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The sum of roots of the equation $x^8=1$ whose real part is positive is

If a tangent to the parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of symmetry of the parabola, then point of contact(s) is/are

If $f\left(x\right)=(x-a)(x-b)(x-c),a<b<c$, then $f^{\prime }\left(x\right)=0$ has exactly one root in

In a examination, $20$ questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true'; if it falls tails, he answers 'false'. The probability that he answers at least $12$ questions correctly is:

If a circle S(x,y)=0 touches the line x + y = 5 at the point (2, 3) and S(1, 2) = 0, then the radius of such circle is .

Show that the perpendicular let fall from any point on the straight line $2x+11y-5=0$ upon the two straight lines $24x+7y=20$ and $4x-3y-2=0$ are equal to each other.

If x=-5+$\sqrt{-4}$, then the value of the expression $x^4$+9$x^3$+35$x^2$-x+4 is

If S is the sample space and P(A) = $\frac{1}{3}$ P(B) and S = $A\cup B$, where A and B are two mutually exclusive events, then P(A) =

Prove the following by using the principle of mathematical induction for all n ∈ N:

Is -1 , -3 ,-5 ,-7 , ... odd numbers

The centroid of a triangle is $(2,7)$ and two of its vertices are $(4,8)$ and $(-2,6).$ The third vertex is

The value of the limit $\underset{x\to 0}{lim}(\frac{x}{\sin x}{)}^{\frac{6}{x^2}}$ is

If the sides of a right angled triangle be in A. P. , then their ratio will be

Find the area of the region bounded by
the curve y² = x and the lines x = 1,
x = 4 and the x-axis.

Let $PQ$ be a focal chord of the parabola $y^2=4ax.$ The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,$ $a>0.$ If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax,$ then $\tan \theta =$

Let $I_1=\underset{0}{\overset{\pi /3}{\int }}(2\sec x{)}^{100}dx,$ and $I_2=\underset{0}{\overset{\ln (2+\sqrt{3})}{\int }}(e^x+e^{-x}{)}^{99}dx$. Then the value of $\frac{I_1}{I_2}$ is

$\text{If}f\left(x\right)=\sqrt{ax}+\frac{a^2}{\sqrt{ax}},\text{then}{f}^{\prime }\left(a\right)=$

If $z_1$ and $z_2$ are two complex numbers, then the inequality $|z_1+z_2{|}^2\le (1+c)|z_1{|}^2+(1+c^{-1})|z_2{|}^2$ is true if

The value of $\underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}\ln (\sin x+\cos x)dx$ is:

Any point on the parabola whose focus is (0, 1) and the directrix is x+2=0 is given by

Let $f:[a,b]\to R$ be a function such that for
$c\epsilon (a,b),f^1\left(c\right)=f^{11}\left(c\right)=f^{111}\left(c\right)=f^{tv}\left(c\right)=f^v\left(c\right)=0$ then

Find the integrals of the functions.
$\int \frac{cosx}{1+cosx}dx.$

The product of all real roots of the equation $x^2-|x|-6=0$ is

If $P$ is a point on the parabola $y=x^2+4$ which is closest to the straight line $y=4x-1,$ then the co-ordinates of $P$ are

The locus of a point $P$, such that the sum of whose distances from $A(4,0,0)$ and $B(-4,0,0)$ is equal to $10,$ is

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 16x² – 9y² = 576

ABCD is a parallelogram and $A_1$ and $B_1$ are the mid-points of the sides BC and CD respectively.

$If\overrightarrow{A{A}_1}+\overrightarrow{A{B}_1}=\lambda \overrightarrow{AC},$ then ${}^{\prime }{\lambda }^{\prime }$ is equal to

If $f:R^{+}\to R^{+}$ is a polynomial function satisfying the functional equation $f\left(f\right(x\left)\right)=6x-f\left(x\right),$ then $f\left(17\right)$ is equal to

Area of the region $\left\{\right(x,y)\in R^2:y\ge \sqrt{|x+3|},5y\le x+9\le 15\}$ is equal to

${lim}_{x\to \infty }\frac{sqrt{x}^2-1}{2x+1}$ is equal to