Let veca = 2 hati+ hatk and vecb = hati+2 hatj+ hatk be two vectors. Consider a vector vecc = vecαa + β vecb, α, β ∈ ℝ If the projection of vecc on the vector ( veca+ vecb) is 3√2, then the minimum value of ( vecc-( veca× vecb))⋅ vecc equals

Questions from JEE Advanced 2019

1. Let be given by

Then which of the following options is/are correct?

2. Let f : ℝ ℝ be given by f(x) = (x - 1)(x - 2)(x - 5). Define

Then which of the following options is/are correct?

3. Three lines

and

are given. For which point(s) on can we find a point P on and a point R on so that P, Q and R are collinear?

4. Let f(x) = sin( cos x) and g(x) = cos(2 sin a) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:
X = {x : f'(x) — 0}, Y = {x : f(x) = 0},
Z = [x : g(x) = 0}, W = [x : g'(x) = 0}.
List -1 contains the sets X, Y, Z and W. List - II contains some information regarding these sets.

List-I List-II

(I) X (p) ⊇

(II) Y (Q) an arithmetic progression

(III) Z (R) NOT an arithmetic progression

(IV) W (S) ⊇

(T) ⊇

(U) ⊇

Question: Which of the following is the only CORRECT combination?

5. Let f(x) = sin( cos x) and g(x) = cos(2 sin a) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:
X = {x : f'(x) — 0}, Y = {x : f(x) = 0},
Z = [x : g(x) = 0}, W = [x : g'(x) = 0}.
List -1 contains the sets X, Y, Z and W. List - II contains some information regarding these sets.

List-I List-II

(I) X (p) ⊇

(II) Y (Q) an arithmetic progression

(III) Z (R) NOT an arithmetic progression

(IV) W (S) ⊇

(T) ⊇

(U) ⊇

Question : Wliich of the following is the only CORRECT combination?

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