiderivative of \( g(x) \) and \[ \int f(x) g(x)(1 + f^2(x)) dx = F(x) \], then \( F(x) = \)
If $f(x) = \cot^{-1}\left( \dfrac{x^n + x^{-n}}{2} \right)$, then $f'(1) = $
The eccentric angle of a point on the ellipse $x^2 + 3y^2 = 6$ lying at a distance of 2 units from its centre is
If a function \( f \) satisfies \( f(x+1) + f(x-1) = \sqrt{2}f(x) \), then \( f(x+2) + f(x-2) = \)
Let \( A = (1, -2, 3) \), \( B = (3, 1, -3) \), \( C = (-3, 1, 3) \) be the vertices of triangle \( ABC \). If \( \cos \theta = \frac{\vec{AB} \cdot \vec{AC}}{|\vec{AB}||\vec{AC}|} \), then \( \cos A \) is:
The least distance from origin to a point on the line $y = x + 3$ which lies at a distance of 2 units from $(0,3)$ is
A true statement among the following identities is
Person A can solve 90% of the problems given in the book and Person B can solve 70%. Then the probability that atleast one of them will solve the problem selected at random from the book is
A person writes letters to 6 friends and addresses envelopes. In how many ways can the letters be placed so that at least two go to wrong envelopes?
In a plane there are 37 straight lines of which 13 pass through point A and 11 pass through point B. Moreover, no three lines (apart from the lines passing through A and B) pass through same point and no two lines are parallel. What is the number of points of intersection of the straight lines?
Let \( \vec{a}, \vec{b} \) be non-collinear vectors, with \( |\vec{a}| = 2\sqrt{2} \), \( |\vec{b}| = 3 \), and the angle between them is \( 45^\circ \). Then the lengths of the diagonals of the parallelogram whose adjacent sides are \( \vec{5a} + \vec{2b} \) and \( \vec{a} - \vec{3b} \) are:
Two vertices of a triangle are at \( -\vec{i} + 3\vec{j} \) and \( 2\vec{i} + 5\vec{j} \). Its orthocenter is at \( \vec{i} + 2\vec{j} \). If the position vector of the third vertex is \( a\vec{i} + b\vec{j} \), then \( (a, b) = \):
Given: $$ 3f(\cos x) + 2f(\sin x) = 5x $$ Find: $$ f'(\cos x) + f'(\sin x) $$
Find the least distance from the point $ (10, 7) $ to the circle: $$ x^2 + y^2 - 4x - 2y - 20 = 0 $$
In a triangle ABC, let \(\angle C = \pi/2\). If r and R are respectively inradius and circumradius of ABC, then R + r =
If $(1 + \tan1^\circ)(1 + \tan2^\circ)\dots(1 + \tan45^\circ) = 2^n$, then $n =$ ?