MHT CET 2021 24th September Evening Shift

The curve \(y=a x^3+b x^2+c x+5\) touches \(X\)-axis at \(P(-2,0)\) and cuts \(Y\)-axis at a point \(Q\), where its gradient is 3, then

The minimum value of the function f(x) = x log x is

The distance 's' in meters covered by a particle in t seconds is given by \(s=2+27 t-t^3\). The particle will stop after _________ distance.

The area of the region bounded by the curve \(y=2 x-x^2\) and X-axis is

If \(y=2 x\) is a chord of circle \(x^2+y^2-10 x=0\), then the equation of circle with this chord as diameter is

If \(\mathrm{\frac{3+2i}{1+i}=\frac{1}{2}(x+iy)}\), then x \(-\) y =

\(\int_\limits0^\pi x \sin x \cos ^4 x d x=\)

\(\int_\limits2^5 2[\mathrm{x}] \mathrm{dx}=\{\text { where }[\mathrm{x}] \text { denotes the greatest integer function } \leq \mathrm{x}\}\)

A population P grew at the rate given by the equation \(\frac{dP}{dt}=0.5P\), then the population will become double in

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

The particular solution of the differential equation \(\left(1+e^{2 x}\right) d y+e^x\left(1+y^2\right) d x=0\) at \(x=0\) and y = 1 is

The differential equation of all parabolas whose axis is \(y\)-axis, is

The order and degree of the differential equation \(\sqrt{\frac{d y}{d x}}-4 \frac{d y}{d x}-7 x=0\) are respectively.

If \(y=1+x e^y\), then \(\frac{d y}{d x}=\)

If \(x=e^t(\sin t-\cos t)\) and \(y=e^t(\sin t+\cos t)\), then \(\frac{d y}{d x}\) at \(t=\frac{\pi}{3}\) is

If \(\sin ^2 x+\cos ^2 y=1\), then \(\frac{d y}{d x}=\)

If \(f(x)=2\{x\}+5 x\), where \(\{x\}\) is fractional part function, then \(f(-1.4)\) is

\(\int \cos ^3 x e^{\log (\sin x)^2} d x=\)

\(\int \frac{d x}{e^x+e^{-x}+2}=\)

\(\int e^x\left(\frac{1+\sin x}{1+\cos x}\right) d x=\)

The value of \(\tan ^{-1} 2+\tan ^{-1} 3\) is

\(\lim _\limits{x \rightarrow 0} \frac{\sqrt{1-\cos x^2}}{1-\cos x}=\)

If \(\mathrm{f}(\mathrm{x})=\mathrm{x}, \quad\) for \(\mathrm{x} \leq 0\)
\(=0,\quad\) for \(x>0\), then the function \(f(x)\) at \(x=0\) is

The region represented by the inequalities \(x \geq 6, y \geq 3,2 x+y \geq 10, x \geq 0, y \geq 0\) is

The negation of '\(\forall x \in N, x^2+x\) is even number' is

If \(\mathrm{p}\) : It is raining.
\(\mathrm{q}\) : Weather is pleasant
then simplified form of the statement "It is not true, if it is raining then weather is not pleasant" is

IF \(A X=B\), where $$A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right], B=\left[\begin{array}{l}4 \\ 0 \\ 2\end{array}\right]$$, then \(2 x+y-z=\)

If $$A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$, and \(A(\operatorname{adj} A)=k I\), then the value of \((k+1)^4\) is

If $$\mathrm{A}=\left[\begin{array}{cc}\lambda & \mathrm{i} \\ \mathrm{i} & -\lambda\end{array}\right]$$ and \(\mathrm{A}^{-1}\) does not exist, then \(\lambda=\) (where \(\mathrm{i}=\sqrt{-1}\))

A committee of 5 is to be formed out of 6 men and 4 ladies. The number of ways this can be done, when at most 2 ladies are included, is

A fair coin is tossed for a fixed number of times. If probability of getting 7 heads is equal to probability of getting 9 heads, then probability of getting 2 heads is

The probability distribution of a random variable X is

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The probability that at least one of the events \(E_1\) and \(E_2\) occurs is 0.6. If the simultaneous occurrence of \(\mathrm{E}_1\) and \(\mathrm{E}_2\) is $$0.2, \mathrm{P}\left(\mathrm{E}_1^{\prime}\right)+\mathrm{P}\left(\mathrm{E}_2^{...

Two dice are thrown simultaneously. If X denotes the number of sixes, then the expectation of X is

In a triangle ABC with usual notations a = 2, b = 3, then value of \(\frac{\cos 2 \mathrm{~A}}{\mathrm{a}^2}-\frac{\cos 2 \mathrm{~B}}{\mathrm{~b}^2}\) is

For the set of 50 observations, the sum of their squares is 3050 , their arithmetic mean is 6. Hence the standard deviation of these observations is

If the polar co-ordinates of a point are \(\left(2, \frac{\pi^{\mathrm{c}}}{4}\right)\), then its Cartesian co-ordinates are

The acute angle between the lines \(\left(x^2+y^2\right) \sin \theta+2 x y=0\) is

If the lines represented by \((k^2+2) x^2+3 x y-6 y^2=0\) are perpendicular to each other, then the values of \(\mathrm{K}\) are

The equation of a line with slope \(-\frac{1}{\sqrt{2}}\) and makes an intercept of \(2 \sqrt{2}\) units on negative direction of \(y\)-axis is

If the lines \(\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}\) and \(\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}\) are perpendicular to each other, then \(\lambda=\)

The equation of the plane which passes through (2, \(-\)3, 1) and is normal to the line joining the points (3, 4, \(-\)1) and (2, \(-\)1, 5) is given by

If the vector equation of the plane \(\bar{r}=(2 \hat{i}+\hat{k})+\lambda \hat{i}+\mu(\hat{i}+2 \hat{j}-3 \hat{k})\) in scalar product form is given by \(\overline{\mathrm{r}} \cdot(3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=\alpha\) then $$\al...

The co-ordinates of the points on the line \(\frac{x+2}{1}=\frac{y-1}{2}=\frac{z+1}{-2}\) at a distance of 12 units from the point A(\(-\)2, 1, \(-\)1) are

If \(G(3,-5, r)\) is the centroid of \(\triangle A B C\), where \(A \equiv(7,-8,1), B \equiv(p, q, 5), C \equiv(q+1,5 p, 0)\) are vertices of the triangle \(A B C\), then the values of \(p, q, r\) are respectively

\(\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+8 \cot 8 \mathrm{~A}=\)

If the angle between the vectors \(\overline{\mathrm{a}}=2 \lambda^2 \hat{\mathrm{i}}+4 \lambda \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overline{\mathrm{b}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}\) is obtuse, then...

If \(\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}\) and \(\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}\), then which of the following is valid.

For any non-zero vectors \(\bar{a}, \bar{b}, \bar{c}\), the value of \(\bar{a} \cdot[(\bar{b} \times \bar{c}) \times(\bar{a}+\bar{b}+\bar{c})]\) is

If \(\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}\) are coterminus edges of a parallelopiped, then its volume is