MHT CET 2020 16th October Morning Shift

If \(f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]\), then value of '\(c\)' by applying LMVT is

The equation of tangent at \(P(-4,-4)\) on the curve \(x^2=-4 y\) is

The area of the region bounded by the curve \(y=4 x^3-6 x^2+4 x+1\) and the lines \(x=1, x=5\) and \(X\)-axis is

The radius of the circle passing through the points \((5,7),(2,-2)\) and \((-2,0)\) is

\(\int_0^a \sqrt{\frac{x}{a-x}} d x=\)

\(\int_\limits2^3 \frac{x}{x^2-1} d x=\)

\(\int_\limits0^{\frac{\pi}{2}} \log \left[\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}\right] d x=\)

The differential equation obtained from the function \(y=a(x-a)^2\) is

The differential equation of all lines perpendicular to the line \(5 x+2 y+7=0\) is

The bacteria increases at the rate proportional to the number of bacteria present. If the original number '\(N\)' doubles in \(4 \mathrm{~h}\), then the number of bacteria in \(12 \mathrm{~h}\) will be

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are \(27 \mathrm{~gm}\) of certain substance and \(3 \mathrm{~h}\) later it is found that \(8 \mathrm{~gm}\) are left, t...

The integrating factor of the differential equation \(\left(1+x^2\right) d t=\left(\tan ^{-1} x-t\right) d x\) is

If \(\frac{x}{\sqrt{1+x}}+\frac{y}{\sqrt{1+y}}=0, x \neq y\), then \((1+x)^2 \frac{d y}{d x}=\)

If \(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4\), then \(\frac{d y}{d x}=\)

If \(f(x)=\log (\sec x+\tan x)\), then \(f^{\prime}\left(\frac{\pi}{4}\right)=\)

The approximate value of the function \(f(x)=x^3-3 x+5\) at \(x=1.99\) is

If \(f(x)=\frac{2 x+3}{3 x-2}, x \neq \frac{2}{3}\), then the function \(f\) of is

If \(\int \frac{\sin \theta}{\sin 3 \theta} d \theta=\frac{1}{2 k} \log \left|\frac{k+\tan \theta}{k-\tan \theta}\right|+c\), then \(k=\)

If \(\int \sqrt{x-\frac{1}{x}}\left(\frac{x^2+1}{x^2}\right) d x=\frac{2}{3}\left(x-\frac{1}{x}\right)^k+c\), then value of \(k\) is

\(\int \cot x \cdot \log [\log (\sin x)] d x=\)

The value of \(\sin ^{-1}\left(-\frac{1}{2}\right)+\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\) is

The points of discontinuity of the function
$$\begin{aligned} f(x) & =\frac{1}{x-1}, \text { if } 0 \leq x \leq 2 \\ & =\frac{x+5}{x+3} \text { if } 2< x \leq 4 \end{aligned}$$
in its domain are

The minimum value of \(Z=5 x+8 y\) subject to \(x+y \geq 5,0 \leq x \leq 4, y \geq 2, x \geq 0, y \geq 0\) is

If \(p \rightarrow(\sim p \vee q)\) is false, then the truth values of \(p\) and \(q\) are respectively

The symbolic form of the following circuit is (where \(p, q\) represents switches \(S_1\) and \(s_2\) closed respectively)

The sum of the cofactors of the elements of second row of the matrix $$\left[\begin{array}{rrr}1 & 3 & 2 \\ -2 & 0 & 1 \\ 5 & 2 & 1\end{array}\right]$$ is

If $$A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$$ and $$A^{-1}=\left[\begin{array}{rrr}3 & -1 & 1 \\ \alpha & 6 & -5 \\ \beta & -2 & 2\end{array}\right]$$, then the values of \(\alpha\) and \(\beta\) ar...

The cartesian co-ordinates of the point on the parabola \(y^2=x\) whose parameter is \(-\frac{4}{3}\) are

The probability that bomb will miss the target is 0.2. Then, the probability that out of 10 bombs dropped exactly 2 will hit the target is

The letters of the word 'LOGARITHM' are arranged at random. The probability that arrangement starts with vowel and end with consonant is

The p.d.f of c.r.v \(X\) is given by \(f(x)=\frac{x+2}{18}\), if $$-2

If the p.m.f of a. r.v. \(X\) is given by
\(P(X=x)=\frac{{ }^5 C_x}{2^5}\)
if \(x=0,1,2, \ldots \ldots . .5=0\),
0 , otherwise,
then which of the following is not true?

In a triangle \(A B C\) with usual notations, if \(\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\), then area of triangle \(A B C\) with \(a=\sqrt{6}\) is

In a triangle \(A B C\), if \(\frac{\sin A-\sin C}{\cos C-\cos A}=\cot B\), then \(A, B, C\), are in

If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is

If \(A=\{x, y, z\}, B=\{1,2\}\), then the total number of relations from set \(A\) to set \(B\) are :

If the equation \(k x y+5 x+3 y+2=0\) represents a pair of lines, then \(k=\)

If \((a,-2 a), a>0\) is the mid-point of a line segment intercepted between the co-ordinate axes, then the equation of the line is

If the angle between the lines given by the equation \(x^2-3 x y+\lambda y^2+3 x-5 y+2=0, \lambda \geq 0\), is \(\tan ^{-1}\left(\frac{1}{3}\right)\), then \(\lambda=\)

If the foot of perpendicular drawn from the origin to the plane is \((3,2,1)\), then the equation of plane is

The angle between the line \(r =(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}})\) and the plane \(\mathbf{r} \cdot(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})=8\) is

The direction cosines of a line which is perpendicular to lines whose direction ratios are \(3,-2,4\) and \(1,3,-2\) are

If the lines given by \(\frac{x-1}{2 \lambda}=\frac{y-1}{-5}=\frac{z-1}{2}\) and \(\frac{x+2}{\lambda}=\frac{y+3}{\lambda}=\frac{z+5}{1}\) are parallel, then the value of \(\lambda\) is

The polar co-ordinates of the point whose cartesian co-ordinates are \((-2,-2)\), are given by

If \(x \cos \theta+y \sin \theta=5, x \sin \theta-y \cos \theta=3\), then the value of \(x^2+y^2=\)

If \(\sin \theta=-\frac{12}{13}, \cos \phi=-\frac{4}{5}\) and \(\theta, \phi\) lie in the third quadrant, then \(\tan (\theta-\phi)=\)

If the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+m \hat{\mathbf{k}}\) are coplanar, then \(m=\)

The angles between the lines
$$\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \text { and } \mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda^{\prime}...

If \([\vec{a}\ \vec{b}\ \vec{c}\ ] \neq 0\), then \(\frac{[\vec{a}\ +\vec{b}\ \vec{b}\ +\vec{c}\ \vec{c}\ +\vec{a}\ ]}{[\vec{b}\ \vec{c}\ \vec{a}\ ]}=\)

In a quadrilateral \(ABCD, M\) and \(N\) are the mid-points of the sides \(A B\) and \(C D\) respectively. If \(\mathbf{A D}+\mathbf{B C}=t \mathbf{M N}\), then \(t=\)