List of top Mathematics Questions asked in MHT CET

In a culture, the bacteria count is 1,00,000 initially. The number increases by 10% in the first 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

The points \( (1, 3) \) and \( (5, 1) \) are two opposite vertices of a rectangle. The other two vertices lie on the line \[ y = 2x + c, \quad \text{where} \quad c \text{ is the constant}, \quad \text{then the co-ordinates of the other two vertices are} \]

Angle between the parabola \[ y^2 = 4(x - 1) \quad \text{and} \quad x^2 + 4(y - 3) = 0 \quad \text{at the common end of their latus rectum is} \]

Evaluate the integral \[ \int \frac{x^3}{(x+1)^2} \, dx. \]

If \( y = ax^r - bx^{r-1} \), then \[ \frac{d^2y}{dx^2} = ? \]

A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

Evaluate the integral \[ \int \cos \left( \frac{\pi x}{6} \right) \cos \left( \frac{\pi x}{3} \right) \, dx. \]

Let \( x \) be the length of each of the equal sides of an isosceles triangle and \( \theta \) be the angle between these sides. If \( x \) is increasing at the rate \( \frac{1}{2} \) m/hour and \( \theta \) is increasing at the rate \( \frac{\pi}{3} \) rad/hour, then the rate at which area of the triangle is increasing when \( x = 12 \) m and \( \theta = \frac{\pi}{4} \) is

The solution set for minimizing the function \( z = x + y \) with constraints \[ x + y \geq 2, \quad x + 2y \leq 8, \quad x \leq 3, \quad x, y \geq 0 \] contains

The abscissae of the points of the curve \[ y = x^3 \quad \text{are in the interval} \quad [-2, 2], \] where the slope of the tangents can be obtained by the mean value theorem for the interval \( [-2, 2] \), are

The distance of the point \( P(3, 8, 2) \) from the line \[ \frac{x}{2} = \frac{y - 3}{4} = \frac{z - 3}{3}, \quad \text{measured parallel to the plane} \quad 3x + 2y - 2z + 15 = 0, \] is

If \( n^+ C_{n+1} - n^3 C_n = 15(n + 2) \), then \( n = \)

The angle between the lines \[ \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-3}{-1}, \quad \frac{x-1}{2} = \frac{y-3}{3} = \frac{z-1}{4}, \] where \( l, m, n \) are roots of the equation \[ x^2 + x^2 - 4x - 4 = 0, \] is

If \( I(\theta) = \cos \theta_1 \cos \theta_2 \cos \theta_3 \dots \cos \theta_n \), then \[ \tan \theta_1 + \tan \theta_2 + \tan \theta_3 + \cdots + \tan \theta_n = ? \]

Evaluate the limit: \[ \lim_{n \to \infty} \left[ \frac{1}{1 - \pi} + \frac{8}{1 + \pi} + \cdots + \frac{x^2}{n\pi} \right]. \]

The correct simplified circuit diagram for the logical statement \[ \left[ (q \land ( \sim q \lor r)) \land (\sim p \lor (p \land r)) \right] \] is

The length of the altitude through the point \( D \) of the tetrahedron where the vertices of the tetrahedron are
\[ A(2, 3, 1), B(4, 1, -2), C(6, 3, 7), D(-5, -4, 8), \] is

The angle between the lines whose direction cosines are \[ \frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2} \quad \text{and} \quad \frac{-\sqrt{3}}{4}, \frac{1}{2}, \frac{1}{\sqrt{3}} \] is

The differential equation of all circles having their centres on the line \( y = 5 \) and touching the X-axis is

In a triangle ABC, with usual notations, \[ \tan \left( \frac{A}{4} \right) = \frac{5}{6}, \quad \tan \left( \frac{C}{2} \right) = \frac{2}{5}, \] then

If \( \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \) has truth value false, then truth values of the statements \( p, q, r \) are respectively

Evaluate the integral \[ \int_0^{\pi/2} \frac{\cos^2 x \sin^2 x}{\cos^2 x + \sin^2 x} \, dx. \]

The volume of the tetrahedron whose coterminous edges are represented by
\[ \mathbf{a} = -12\hat{i} + p\hat{k}, \quad \mathbf{b} = -3\hat{j} - \hat{k}, \quad \mathbf{c} = -2\hat{i} + \hat{j} - 15\hat{k}, \] is 570 cu. units, then \( p = \)