List of top Mathematics Questions

The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ (with $a \ne 0$) from the set $\{0,1,2,4\}$ is
Find the vector equation of a straight line passing through the point (5, 2, -4) and parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).
The number of ways of dividing 15 persons into 3 groups containing 3, 5 and 7 persons so that two particular persons are not included into the 5 persons group is
Raina is 1.5 m tall. At an instant, his shadow is 1.8 m long. At the same instant, the shadow of a pole is 9 m long. How tall is the pole?
If \( 3\overline{i} + \overline{j} + \overline{k} \), \( 2\overline{i} + \overline{k} \), and \( \overline{i} + 5\overline{j} \) are the position vectors of three non-collinear points A, B, C respectively. If the perpendicular drawn from C onto \( \overline{AB} \) meets \( \overline{AB} \) at the point \( a\overline{i} + b\overline{j} + c\overline{k} \), then \( a + b + c = \)
Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} |x|^{1/8} \sin \tfrac{1}{|x|} \cos x & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} e^x \cos \tfrac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
The solution of \( x - 2y = 0 \) and \( 3x + 4y - 20 = 0 \) is:

A quadratic polynomial \( (x - \alpha)(x - \beta) \) over complex numbers is said to be square invariant if \[ (x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). \] Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is ___________. (rounded off to one decimal place) 

If the equation of the circle having the common chord to the circles $$ x^2 + y^2 + x - 3y - 10 = 0 $$ and $$ x^2 + y^2 + 2x - y - 20 = 0 $$ as its diameter is $$ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, $$ then find $ \alpha + 2\beta + \gamma $.
The equation of chord AB of ellipse \(2x^2 + y^2 = 1\) is \(x - y + 1 = 0\). If O is the origin, then \(\angle AOB =\)
The domain of the real valued function $ f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x) $ is
What is the next number in the sequence? 2, 6, 12, 20, 30, ?

For the curve \( \sqrt{x} + \sqrt{y} = 1 \), find the value of \( \frac{dy}{dx} \) at the point \( \left(\frac{1}{9}, \frac{1}{9}\right) \).

A batch of 36 school children goes for an excursion trip to Nainital. When one 10-year-old child is replaced with another child, the average age of the children in the batch increases by \( 2\frac{1}{2} \) months. What is the age of the new child?
A number is selected randomly from each of the following two sets:
{1, 2, 3, 4, 5, 6, 7, 8}, {2, 3, 4, 5, 6, 7, 8, 9}
What is the probability that the sum of the numbers is 9?
If $ f(x) = \log 3 - \sin x $, $ y = f(f(x)) $, find $ y(0) $.
A prime number \( p \) divides \( a^2 \) where \( a \) is a positive integer, then
Solve: \( \int \frac{(3x + 5)dx}{x^3 - x^2 - x + 1} \)
Find the unit vector along the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).
If the locus of a point which is equidistant from the coordinate axes forms a triangle with the line \(y = 3\), then the area of the triangle is
If \( x = p \cos^3 \alpha \) and \( y = q \sin^3 \alpha \), then the value of
\( \left( \frac{x}{p} \right)^{2/3} + \left( \frac{y}{q} \right)^{2/3} \text{ is:} \)

In the given figure, a circle inscribed in \( \triangle ABC \) touches \( AB, BC, \) and \( CA \) at \( X, Z, \) and \( Y \) respectively.

If \( AB = 12 \, \text{cm}, AY = 8 \, \text{cm}, \) and \( CY = 6 \, \text{cm} \), then the length of \( BC \) is:
 

Area of a sector of a circle with radius 4 cm and angle 30° is (use \( \pi = 3.14 \)):
If assumed mean of a data is 47.5, \( \sum f_i d_i = 435 \) and \( \sum f_i = 30 \), then mean of that data is:
Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).