List of top Mathematics Questions

Given: $$ 3f(\cos x) + 2f(\sin x) = 5x $$ Find: $$ f'(\cos x) + f'(\sin x) $$

Evaluate: $$ \lim_{x \to \infty} \log_e(\cosh x) + x $$

If $ a, b, c $ are distinct real numbers and: $$ \lim_{x \to \infty} \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} = \frac{1}{2} $$ then what is the value of $ a + 2c $?

Evaluate: $$ \lim_{x \to \infty} \frac{3|x| - x}{|x| - 2x} - \lim_{x \to 0} \frac{\log(1 + x^3)}{\sin^3 x} $$

Assertion (A): $$ \frac{d}{dx} \left( \frac{x^2 \sin x}{\log x} \right) = \frac{x^2 \cos x \cdot \log x - x^2 \sin x \cdot \frac{1}{x}}{(\log x)^2} + \frac{2x \sin x}{\log x} \Rightarrow \text{(matches the structure of product + quotient rule)} $$ Reason (R): $$ \frac{d}{dx} \left( \frac{uv}{w} \right) = \frac{uv'}{w} + \frac{u'v}{w} - \frac{uvw'}{w^2} \Rightarrow \text{(correct general derivative formula)} $$

If the direction cosines of a line are: $$ \left( \frac{a}{\sqrt{83}},\ \frac{5}{\sqrt{83}},\ \frac{c}{\sqrt{83}} \right) $$ and it is given that $ c - a = 4 $, find $ ca $.

Which of the following equations represents a **parabola**?

If the angle between the lines joining the foci and the ends of the minor axis of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $ 90^\circ $, then its **eccentricity** is:

Find the locus of the point of intersection of tangents drawn at the ends of a **normal chord** of the hyperbola: $$ x^2 - y^2 = a^2 $$

If $ e_1 $ and $ e_2 $ are the eccentricities of the hyperbola $$ 16x^2 - 9y^2 = 1 $$ and its conjugate respectively, then find $ 3e_1 $ in terms of $ e_2 $.

Find the radical centre of the three circles: $$ C_1: x^2 + y^2 - 1 = 0,\quad C_2: x^2 + y^2 - 8x + 15 = 0,\quad C_3: x^2 + y^2 + 10y + 24 = 0 $$

Let point $ P $ divide the line segment joining $ A(1, 2, -1) $ and $ B(-1, 0, 1) $ externally in the ratio $ 1:2 $. Given point $ Q = (1, 3, -1) $, find $ PQ $.

Find the radius of the circle that has the lines: $$ 3x - 4y + 4 = 0 \quad \text{and} \quad 6x - 8y - 7 = 0 $$ as its tangents.

Find the least distance from the point $ (10, 7) $ to the circle: $$ x^2 + y^2 - 4x - 2y - 20 = 0 $$

Three points $ A, B, C $ in the plane have: - x-coordinates in G.P. - y-coordinates in the same G.P. Then, the points $ A, B, C $ lie on:

Sides passing through vertex $ (\alpha, \beta) $ of a triangle are bisected at right angles by the lines: $$ y^2 - 8xy - ax^2 = 0 $$ Find the coordinates of the centroid of the triangle.

Let the coordinates of points A and B satisfy:
- $ x^2 + 2x - a^2 = 0 $
- $ y^2 + 4y - b^2 = 0 $ Find the equation of the circle having AB as diameter.

Let the slopes $ m_1 $ and $ m_2 $ of the lines represented by: $$ ax^2 + 2hxy + by^2 = 0 $$ satisfy: $$ 3(m_1 - m_2) - 7 = 0 \quad \text{and} \quad m_1 m_2 - 2 = 0 $$ Then find the correct relation among $ a, b, h $.

A circle is such that it is tangent to every line of the form: $$ (x - 2)\cos\theta + (y - 2)\sin\theta = 1 $$ for all $ \theta $. Find the equation of the circle.

Find the length of the intercept made by the line $ x + 1 = 0 $ between the lines: $$ 3x + 2y = 5 \quad \text{and} \quad 3x + 2y = 3 $$

Points $ P $ and $ Q $ lie on the lines: $$ 3x + 4y - 4 = 0 \quad \text{and} \quad 5x - y - 4 = 0 $$ The midpoint of segment $ PQ $ is $ (1, 5) $. Find the slope of the line $ PQ $.

For a Poisson distribution:
- Mean $ = \ell $
- Variance $ = m $
- $ \ell + m = 8 $
Then evaluate: $$ e^4 \cdot \left[1 - P(X>2)\right] $$

There are two bags:
- Bag 1: 4 red, 3 black → total = 7 balls
- Bag 2: 2 red, 3 black → total = 5 balls
A bag is chosen at random and one ball is drawn. What is the probability that the ball is red?

In a binomial distribution, if:
- Number of trials $ n $
- Mean $ = 4 $
- Variance $ = 3 $
Then find: $$ 2^{32} \cdot P\left(X = \frac{n}{2}\right) $$

Find the locus of midpoints of points of intersection of the line $$ x \cos \theta + y \sin \theta = 1 $$ with the coordinate axes.