MHT CET 2021 20th September Morning Shift

A wire of length 20 units is divided into two parts such that the product of one part and cube of the other part is maximum, then product of these parts is

A particle is moving on a straight line. The distance \(\mathrm{S}\) travelled in time \(\mathrm{t}\) is given by \(\mathrm{S=a t^2+b t+6}\). If the particle comes to rest after 4 seconds at a distance of \(16 \mathrm{~m}\). from the starti...

The equation of the tangent to the curve \(y = 4x{e^x}\) at \(\left( { - 1,{{ - 4} \over e}} \right)\) is

The area of the region bounded by the parabola x\(^2\) = y and the line y = x is

If the lines \(3x - 4y + 4 = 0\) and \(6x - 8y - 7 = 0\) are tangents to a circle, then the radius of the circle is

The value of (1 + i)\(^5\) (1 \(-\) i)\(^7\) is

\(\int_\limits0^{\pi / 4} \log (1+\tan x) d x=\)

If \(\int_\limits0^{\frac{\pi}{2}} \frac{d x}{5+4 \sin x}=A \tan ^{-1} B\), then A + B =

A differential equation for the temperature 'T' of a hot body as a function of time, when it is placed in a bath which is held at a constant temperature of 32\(^\circ\) F, is given by (where k is a constant of proportionality)

The general solution of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+y-1}\) is given by

The general solution of the differential equation \(x+y \frac{d y}{d x}=\sec \left(x^2+y^2\right)\) is

The differential equation of all circles which pass through the origin and whose centre lie on Y-axis is

An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half the quantity of ice melts in 20 minutes, \(x_0\) is the initial quantity of ice. If after 40 minutes the amount of ice left is \(\mathrm{Kx}_0\),...

If \(y=\log \tan \left(\frac{x}{2}\right)+\sin ^{-1}(\cos x)\), then \(\frac{d y}{d x}=\)

If \(h(x)=\sqrt{4 f(x)+3 g(x)}, f(1)=4, g(1)=3, f^{\prime}(1)=3, g^{\prime}(1)=4\), then \(h^{\prime}(1)=\)

If \(x=a \cos \theta, y=b \sin \theta\), then \(\left[\frac{d^2 y}{d x^2}\right]_{\theta=\frac{\pi}{4}}=\)

The domain of the function \(f(x)=\frac{1}{\sqrt{x+|x|}}\) is

\(\int \tan ^{-1}(\sec x+\tan x) d x=\)

If \(\int \frac{1+x^2}{1+x^4} d x=\frac{1}{\sqrt{2}} \tan ^{-1}\left[\frac{f(x)}{\sqrt{2}}\right]+c\), then \(f(x)=\)

\(\int \frac{x+\sin x}{1+\cos x} d x=\)

If \(4 \sin ^{-1} x+6 \cos ^{-1} x=3 \pi\), where \(-1 \leq x \leq 1\), then \(x=\)

\(\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} + 5x - 7} - x} \right) =\)

If f(x) = |x|, for x \(\in\) (\(-1,2\)), then f is discontinuous at (where [x] represents floor function)

The shaded part of the given figure indicates the feasible region. Then the constraints are

The negation of a statement 'x \(\in\) A \(\cap\) B \(\to\) (x \(\in\) A and x \(\in\) B)' is

The logical expression \(\mathrm{p} \wedge(\sim \mathrm{p} \vee \sim \mathrm{q}) \equiv\)

$$A(\propto)=\left[\begin{array}{cc}\cos \propto & \sin \propto \\ -\sin \propto & \cos \propto\end{array}\right]$$, then \(\left[A^2(\propto)\right]^{-1}=\)

If inverse of $$\left[\begin{array}{ccc}1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6\end{array}\right]$$ does not exist, then \(x=\)

If \(A = \left[ {\matrix{ 3 & 2 & 4 \cr 1 & 2 & 1 \cr 3 & 2 & 6 \cr } } \right]\) and A\(_{ij}\) are cofactors of the elements a\(_{ij}\) of A, then \({a_{11}}{A_{11}} + {a_{12}}{A_{12}} + {a_{13}}{A_{13}}\) is equal to

All the letters of the word 'ABRACADABRA' are arranged in different possible ways. Then the number of such arrangements in which the vowels are together is

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number, is

A random variable X has the following probability distribution

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Rajesh has just bought a VCR from Maharashtra Electronics and the shop offers after sales service contract for Rs. 1000 for the next five years. Considering the experience of VCR users, the following distribution of maintenance expenses for...

If \(X \sim B(4, p)\) and \(P(X=0)=\frac{16}{81}\), then \(P(X=4)=\)

With usual notations if the angles of a triangle are in the ratio 1 : 2 : 3, then their corresponding sides are in the ratio.

If 1 is added to first 10 natural numbers, then variance of the numbers so obtained is

If the acute angle between the lines given by \(\mathrm{a x^2+2 h x y+b y^2=0}\) is \(\frac{\pi}{4}\), then \(\mathrm{4 h^2=}\)

The joint equation of the pair of lines through the origin and making an equilateral triangle with the line \(x=3\) is

The slope of the line through the origin which makes an angle of 30\(^\circ\) with the positive direction of Y-axis measured anticlockwise is :

The Cartesian equation of the plane passing through the point \((0,7,-7)\) and containing the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\) is

If the lines \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}\) intersect, then the values of \(k\) is

The parametric equations of a line passing through the points \(\mathrm{A}(3,4,-7)\) and \(\mathrm{B}(1,-1,6)\) are

The angle between a line with direction ratios 2, 2, 1 and a line joining (3, 1, 4) and (7, 2, 12) is

If the line \(\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6}\) lies in the plane \(3 x-14 y+6 z+49=0\), then the value of \(m\) is

If \(x \in\left(0, \frac{\pi}{2}\right)\) and \(x\) satisfies the equation \(\sin x \cos x=\frac{1}{4}\), then the values of \(x\) are

The value of \(\sin 18^{\circ}\) is

If the volume of a tetrahedron whose conterminous edges are \(\vec{\mathrm{a}}+\vec{\mathrm{b}}, \vec{\mathrm{b}}+\vec{\mathrm{c}}, \vec{\mathrm{c}}+\vec{\mathrm{a}}\) is 24 cubic units, then the volume of parallelopiped whose coterminous e...

If \(\overline{\mathrm{e}}_1, \overline{\mathrm{e}}_2\) and \(\overline{\mathrm{e}}_1+\overline{\mathrm{e}}_2\) are unit vectors, then the angle between \(\overline{\mathrm{e}}_1\) and \(\overline{\mathrm{e}}_2\) is

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}} , \overline{\mathrm{c}}\) are three vectors which are perpendicular to \(\overline{\mathrm{b}}+\overline{\mathrm{c}}, \overline{\mathrm{c}}+\overline{\mathrm{a}}\) and $$\overline{\mathrm{a}...

\((2 \hat{\mathrm{i}}+6 \hat{\mathrm{i}}+27 \hat{\mathrm{k}}) \times(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\mu \hat{\mathrm{k}})=\overline{0}\), then \(\lambda\) and \(\mu\) are respectively