MHT CET 2023 9th May Evening Shift

If slope of a tangent to the curve \(x y+a x+b y=0\) at the point \((1,1)\) on it is 2, then a - b is

The equation \(x^3+x-1=0\) has

The range of values of \(x\) for which \(f(x)=x^3+6 x^2-36 x+7\) is increasing in

The maximum value of the function \(f(x)=3 x^3-18 x^2+27 x-40\) on the set \(\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}\) is

A water tank has a shape of inverted right circular cone whose semi-vertical angle is \(\tan ^{-1}\left(\frac{1}{2}\right)\). Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/ minute at which level of wate...

Let \(\mathrm{f}(0)=-3\) and \(\mathrm{f}^{\prime}(x) \leq 5\) for all real values of \(x\). The \(\mathrm{f}(2)\) can have possible maximum value as

The area (in sq. units) bounded by the curve \(y=x|x|, \mathrm{X}\)-axis and the lines \(x=-1\) and \(x=1\) is

The sides of a rectangle are given by the equations \(x=-2, x=4, y=-2\) and \(y=5\)
Then the equation of the circle, whose centre is the point of intersection of the diagonals, lying within the rectangle and touching only two opposite sides...

If \(Z_1=2+i\) and \(Z_2=3-4 i\) and \(\frac{\overline{Z_1}}{\overline{Z_2}}=a+b i\), then the value of \(-7 a+b\) is (where \(i=\sqrt{-1}\) and \(a, b \in R)\)

If \(\mathrm{f}^{\prime}(x)=x-\frac{5}{x^5}\) and \(\mathrm{f}(1)=4\), then \(\mathrm{f}(x)\) is

\(\int_\limits1^2 \frac{\mathrm{d} x}{\left(x^2-2 x+4\right)^{\frac{3}{2}}}=\frac{\mathrm{k}}{\mathrm{k}+5} \text {, then } \mathrm{k} \text { has the value }\)

General solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=a x+b y\) is

The differential equation of all circles which pass through the origin and whose centres lie on \(\mathrm{Y}\)-axis is

If \(y\) is a function of \(x\) and \(\log (x+y)=2 x y\), then the value of \(y^{\prime}(0)\) is

If \(x^{\mathrm{k}}+y^{\mathrm{k}}=\mathrm{a}^{\mathrm{k}}(\mathrm{a}, \mathrm{k}>0)\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0\), then \(\mathrm{k}\) has the value

If \(\mathrm{g}\) is the inverse of \(\mathrm{f}\) and \(\mathrm{f}^{\prime}(x)=\frac{1}{1+x^3}\), then \(\mathrm{g}^{\prime}(x)\) is

If \(\int \frac{\sin x}{3+4 \cos ^2 x} \mathrm{~d} x=\mathrm{A} \tan ^{-1}(\mathrm{~B} \cos x)+\mathrm{c}\), (where \(\mathrm{c}\) is a constant of integration), then the value of \(\mathrm{A}+\mathrm{B}\) is

\(\int(\sqrt{\tan x}+\sqrt{\cot x}) d x=\)

Let \(\alpha \in\left(0, \frac{\pi}{2}\right)\) be fixed. If the integral
\(\int \frac{\tan x+\tan \alpha}{\tan x-\tan \alpha} \mathrm{d} x=\mathrm{A}(x) \cos 2 \alpha+\mathrm{B}(x) \sin 2 \alpha+\mathrm{c},\)
(where \(\mathrm{c}\) is a con...

The value of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right), |x| < \frac{1}{2}, x \neq 0\)

\(\lim _\limits{x \rightarrow 0} \frac{(1-\cos 2 x) \cdot \sin 5 x}{x^2 \sin 3 x}\) is

$$f(x)=\left\{\begin{array}{ll} \frac{1-\cos k x}{x^2}, & \text { if } x \leq 0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text { if } x>0 \end{array}\right. \text { is continuous at }$$
\(x=0\), then the value of \(\mathrm{k}\) is

The graphical solution set for the system of inequations
\(x-2 y \leq 2,5 x+2 y \geq 10,4 x+5 y \leq 20, x \geq 0, y \geq 0\) is given by

If \(\log _2 x+\log _4 x+\log _8 x+\log _{16} x=\frac{25}{36}\) and \(x=2^{\mathrm{k}}\) then \(\mathrm{k}\) is

Negation of the statement
"The payment will be made if and only if the work is finished in time." Is

Let \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) be three statements, then \([p \rightarrow(q \rightarrow r)] \leftrightarrow[(p \wedge q) \rightarrow r]\) is

If $$\left|\begin{array}{ccc}\cos (A+B) & -\sin (A+B) & \cos (2 B) \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B\end{array}\right|=0$$, then the value of \(B\) is

Let $$A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1\end{array}\right], B=\left[\begin{array}{c}6 \\ 11 \\ 0\end{array}\right]$$ and $$X=\left[\begin{array}{l}a \\ b \\ c\end{array}\right]$$, if \(\mathrm{AX}=\mathrm{B}\), t...

A linguistic club of a certain Institute consists of 6 girls and 4 boys. A team of 4 members to be selected from this group including the selection of a Captain (from among these 4 members) for the team. If the team has to include atmost on...

A man takes a step forward with probability 0.4 and backwards with probability 0.6 . The probability that at the end of eleven steps, he is one step away from the starting point is

A problem in statistics is given to three students A, B and C. Their probabilities of solving the problem are \(\frac{1}{2}, \frac{1}{3}\) and \(\frac{1}{4}\) respectively. If all of them try independently, then the probability, that proble...

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards, then mean of number of queens is

In \(\triangle \mathrm{PQR}, \sin \mathrm{P}, \sin \mathrm{Q}\) and \(\sin \mathrm{R}\) are in A.P., then

Let \(a, b, c\) be the lengths of sides of triangle \(A B C\) such that \(\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}=k\). Then \(\frac{(\mathrm{A}(\triangle \mathrm{ABC}))^2}{\mathrm{k}^4}=\)

In \(\triangle \mathrm{ABC}\), with usual notations, \(\mathrm{m} \angle \mathrm{C}=\frac{\pi}{2}\), if \(\tan \left(\frac{A}{2}\right)\) and \(\tan \left(\frac{B}{2}\right)\) are the roots of the equation $$a_1 x^2+b_1 x+c_1=0\left(a_1 \ne...

If the mean and S.D. of the data \(3,5,7, a, b\) are 5 and 2 respectively, then \(a\) and \(b\) are the roots of the equation

A random variable \(\mathrm{X}\) assumes values 1, 2, 3, ....., n with equal probabilities, if \(\operatorname{var}(X)=E(X)\), then \(\mathrm{n}\) is

If the slope of one of the lines represented by \(a x^2+(2 a+1) x y+2 y^2=0\) is reciprocal of the slope of the other, then the sum of squares of slopes is

The equation of a line, whose perpendicular distance from the origin is 7 units and the angle, which the perpendicular to the line from the origin makes, is \(120^{\circ}\) with positive \(\mathrm{X}\)-axis, is

If a line \(\mathrm{L}\) is the line of intersection of the planes \(2 x+3 y+z=1\) and \(x+3 y+2 z=2\). If line \(\mathrm{L}\) makes an angle \(\alpha\) with the positive \(\mathrm{X}\)-axis, then the value of \(\sec \alpha\) is

The shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}\) is

The co-ordinates of the point, where the line \(\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}\) meets the plane \(2 x+4 y-\mathrm{z}=3\), are

The equation of a plane, containing the line of intersection of the planes \(2 x-y-4=0\) and \(y+2 z-4=0\) and passing through the point \((2,1,0)\), is

If \(\cos 2 B=\frac{\cos (A+C)}{\cos (A-C)}\). Then \(\tan A, \tan B, \tan C\) are in

\(\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}\), then vector \(\overline{\mathrm{r}}\) satisfying $$\overline{\mathrm{a}} \times \overline{\mathrm{r}}=\ov...

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

If \(\bar{a}, \bar{b}\) and \(\bar{c}\) are any three non-zero vectors, then \((\bar{a}+2 \bar{b}+\bar{c}) \cdot[(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})]=\)

Vectors \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are such that \(|\overline{\mathrm{a}}|=1 ;|\overline{\mathrm{b}}|=4\) and \(\bar{a} \cdot \bar{b}=2\). If \(\bar{c}=2 \bar{a} \times \bar{b}-3 \bar{b}\), then the angle betwee...

Two adjacent sides of a parallelogram are \(2 \hat{i}-4 \hat{j}+5 \hat{k}\) and \(\hat{i}-2 \hat{j}-3 \hat{k}\), then the unit vector parallel to its diagonal is

If \(\mathrm{D}, \mathrm{E}\) and \(\mathrm{F}\) are the mid-points of the sides \(\mathrm{BC}\), \(\mathrm{CA}\) and \(\mathrm{AB}\) of triangle \(\mathrm{ABC}\) respectively, then $$\overline{\mathrm{AD}}+\frac{2}{3} \overline{\mathrm{BE}...