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Find the angles between the lines $\sqrt{3}x+y=1$ and $x+\sqrt{3}y=1$

In the world of matrices if null matrix represents a zero ,then ______ represents a one.

If $\sin (x+{20}^{\circ })=2\sin x\cos {40}^{\circ }$ where $x\in (0,\frac{\pi }{2})$, then which of the following is/are correct ?

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$1^2+2^2+3^2+......+n^2=\frac{n(n+1)(2n+1)}{6}$

If the sum of the slopes of the normal from a point $P$ to the rectangular hyperbola $xy=c^2$ is equal to $\lambda (\lambda \in R^{+})$, then locus of $P$ is

Observe the graph:





Where the output has the highest increase, the required domain is

Let $f\left(x\right)=\frac{1}{2}{a}_0+{\sum }_{i-1}^n{a}_{i}cos\left(ix\right)+{\sum }_{j-1}^n{b}_{j}sin\left(jx\right),then{\int }_{-\pi }^{\pi }f\left(x\right)coskxdx$ then is equal to

If the lines$x+ay+a=0,bx+y+b=0$ and $cx+cy+1=0$ are concurrent,then write the value of $2abc-ab-bc-ca.$

If $(1-i)z=(1+i)\overline{z}$, then $i\overline{z}$ is

Locus of a point through which three normals of parabola $y^2=4ax$ are passing, two of which are making angles $\alpha$ and $\beta$ with positive $x-$ axis and $\tan \alpha \cdot \tan \beta =2$ is

Find the vector
equation of the plane passing through the intersection of the planes

and through the point (2, 1, 3)

A parabola $y=a{x}^2+bx+c$ crosses the x-axis at $(\alpha ,0)(\beta ,0)$ both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is

If the domain of the function $y=f\left(x\right)$ is $[-3,2],$ then the domain of $f\left(|\right[x\left]|\right)$ is

(where [.] denotes the greatest integer function)

The maximum of $f\left(x\right)=\frac{\log x}{x^2}(x>0)$ occurs, when $x$ is equal to

If $A=\left[\begin{array}{cc}0& x\\ y& 0\end{array}\right]$ and $A^3+A=O,$ then which of the following is correct

The ratio in which the area bounded by the curves $y^2=12x$ and $x^2=12y$ is divided by the line $x=3,$ is

Let $x,x^{lo{g}_{10}x},y^{lo{g}_{10}y}$ and $(xy{)}^{lo{g}_{10}\left(xy\right)}$ are four consecutive terms of a geometric progression (x, y > 0).

Number of ordered pair (x, y) is

Consider an infinte ladder network shown in figure. A voltage $V$ is applied between the points $A$ and $B$. This applied value of voltage is halved after each section. Then:

The set of all real numbers x for which $x^2-|x+2|+x>0$ is

Let $[.]$ and $\{.\}$ represent the greatest integer function and the fractional part function respectively. The number of value(s) of $x$ satisfying the equation $|2x-1|=3\left[x\right]+2\left\{x\right\}$ is

A person cut two cakes into two different shapes and eat remaining part. Find out how much cake left with him?

Find the equations of all lines having
slope 0 which are tangent to the curve
.

The differential equation of all circles in the first quadrant which touch the coordinate axes is of order

Suppose $ABCDEF$ is a hexagon such that $AB=BC=CD=1\text{and}DE=EF=FA=2.$ If the vertices $A,B,C,D,E,F$ are concyclic, the radius of the circle passing through them is

If the angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda }$ and the plane $x+2y+3z=4$ is ${\cos }^{-1}\left(\sqrt{\frac{5}{14}}\right)$, then