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If every pair of equations $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $x^2+cx+ab=0$ has a common root, then product of these common roots is

If $sec\theta =x+\frac{1}{4x}$, then $sec\theta +tan\theta =$

If f(x) = |x|, then f’(x), where x $\ne$ 0 is equal to

The value of $\int \frac{dx}{12\sin x+5\cos x}$ is

(where $C$ is constant of integration)

The sum of real value(s) of $\alpha$ for which the system of equations

$x+3y+5z=\alpha x$

$5x+y+3z=\alpha y$

$3x+5y+z=\alpha z$

has infinite number of solutions is

$Thedomainofthefunctionf\left(x\right)=\frac{1}{x}+si{n}^{-1}x+\frac{1}{\sqrt{x-2}}is$

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4(x-a^2)=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is $250$ sq. units, then a value of '$a$' is :

Find the equation of the parabola that satisfies the given conditons:
Focus
Vertex (0,0) passing through (2,3) and axis is along x - axis.

Find the equation of the line which passes through the point (-4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5:3 by this point.

If x and y
are connected parametrically by the equation, without eliminating the
parameter, find.

Find the domain and range of each of the following real valued functions:
(i) $f\left(x\right)=\frac{ax+b}{bx-a}$
(ii) $f\left(x\right)=\frac{ax-b}{cx-d}$
(iii) $f\left(x\right)=\sqrt{x-1}$
(iv) $f\left(x\right)=\sqrt{x-3}$
(v) $f\left(x\right)=\frac{x-2}{2-x}$
(vi) $f\left(x\right)=|x-1|$
(vii) $f\left(x\right)=|x|$
(viii) $f\left(x\right)=\sqrt{9-x^2}$

$\frac{cos\theta }{1-sin\theta }=\frac{1+cos\theta +sin\theta }{1+cos\theta -sin\theta }$

Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of ${}^{14}C$ by neutron capture in the upper atmosphere.
${}_7^{14}N{+}_0^{1}n{\to }_6^{14}C{+}_1{P}^1$
${}^{14}C$ is absorbed by living organisms during photosynthesis. The ${}^{14}C$ content is constant in living organisms once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of ${}^{14}C$ in the dead being, falls due to the decay which C-14 undergoes ${}_6^{14}C{\to }_7^{14}N+{\beta }^{-}$
The half-life period of ${}^{14}Cis5770yr.$
The decay constant $\left(\lambda \right)$ can be calculated by using the following formula $\lambda =\frac{0.693}{\frac{t_1}{2}}.$
The comparison of the ${\beta }^{-}$ activity of the dead matter with that of the carbon still in circulation
enables measurement of the period of the isolation of the material from the living cycle. The
method however, ceases to be accurate over periods longer than 30,000 yr. The proportion of ${}^{14}C$ to
${}^{12}C$ in living matter is $1:{10}^{12}.$
What should be the age of fossil for meaningful determination of its age?

The points A(5, -1, 1); B(7, -4, 7); C(1, -6, 10) and D(-1, -3, 4) are vertices of a [RPET 2000]

If $\alpha$ and $\beta$ are different complex numbers with $|\beta |=1,$ then the value of $\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|$ is

If $f\left(x\right)=2\cos x+ax+4$ is strictly increasing over $R,$ then



[1 mark]

The middle term(s) in the expansion of ${(3x-\frac{x^3}{6})}^7$ is/are:

Locus of the point whose sum of distances from the origin and the $x-$ axis is $4$ units is

Write down all possible subsets of $A=\{1,\{2,3\}\}$.

Evaluate the Given limit:

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by

, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Which of the following is a lower triangular matrix

$\frac{c-bcosA}{b-ccosA}=\frac{cosB}{cosC}$

If $\sin x+\sin y=3(\cos y-\cos x)$, then the value of $\frac{\sin 3x}{\sin 3y}$ is

If a circle passes through the points $(1,-6),(2,1)$ and $(5,2),$ then the equation of the circle is

Find all points of discontinuity of f,
where f is
defined by

If point $P(x,y)$ completely lies in the first quadrant, then the range of ${\cos }^{-1}x+{\cot }^{-1}(-\frac{1}{y})+{\tan }^{-1}y$ is