If `f(x)=(a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_n x+b_n)^2` , then prove that `(a_1b_1+a_2b_2++a_n b_n)^2lt=(a1 2+a2 2++a n2)^(b1 2+b2 2++b n2)dot`

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec b . If the angle between aa n db is pi/6, then prove that |(a_1 a_2a_3)(b_1b_2b_3)(c_1c_2c_3)|=1/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)

If a_1, a_2, ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)++(a_(n-1))/(a_n)+(a_n)/(a_1)> n

If sum_(r=0)^n{a_r(x-alpha+2)^r-b_r(alpha-x-1)^r}=0, then prove that b_n-(-1)^n a_n=0.

If |a_1sinx+a_2sin2x++a_nsinn x|lt=|sinx| for x in R , then prove that |a_1+2a_1+3a+3+n a_n|lt=1

If b_1, b_2 b_n are the nth roots of unity, then prove that ^n C_1dotb_1+^n C_2dotb_2++^n C_ndotb_n=b_1/b_2{(1+b_2)^n-1}^dot

If f(x)=|x+a^2 a b ac a b x+b^2 bc a c b c x+c^2|, t h e n prove that f^(prime)(x)=3x^2+2x(a^2+b^2+c^2)dot

The value of the determinant (a_1-b_1)^2(a_1- b_2)^2(a_1-b_3)^2(a_1-b_4)^2(a_2-b_1)^2(a_2-b_2)^2(a_2-b_3)^2(a_2- b_4)^2(a_3-b_1)^2(a_3-b_2)^2(a_3-b_3)^2(a_3-b_4)^2(a_4-b_1)^2(a_4-b_2)^2(a_4-b_3)^2(a_4-b_4)^2 is dependant on a_i , i=1,2,3,4 dependant on b_i , i=1,2,3,4 dependant on a_(i j), b_i i=1,2,3,4

Suppose p(x)=a_0+a_1x+a_2x^2++a_n x^ndot If |p(x)|lt=e^(x-1)-1| for all xgeq0, prove that |a_1+2a_2++n a_n|lt=1.

If (1+x+x^2++x^p)^n=a_0+a_1x+a_2x^2++a_(n p)x^(n p), then find the value of a_1+2a_2+3a_3+ddot+n pa_(n p)dot

If a ,b ,a n dc are in G.P. then prove that 1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)dot