Q1. 1. Answer the following questions : 6 × 5 = 30
() (a) Suppose the x and y-axes in the plane R2 are rotated counterclockwise 00 45°, so that the new x' and y'-axes are along the line y = x and the line y = -x respectively. Find the change-of-basis matrix P and the coordinates of the point A(5,6) under the given rotation. मान लें कि R2 समतल में x एवं y-अक्षों को वामवर्त रूप से 45° घुमाया जाता है ताकी नए x' और y'-अक्ष, y = x और y = -x के ऊपर क्रमशः पड़ते हैं। चेंज-ऑफ-बेसीस मैट्रीक्स P पता करें और दिए गए घूर्णन के अंतर्गत बिन्दु A(5,6) के कोआर्डिनेट पता करें。
() (b) Evaluate the limit lim_{(x,y)→(2,-4)}(y+4)/(x2 y - x y + 4x2 - 4x) where y ≠ -4 and x ≠ 0 and x ≠ 1. The expression factors as x(x-1)(y+4) in the denominator, so the limit simplifies to lim_{(x,y)→(2,-4)} 1/[x(x-1)] = 1/2.
() (c) A force F = 2i + j - 3k is applied to a spacecraft with velocity v = 3i - j. Express F as a sum of a vector parallel to v and a vector orthogonal to v.
() (d) If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, find dy/dx.
