List of top Mathematics Questions asked in MHT CET

If \[ y = \left( \frac{x^2 + 1}{x} \right)^x \text{ and } \frac{dy}{dx} = y \left[ g(x) + \log \left( \frac{x^2}{x+1} \right) \right], \text{ then } g(x) = \]

The solution of \( rdx + (x - r^2) dr = 0 \) is

A metal has half-life period of 10 days. A sample originally has a mass of 1000 mg, then the mass remaining after 50 days is

If \[ A = \{ x / x \text{ is a prime number}, 0 \leq x \leq 9 \}, \text{ then the number of elements of the power set of } A \text{ is} \]

The general solution of \( \tan \theta + \tan 2\theta = \tan 3\theta \)

The equations of the lines which make intercepts on the axes whose sum is 8 and product is 15 are

The symbolic form of the following circuit is (where \( p, q, r \) represent switches \( s_1, s_2, s_3 \) which are closed respectively)

If the function \[ f(x)= \begin{cases} \dfrac{1-\sin2x+\cos2x}{1+\sin2x+\cos2x}, & x\neq \dfrac{\pi}{2}
k, & x=\dfrac{\pi}{2} \end{cases} \] is continuous at \( x=\dfrac{\pi}{2} \), then \( k \) is

The equation of a normal to the curve \[ x = 4\sec\theta,\qquad y = 4\tan^2\theta \] at \( \theta = \frac{\pi}{4} \) is

If \( \log_{10}\!\left(\dfrac{x^3-y^3}{x^3+y^3}\right)=2 \), then find \( \dfrac{dx}{dy} \).

If a line in octant OXYZ makes equal angles with the coordinate axes, then

If \( \vec{AB} = 3\hat{i} + 5\hat{j} + 4\hat{k} \), \( \vec{AC} = 5\hat{i} - 5\hat{j} + 2\hat{k} \) represent the sides of triangle \( ABC \), then the length of the median through \( A \) is

A tangent to the curve \( x = at^2,\; y = 2at \) is perpendicular to the X-axis. Then the point of contact is

If \( P(A') = 0.6 \), \( P(B) = 0.8 \) and \( P(B|A) = 0.3 \), then find \( P(A|B) \).

Given below is the probability distribution of a discrete random variable \( X \):
\[ \begin{array}{c|cccccc} X=x & 1 & 2 & 3 & 4 & 5 & 6
\hline P(X=x) & k & 0 & 2k & 5k & k & 3k \end{array} \] Then find \( P(X \ge 4) \).

If the function \( f \) defined by \[ f(x) = \begin{cases} K(x - x^2), & 0<x<1
0, & \text{otherwise} \end{cases} \] is the p.d.f. of a random variable \( X \), then the value of \( P(X<\tfrac{1}{2}) \) is

If the vectors \( \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} - 5\hat{j} + p\hat{k} \) and \( \vec{c} = 5\hat{i} - 9\hat{j} + 4\hat{k} \) are coplanar, then the value of \( p \) is

Evaluate the integral: \[ \int_{-a}^{a} x^2 \left( \frac{e^{x^3} - e^{-x^3}}{e^{x^3} + e^{-x^3}} \right) dx \]

If \( A = \left[ \begin{array}{cc} 1 & 2 \\ 1 & 0 \end{array} \right] \), \( B = \left[ \begin{array}{cc} 1 & 2\\ 2 & 1 \end{array} \right] \), then \( (AB)^{-1} \) is

If the function given by \( f(x) = \left( \frac{4x + 1}{1 - 4x} \right)^{\frac{1}{3}} \) for \( x \neq 0 \) is continuous at \( x = 0 \), the value of \( f(0) \) is

Evaluate \[ \int_0^1 \frac{x^2 - 2}{x^2 + 1} \, dx \]

If \( a = \sin 175^\circ + \cos 175^\circ \), then

The negation of the statement 'He is poor but happy' is

If a die is thrown at random, then the expectation of the number on it is

The p.d.f. of a continuous r.v. \( X \) is given by \[ f(x) = \frac{x}{8}, \quad 0<x<4 \quad \text{and} \quad f(x) = 0, \text{ otherwise,} \] then \( P(X \leq 2) \) is