answer:To find angle AOB in triangle PAB, where PAB is formed by three tangents to circle O and angle APB = 40^circ, we proceed as follows: 1. **Identify Angles in Triangle ABO**: - We start by noting that angle AOB = 180^circ - angle BAO - angle ABO. This is because the sum of angles in any triangle is 180^circ, and angle AOB is the exterior angle for triangle ABO. 2. **Bisecting Angles**: - Recognizing that AO bisects angle BAT (this can be seen by drawing radii from O to AB and AT, forming two congruent right triangles), we have angle BAO = frac{angle BAT}{2}. Similarly, angle ABO = frac{angle ABR}{2}. 3. **Relating Angles**: - We also know that angle BAT = 180^circ - angle BAP and angle ABR = 180^circ - angle ABP. This is due to the fact that the sum of angles at a point is 360^circ, and considering the right angles formed at the points of tangency. 4. **Calculating angle AOB**: - Substituting the expressions for angle BAO and angle ABO into our formula for angle AOB, we get: [ angle AOB = 180^circ - frac{angle BAT}{2} - frac{angle ABR}{2} = 180^circ - frac{180^circ - angle BAP}{2} - frac{180^circ - angle ABP}{2} ] - Simplifying, this becomes: [ angle AOB = frac{angle BAP + angle ABP}{2} ] 5. **Using Triangle ABP**: - In triangle ABP, we know that angle BAP + angle ABP + angle APB = 180^circ. Given angle APB = 40^circ, we find angle BAP + angle ABP = 180^circ - 40^circ = 140^circ. 6. **Final Calculation**: - Substituting the sum of angle BAP and angle ABP into our formula for angle AOB, we get: [ angle AOB = frac{140^circ}{2} = 70^circ ] Therefore, the measure of angle AOB is boxed{70^circ}.

answer:To find the solution set for the inequality x^2 - 3x + 2 < 0, we can factor the quadratic expression on the left-hand side. The factored form of the quadratic is (x - 2)(x - 1) < 0. For this inequality to hold, one of the following conditions must be met: 1. (x - 2) < 0 and (x - 1) > 0 ...① 2. (x - 2) > 0 and (x - 1) < 0 ...② Solving condition ①, we get x < 2 and x > 1, which implies 1 < x < 2. Solving condition ②, we end up with x > 2 and x < 1, which is impossible, as no x can simultaneously be greater than 2 and less than 1. Therefore, the only possible solution for the inequality is the interval where 1 < x < 2. Hence, the correct answer is boxed{D: (1, 2)}.

answer:Let's denote the cost price of the apple as C. According to the information given, the seller loses 1/6th of the cost price when he sells the apple for Rs. 20. This means that the selling price is 1/6th less than the cost price. So, we can write the following equation: Selling Price (SP) = Cost Price (CP) - (1/6 * CP) Given that the Selling Price (SP) is Rs. 20, we can substitute SP with 20 in the equation: 20 = CP - (1/6 * CP) To solve for CP, we first need to get rid of the fraction. To do this, we can multiply both sides of the equation by 6 to eliminate the denominator: 6 * 20 = 6 * CP - CP 120 = 6CP - CP Now, combine like terms: 120 = 5CP Next, divide both sides by 5 to solve for CP: 120 / 5 = CP CP = 24 Therefore, the cost price of the apple is Rs. boxed{24} .

answer:Since f^{-1}(x) = 2^{x+1}, then f(x) = log_{2}x - 1. Therefore, f(1) = -1. Hence, the correct option is boxed{D}.