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NCERT Exemplar · Class 12 Mathematics Linear Programming

45 questions · 45 still being checked

EXERCISE 12.3 41–45 (part 5 of 5)

  1. Fill in the blanks in each of the Exercises $\displaystyle 35$ to 41.

    Exercise 41

    The feasible region for an LPP is always a ____ polygon.

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    NCERT’s answer
    Convex
    convexEach constraint has the form \(\displaystyle a_i x + b_i y \le c_i\). Take \(\displaystyle P(x_1, y_1)\) and \(\displaystyle Q(x_2, y_2)\) in the region, and a point \(\displaystyle R\) on segment \(\displaystyle PQ\):\[R = \big(t x_1 + (1-t)x_2,\; t y_1 + (1-t)y_2\big), \quad 0 \le t \le 1 \]\[a_i x_R + b_i y_R = t\,(a_i x_1 + b_i y_1) + (1-t)\,(a_i x_2 + b_i y_2) \]\[a_i x_R + b_i y_R \le t c_i + (1-t)c_i = c_i \]NCERT_Solution_Class12_Maths_Exemplar_Ch12_Ex12-3_Q41\(\displaystyle R\) satisfies every constraint, so the whole segment \(\displaystyle PQ\) lies in the region.
  2. State whether the statements in Exercises $\displaystyle 42$ to $\displaystyle 45$ are True or False.

    Exercise 42

    If the feasible region for a LPP is unbounded, maximum or minimum of the objective function Z=ax+by\displaystyle \mathrm{Z}=a x+b y may or may not exist.

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    NCERT’s answer
    True
    True: an unbounded region can give a minimum but no maximum.Take \(\displaystyle Z = x + y\) on \(\displaystyle x \ge 0,\ y \ge 0,\ x + y \ge 2\).\[Z(2, 0) = 2, \qquad Z(0, 2) = 2 \]\[Z = x + y \ge 2 \quad \text{throughout the region} \]\[Z(t, 0) = t \quad (t \ge 2) \]NCERT_Solution_Class12_Maths_Exemplar_Ch12_Ex12-3_Q42The minimum \(\displaystyle 2\) exists; \(\displaystyle Z\) grows without limit, so there is no maximum.

  3. Exercise 43

    Maximum value of the objective function Z=ax+by\displaystyle \mathrm{Z}=a x+b y in a LPP always occurs at only one corner point of the feasible region.

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    NCERT’s answer
    False
    False: the maximum can occur at two adjacent corner points, and then at every point of the segment joining them.Take \(\displaystyle Z = x + y\) on \(\displaystyle x \ge 0,\ y \ge 0,\ x + y \le 4\).\[Z(0, 0) = 0, \qquad Z(4, 0) = 4, \qquad Z(0, 4) = 4 \]\[Z(t, 4-t) = t + (4 - t) = 4 \quad (0 \le t \le 4) \]NCERT_Solution_Class12_Maths_Exemplar_Ch12_Ex12-3_Q43

  4. Exercise 44

    In a LPP, the minimum value of the objective function Z=ax+by\displaystyle \mathrm{Z}=a x+b y is always 0\displaystyle 0 if origin is one of the corner point of the feasible region.

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    NCERT’s answer
    False
    False: the origin gives \(\displaystyle Z = 0\), but another corner point can give a smaller value.Take \(\displaystyle Z = x - y\) on \(\displaystyle x \ge 0,\ y \ge 0,\ x + y \le 4\).\[Z(0, 0) = 0, \qquad Z(4, 0) = 4, \qquad Z(0, 4) = -4 \]NCERT_Solution_Class12_Maths_Exemplar_Ch12_Ex12-3_Q44The minimum is \(\displaystyle -4\), not \(\displaystyle 0\).

  5. Exercise 45

    In a LPP, the maximum value of the objective function Z=ax+by\displaystyle \mathrm{Z}=a x+b y is always finite.

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    NCERT’s answer
    True
    True: a maximum, when it exists, occurs at a corner point \(\displaystyle (x_0, y_0)\) of the feasible region.\[Z_{\max} = a x_0 + b y_0 \]\[a,\ b,\ x_0,\ y_0 \in \mathbb{R} \ \Rightarrow\ Z_{\max} \in \mathbb{R} \]So the maximum value is a finite number; if \(\displaystyle Z\) is unbounded, it has no maximum at all.