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If $\frac{x(4^x-1)(3^x-9)}{({\log }_{2}x-1)(|x|-2)}\ge 0$, then $x\in$

An urn contains 25 balls of which 10 balls bear a mark X and the remaining 15 bear a mark Y. A ball is drawn at random from the urn, its mark note down and it is replaced. If 6 balls are drawn in this way, find the probability that

all will bear X mark

not more than 2 will bear Y mark

atleast one ball will bear Y mark

The number of balls with X mark and Y mark will be equal.

If $\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors satisfying the relation, $\overrightarrow{A}.\overrightarrow{B}=|\overrightarrow{A}\times \overrightarrow{B}|$. Then the value of $|\overrightarrow{A}-\overrightarrow{B}|$ will be -

The number of ways in which n different prizes can be distributed among m (<n) persons if each is entitled to receive at most n - 1 prizes, is_______.

The general solution(s) of the equation $4{\cos }^{2}x+6{\sin }^{2}x=5$ is/are

If $a=sec\theta -tan\theta$ and $b=cosec\theta +cot\theta$, then show that ab+a-b+1=0.

Let $y=a{x}^n,a\ne 0$ has a point of inflection at $x=0,$ then which of the following is correct

If $co{s}^2\frac{\pi }{8}$ is a root of equation $x^2$ + ax + b = 0 where a, b $ϵ$ Q then a + b = ___

If $\varphi$ denotes the empty set, then which of the folowing is correct

An article manufactured by a company consists of two parts X and Y. In the process of manufacture of part X, 9 out of 104 parts may be defective. Similartly, 5 out of 100 are likely to be defective in the manufacture of the part Y. Calculate the probability that the assembled product will not be defective.

A straight line through the vertex P of a traingle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then

Find the
derivative of the following functions from first principle:

(i) –x (ii) (–x)^–1 (iii) sin
(x + 1)

(iv)

$\int \frac{xdx}{2-x^2+\sqrt{2-x^2}}$ equals to

(where $C$ is constant of integration)

With respect to position of two circles which of the following statements is/are correct?

1. If one circle lies completely outside the other circle,

Number of direct common tangents = 2

Number of transverse common tangents = 2

2. If two circles touch each other externally

Number of direct common tangents = 2

Number of transverse common tangents = 1

3. If two circles touch each other internally

Number of direct common tangents = 1

Number of transverse common tangents = 0

4. If two circles intersect each other at two points

Number of direct common tangents = 2

Number of transverse common tangents = 0

5. If one circle lies completely inside the other circle

Number of direct common tangents = 0

Number of transverse common tangents = 0

Which of the following pair of linear equations is inconsistent?

The solution set of $x^2+1<10$ is

Match the following for system of linear equations
2x -3y + 5z =12
$3x+y+\lambda z=\mu$
x -7y + 8z =17
$\begin{array}{cccc}& \text{Column - I}& & \text{Column - I}\\ \left(P\right)& \text{Unique solution}& \left(1\right)& \lambda =2,\mu =7\\ \left(Q\right)& \text{Infinite solution}& \left(2\right)& \lambda \ne 2,\mu =7\\ \left(R\right)& \text{No solution}& \left(3\right)& \lambda \ne 2,\mu \ne 7\\ \left(S\right)& \text{Consistent system}& \left(4\right)& \lambda \in R,\mu \ne 7\\ & \text{equation}& & \\ & & \left(5\right)& \lambda =2,\mu \ne 7\end{array}$

If $y=e^{\tan 3x},$ then $\frac{dy}{dx}=$

Show that for a $\ge 1,f\left(x\right)=\sqrt{3}sinx-cosx-2ax+b$ is decreasing in R.

The set of values of $a$ for which the function $f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$ possesses a negative point of inflection is

If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,$ and $P$ is any point on it then range of values of $SP\cdot S^{\prime }P$ is

Let a, b, c be non-zero real numbers such that ${\int }_0^3(3a{x}^2+2bx+c)dx={\int }_1^3(3a{x}^2+2bx+c)dx.$ Then

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = –9y

Divide $4x^3+12x^2+11x+3byx+1$ and then find the quotient.

The number of triplets $(a,b,c)$ of positive integers satisfying the equation $\left|\begin{array}{ccc}{a}^3+1& a^{2}b& a^{2}c\\ a{b}^2& b^3+1& b^{2}c\\ a{c}^2& b{c}^2& c^3+1\end{array}\right|=11$ is equal to

$co{s}^{-1}x\{-cos\left(\frac{-13\pi }{6}\right)\}$ is equal to

Let $f,g,h$ be the length of the perpendiculars from the circumcentre of the $\Delta ABC$ on the sides $a$, $b$, and $c$, respecitively then the value of $k$ for which $\frac{a}{f}+\frac{b}{g}+\frac{c}{h}=k\frac{abc}{fgh},$ is