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If $\alpha ,\beta ,\gamma$ are the roots of $x^3-x^2-1=0,$ then the value of $\frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }=$

Integrate the following functions.
$\int$ cot x log sin x dx.

The locus of a point on the variable parabola $y^2=4ax$, whose distance from focus is always equal to $k$, is equal to

The set of values of $k$ for which the equation $(k+2)x^2-2kx-k=0$ has two roots on the number line symmetrically placed about $1$ is

Show that
the points A, B and C with position vectors,,
respectively
form the vertices of a right angled triangle.

Let $\overrightarrow{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}.$ If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}\cdot \overrightarrow{c}=|\overrightarrow{c}|\text{and}|\overrightarrow{c}-\overrightarrow{a}|=2\sqrt{2}$ and the angle between $(\overrightarrow{a}\times \overrightarrow{b})$ and $\overrightarrow{c}$ is ${30}^0$, then $|(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|$ is equal to

The eccentric angle of point of intersection of the ellipse $x^2+4y^2=4$ and the parabola $x^2+1=y$ is

Evaluate $\int \frac{\cos 3x}{\cos 2x-2{\sin }^{2}x}dx$

(where $C$ is constant of integration)

The mean of $n$ items is $\overline{x}$. If each item is successively increased by $3,3^2,3^3,...,3^n,$ then new mean equals

Maximise Z
= − x + 2y, subject to the constraints:

.

A unit vector perpendicular to the plane determined by the vectors $4\hat{i}+3\hat{j}-\hat{k}and2\hat{i}-6\hat{j}-3\hat{k}$

The negation of $\sim s\vee (\sim r\wedge s)$ is equivalent to

Find the sum of the multiplicative inverses of $\frac{1}{32},-\frac{1}{43}$ and $\frac{1}{100}$. Also find the additive inverse of the sum obtained.

If b+c=3a, then cot B/2 cot C/2 is equal to

What is the probability of finding numbers from 1 to 1000 such that they are squares, cubes or both of a natural number?

$r^{th}$ term in the expansion of $(a+2x{)}^n$ is

Expand
the expression (1– 2x)⁵

The sides of a triangle are 3x+4y, 4x+3y, 5x+5y where x,y>0 then Triangle is

1.Right angle

2.Obtuse angle

3.Equilateral

4.None of the above

If each element of a second order determinant is zero or one , what is the probability that the values of the determinant is positive?(Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability$\frac{1}{2}$

Find the sum of the following series to infinity :

(i) $1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+\frac{1}{3^4}+....\infty$

(ii) $8+4\sqrt{2}+4+....\infty$

(iii) $\frac{2}{5}+\frac{3}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+.....\infty$.

(iv) 10 - 9 + 8.1 - 7.29 + ...... $\infty$

(v) $\frac{1}{3}+\frac{1}{5^2}+\frac{1}{3^3}+\frac{1}{5^4}+\frac{1}{3^5}+\frac{1}{5^6}+.....\infty$

The solution set of the inequality $|3x-2|>|x+4|$ is

If circles with radii $a$ units and $b$ units touch each other externally and the angle between their direct common tangents is $\theta$, where $a>b\ge 2$, then the value of $\sin \theta$ is

$f\left(x\right)=\left|\begin{array}{ccc}3& 1& 2\\ a^3& 2a& a^2\\ 3x^2& 2& 2x\end{array}\right|then{\int }_0^{a}f\left(x\right)is......$

$ifi=\sqrt{-1},then4+5{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{334}+3{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{365}isequalto$

For the two functions



$f(x,y)=x^3$-$3x{y}^2$ and $g(x,y)=3x^{2}y$ -$y^2$



Which one of the following options is correct?

The locus of centre of a circle which passes through the origin and cuts off a length of $4$ units on the line $x=3$ is

If $z=z\left(x\right)$ and $(2+\cos x)\frac{dz}{dx}+\left(\sin x\right)\cdot z=\sin x$, $z\left(x\right)>0$ and $z\left(\frac{\pi }{2}\right)=3$, then $z\left(\frac{\pi }{3}\right)$ is

The number of real circles cutting orthogonally the circle $x^2+y^2+2x–2y+7=0$ is