27 to 29 please answer

Q27. The distanced between the two parallel lines is 1 unit. A point 'A' is chosen to lie between the lines at a distance d form one of them. Triangle ABC is equilateral with B on one line and C on the other parallel line. The length of the side of the equilateral triangle is :


    $\left(\mathrm{A}\right)\frac{2}{3}\sqrt{{\mathrm{d}}^2 + \mathrm{d} +1} \left(\mathrm{B}\right) 2\sqrt{\frac{{\mathrm{d}}^2- \mathrm{d} + 1}{3}} \left(\mathrm{C}\right) 2\sqrt{{\mathrm{d}}^2- \mathrm{d} + 1} \left(\mathrm{D}\right) \sqrt{{\mathrm{d}}^2 - \mathrm{d} +1}$

Q28. Given A (0,0) and B (x, y) with x $\in$(0, 1) and y > 0. Let the slope of the line AB equals m ₁ point Clines on the line x = 1 such that the slope of BC equals m₂ where 0 < m₂ < m ₁. If the area of the triangle ABC can be expressed as (m₁– m₂ ​) f(x) , then the largest possible value of f(x) is :

    (A) 1                                 (B) 1/2                               (C) 1/4                                  (D) 1/8



Q29. If the vertices P and Q of a triangle PQR are given by (2,5) and (4, – 11) respectively, and the point R moves along the line N : 9x + 7y + 4 = 0 , then the locus of the centroid of the triangle PQR is a straight: line parallel to :

    (A) PQ                             (B) QR                                (C) RP                                     (D) N