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If $\alpha ,\beta$ are roots of the equation $a{x}^2+bx+c=0,$ then the quadratic equation whose roots are $\frac{1}{(a\alpha +b{)}^2},\frac{1}{(a\beta +b{)}^2}$, is

For a standard hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Match the following.

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ 1.a^2>b^2& \text{P.Director circle is real}\\ 2.a^2=b^2& \text{Q.Director circle is imaginary}\\ 3.a^2<b^2& \text{R.Centre is the only point from which}\\ & \text{two perpendicular tangents can be drawn on the}\\ & \text{hyperbola}\end{array}$

The distance (in units) between the parallel lines $\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+\hat{k})$ and $\overrightarrow{r}=2\hat{i}-\hat{j}-\hat{k}+\mu (\hat{i}-\hat{j}+\hat{k})$ is

I had a doubt that while practising a chapter in maths, after completing all the NCERT problems in that chapter , should I first complete seeing the board questions of previous year of that particular chapter OR should I first complete all the NCERT sums of all subjects and then on the whole refer to questions on other resources. Please give me a proper method of learning for class CBSE 12 to score above 90 . Help me with your guidance

The anti-derivative of $\frac{cos5x+cos4x}{1-2cos3x}$ is

Let z,w be complex numbers such that $z+iw=0$ and arg $zw=\pi$ then arg z equals

State whether the two lines in each of the following are parallel, perpendicular or neither:

(i) Through (5, 6) and (2, 3); through (9, -2) and (6, -5)

(ii) Through (9, 5) and (-1, 1); through (3, -5) and (8, -3)

(iii) Through (6, 3) and (1, 1); through (-2, 5) and (2, -5)

(iv) Through (3, 15) and (16, 6); through (-5, 3) and (8, 2).

If $x\in [-4,-1]$, then $\frac{1}{x^2+4x+7}$ belongs to

If $(h,k)$ is the centre of the circle touches $y-\text{axis}$ at a distance of $12\text{units}$ from the origin and makes an intercept of $10\text{units}$ on $x-\text{axis}$, then the equation of circle for which $(h+k)$ is minimum, is

1. 2.7

Simplified form of $\frac{(\sqrt{3}-i{)}^6}{(1+i{)}^8}$ is

There are two perpendicular straight lines touching the parabola $y^2=4a(x+a)$ and $y^2=4b(x+b)$, then the point of intersection of these two lines lie on the line given by

Find the equation of a line perpendicular to the line $\sqrt{3}x-y+5=0$ and at a distance of 3 units from the origin.

If the imaginary part of $(z-1)(\cos \alpha -i\sin \alpha )+(z-1{)}^{-1}\times (\cos \alpha +i\sin \alpha )$ is zero, then which of the following can be correct?

If centre of circles $x^2+y^2=25$ and $x^2+y^2-4x+9y+3=0$ are the endpoints of the diameter of a circle $S$, then equation of the circle $S$ is

f(x) = $\{\begin{array}{c}3,if0\le x<1\\ 4,if1<x<3\\ 5,if3\le x\le 10\end{array}$

$\text{If}y=2^{2^x},then\frac{dy}{dx}=$

If the curves, $\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ intersect each other at an angle of ${90}^{\circ }$, then which of the following relations is $\text{TRUE}?$

The set of all real values of ${}^{\prime }{a}^{\prime }$ so that the range of function $y=\frac{x^2+a}{x+1},x\in R-\{-1\}$ is $R$, is

If $f\left(x\right)=alo{g}_e|x|+b{x}^2+x$ has extremum at x = 1 and x = 3, then

Length of intercepts made by circle $x^2+y^2-10x-8y+4=0$ on the $X$ and $Y$ axes respectively are

A man draws a card from a pack of 52 playing cards, replaces it and shuffles

the pack. He continues this processes until he gets a card of spade. The

probability that he will fail the first two times is [MNR 1980]

Differentiate the
functions with respect to x.

The trigonometric form of $z=(1-icot8{)}^3$ (where $i=\sqrt{-1}$) is

If the major axis is "n” times the minor axis of the ellipse, then its eccentricity is

The slope of the normal to the curve y = 2x²
+ 3 sin x at x = 0 is

(A) 3 (B) (C) −3 (D)

If $\int \frac{se{c}^{2}x-2010}{si{n}^{2010}x}dx=\frac{P\left(x\right)}{si{n}^{2010}x}+C,$ then value of $P\left(\frac{\pi }{3}\right)$ is