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Find the value of k if x + y + 5 = 0 is a tangent to the circle $x^2+y^2+10x+2ky+10=0$

If $\sum _{k=1}^{n}f\left(k\right)=\frac{n}{2(n+2)}$, then the value of $\sum _{k=1}^{10}\frac{1}{f\left(k\right)}$ is equal to

The plane ax+by =0 is rotated through an angle $\alpha$ about its line of intersection with the plane z=0. Then the equation of the plane in the new position is

If $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{c}=\hat{i}+3\hat{j}-({\lambda }^2+3\lambda )\hat{k}$, where $\lambda$ is a constant and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}-\lambda \overrightarrow{b}$, then sum of the different values of $\lambda$ is

If z = $i^i$ . Express ${log}_e$z in A+iB form.Find the value of A and B.

For any two sets $A\&B,n\left(A\right)=20;n\left(B\right)=6;$$n(A\cup B)=22,$ then match the following:

If $|z_1+z_2|=|z_1-z_2|$ then $\arg z_1-\arg z_2=$

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to the n^th power of the other root, then the value of $(a{c}^n{)}^{\frac{1}{n+1}}$ + $(a^{n}c{)}^{\frac{1}{n+1}}$

If three points $A,B$ and $C$ lie on a line and $A\equiv (3,4),$ $B\equiv (7,7)$ and $AC=10$, then the coordinates of the point $C$ can be

If the focus of the parabola $x^2-ky+3=0$ is $(0,2)$, then the value(s) of $k$ is/are

The equation of the chord of the circle $x^2+y^2=r^2$ passing through $(2,3)$ and farthest from the centre is

The term independent of $x$ in expansion of ${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$ is :

Set of values of $x$ in $(0,\pi )$ satisfying $1+{\log }_2\sin x+{\log }_2\sin 3x\ge 0$ is

Two integers are selected at random from the set $\{1,2,\dots ,11\}.$ Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

The value of $\tan {20}^{\circ }\tan {80}^{\circ }\cot {50}^{\circ }$ is

A straight line L through the poit $(3,-2)$ is inclined at an angle ${60}^0$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, then the equation of L is

If $\frac{a+ib}{c+id}=x+iy$, then $\frac{a^2+b^2}{c^2+d^2}$ is equal to

If $p^{th},q^{th},r^{th}$ terms of an A.P. are $a,b,c$ respectively, then the value of $a(q-r)+b(r-p)+c(p-q)$ is

Equation of the chord of contact, drawn to the ellipse $4x^2+9y^2=36$ from the point $(m,n)$ where $m\cdot n=m+n$ and $m,n\in I^{+}$ is

If $\frac{cos(x-y)}{cos(x+y)}+\frac{cos\left(7_t\right)}{cos(7-t)}=0$ then tan x tan y tan 7 tan t=

Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद



Roots of x² + 6x + 12 = 0 are α and β, where α and β are complex numbers, then



समीकरण x² + 6x + 12 = 0 के मूल α व β हैं, जहाँ α व β सम्मिश्र संख्याएं हैं, तब



Q. |α² – β²| is



प्रश्न - |α² – β²| का मान है

If in the expansion of $(a-2b{)}^n$, the sum of $5^{\text{th}}$ and $6^{\text{th}}$ terms is $0$, then the value of $\frac{a}{b}$ is equal to

The solution lof $\frac{dy}{dx}-xtan(y-x)=1$

If $x=3\sec \theta -2$ and $y=3\tan \theta +2$, then which of the following equation in $x,y$ is correct?

If both the roots of the quadratic equation $x^2$-2kx+$k^2$+k-5=0 are less than 5, then k lies in the interval.

If $li{m}_{x\to 2}$ $\frac{\left(x^n\right)-\left(2^n\right)}{x-2}$ =80 , where n is a positive integer,

then n=

Shortest distance (in units) between the two parabolas $y^2=x-2,x^2=y-2$ is

If $C_0,C_1,C_2,\dots ,C_n$ denote the binomial coefficients of the expansion $(1+x{)}^n$ and $\sum _{r=0}^n(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\dots \text{upto}m\text{terms}]$$=\frac{a^{mn}-1}{b^{mn}(c^n-1)},$ then

Let $A,B,C$ are three sets such that



Then $A^C\cap B\cap C^C=$

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

A particle $P$ starts from the point $z_0=1+2i,$ where $i=\sqrt{-1}.$ It moves first horizontally away from origin by $5$ units and then vertically away from origin by $3$ units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{2}$ units away from origin in the direction of $x=y$ and then it moves through an angle $\frac{\pi }{2}$ in anticlockwise direction on a circle with centre at origin to reach a point $z_2$. Then point $z_2$ is given by

The inverse function of $f\left(x\right)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x\in (-1,1)$, is

If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right)$ and $\beta =3{\cos }^{-1}\left(\frac{4}{9}\right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is/are

The value(s) of $\theta$ for which $\cos \theta =-\frac{1}{2}$ is/are

If a chord of the circle $x^2+y^2=8$ makes equal intercepts $a$ on the coordinate axes, then

Let $P$ be any moving point on the circle $S_1:x^2+y^2-2x-1=0.$ A chord of contact is drawn from the point $P$ to the circle $S:x^2+y^2-2x=0.$ If $C$ is the centre and $A,B$ are the points of contact of circle $S,$ then the locus of the circumcentre of $△CAB$ is

$C_0-C_1+C_2-C_3+........+(-1{)}^n{C}_n$ is equal to

If $f\left(x\right)=\int (\cot \frac{x}{2}-\tan \frac{x}{2})dx,$ where $f\left(\frac{\pi }{2}\right)=0,$ then which of the following statements is (are) CORRECT ?



$($Note : $\text{sgn}\left(y\right)$ denotes the signum function of $y.)$

The solution of $(y+x+5)dy=(y-x+1)dx$ is

Equation of the hyperbola passing through the point $(1,-1)$ and having asymptotes $x+2y+3=0$ and $3x+4y+5=0$ is :

If the vertices of a variable triangle are $(3,4),$ $(5\cos \theta ,5\sin \theta )$ and $(5\sin \theta ,-5\cos \theta ),$ then the locus of its orthocentre is

The circle $C_1:x^2+y^2=8$ cuts orthogonally the circle $C_2$ whose centre lies on the line $x-y-4=0$ then, the circle $C_2$ passes through a fixed point, which lies on

If X and Y are two sets and $X^{\prime }$ denotes the complement of X, then

$X\cap (X\cup Y{)}^{\prime }$ is equal to

If $S_{2n}=3S_n$ and $S_{5n}=k{S}_{3n}$, where $S_n$ is the sum of $n$ terms of an A.P., then the value of $k$ is