Let `V_1 = 3ax^2 i - 2(x-1)j` and `V_2 = b(x-1)i + x^2 j` where, `ab < 0` . The vector `V_1` and `V_2` are linearly dependent for

Let vec V_1=3a x^2 hat i-2(x-1) hat j and vec V_2=b(x-1) hat i+x^2 hat j , where a b<0. The vector vec V_1 and vec V_2 are linearly dependent for atleast one x in (0, 1) atleast one x in (-1,0) atleast one x in (1,2) no value of x in (0, 1)

If (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where , S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n)a_(i) a_(j) a_(k) and so on . Coefficient of x^(7) in the expansion of (1 + x)^(2) (3 + x)^(3) (5 + x)^(4) is

If (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where , S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n)a_(i) a_(j) a_(k) and so on . Coefficient of x^(7) in the expansion of (1 + x)^(2) (3 + x)^(3) (5 + x)^(4) is

If (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where , S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n)a_(i) a_(j) a_(k) and so on . Coefficient of x^(7) in the expansion of (1 + x)^(2) (3 + x)^(3) (5 + x)^(4) is

Find λ such that the vectors vec v_1 =2 hat i - hat j +hat k, vec v_2= hat i +2hat j -3 hat k and vec v_3= 3 hat i +λ hat j +5 hat k are coplanar.

Two particles of equal mass have velocities v1=2 i ms^(-1) and v2= 2 j ms^(-1) First particle has an acceleration a= (3i + 3j) ms^(-2) while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of