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The set of real values of x satisfying the equation

${|x-1|}^{lo{g}_3\left(x^2\right)-2lo{g}_x\left(9\right)}=(x-1{)}^7$

is

Show that the points $\hat{i}-\hat{j}+3\hat{k}$ and $3(\hat{i}+\hat{j}+\hat{k})$ are equidistant from the plane $\overrightarrow{r}.(5\hat{i}+2\hat{j}-7\hat{k})+9=0$ and lies on opposite side of it.

The point on the curve $y=e^x$ which is at shortest distance from the line $y=x-4,$ is

The letters of word $OUGHT$ are written in all possible orders and these words are written out as in a dictionary. Then the rank of the word $TOUGH$ is

Let $y=y\left(x\right)$ be the solution of the differential equation $(y+1){\tan }^{2}xdx+\tan xdy+ydx=0,x\in (0,\frac{\pi }{2}).$ If $\underset{x\to 0^{-}}{lim}xy\left(x\right)=1,$ then the value of $y\left(\frac{\pi }{4}\right)$ is

[(1+cosA)/sinA] = [sinA/(1-cosA)]

${lim}_{x\to 0}\frac{a^{mx}-1}{b^{nx}-1},n\ne 0$

The
probability of obtaining an even prime number on each die, when a
pair of dice is rolled is

(A) 0 (B) (C) (D)

If a chord of the circle $x^2+y^2-4x-2y-c=0$ is trisected at the points (1/3, 1/3) and (8/3, 8/3), then

The circle passing through $(1,-2)$ and touching the axis of $x$ at $(3,0)$ also passes through the point

If $\alpha ,\beta$ be the roots of $x^2$+px+q=0 and $\alpha +h,\beta +h$ are the roots of $x^2$+rx+s=0 , then

If $A,B,C$ are the angles of triangle such that $0<A\le \frac{\pi }{3}$, then the range of $\frac{\tan B+\tan C}{\tan B\tan C-1}$ is

Solve $\sqrt{3}cosx+sinx=\sqrt{2}.$

$\text{Solve the following equations :}\left(i\right)cot\theta +tan\theta =2\left(ii\right)2si{n}^2\theta =3cos\theta ,0\le \theta \le 2\pi \left(iii\right)sec\theta cos5\theta +1=0,0<\theta <\frac{\pi }{2}\left(iv\right)5co{s}^2\theta +7si{n}^2\theta -6=0\left(v\right)sinx-3sin2x+sin3x=cosx-3cos2x+cos3x$

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below:



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}{E}_1\text{and}{E}_2\text{be mutually exclusive events, then}& 1& E_1\cap E_2=E_1\\ \text{(B)}& \text{If}{E}_1\text{and}{E}_2\text{are mutually exclusive and exhaustive events, then}& 2& (E_1-E_2)\cup (E_1\cap E_2)=E_1\\ \text{(C)}& \text{If}{E}_1\text{and}{E}_2\text{have common outcomes, then}& 3& E_1\cap E_2=\varphi ,E_1\cup E_2=S\\ \text{(D)}& \text{If}{E}_1\text{and}{E}_2\text{are events such that}{E}_1\subset E_2\text{, then}& 4& E_1\cap E_2=\varphi \end{array}$



Which of the following is the only CORRECT combination?

If $\omega \ne 1$ is a cube root of unity and $\left|\begin{array}{ccc}x+{\omega }^2& \omega & 1\\ \omega & {\omega }^2& 1+x\\ 1& x+\omega & {\omega }^2\end{array}\right|=0$, then value of x is

Minor $M_{33}$ $($Minor of the element of $i^{\text{th}}$ row and $j^{\text{th}}$ column$)$ of the determinant $\left|\begin{array}{ccc}2& 3& 5\\ 2& -1& 8\\ 1& 2& 4\end{array}\right|$ is

Let P be the point on the parabola, $y^2=8x$, which is at a minimum distance from the centre C of the circle,$x^2+(y+6{)}^2=1$. Then, the equation of the circle, passing through C and having its centre at P is

What is Cos(225)°?

In a $\Delta ABC$, side b is equal to

Let $f$ be a one-one function with domain $\{x,y,z\}$ and range $\{1,2,3\}.$ It is given that only one of the three conditions given below is true and remaining two are false. $f\left(x\right)=1,f\left(y\right)\ne 1,f\left(z\right)\ne 2.$ Then

Examine if Rolle’s
Theorem is applicable to any of the following functions. Can you say
some thing about the converse of Rolle’s Theorem from these
examples?

(i)

(ii)

(iii)

If the projections of the line segment $AB$ on the coordinate axes are $2,k,4$ and $AB=\frac{7}{\sqrt{2}},$ then $2k^2-\sqrt{2}k+1$ is equal to

${lim}_{x\to 0}\frac{si{n}^{2}4x^2}{x^4}$

Find the
intervals in which the function f given by f(x)
= 2x³ − 3x² − 36x
+ 7 is

(a) strictly
increasing (b) strictly decreasing

The range of $p$ for which the number $6$ lies between the roots of $x^2+2(p-3)x+9=0$ is

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1-z_2|=||z_1|-|z_2||$ then $\arg \left(z_1\right)-\arg \left(z_2\right)=$