Evaluate the following limit: If _ x 0 kx cosec x = _ x 0 x cosec kx, find k.

_ x 0 kx cosec x = _ x 0 x cosec kx, _ x 0 kx1 = _ x 0 x1 k _ x 0 ( x )=1 k _ x 0 ( kx ) k=1 k k^2=1 k= 1

Evaluate the following limit: If _ x 0 kx cosec x = _ x 0 x cosec…

Prove that the radii of the circles $x^2 + y^2 = 1, x^2 + y^2 - 2x − 6y - 6 = 0$ and $x^2 + y^2 - 4x - 12y - 9 = 0$ are in A.P.

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A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that,

(i) number on the ticket is a perfect square or divisible by 4.

(ii)number on the ticket is a prime number or greater than 40.

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Tangents to the circle $x^2+y^2=a^2$ with inclinations, $\theta_1$ and $\theta_2$ intersect in $P$. Find the locus of P such that

1.$\tan \theta_1+\tan \theta_2=0$

2. $\cot \theta_1+\cot \theta_2=5$

3. $\cot \theta_1 \cdot \cot \theta_2=c$

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Find the coordinates of the vertices of a triangle, the equations of whose sides are:
x + y - 4 = 0, 2x - y + 3 = 0 and x - 3y + 2 = 0

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Show that $\sin100^\circ-\sin10^\circ$ is positive.

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At What point the origin be shifted so that the equation $x^2+ xy - 3x - y + 2 = 0$ cantain any first degree term and constand term?

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Evaluate the following limits:
$\lim _{x \rightarrow \pi}\left[\frac{\sqrt{5+\cos x-2}}{(\pi-x)^2}\right]$

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Two tangents to the parabola $y^2=8 x$ meet the tangents at the vertex in the points $P$ and

Q. If PQ = 4, prove that the equation of the locus of the point of intersection of two

tangents is $y^2=8(x+2)$.

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Prove that $\text{x}^{2\text{n}-1}+\text{y}^{2\text{n}-1}$ is divisible by x + y for all $\text{n}\in\text{N}.$

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Solve the following system of equations in R.
$2\text{x}+5\leq0,\text{x}-3\leq0$

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