MHT CET 2020 16th October Evening Shift

For every value of \(x\), the function \(f(x)=\frac{1}{a^x}, a>\) 0 is,

The equation of normal to the curve \(y=\sin \left(\frac{\pi x}{4}\right)\) at the point \((2,5)\) is

\(\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=\)

The area included between the parabolas \(y^2=5 x\) and \(x^2=5 y\) is

\(\int_\limits{-5}^5 \log \left(\frac{7-x}{7+x}\right) d x=\)

\(\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}}\left[\frac{\tan x}{\tan x+\cot x}\right] d x=\)

\(\int_\limits0^1\left(\frac{x^2-2}{x^2+1}\right) d x=\)

The micro-organisms double themselves in 3 h. Assuming that the quantity increases at a rate proportional to it self, then the number of times it multiplies themselves in 18 yr is

The integrating factor of the differential equation \(\sin y\left(\frac{d y}{d x}\right)=\cos y(1-x \cos y)\) is

The particular solution of the differential equation \(y\left(\frac{d x}{d y}\right)=x \log x\) at \(x=e\) and \(y=1\) is

The order and degree of the differential equation
\(\left[1+\left[\frac{d y}{d x}\right]^3\right]^{\frac{7}{3}}=7 \frac{d^2 y}{d x^2}\) are respectively.

The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 10 min , then the time requi...

If \(f(x)=\sin ^{-1}\left(\sqrt{\frac{1-x}{2}}\right)\), then \(f^{\prime}(x)=\)

The eccentricity of the ellipse \(y^2+4 x^2-12 x+6 y+14=0\) is

The domain of the function \(f(x)=\sqrt{x}\) is

For \(f(x)=[x]\), where \([x]\) is the greatest integer function, which of the following is true, for every \(x \in \mathbf{R}\)

\(\int\left[-\frac{\log x-1}{1+(\log x)^2}\right]^2 d x=\)

\(\int \frac{1+2 e^{-x}}{1-2 e^{-x}} d x=\)

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

The value of \(\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{8}\right)\) is

\(\left[\sin \left(\tan ^{-1} \frac{3}{4}\right)\right]^2+\left[\sin \left(\tan ^{-1} \frac{4}{3}\right)\right]^2=\)

The function \(f(x)=\frac{x+1}{9 x+x^3}\) is

If \(f(x)=\frac{|x|}{x}\), for \(x \neq 0\)
\(=1\), for \(x=0\), then tre function is

The maximum value of \(Z=3 x+5 y\), subject to \(3 x+2 y \leq 18, x \leq 4, y \leq 6, x, y \geq 0\) is

If \(p, q\) are true statement and \(r\) is false statement, then which of the following statements is a true statement.

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

The matrix $$A=\left[\begin{array}{rrr}a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2\end{array}\right]$$ is not invertible only if \(a=\)

If $$A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], \quad B=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]$$, then \(B^{-1} A^{-1}=\)

The length of latus-rectum of the parabola \(x^2+2 y=8 x-7\) is

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

The odds in favour of drawing a king from a pack of 52 playing cards is

Out of 100 people selected at random, 10 have common cold. If five persons selected at random from the group, then the probability that at most one person will have common cold is

The pdf of a continuous r.v. \(X\) is given by \(f(x)=\frac{x}{8}, 0 < x < 4=0\), otherwise, then \(P(X \leq 2)\) is

In a \(\triangle A B C\) if \(2 \cos C=\sin B \cdot \operatorname{cosec} A\), then

The rational form of a number 1.$\overline{41}$ is

The straight lines represented by the equation \(9 x^2-12 x y+4 y^2=0\) are

If the equation \(a x^2+2 h x y+b y^2+2 g x+2 f y=0\) has one line as the bisector of the angle between co-ordinate axes, then

The points \(A(-a,-b), B(0,0), C(a, b)\) and \(D\left(a^2, a b\right)\) are

The angle between the lines \(\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}\) and \(\frac{x-2}{2}=\frac{y+1}{2}=\frac{z-4}{1}\) is

The cosine of the angle included between the lines \(\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\) and $$\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}+3...

If the line \(r=(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) is parallel to the plane \(r \cdot(3 \hat{i}-2 \hat{\mathbf{j}}+m \hat{\mathbf{k}})=10\), then the va...

If the plane \(2 x+3 y+5 z=1\) intersects the co-ordinate axes at the points \(A, B, C\), then the centroid of \(\triangle A B C\) is

The direction co-sines of the line which bisects the angle between positive direction of \(Y\) and \(Z\) axes are

\(\tan A+2 \tan 2 A+4 \tan 4 A+8 \cot 8 A=\)

If \(x+y=\frac{\pi}{2}\), then the maximum value of \(\sin x \cdot \sin y\) is

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

$$\begin{aligned} & \cos \left(36^{\circ}-A\right) \cos \left(36^{\circ}+A\right)+\cos \left(54^{\circ}+A\right) \cos \\ & \left(54^{\circ}-A\right)= \end{aligned}$$

Let \(G\) be the centroid of a \(\triangle A B C\) and \(\mathrm{O}_{b_\theta}\) other point in that plane, then \(\mathrm{OA}+\mathrm{OB}+\mathrm{OC}+\mathrm{CG}=\)

If the volume of the parallelopiped whose conterminus edges are along the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) is 12, then the volume of the tetrahedron whose conterminus edges are \(\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}\) ...

For any non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\),