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The normal to the rectangular hyperbola $xy=c^2$ at the point '$t_1$' meets the curve again at the point '$t_2$'. Then the value of $t_1^3{t}_2$is

Calculate the mean deviation of the following income groups of five and seven members from their medians :

$\begin{array}{cc}\text{I}& \text{II}\\ \text{Income in Rs}& \text{Income in Rs}\\ 4000& 3800\\ 4200& 4000\\ 4400& 4200\\ 4600& 4400\\ 4800& 4600\\ & 4800\\ & 5800\end{array}$

Find the area bounded by curves $\left\{\right(x,y):y\ge x^{2}andy\le |x|$

If $sin\left[2co{s}^{-1}cot\right(2ta{n}^{-1}x\left)\right]=0$, then the value of x=

Two natural numbers are chosen at random from the first one hundred natural numbers. The probability that the product of the chosen numbers is a multiple of 7 is.

If $|\overrightarrow{F_1}\times \overrightarrow{F_2}|=\overrightarrow{F_1}.\overrightarrow{F_2}$, then $|\overrightarrow{F_1}+\overrightarrow{F_2}|$ is

If $f\left(x\right)=\left|\begin{array}{ccc}\cos x& e^{x^2}& 2x{\cos }^2\frac{x}{2}\\ x^2& \sec x& \sin x+x^3\\ 1& 2& x+\tan x\end{array}\right|$, then the value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}(x^2+1)\left[f\right(x)+f^{\prime \prime }(x\left)\right]dx$

The sum of the series ${20}_{c_0}-{20}_{c_1}+{20}_{c_2}-{20}_{c_3}+...{20}_{c_{10}}is$

The complete solution set of the inequality $[co{s}^{-1}x{]}^2-6[co{t}^{-1}x]+\le 0$, where [.] denotes the greatest integer function, is :

For a set $Y,$ if $P\left(Y\right)=\{\varnothing ,\{2\},\{\left\{4\right\}\},\{2,\left\{4\right\}\left\}\right\},$ then $Y=$

Consider
a binary operation * on
N defined
as a * b
= a³
+ b³.
Choose the correct answer.

(A) Is
* both
associative and commutative?

(B) Is
* commutative
but not associative?

(C) Is
* associative
but not commutative?

(D) Is
* neither
commutative nor associative?

Choose the correct Set-builder representation of the interval $(4,12)$.

The vertices of a hyperbola are (2, 0), (–2, 0) and the foci are (3, 0), (–3, 0). The equation of the hyperbola is

In the matrix $\left[\begin{array}{cccc}2& 5& 19& -7\\ 35& -2& \frac{5}{2}& 12,\\ \sqrt{3}& 1& -5& 17\end{array}\right]$,write
(i)the order of the matrix

(ii)The number of elements,

(ii)The elements $a_{13},a_{21},a_{33},a_{24},a_{23}.$

If the lines joining the points $(K,2),(3,6)$ and $(2,3),(K,7)$ are perpendicular to each other, then the possible value of $K$ is

The number of the points on the curve $y=x^3–11x+5$ at which the tangent are parallel to the line y = x + 5 is

If $f\left(x\right)=1-\frac{1}{x}$, then write the vale of $f\left(f\left(\frac{1}{x}\right)\right)$.

Which
of the following statements are true and which are false? In each
case give a valid reason for saying so.

(i) p:
Each radius of a circle is a chord of the circle.

(ii) q:
The centre of a circle bisects each chord of the circle.

(iii) r:
Circle is a particular case of an ellipse.

(iv) s:
If x
and
y
are integers such that x
> y,
then –x
< –y.

(v) t:
is a rational number.

A juggler keeps on moving four balls in the air throwing the balls after regular intervals. When one ball leaves his hand (speed =20 m $s^{-1}$), the position of other balls (height in m) will be (table g= 10 m $s^{-2}$)

Find
the inverse of each of the matrices, if it exists.

If tangents $OQ$ and $OR$ are drawn to variable circles having radius $r$ and the centre lying on the rectangular hyperbola $xy=1,$ then locus of circumcentre of triangle $OQR$ is $(O$ being the origin$).$

Find out the appropriate word which fits the 3rd blank.

The interval of $f\left(x\right)=x^3+2x^2+5x,x<0$ for which it is concave upwards is

The locus of the centre of a circle which touch the circles $x^2+y^2-6x-6y+14=0$ externally and also the Y - axis is given by

Area enclosed by $x^2=4y$ and $y=\frac{8}{x^2+4}$ is ?

If the sequence is given by $T_n=\frac{a_{n+3}}{a_{n-1}}$, where $a_n=\frac{(2n-3)}{6}$, then the third term is

Let $f:R\to R$ be a function satisfying $f(x+y)=f\left(x\right)+2y^2+kxy$ for all $x,yϵR$. If f(1) = 2 and f(2) = 8, then f(x) is equal to.

Show that the products
of the corresponding terms of the sequences

form a G.P, and find the common ratio.

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