Comprehension: In a triangle ABC, the coordinates of the vertices are A= (1, 2), B = (4, 5) and C = (2, 8). The points D, E and F are the mid-points of the sides BC, CA and AB respectively. The points P, Q and R are the feet of the altitude from A, B and C to the sides BC, CA and AB respectively.

Now, answer the following questions

1. The equation of the line parallel to DE and passing through A is
A) 2x – y = 0 B) 3x + 2y = 7
C) 2x + 5y = 12 D) None

2. The equation of the line AP is
A) 3x + 2y = 7 B) 2x − 3y = 5
C) 2x − 3y + 4 = 0 D) 3x + 2y = 8

3. The equation of the median CF is
A) 9x + y = 10 B) 18x + 2y = 15
C) 2x − 7y = 10 D) 9x + y = 26

Comprehension: The vertices of a triangle are the points A(p, p tanα) B(q, q tan β) and C(r, r tan γ) where α + β + γ = π. The circumcentre is at the origin and the orthocenter is H(x, y).

4. The coordinates of the vertex ‘A’ If ‘R’ is the circumradius of the ΔABC.
A) (R sin α, R cos α)
B) (R cos α, R sin α)
C) (R cos α/2, R sin α/2)
D) none of these

Comprehension: A(3, 0), B(6, 0) are two fixed points and U (α, β) is a variable point on the plane. AU and BU meet the y-axis at C and D respectively, and AD meets OU at V.

5. The equation of CV always passes through
the point for any position of U in the plane
A) (3, 0) B) (2, 0) C) (6, 0) D) (4, 0)

Comprehension: A cylinder is inscribed in a sphere of radius R. The volume V of the cylinder is written as V = f(x), where x is the height of the cylinder.

6. The function V/x represents
A) a straight line B) an increasing function
C) a circle D) a decreasing function

Comprehension: The complex number z satisfying the condition z-25i| ≤ 15 represented by the points in side and on the circle of radius 15, centre (0, 25) and the complex number having least positive
argument and maximum positive argument in this regions are the points of contacts of tangents drawn from the origin to the circle.

7. The complex number z having least positive argument
A) -12 + 6i B) 12 + 6i C) -12-6i D) none

8. The complex number z having least modulus
A) 40i B) 20i C) 25i D) 10i

Comprehension: Suppose z and w be two complex numbers. Such that |z| ≤ 1, |w| ≤ 1 and |z + iw| = |z – iw| = 2. Use the result |z|2 = z z and |z + w| ≤ |z| + |w|.

9. Number of complex numbers z satisfying the above conditions is
A) 1 B) 2 C) 4 D) none.

KEY
1) D 2) C 3) D 4) B 5) B 6) D 7) D 8) D 9) D.

1. The diagonals of parallelogram PQRS are along the lines x + 3y = 4 and 6x – 2y = 7. Then PQRS can be a
(A) rectangle (B) square
(C) cyclic quadrilateral (D) rhombus

2. The equation of the line perpendicular to x + y + 1 = 0 can be
(A) y − x +1 = 0 (B) y − x −1 = 0
(C) y − 2x + 2 = 0 (D) y − x − 2 = 0

3. If the co-ordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will always have rational co-ordinates
(A) centroid (B) incentre
(C) circumcentre (D) orthocentre

4. The diagonal of a parallelogram PQRS are along the lines x + 3y = 4 and 6x − 2y = 7 then PQRS must be a
A) rectangle B) square
C) cyclic quadrilateral D) rhombus

5. The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal
A) is linear
B) is homogeneous of first degree
C) has separable variable
D) is second order

6. The numbers 1, 4, 16 can be three terms, not necessarily consecutive, of
A) no A.P. B) only one G.P.
C) infinite number of A.Ps.
D) infinite number of G.Ps.

KEY: 1) BD 2) ABD 3) ACD 4) BD 5) AB 6) CD