Let `P(x)=x^2+(4x)/3+log_10(4. bar9), A=prod_(i=1)^12 P(a_i)` where `a_1,a_2,........,a_12` are positive reals and `B=prod_(j=1)^13P(b_j)` where `b_1,b_2,......,b_13` are non-positive reals ,then which one of the fellowing is always correct?

Let P(x)=x^(2)+(4x)/(3)+log_(10)(4.bar(9)),A=prod_(i=1)^(12)P(a_(i)) where a_(1),a_(2),.......,a_(12) are positive reals and B=prod_(j=1)^(13)P(b_(j)) where b_(1),b_(2),......,b_(13) are non- positive reals,then which one of the fellowing is always correct?

For any three sets A_1,A_2, A_3 , let B_1=A_1, B_2 = A_2-A_1 and B_3 = A_3-(A_1uuA_2) , then which one of the following statement is always true

For any three sets A_1 , A_2 ,A_3 . Let B_1 = A_1 , B_2 = A_2 - A_1 and B_3 = A_3 - A_1 cup A_2 , then which of the following statement is always true ?

If x is real, find the real values of p which make 4x^2+px+1 always positive.

If f(x)= prod_(i=1)^3 (x-a_i) +sum_(i=1)^3a_i-3x where a_i lt a_(i+1) forall i=1,2,. . . then f(x)=0 has

If a, a_1 ,a_2----a_(2n-1),b are in A.P and a,b_1,b_2-----b_(2n-1),b are in G.P and a,c_1,c_2----c_(2n-1),b are in H.P (which are non-zero and a,b are positive real numbers), then the roots of the equation a_nx^2-b_nx |c_n=0 are

If a, a_1 ,a_2----a_(2n-1),b are in A.P and a,b_1,b_2-----b_(2n-1),b are in G.P and a,c_1,c_2----c_(2n-1),b are in H.P (which are non-zero and a,b are positive real numbers), then the roots of the equation a_nx^2-b_nx +c_n=0 are

We know that, if a_1, a_2, ..., a_n are in H.P. then 1/a_1,1/a_2,.....,1/a_n are in A.P. and vice versa. If a_1, a_2, ..., a_n are in A.P. with common difference d, then for any b (>0), the numbers b^(a_1),b^(a_2),b^9a_3),........,b^(a_n) are in G.P. with common ratio b^d. If a_1, a_2, ..., a_n are positive and in G.P. with common ration, then for any base b (b> 0), log_b a_1 , log_b a_2,...., log_b a_n are in A.P. with common difference logor.If x, y, z are respectively the pth, qth and the rth terms of an A.P., as well as of a G.P., then x^(z-x),z^(x-y) is equal to

Let f_1 : (0, oo) to R and f_2 : (0, oo) to R be defined by f_1(x) = int_0^xprod_(j= 1)^21((t − j)^j)dt, x gt 0 and f_2(x) = 98(x − 1)^50 − 600(x − 1)^49 + 2450, x gt 0 , where, for any positive integer n and real numbers a_1, a_2, … , a_n, prod_(i=1)^n(a_i) denotes the product of a_1 , a_2, … , a_n . Let m_i and n_i , respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 , in the interval (0, oo) The value of 6m_2+4n_2+8m_2n_2 is ___.

Let f_1 : (0, oo) to R and f_2 : (0, oo) to R be defined by f_1(x) = int_0^xprod_(j= 1)^21((t − j)^j)dt, x gt 0 and f_2(x) = 98(x − 1)^50 − 600(x − 1)^49 + 2450, x gt 0 , where, for any positive integer n and real numbers a_1, a_2, … , a_n, prod_(i=1)^n(a_i) denotes the product of a_1 , a_2, … , a_n . Let m_i and n_i , respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 , in the interval (0, oo) The value of 2m_1+3n_1+m_1n_1 is ___.