answer:Let's denote the equation of the line parallel to 2x - y - 1 = 0 as 2x - y + c = 0. Since the line must pass through the point (-1, 1), we substitute these coordinates into the equation: 2(-1) - 1 + c = 0 -2 - 1 + c = 0 c = 3 Therefore, the equation of the line passing through (-1, 1) and parallel to 2x - y - 1 = 0 is 2x - y + 3 = 0. So the final answer is boxed{2x - y + 3 = 0}.

answer:1. **Understanding the Problem**: - We are given a rectangular prism (ABCD - A_1B_1C_1D_1). - (P) is a moving point on the edge (AB). - We need to find the minimum value of the dihedral angle between plane (PDB_1) and plane (AD D_1A_1). 2. **Geometric Construction**: - Consider extending (B_1P) to intersect (A_1A) at point (Q). Then, (Q) lies on both planes (PDB_1) and (AD D_1A_1), making (DQ) the line of intersection (common line) between the two planes. 3. **Properties of Planes and Perpendiculars**: - Since (PA) is perpendicular to the plane (AD D_1A_1), let’s establish a perpendicular from (A) to (DQ). Denote this perpendicular as (AH), where (H) is on (DQ). - Connect (P) and (H); then, (angle PHA) is the dihedral angle’s plane angle that we seek. 4. **Setting Up Coordinates**: - Let the edge lengths of the rectangular prism be 1 unit. Assume (PA = x). - Using similar triangles and geometric properties, compute distances: [begin{array}{l} AQ = frac{x}{1 - x}, DQ = frac{sqrt{1 - 2x + 2x^2}}{1 - x}, AH = frac{x}{sqrt{1 - 2x + 2x^2}}. end{array}] 5. **Calculate Tangent of the Angle**: - The tangent of (angle PHA) is given by: [ tan angle PHA = frac{PA}{AH} = sqrt{1 - 2x + 2x^2}. ] Simplifying, we get: [ tan angle PHA = sqrt{2left(x - frac{1}{2}right)^2 + frac{1}{2}}. ] 6. **Finding the Minimum Tangent Value**: - The minimum value of (sqrt{2left(x - frac{1}{2}right)^2 + frac{1}{2}}) occurs when (left(x - frac{1}{2}right) = 0). Hence, (x = frac{1}{2}). begin{align*} tan angle PHA &= sqrt{2left(0right)^2 + frac{1}{2}} = sqrt{frac{1}{2}} = frac{sqrt{2}}{2}. end{align*} Conclusion: The minimum value of the dihedral angle’s tangent between the given planes is: [ boxed{arctanfrac{sqrt{2}}{2}}. ]

answer:1. **Define Variables**: Let ( x ) be the number of three-point baskets and ( y ) the number of two-point baskets. The total number of baskets made is ( 7 ). 2. **Formulate the Equation**: The relationship between the three-point and two-point baskets can be expressed as: [ x + y = 7 ] The total points ( P ) scored can be given by: [ P = 3x + 2y ] 3. **Substitute and Simplify**: Using the equation ( y = 7 - x ) and substituting into the points equation: [ P = 3x + 2(7 - x) = 3x + 14 - 2x = x + 14 ] 4. **Calculate Possible Scores**: Varying ( x ) from 0 to 7, we get ( P ) as: - ( x = 0 ): ( P = 14 ) - ( x = 1 ): ( P = 15 ) - ( x = 2 ): ( P = 16 ) - ( x = 3 ): ( P = 17 ) - ( x = 4 ): ( P = 18 ) - ( x = 5 ): ( P = 19 ) - ( x = 6 ): ( P = 20 ) - ( x = 7 ): ( P = 21 ) 5. **Count Distinct Scores**: Distinct scores are 14, 15, 16, 17, 18, 19, 20, 21, totaling 8 different scores. 6. **Conclusion**: The number of distinct possible scores is (8). The correct answer is boxed{D. 8}.

answer:Since f(1+x) = f(1-x), we know that the function is symmetric about x=1. When x < 1, 2-x > 1, So, f(x) = f(2-x) = ln[(2-x) + 1] = ln(3-x). Therefore, the answer is boxed{ln(3-x)}.