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The continued product of the four values of ${[\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right)]}^{3/4}$ is

Find x, if [x -5 -1] $\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]\left[\begin{array}{c}x\\ 4\\ 1\end{array}\right]=0$.

Any complex number in the polar form can be expressed in Euler's form as $\cos \theta +i\sin \theta =e^{i\theta }$. This form of the complex number is useful in finding the sum of series $\sum _{r=0}^n{}^n{C}_r(\cos \theta +i\sin \theta {)}^r$.

$\sum _{r=0}^n{}^n{C}_r(\cos r\theta +i\sin r\theta )=\sum _{r=0}^n{}^n{C}_r{e}^{ir\theta }=\sum _{r=0}^n{}^n{C}_r(e^{i\theta }{)}^r=(1+e^{i\theta }{)}^n$

Also, we know that the sum of binomial series does not change if $r$ is replaced by $n-r$. Using these facts, answer the following questions.



The value of $\sum _{r=0}^{100}{}^{100}{C}_r\left(\sin rx\right)$ is equal to

If the plane ax+by=0 is rotated about its line of intersection with the plane z=0 through an angle $\alpha$,then prove that the equation of the plane in its new position is $ax+by\pm \left(\sqrt{a^2+b^2}tan\alpha \right)z=0.$

If the circles $x^2+y^2-4x-6y-12=0$ and $5(x^2+y^2)-8x-14y-32=0$ touch each other then their point of contact is

Let ${\Delta }_r=\left|\begin{array}{ccc}r-1& n& 12\\ (r-1{)}^2& 2n^2& 8n-4\\ (r-1{)}^3& 3n^3& 6n^2-6n\end{array}\right|$. Then the value of $\sum _{r=1}^n{\Delta }_r$ is:

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If $x^2-2hxy+y^2=0$ represents the equation of pair of straight lines both of which make an angle $\theta$ with the straight lines $x+y=2,$ then

The product of complex numbers $(3-2i)$ and $(3+i4)$ results in

Determine n if

$\left(i\right){}^{2n}{C}_3:{}^n{C}_2=12:1\left(ii\right){}^{2n}{C}_3:{}^n{C}_3=11:1$

If A = $\left[\begin{array}{ccc}2& 4& 8\\ 6& 9& 1\\ 8& 7& 5\end{array}\right]$ and B = $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then the third element of the first row of A - B = -------

Maximise Z=x+y subject to $x+4y\le 8,2x+3y\le 12,3x+y\le 9,x\ge 0$ and $y\ge 0$

Evaluate $\underset{0}{\overset{\pi }{\int }}{e}^{|\cos x|}[2\sin \left(\frac{1}{2}\cos x\right)+3\cos \left(\frac{1}{2}\cos x\right)]\sin xdx$

If f is the identity function and g is the modulus function, then find f + g is

The coordinates of the foot of the perpendicular from the point (2, 3) on the line $x+y-11=0$ are

If $x$ is real and $k=\frac{x^2-x+1}{x^2+x+1}$, then

The value of $\underset{1}{\overset{a}{\int }}\left[x\right]f^{\prime }\left(x\right)dx,a>1,$ where $\left[x\right]$ denotes the greatest integer not exceeding $x$ is

If PM is the perpendicular from P (2,3) on to the line x+y=3 then the co-ordinates of M are

If $x+y=1,x\ne \{0,1\}$, then for which value of $x$, $\sum _{r=0}^{10}{r}^2\left({}^{10}{C}_r\right)x^r{y}^{10-r}=0$

find the area of the triangle formed by the positive x axis and the tangent and normal to the curve x^2 + y^2 =9 at(2,√5).

Find the equation of two straight lines which are parallel to $x+7y+2=0$ and at unit distance from the point (1, -1).

For $x\in (0,\frac{3}{2}),$ let $f\left(x\right)=\sqrt{x},g\left(x\right)=\tan x$ and $h\left(x\right)=\frac{1-x^2}{1+x^2}.$ If $\varphi \left(x\right)=\left(\right(hof\left)og\right)\left(x\right),$ then $\varphi \left(\frac{\pi }{3}\right)$ is equal to

1. 0

Let $S_n=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\cdots +(n-1)\cdot 1,n\ge 4.$ The sum $\sum _{n=4}^{\infty }(\frac{2S_n}{n!}-\frac{1}{(n-2)!})$ is equal to

Find the sum to
indicated number of terms in each of the geometric progressions in
Exercise 7 to 10:

If $f\left(x\right)=x-\left[x\right]$ and $g\left(x\right)=\underset{0}{\overset{x}{\int }}f(t+n)dt\forall n\in N,$ then $g^{\prime }\left(\frac{5}{2}\right)$ is equal to

If the integral $\int e^x\frac{1+sinxcosx}{co{s}^{2}x}dx$ is the of the form $\int e^x$ [f(x) + f'(x)]dx then the appropriate f(x) would be -

Prove that