Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsVector Algebra

Let S be the set of all a R for which the angle between the vectors p = a i - j + k and q = i + a j - 3 k is acute for at least one value of [0, ] . Then S is equal to

Options

  1. A(- , -12) ( 3 2 , )
  2. B(- , -12) (0, )
  3. C( 3 2 , )

Correct answer

A. (- , -12) ( 3 2 , )

Step-by-step solution

For the angle between p and q to be acute, their dot product must be strictly positive. p q > 0 (a )( ) + (-1)(a ) + (1)(-3) > 0 a ^2 - a - 3 > 0 Let t = . Since [0, ] , we have t [-1, 1] . The condition becomes f(t) = at^2 - at - 3 > 0 for at least one t [-1, 1] . This requires the maximum value of f(t) on [-1, 1] to be strictly positive. Case 1: a > 0 The parabola opens upwards. The vertex is at t = -(-a) 2a = 1 2 . The maximum value on [-1, 1] occurs at the point furthest from the vertex, which is t = -1 . f(-1)

Practice Vector Algebra on Quantrex Academy →

More from Vector Algebra

Let a , b be two vectors, and let P, Q and R be the points with position vectors a , b and a + b , respectively, with respect to the origin O . If | a + b | = 21 , | a - b | = 3 , 2026Let a = 4 i - j + 3 k , b = 10 i + 2 j - k and a vector c be such that 2( a b ) + 3( b c ) = 0 . If a c = 15 , then c ( i + j -3 k ) is equal to: 2026Let a = 2 i + 3 j + 3 k and b = 6 i + 3 j + 3 k . Then the square of the area of the triangle with adjacent sides determined by the vectors (2 a + 3 b ) and ( a - b ) is : 2026If a = i + j + k , b = j - k and c be three vectors such that a c = b and a c = 3 , then c ( a - 2 b ) is equal to _______. 2026Let O be the origin, OP = a and OQ = b . If R is the point on OP such that OP = 5 OR , and M is the point such that OQ = 5 RM , then PM is equal to : 2026Let a = 7 i + j - k and b = j + 2 k . If r is a vector such that r a + a b = 0 and r a = 0 , then |3 r |^2 is equal to: 2026Let u and v be unit vectors inclined at an acute angle such that | u v |= 3 2 . If A = u + v +( u v ) , then is equal to: 2026Let a_k = ( _k) i + j and b_k = i - ( _k) j , where _k = 2^ k-1 2^n + 1 , for some n N , n > 5 . Then the value of _ k=1 ^ n | a_k |^2 _ k=1 ^ n | b_k |^2 is _____. 2026 Full Vector Algebra list All JEE Main PYQs