Exercise 11.1 Class 12 Math Solutions - Unit 11 Vector Valued Functions, Punjab Board
Step-by-step solutions to Exercise 11.1, Unit 11: Vector Valued Functions and Their Derivations, for Class 12 Punjab Board students. Covers distinguishing between scalar and vector-valued functions of a single variable, constructing parametric vector functions for physical paths such as projectile motion and drone flight trajectories, evaluating component domain intersections to find the domain and range of vector functions, and applying key identification criteria, worked out question by question.
Exercise Overview
Exercise 11.1 introduces the foundational concepts of Unit 11: Vector Valued Functions and Their Derivations in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum. In earlier calculus units, students worked almost exclusively with scalar-valued functions (y = f(x)), where a single real number input generates a single real number output ((f: \mathbb{R} \to \mathbb{R})). Unit 11 expands mathematical analysis to vector-valued functions (also called vector functions), where a single real scalar parameter (typically time (t)) is mapped to a vector in 2D space ((\mathbf{r}: \mathbb{R} \to \mathbb{R}^2)) or 3D space ((\mathbf{r}: \mathbb{R} \to \mathbb{R}^3)).
A vector-valued function in three-dimensional Cartesian space is expressed in component form as: \[ \mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k} = \langle f(t), g(t), h(t) \rangle \] where (f(t)), (g(t)), and (h(t)) are real-valued scalar component functions. Geometrically, as the parameter (t) varies over its domain, the terminal point of the position vector (\mathbf{r}(t)) traces out a continuous curve in space known as a space curve.
A crucial analytical skill developed in Exercise 11.1 is finding the Domain of a Vector-Valued Function. The domain of (\mathbf{r}(t)) is defined as the largest set of real numbers for which all component functions (f(t)), (g(t)), and (h(t)) are simultaneously defined and real. Mathematically: \[ \text{Dom}(\mathbf{r}) = \text{Dom}(f) \cap \text{Dom}(g) \cap \text{Dom}(h) \] Students systematically determine individual domain restrictions-such as non-negative radicands for even roots ((g(t) \ge 0)), non-zero denominators, and positive arguments for logarithms-and compute their set intersection on the real number line.
Exercise 11.1 also emphasizes real-world vector modeling, specifically constructing position vector functions for Projectile Motion under uniform gravitational acceleration ((\mathbf{r}(t) = (v_0 \cos\alpha \cdot t)\mathbf{i} + (v_0 \sin\alpha \cdot t - \frac{1}{2}gt^2)\mathbf{j})) and modeling 3D spatial trajectories such as the Flight Path of a Drone navigating complex coordinate routes. For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 11.1 provides high-frequency Section B short questions and conceptual MCQs. These step-by-step solutions provide rigorous component domain evaluations, intersection intervals, and parametric vector formulations for every textbook exercise problem.