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If $z=\cos \theta +i\sin \theta$ be a root of the equation $a_0{z}^n+a_1{z}^{n-1}+a_2{z}^{n-2}+\dots \dots +a_{n-1}z+a_n=0$, then

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors of equal magnitude $5$ uints. Let $\overrightarrow{p},\overrightarrow{q}$ be vectors such that $\overrightarrow{p}=\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{q}=\overrightarrow{a}+\overrightarrow{b}$. If $|\overrightarrow{p}\times \overrightarrow{q}|=2\{\gamma -(\overrightarrow{a}.\overrightarrow{b}{)}^2{\}}^{1/2}$, then the value of $\gamma$ is

The point $A$ divides the line segment joining the points $(-5,1)$ and $(3,5)$ in the ratio $k:1$. The coordinates of points $B$ and $C$ are $(1,5)$ and $(7,-2)$ respectively. If the area of $△ABC$ is $2$ square units, then the number of values of $k$ is

The function $f\left(x\right)-p[x+1]+q[x-1]$, where is the greatest integer function is continuous at x =1, if

P is point lying outside a cicle. 3 lines are drawn from P to the cicle so that they intersect the circle at 6 points as shown in the figure.

$P_1,P_2,P_3$ are points formed by intersections of tangents at A and B, C and D and E and F respectively.

The coefficient of $x^{28}$ in the expansion of $(1+x^3-x^6{)}^{30}$ is

If $\overrightarrow{u}=3\hat{i}-5\hat{j}+9\hat{k}\text{and}\overrightarrow{v}=3\hat{i}+4\hat{j}+0k;$ What is the component of $\overrightarrow{u}$ along the direction of $\overrightarrow{v}$?

If $2\hat{i}+3\hat{j}+4\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ are two adjacent sides of a parallelogram, then the area of the parallelogram will be

The number of ways of selecting two squares on a chess board such that they have a side in common is

A man walks a distance of 3 units from the origin towards the north-east $\left(N{45}^{o}E\right)$ direction. From there, he walks a distance of 4 units towards the north-west $\left(N{45}^{o}W\right)$ direction to reach a point P. Then the position of P in the Argand plane is

The number of non-negative integral values of $b$ for which the origin and point $(1,1)$ lie on the same side of straight line $a^{2}x+aby+1=0,\forall a\in R-\left\{0\right\},$ is

Given A and C are coefficient and augmented matrices respectively for a system of linear equations. Which of the following cases tells if the equations are consistent?

The co-ordinates of the extremities of the latus rectum of the parabola $5y^2=4x$ are

Let $z$ be a complex number such that $|z|=z+32-24i$. Then $Re\left(z\right)+Im\left(z\right)$ is equal to

There are $2$ brothers among a group of $20$ persons. The number of ways the group can be arranged around a circle so that there is exactly one person between the two brothers is

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n,$ then

the value of $\underset{0\le r<s\le n}{\sum \sum }(r+s)C_r{C}_s$ is

If $I={\int }_0^1\frac{sinx}{\sqrt{x}}dxandJ={\int }_0^1\frac{cosx}{\sqrt{x}}dx,$ then, which one of the following is true?

The ratio in which a point $P(2,3)$ divides the line segment joining $A(-2,-7)$ and $B(4,8)$ is

The derivative of $si{n}^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to $ta{n}^{-1}\left(\frac{2x}{1-x^2}\right)$ is

If the point $P(\alpha ,-\alpha )$ lies inside the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1,$ then

The general solution of the equation $\tan 5\theta =\cot 3\theta$ is given by

The range of values of $x$ that satisfies the inequation ${\log }_2{\log }_{0.5}\left(2^x\frac{15}{16}\right)\le 2$ is

The solution set of $x^2+4x+9\ge 0$ is

Let $P_i$ and $P_i^{\prime }$ be the feet of perpendiculars drawn from foci $S,S^{\prime }$ on a tangent $T_i$ to an ellipse whose length of semi major axis is $20,$ if $\sum _{i=1}^{10}\left(S{P}_i\right)\left(S{P}_i^{\prime }\right)=2560,$ then the value of eccentricity is

If A and B are two matrices of order '3' such that $3A+4B{B}^T=I$ and $B^{-1}=A^T$, then identify which of the following statements is/are correct?

Which of the following is the inverse pair of 1 under the operations multiplication and subtraction?

Locus of point z so that z, i, and iz are collinear, is

If $|x{|}^2-6|x|+9\le 4$, then $x\in$

If $6^{th}$ term in the expansion of $(\frac{3}{2}+\frac{x}{3}{)}^n$ is the numerically greatest term when x=3, then find the sum of all possible values of n

The angle between the tangents to the curve $y^2=2ax$ at the points where $x=\frac{a}{2}$, is

Let the coefficients of powers of $x$ in the second, third and fourth terms in the binomial expansion of $(1+x{)}^n,$ where $n$ is a positive integer, be in arithmetic progression. The sum of the coefficients of odd powers of $x$ in the expansion is

If U = {2, 4, 6, 8, 10, 12, 14} and

A = {2, 4, 10}, where U is the Universal set .

Which of the following is A^C?

The distance of the point (1, 3, -7) from the plane passing through the point (1, -1, -1) having normal perpendicular to both the lines

$\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$, is

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

$\sin A+\sin B+\sin C=0,$ then the value of $\frac{\sin A.\sin B.\sin C}{\sin 3A+\sin 3B+\sin 3C}$ (wherever defined) is

Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद



Given f(x) = ax² + bx + c, a, b, c ∈ R and a ≠ 0. α and β are roots of f(x) = 0.



दिया है f(x) = ax² + bx + c, a, b, c ∈ R तथा a ≠ 0 है। α तथा β, f(x) = 0 के मूल हैं।



Q. The value of f(α + β) is



प्रश्न - f(α + β) का मान है

If $x\in [-4,3]$, then $x^2$ lies in

The least value of $\alpha ϵR$ for which $4\alpha x^2+\frac{1}{x}\ge 1$, for all x > 0, is

If a triangle is formed by any three tangents of the parabola $y^2=4ax$ whose two of its vertices lie on $x^2=4by$, then third vertex lie on

If f(x) be a continuous function defined for $1\le x\le 3.f\left(x\right)ϵQ\forall xϵ[1,3]andf\left(2\right)=10$ (Where Q is a set of all rational numbers). Then, f(1.8) is

Let set $R=\{P:B\subseteq P\subseteq A\}$

If $A=\{1,2,3,4,5\}$ and $B=\{1,2\}$, then number of elements in set $R$ is

If $x_1,x_{2}and{x}_3$ as well as $y_1,y_{2}and{y}_3$ are in GP with same common ratio, the points

$P(x_1,y_1),Q(x_2,y_2)andR(x_3,y_3)$

If cos x +cosy = $\frac{1}{3}$, sin x + sin y = $\frac{1}{4}$ then sin (x + y) =

The maximum number of permutations of $2n$ letters in which there are only $a^{\prime }s$ and $b^{\prime }s,$ taken all at a time is given by

Let $f:R\to R$ be a mapping, such that $f\left(x\right)=\frac{x^2}{1+x^2}$ Then, f is