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A non zero polynomial f(x) with real coefficients satisfy f(x) = $f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)$ and f(0) = 96. If 'a' is the coefficient of highest degree term of f(x), then

The equation of straight line passing through the points (a, b, c) and (a - b, b- c, c - a), is

A rod of length $l$ moves such that its ends $A$ and $B$ always lie on the lines $3x-y+5=0$ and $y+5=0$ respectively. Then the locus of the point $P$ which divides $AB$ internally in the ratio of $2:1,$ is

The simplified form of ${\tan }^{-1}\frac{x}{y}-{\tan }^{-1}\frac{x-y}{x+y}$ is equal to

If X is Poisson variate with P(X=0)=P(X=1), then P(X=2)=

Let $\overrightarrow{a}$ and $\overrightarrow{c}$ be unit vectors and $|\overrightarrow{b}|=4$ with $\overrightarrow{a}\times \overrightarrow{b}=2\overrightarrow{a}\times \overrightarrow{c}.$ The angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is ${\cos }^{-1}\left(\frac{1}{4}\right).$ If $\overrightarrow{b}-2\overrightarrow{c}=\lambda \overrightarrow{a},$ then $\lambda$ is equal to

The number of ways in which the letters of the word Article can be arranged so that even places are always occupied by consonants is

Area enclosed by curve $y^3-9y+x=0$ and Y - axis is -

The number of ordered pairs (x,y) of real numbers that satisfy the simultaneous equations $x+y^2=x^2+y=12$ is

The value of $\frac{dy}{dx}ify=(x-1{)}^2(x+2{)}^3$ is equal to

If sin x+siny =$\frac{1}{2}$ and cosx+cosy = 1, then tan (x+y) = ……..

If A = $\left[\begin{array}{ccc}0& 2& 6\\ 3& 5& 4\\ 5& 7& 10\end{array}\right]$, B = $\left[\begin{array}{ccc}3& 4& 5\\ 2& 3& 1\\ 1& 3& 7\end{array}\right]$ then A + B =

Solve the differential equation $\frac{dy}{dx}-3ycotx=sin2x,y=2\text{when}x=\frac{\pi }{2}$ .

There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of them are collinear, is

$\frac{5x}{2}+\frac{3x}{4}\ge \frac{39}{4}$

An integrating factor of the differential equation $(1+y+x^{2}y)dx+(x+x^3)dy=0$ is

Which among the following relations defined on $R$ is/are functions

Which of the following options is/are true?
$[A\in (0^{\circ },{90}^{\circ })]$

Solve

$\frac{x-3}{x+1}<0,xϵR$

Maximum value of the expression $\sqrt{{\cos }^{2}x-10\cos x+25}+3$ is

The area of the figure bounded by the parabola $x=-2y^2,x=1-3y^2$ is

The equation of the plane containing the straight line $\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z}{5}$ and perpendicular to the plane $x-y+z+2=0$ is

In figure, AB is a chord of the circle and AOC is its diameter such that $\angle ACB={50}^{\circ }$. If AT is the tangent to the circle at the point A, then $\angle BAT$ is equal to:

The probability function of a binomial distribution is $P\left(x\right)={{}^{6}C}_x{p}^x{q}^{6-x},x=0,1,2,...,6$. If $2P\left(2\right)=3P\left(3\right)$, then $p=$____

If $\alpha$ and $\beta$ satisfies the equations $5\sin 2\beta =3\sin 2\alpha$ and $3\tan \alpha =\tan \beta$ simultaneously, where $0<\alpha <\pi ,0<\beta <\pi ,\alpha ,\beta \ne \frac{\pi }{2}$, then the possible value(s) of $\tan \alpha +\tan \beta$ is/are

$\begin{array}{cccc}& ColumnI& & ColumnII\\ & \left(\text{Vascular bundles}\right)& & \left(\text{Organs}\right)\\ 1.& \text{Closed collateral, endarch}& p& \text{Dicot root}\\ 2.& \text{Open collateral, endarch}& q& \text{Monocot root}\\ 3.& \text{Radial, tetrarch, exarch}& r& \text{Dicot stem}\\ 4.& \text{Radial, polyarch, exarch}& s& \text{Monocot stem}\\ & & t& \text{Dicot leaf}\end{array}$

From the following information of Lions' Club, Show the sports material items in the Income and Expenditure Account for the year ending 31-03-2018.

$\begin{array}{ccc}& & \\ & 2017\left(Rs\right)& 2018\left(Rs\right)\\ \text{Sports Material}& 22,000& 58,000\\ \text{Creditors for Sports Material}& 78,000& 92,000\\ \text{Advance to Suppliers for Sports Material}& 15,000& 25,000\\ \text{Payment to suppliers for the sports material during the year war Rs. 1,20,000.}& & \end{array}$

If the equation $x^2+y^2=2x$ is written in polar coordinate system, then $r$ can be

Write the range of the funciton $f\left(x\right)=sin\left[x\right]$, where $\frac{-\pi }{4}\le x\le \frac{\pi }{4}$.

If $\overrightarrow{a}=2\hat{i}+\lambda \hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k}$ are orthogonal, then value of $\lambda$ is

Find the equation of the hyperbola with centre at the origin, length of the transverse axis 6 and one focus at (0, 4).

Choose the correct answer in the following question:

If A,B are symmetric matrices of same order, then AB-BA is a
(a)skew-symmetric matrix
(b)symmetric matrix
(c)zero matrix
(d)identity matrix

Choose the correct answer in the following question:

If $A=\left[\begin{array}{cc}cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{array}\right]$, then A+A'=I, if the value of $\alpha$ is

$\left(a\right)\frac{\pi }{6}\left(b\right)\frac{\pi }{3}\left(c\right)\pi \left(d\right)\frac{3\pi }{2}$

Find the integrals of the functions.
$\int si{n}^3(2x+1)dx$

Volume of parallelopiped formed by vectors $\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{b}\times \overrightarrow{c}$ and $\overrightarrow{c}\times \overrightarrow{a}$ is $36$ cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

$\begin{array}{cc}Column1& Column2\\ \text{a. Volume of parallelopiped formed by vectors}& \text{p.}0\text{cubic units}\\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}& \\ \text{b. Volume of tetrahedron formed by vectors}& \text{q.}12\text{cubic units}\\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}& \\ \text{c. Volume of parallelopiped formed by vectors}& \text{r.}6\text{cubic units}\\ \overrightarrow{a}+\overrightarrow{b},\overrightarrow{b}+\overrightarrow{c}\text{and}\overrightarrow{c}+\overrightarrow{a}\text{is}& \\ \text{d. Volume of parallelopiped formed by vectors}& \text{s.}1\text{cubic unit}\\ \overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c}\text{and}\overrightarrow{c}-\overrightarrow{a}\text{is}& \end{array}$

Find the equation of the hyperbola whose vertices are $(\pm 7,0)$ and the eccentricity is $\frac{4}{3}.$

Find the real solution of

$ta{n}^{-1}\sqrt{x(x+1)}+si{n}^{-1}\sqrt{x^2+x+1}=\frac{x}{2}$.

A homogeneous equation of the form $\frac{dy}{dx}=h\left(\frac{x}{y}\right)$ can be solved making the substitution
(a) y=vx
(b) v=yx
(c) x=vy
(d) x=v

Find the number of solutions for $4\left\{x\right\}=x+\left[x\right]$, where $\{.\}$, [.] represents fractional part and greatest integer function.

Find the number of integers in the domain of $\sqrt{5-x^2}$