List of top Questions asked in MHT CET

The third overtone of a closed pipe is in unison with the second overtone of an open pipe. Hence the ratio of the length of the closed pipe to that of the open pipe is:
The direction cosines of a line which lies in the \( ZOX \) plane and makes an angle of \( 30^\circ \) with the \( Z \)-axis are

If \[ y = \cot^{-1}\!\left(\sqrt{\frac{1-\sin x}{1+\sin x}}\right), \quad \text{then find} \quad \frac{dy}{dx}. \]

If \[ \begin{vmatrix} a + b & b + c & c + a \\ b & c & a \\ c & a & b \end{vmatrix} \neq 0, \quad \text{then} \quad \begin{vmatrix} a + b & b + c & c + a \\ b & c & a \\ c & a & b \end{vmatrix} = \]
If the vectors \( \hat{i} + 2\hat{j} + \hat{k} \) and \( \hat{i} + 6\hat{j} + 4\hat{k} \) are collinear, then the values of \( x \) and \( y \) are respectively
The distance of the point \( (3,4,5) \) from the point of intersection of the line \[ \frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2} \] and the plane \( x+y+z=2 \) is
If $A$ and $B$ are two angles such that $A,B\in(0,\pi)$ and they are not supplementary angles such that $\sin A-\sin B=0$, then
If the foot of perpendicular drawn from the origin to the plane is \( (3, 2, 1) \), then the equation of the plane is
The co-ordinates of the mid-point of the chord cut off by the line \( 2x - 5y + 18 = 0 \) by the circle \( x^2 + y^2 - 6x + 2y - 54 = 0 \) are
The rate of decay of mass of a certain substance at time \( t \) is proportional to the mass at that instant. The time during which the original mass of \( m_0 \) gm will be left to \( m_1 \) gm is
If \[ |3x - 2| \leq \frac{1}{2}, \] then \( x \in \)
Evaluate the integral \[ \int_0^\infty \frac{dx}{(x^2 + 4)(x^2 + 9)} = \]
Simplify \( \cos\left(\dfrac{3\pi}{4}+x\right) - \sin\left(\dfrac{\pi}{4}-x\right) \)
If \( A(-1,2,3) \), \( B(3,-2,1) \), \( C(2,1,3) \) and \( D(-1,-2,4) \) are the vertices of a tetrahedron, then its volume is
If \( n(X) = 700 \), \( n(A) = 200 \), \( n(B) = 300 \), \( n(A \cap B) = 100 \), where \( X \) is the universal set and \( A \) and \( B \) are subsets of \( X \), then \( n(A' \cap B') = \)
If \( \frac{1-\tan\theta}{1+\tan\theta} = \frac{1}{\sqrt{3}}, \) where \( \theta \in \left(0,\frac{\pi}{2}\right) \), then \( \theta = \)
The dual of the statement pattern \( \sim p \land (q \lor t) \) is (where \( t \) is a tautology and \( c \) is a contradiction)
Evaluate the sum of the series: \[ \frac{1^2}{2} + \frac{1^2 + 2^2}{3} + \frac{1^2 + 2^2 + 3^2}{4} + \frac{1^2 + 2^2 + 3^2 + 4^2}{5} + \cdots \quad \text{upto 8 terms} \]
The equation of a plane passing through the intersection of the planes \(x + 2y - 3z + 2 = 0\) and \(6x + y + z + 1 = 0\) and parallel to the line \(x - 1 = y + 2 = 7 - z\) is
If the p.m.f. is given by \( P(X) = k \left( \frac{4}{x} \right) \) for \( x = 0, 1, 2, 3, 4 \) and \( k>0 \), then the value of \( k \) is
Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x} + \sqrt[3]{\csc x}} \, dx \]
If \[ x \cos \theta + y \sin \theta = 5, \quad x \sin \theta - y \cos \theta = 3, \quad \text{then the value of} \, x^2 + y^2 \, \text{is} \]

If
\[ A = \begin{bmatrix} \cos\theta & -\sin\theta \\ -\sin\theta & -\cos\theta \\ \end{bmatrix} \] then \(A^{-1} =\) 

The rate of decay of a certain substance is directly proportional to the amount present at that instant. Initially, there are 27 grams of the substance and 3 hours later it is found that 8 grams are left, then the amount left after one more hour is
The probability mass function of a random variable \(X\) is given by \[ P(X = x) = \frac{5cx}{2^5}, \quad x = 0,1,2,3,4,5 \] \[ P(X = x) = 0, \quad \text{otherwise} \] Then find \(P(X \le 2)\).