Vector Algebra - JEE Advanced PYQ | Last 5 years | PYQ Khazana #jeeadvancedPYQ

Let OP= α-1/α i+j+k, OQ= î+β-1/βj+k and OR=î+j+2k be three vectors, where
α,βR -{O} and O denotes the origin. If (OP×OQ). OR = 0 and the point (a, ß,2) lies on
the plane 3x+3y -z +l = 0, then the value of l is____.

Let p =2i + j +3k and q = i - j + k. If for some real numbers a, ß, and y, we have
15î+10j+6k=a(2p+q)+ß(p-2q)+y(pxq), then the value of yJis____.

Let the position vectors of the points P,Q,R and S be a= 1+2j-5k, b=31+6j+3k, c=7i+16j+7k
and d=2î + j+k, respectively. Then which of the following statements is true?
(A) The points P,Q,R and S are NOT coplanar
(B) b+2d3is the position vector of a point which divides PR internally in the ratio 5 : 4
(C) b+2d3is the position vector of a point which divides PR externally in the ratio 5 :
(D) The square of the magnitude of the vector bxd is95

Let î, j and k be the unit vectors along the three positive coordinate axes.

Let ü, v and w be vectors in three-dimensional space, where u and v are unit vectors which are not
perpendicular to each other and u.w=1, v.w=1, w.w =4
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors u, v and w,
is v2 , then the value of 3u+5v is.

Let O be the origin and OA =21+2j+k, OB= 1-2j+2k and OC =- (OB-AOA) for some a 0. If
OBxOc|-2, then which of the following statements is (are) TRUE?

Let a and b be positive real numbers. Suppose PQ=ai+bj and PS=ai-bj are adjacent sides of a
parallelogram PQRS. Let u and v be the projection vectors of w = i + j along PQ and PS, respectively.
If u +v = w and if the area of the parallelogram PQRS is 8, then which of the following statements
is/are TRUE ?

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