MHT CET 2022 11th August Evening Shift

If the normal to the curve \(y=f(x)\) at the point \((3,4)\) makes an angle \(\left(\frac{3 \pi}{4}\right)^c\) with positive \(X\)-axis, then \(f^{\prime}(3)\) is equal to

If \(y=\cos \left(\sin x^2\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=\sqrt{\frac{\pi}{2}}\) is

A firm is manufacturing 2000 items. It is estimated that the rate of change of production \(P\) with respect to additional number of workers \(x\) is given by \(\frac{\mathrm{d} P}{\mathrm{~d} x}=100-12 \sqrt{x}\). If the firm employs 25 mo...

If the function \(f(x)=x^3-3(a-2) x^2+3 a x+7\), for some \(a \in I R\) is increasing in \((0,1]\) and decreasing in \([1,5)\), then a root of the equation \(\frac{f(x)-14}{(x-1)^2}=0(x \neq 1)\) is

A spherical iron ball of \(10 \mathrm{~cm}\) radius is coated with a layer of ice of uniform thickness that melts at the rate of \(50 \mathrm{~cm}^3 / \mathrm{min}\). If the thickness of ice is \(5 \mathrm{~cm}\), then the rate at which the...

The area (in sq. units) of the region described by \(A=\left\{(x, y) / x^2+y^2 \leq 1\right.\) and \(\left.y^2 \leq 1-x\right\}\) is

If the lines \(3 x-4 y-7=0\) and \(2 x-3 y-5=0\) pass through diameters of a circle of area \(49 \pi\) square units, then the equation of the circle is

Let \(z\) be a complex number such that \(|z|+z=3+i, i=\sqrt{-1}\), then \(|z|\) is equal to

\(\int_\limits{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{\mathrm{d} x}{1+\cos x}\) is equal to

The value of the integral \(\int_\limits0^1 \sqrt{\frac{1-x}{1+x}} \mathrm{~d} x\) is

\(\int_\limits0^2[x] \mathrm{d} x+\int_\limits0^2|x-1| \mathrm{d} x=\)
(where \([x]\) denotes the greatest integer function.)

The differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles with

The general solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 x+y}{x-y}\) is (where \(C\) is a constant of integration.)

If \(y^{\frac{1}{m}}+y^{\frac{-1}{m}}=2 x, x \neq 1\), then \(\left(x^2-1\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2\) is equal to

If \(y=\log \sqrt{\frac{1+\sin x}{1-\sin x}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=\frac{\pi}{3}\) is

If \(y=\sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to

If \([x]\) is greatest integer function and \(2[2 x-5]-1=7\), then \(x\) lies in

\(\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x=\)
(where \(C\) is a constant of integration.)

If \(f(x)=\sqrt{\tan x}\) and \(g(x)=\sin x \cdot \cos x\) then \(\int \frac{f(x)}{g(x)} \mathrm{d} x\) is equal to (where \(C\) is a constant of integration)

\(\text { If } \int e^{x^2} \cdot x^3 \mathrm{~d} x=e^{x^2} \cdot[f(x)+C]\)
(where \(C\) is a constant of integration.) and \(f(1)=0\), then value of \(f(2)\) will be

\(\int \frac{3 x-2}{(x+1)(x-2)^2} \mathrm{~d} x=\)
(where \(C\) is a constant of integration)

The principal value of \(\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)\) is

The value of \(\sin \left(2 \sin ^{-1} 0.8\right)\) is equal to

If \(\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x\), then \(x\) has the value

\(\matrix{ {f(x) = a{x^2} + bx + 1,} & {if} & {\left| {2x - 3} \right| \ge 2} \cr { = 3x + 2,} & {if} & {{1 \over 2} < x < {5 \over 2}} \cr }\)
is continuous on its domain, then \(a+b\) has the value

If \(\lim _\limits{x \rightarrow 1} \frac{x^2-a x+b}{(x-1)}=5\), then \((a+b)\) is equal to

Maximum value of \(Z=5 x+2 y\), subject to \(2 x-y \geq 2, x+2 y \leq 8\) and \(x, y \geq 0\) is

The negation of the statement pattern \(p \vee(q \rightarrow \sim r)\) is

If \(p: \forall n \in I N, n^2+n\) is an even number \(q: \forall n \in I N, n^2-n\) is an odd numer, then the truth values of \(p \wedge q, p \vee q\) and \(p \rightarrow q\) are respectively

The negation of the statement, "The payment will be made if and only if the work is finished in time" is

Given $$A=\left[\begin{array}{ccc}x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z\end{array}\right]$$, if \(x y z=60\) and \(8 x+4 y+3 z=20\), then \(A\).(adjA)

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

If a question paper consists of 11 questions divided into two sections I and II. Section I consists of 6 questions and section II consists of 5 questions, then the number of different ways can student select 6 questions, taking at least 2 q...

If \(P(A \cup B)=0.7, P(A \cap B)=0.2\), then \(P\left(A^{\prime}\right)+P\left(B^{\prime}\right)\) is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of kings is

The incidence of occupational disease in an industry is such that the workmen have a \(10 \%\) chance of suffering from it. The probability that out of 5 workmen, 3 or more will contract the disease is

The variance and mean of 15 observations are respectively 6 and 10 . If each observation is increased by 8 then the new variance and new mean of resulting observations are respectively

The joint equation of pair of lines through the origin and making an equilateral triangle with the line \(y = 5\) is

The equations of the lines passing through the point \((3,2)\) and making an acute angle of \(45^{\circ}\) with the line \(x-2 y-3=0\) are

A line makes the same angle '\(\alpha\)' with each of the \(x\) and \(y\) axes. If the angle '\(\theta\)', which it makes with the \(z\)-axis, is such that \(\sin ^2 \theta=2 \sin ^2 \alpha\), then the angle \(\alpha\) is

The distance between parallel lines \(\frac{x-1}{2}=\frac{y-2}{-2}=\frac{z-3}{1}\) and \(\frac{x}{2}=\frac{y}{-2}=\frac{z}{1}\) is :

The Cartesian equation of a line passing through \((1,2,3)\) and parallel to \(x-y+2 z=5\) and \(3 x+y+z=6\) is

A tetrahedron has verticles \(P(1,2,1), Q(2,1,3), R(-1,1,2)\) and \(O(0,0,0)\). Then the angle between the faces \(O P Q\) and \(P Q R\) is

The equation of the plane passing through the points \((2,3,1),(4,-5,3)\) and parallel to \(X\)-axis is

If \(\cot (A+B)=0\), then \(\sin (A+2 B)\) is equal to

If \(\bar{a}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{k}}, \bar{b}=x \hat{\boldsymbol{i}}+\hat{\boldsymbol{j}}+(1-x) \hat{\boldsymbol{k}}\) and \(\bar{c}=y \hat{\boldsymbol{i}}+x \hat{\boldsymbol{j}}+(1+x-y) \hat{\boldsymbol{k}}\), then $$[\ba...

The polar co-ordinates of the point, whose Cartesian coordinates are \((-2 \sqrt{3}, 2)\), are

If \(\bar{a}=\hat{\boldsymbol{i}}+\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}, \bar{b}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}\) and \(\bar{c}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}-\hat{\boldsymbol{k}}\) are three v...

Let \(\bar{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\bar{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) be two vectors. If \(\bar{c}\) is a vector such that \(\bar{b} \times \bar{c}=\bar{b} \times \bar{a}\) a...

The magnitude of the projection of the vector \(2 \hat{\mathbf{i}}+ 3\hat{\mathbf{j}}+\hat{\mathbf{k}}\) on the vector perpendicular to the plane containing the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and $$\hat{\math...