This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. 1. Prove that: (^3 A - ^3 A)/( A - A) = 1 + (1)/(2) 2A Step 1: Start with the Left Hand Side (LHS). LHS = (^3 A - ^3 A)/( A - A) Step 2: Apply the algebraic identity a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here, a = A and b = A. LHS = (( A - A)(^2 A + A A + ^2 A))/( A - A) Step 3: Cancel out the common term ( A - A) from the numerator and denominator. LHS = ^2 A + A A + ^2 A Step 4: Use the trigonometric identity ^2 A + ^2 A = 1. LHS = 1 + A A Step 5: Use the double angle identity 2A = 2 A A, which implies A A = (1)/(2) 2A. LHS = 1 + (1)/(2) 2A Step 6: Compare with the Right Hand Side (RHS). RHS = 1 + (1)/(2) 2A Since LHS = RHS, the identity is proven. (^3 A - ^3 A)/( A - A) = 1 + (1)/(2) 2A 2. Solve: 4^2 = 3 (0^ ≤ ≤ (^c)/(2)) Step 1: Isolate ^2 . 4^2 = 3 ^2 = (3)/(4) Step 2: Take the square root of both sides. = ± sqrt((3)/(4)) = ± sqrt(3)2 Step 3: Consider the given range for . The range is 0^ ≤ ≤ (^c)/(2), which means 0^ ≤ ≤ 90^. In this first quadrant, the cosine function is positive. Therefore, we only consider the positive value: = sqrt(3)2 Step 4: Find the value of . We know that 30^ = sqrt(3)2. In radians, 30^ = ()/(6) radians. So, = 30^ or = ()/(6). The value of is 30^ or ()/(6). 3. The position vectors of the points A and B are 2i + 3j and -3i + 8j respectively. If M divides the line segment AB externally in the ratio 5:2, then find the position vector of the point M. Step 1: Write down the given position vectors and the ratio. Position vector of A: a = 2i + 3j Position vector of B: b = -3i + 8j Ratio of external division m:n = 5:2. So, m=5 and n=2. Step 2: Use the formula for the position vector of a point M that divides a line segment AB externally in the ratio m:n. m = mb - nam - n Step 3: Substitute the given values into the formula. m = 5(-3i + 8j) - 2(2i + 3j)5 - 2 Step 4: Simplify the expression. m = (-15i + 40j) - (4i + 6j)3 m = -15i + 40j - 4i - 6j3 m = (-15 - 4)i + (40 - 6)j3 m = -19i + 34j3 m = -(19)/(3)i + (34)/(3)j The position vector of point M is -(19)/(3)i + (34)/(3)j. 4. In a series, the first quartile is 45 and the quartile deviation is 15. Find the coefficient of quartile deviation. Step 1: Write down the given values. First quartile (Q_1) = 45 Quartile Deviation (QD) = 15 Step 2: Use the formula for Quartile Deviation to find the third quartile (Q_3). QD = (Q_3 - Q_1)/(2) 15 = (Q_3 - 45)/(2) 15 × 2 = Q_3 - 45 30 = Q_3 - 45 Q_3 = 30 + 45 Q_3 = 75 Step 3: Use the formula for the coefficient of quartile deviation. Coefficient of QD = (Q_3 - Q_1)/(Q_3 + Q_1) Step 4: Substitute the values of Q_1 and Q_3 into the formula. Coefficient of QD = (75 - 45)/(75 + 45) Coefficient of QD = (30)/(120) Coefficient of QD = (1)/(4) Coefficient of QD = 0.25 The coefficient of quartile deviation is 0.25.