The first question ---**Objective Mathematics XIIB-1** **Practice Sheet-3** (Miscellaneous models) **Section - A** **More than One correct Answer Type Questions** **Question 1:** The equation e^x - ax - b = 0 has A) one real root if a ≤ 0 B) one real root if b > 0, a ≤ 0 C) two real roots if a > 0, a ln a ≥ a - b D) no real root if a > 0, a log a < a - b **Question 2:** If f(x) and g(x) are two functions such that f(x+y) = f(x) g(y) + g(x) f(y) then | f(α) g(α) f(α+θ) | | f(β) g(β) f(β+θ) | | f(γ) g(γ) f(γ+θ) | is independent of A) α B) β C) γ D) θ **Question 3 (Partially Visible):** for x ∈ R, |x^2 + 6x + 8| = |x^2 + 4x + 5| + |2x + 3| is.

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Let's analyze the equation e^x minus ax minus b equals zero. We need to determine when this equation has one, two, or no real roots. The key insight is to consider this as finding intersections between the exponential function e^x and the linear function ax plus b. When a is less than or equal to zero, the linear function has a non-positive slope, ensuring exactly one intersection point with the always-increasing exponential curve. Now let's analyze this mathematically. We define f of x equals e to the x minus ax minus b. Taking the derivative, we get f prime of x equals e to the x minus a. When a is less than or equal to zero, since e to the x is always positive, f prime of x is always positive. This means f of x is strictly increasing. A strictly increasing continuous function can cross the x-axis at most once, guaranteeing exactly one real root. Let's examine the behavior at the limits. As x approaches negative infinity, e to the x approaches zero. When a is less than or equal to zero, negative ax is non-negative, so f of x approaches negative infinity or negative b. As x approaches positive infinity, e to the x grows without bound, making f of x approach positive infinity. Since f is continuous and strictly increasing, going from negative values to positive values, it must cross the x-axis exactly once by the Intermediate Value Theorem. Let's examine the other options. Option B adds the condition b greater than zero, making it more restrictive than option A but still correct. Options C and D deal with the case when a is positive, where the linear function has positive slope. Option C describes when there are two intersections, and option D when there are none. However, option A provides the most general and fundamental condition. When a is less than or equal to zero, we always get exactly one real root regardless of the value of b. In conclusion, the equation e to the x minus ax minus b equals zero has one real root if a is less than or equal to zero. This is because when a is non-positive, the derivative f prime of x equals e to the x minus a is always positive, making f of x strictly increasing. A strictly increasing continuous function that ranges from negative infinity to positive infinity must cross the x-axis exactly once. Therefore, the answer is A.