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Find the
values of x and y so that the vectors
are
equal

The number 916238457 is an example of nine digit number which contains each of the digit 1 to 9 exactly once. It also has the property that the digits 1 to 5 occur in their natural order, while the digits 1 to 6 do not. Number of such numbers are

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

​​​​​​$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)x^2+y^2=a^2& \left(i\right)my=m^{2}x+a& \left(P\right)(\frac{a}{m^2},\frac{2a}{m})\\ \left(II\right)x^2+a^2{y}^2=a^2& \left(ii\right)y=mx+a\sqrt{m^2+1}& \left(Q\right)(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}})\\ \left(III\right)y^2=4ax& \left(iii\right)y=mx+\sqrt{a^2{m}^2-1}& \left(R\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}})\\ \left(IV\right)x^2-a^2{y}^2=a^2& \left(iv\right)y=mx+\sqrt{a^2{m}^2+1}& \left(S\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}})\end{array}$



The tangent to a suitable conic (Column 1) at $(\sqrt{3},\frac{1}{2})$ is found to be $\sqrt{3}x+2y=4$, then which of the following options is the only CORRECT combination?

If $A(\alpha ,\beta )=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ -\sin \alpha & \cos \alpha & 0\\ 0& 0& e^{\beta }\end{array}\right],$ then

The domain of the real function $f\left(x\right)=\frac{1}{\sqrt{25-4x^2}}$ is

If $\underset{0}{\overset{1}{\int }}{e}^{x^2}(x-\alpha )dx=0,$ then which of the following is true for $\alpha$?

Find the angle between each of the following pairs of straight lines:

(i) $3x+y+12=0$ and $x+2y-1=0$

(ii) $3x-y+5=0$ and $x-3y+1=0$

(iii) $3x+4y-7=0$ and $4x-3y+5=0$

(iv) $x-4y=3$ and $6x-y=11$

(v) $(m^2-mn)y=(mn+n^2)x+n^2$ and $(mn+m^2)y=(mn+n^2)x+m^3$.

The solution of the differential equation $(x^{2}si{n}^{3}y-y^{2}cosx)dx+(x^{3}cosysi{n}^{2}y-2ysinx)dy=0$

If $\int \frac{2-3\sin x}{{\cos }^{2}x}dx=a\sec x+b\tan x+C,$ then which of the following is/are true (where $a$ and $b$ are fixed constants and $C$ is constant of integration)

The number of real solutions of
$ta{n}^{-1}\sqrt{x(x+1)}+si{n}^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$ is

[IIT 1999]

${lim}_{x\to 0}\frac{sin2x}{e^x-1}$

Let a vector $\overrightarrow{a}$ be coplanar with vectors $\overrightarrow{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+\hat{k}.$ If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{d}=3\hat{i}+2\hat{j}+6\hat{k}$, and $|\overrightarrow{a}|=\sqrt{10}.$ Then a possible value of $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]+\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}\right]+\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]$ is equal to

Cards are drawn one by one without replacement from a pack of 52 cards until two aces are drawn. Let $P\left(m\right)$ be the probability that the event occurs in exactly $m$ trials then $P\left(m\right)$ must be zero at

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find :

(i) how many drink tea and coffee both

(ii) how many drink coffee but not tea.

The solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ satisfying the condition $y\left(1\right)=1$ is

Find the set of values of x for which the functions $f\left(x\right)=3x^2-1$ and $g\left(x\right)=3+x$ are equal.

The
approximate change in the volume of a cube of side x metres
caused by increasing the side by 3% is

A.
0.06 x³ m³ B. 0.6 x³
m³ C. 0.09 x³ m³ D.
0.9 x³ m³

If $y=\frac{(x-a)(x-c)}{x-b}$ assumes all real values for $x\in R-\left\{b\right\},$ then

, then c is equal to

If the curves $a{x}^2+4xy+2y^2+x+y+5=0$ and $a{x}^2+6xy+5y^2+2x+3y+8=0$ intersect at four concyclic points, then the value of $a$ is

Let n be a positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $(1+x{)}^n$ are in AP, then the value of n is___

The solution set of $({\cos }^{-1}x{)}^4-({\sin }^{-1}x{)}^4>0$ is

The value of $\underset{x\to \frac{\pi }{2}}{lim}\frac{(1-\sin x)(8x^3-{\pi }^3)\cos x}{(\pi -2x{)}^4}$ is