Mathematics Test 69

A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi-major axis is

In a shop there are five types of ice-creams available. A child buys six ice-creams.

Statement-1: The number of different ways the child can buy the six ice-creams is 10C₅.

Statement-2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4B’s in a row.

Statement-1: The point A (3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x−y+z=5.

Statement-2: The plane x−y+z=5 bisects the line segment joining A (3, 1, 6) and B (1, 3, 4).

Let A be a 2×2 matrix with non-zero entries and let A²=I, where I is 2×2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A||=determinant of matrix A.

Statement-1: Tr(A) = 0
Statement-2: |A|=1

For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is

The number of complex numbers z such that |z−1|=|z+1|=|z−i| equals (where i = √i1)

A line AB in three-dimensional space makes angles 45^∘ and 120^∘ with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z-axis, then θ equals