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Let $g\left(x\right)$ is a continuous and differentiable function and $f\left(x\right)=x\cdot g\left(x\right)$ has a positive root at $x=\alpha ,$ then one root of $f^{\prime }\left(x\right)=0$ can be

The point(s) on the curve $y^3+3x^2=12y$ where the tangent is vertical, is(are)

If the equation, $x^2+bx+45=0(b\in R)$ has conjugate complex roots and they satisfy $|z+1|=2\sqrt{10}$, then:

How many 5 digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digits appears more than once ?

If $f\left(x\right)=\underset{x}{\overset{x^2}{\int }}\frac{1}{\log t}dt$ (where $x>0$), then the value of $f^{\prime }\left(x\right)$ is

The minimum value of the function $f\left(x\right)=\left(\frac{x^3}{3}\right)-x$ occurs at :

If $f\left(x\right)=\left|\begin{array}{cc}6x^3& x^2\\ 2& x^6\end{array}\right|,$ then which of the following(s) is/are true?

If $\left[x^2\right]=[x+2]$, where [x] = the greatest integer less than or equal to x, then x must be such that

Let a, b and c be real numbers such that a+ 2b + c = 4 then the maximum value of ab + bc + ca is

Differentiate the given functions w.r.t. x.

$(xcosx{)}^x+(xsinx{)}^{\frac{1}{x}}.$

If the system of equations $ax+y=3,x+2y=3$ and $3x+4y=7$ is consistent, then value of $a$ is

Find the value of b, if one root of the equation $x^2$ + ax + 2b = 0 is 2 + 4i, where a, b ∈ R

The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4ax$ to a moving point of the parabola, is another parabola whose directrix is:

If the product of first $n$ odd natural numbers is $\frac{4040!}{2^{2020}\cdot 2020!}$, then the sum of these odd natural numbers is

Tangent are drawn from $P$ to $y^2=4ax.$ If the difference of the ordinates of the point of contact of the two tangents is $l$, then the equation of the locus of $P$ is

For any angle $\theta ,$ In a $\Delta ABC,b\cos (C+\theta )+c\cos (B-\theta )$ is equal to

If tangents are drawn at the points $A(1,2),B(4,-4)$ and a variable point ${}^{\prime }{C}^{\prime }$ lies on the parabola $y^2=4x$, then the locus of the orthocentre of triangle formed by these tangents is

Three lines $L_1:\overrightarrow{r}=\lambda \hat{i},\lambda \in R,$$L_2:\overrightarrow{r}=\hat{k}+\mu \hat{j},\mu \in R,$ and $L_3:\overrightarrow{r}=\hat{i}+\hat{j}+\nu \hat{k},\nu \in R$ are given.

For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P,Q$ and $R$ are collinear ?

For all sets $A$ and $B$, $A-(A\cap B)$ is equal to

If $cose{c}^2\theta =\frac{4xy}{(x+y{)}^2}$, then

Find the slope of the tangent to curve
y = x³ − x + 1 at the point
whose x-coordinate is 2.

Let $y=y\left(x\right)$ satisfies the equation $\frac{dy}{dx}-|A|=0,$ for all $x>0,$ where $A=\left[\begin{array}{ccc}y& \sin x& 1\\ 0& -1& 1\\ 2& 0& \frac{1}{x}\end{array}\right]$.

If $y\left(\pi \right)=\pi +2,$ then the value of $y\left(\frac{\pi }{2}\right)$ is:

The letters of the word $\text{PROBABILITY}$ are written down at random in a row. Let $E_1$ denote the event that two $I^{\prime }s$ are together and $E_2$ denote the event that two $B^{\prime }s$ are together. Then which of the following is (are) CORRECT?

Wrtie the number of terms in the expansion of $\left[\right(2x+y^3{)}^4{]}^7$

The set of solutions satisfying both $x^2+5x+6\ge 0and{x}^2+3x-4<0is$

Geometrically Rolle's theorem ensures that there is at least one point on the curve f(x) , whose abscissa lies in (a, b) at which the tangent is

Domain of $\frac{1}{\ln (x-4)}$ is