The polynomials `a x^3+3x^2-3` and `2x^3-5x+a` when divided by `(x-4)` leave the remainders `R_1 a n d R_2` respectively. Find the values of a in each of the following cases, if (i) `R_1=R_2` (ii) `R_1+R_2=0` (iii) `2R_1-R_2=0`

The polynomials a x^3+3x^2-3 and 2x^3-5x+a when divided by (x-4) leave the remainders R_1a n dR_2 respectively. Find the values of a in each of the following cases, if R_1=R_2 (ii) R_1+R_2=0 (iii)2R_1-R_2=0

Let R_1 and R_2 are the remainders when the polynomials x^3+2x^2-5a x-7 and x^3+a x^2-12 x+6 are divided by x+1 and x-2 respecti-vely. If 2R_1+R_2=6, find the value of adot

Let R_1 and R_2 are the remainders when the polynomials x^3+2x^2-5a x-7 and x^3+a x^2-12 x+6 are divided by x+1 and x-2 respecti-vely. If 2R_1+R_2=6, find the value of adot

On dividing the polynomial f(x)=x^3-3x^2+x+2 by a polynomial g(x) , the quotient q(x) and remainder r(x) where q(x)=x-2 and r(x)=-2x+4 respectively. Find the polynomial g(x) .

Consider an unknow polynomial which divided by (x - 3) and (x-4) leaves remainder 2 and 1, respectively. Let R(x) be the remainder when this polynomial is divided by (x-3)(x-4) . If equations R(x) = x^(2)+ ax +1 has two distint real roots, then exhaustive values of a are.

Assertion (A) : The remainder of x^(3)+2x^(2)-5x-3 which is divided by x-2 is 3. Reason (R) : The remainder of the polynomial f(x) when divided by x-a is f(a)

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1 r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. Number of common tangents to the circles x^2 + y^2 - 6x = 0 and x^2 + y^2 + 2x = 0 is (A) 1 (B) 2 (C) 3 (D) 4

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1+ r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. If circles (x-1)^2 + (y-3)^2 = r^2 and x^2 + y^2 - 8x + 2y + 8=0 intersect each other at two different points, then : (A) 1ltrlt5 (B) 5ltrlt8 (C) 2ltrlt8 (D) none of these

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1 r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. circles x^2 + y^2 + 2ax + c^2 = 0 and x^2 + y^2 + 2by + c^2 = 0 touche each other if (A) 1/a^2 + 1/b^2 = 2/c^2 (B) 1/a^2 + 1/b^2 = 2/c^2 (C) 1/a^2 - 1/b^2 = 2/c^2 (D) 1/a^2 - 1/b^2 = 4/c^2

Let f:[-1,2]->[0,oo) be a continuous function such that f(x)=f(1-x)fora l lx in [-1,2]dot Let R_1=int_(-1)^2xf(x)dx , and R_2 be the area of the region bounded by y=f(x),x=-1,x=2, and the x-axis . Then (a) R_1=2R_2 (b) R_1=3R_2 (c) 2R_1 (d) 3R_1=R_2