Q.For the linear programming problem (LPP), the objective function is $Z=4x+3y$ and the feasible region determined by a set of constraints is shown in the graph: (Note: The figure is not to scale.) Which of the following statements is true?
(A) Maximum value of $Z$ is at $R(40,0)$.
(B) Maximum value of $Z$ is at $Q(30,20)$.
(C) Value of $Z$ at $R(40,0)$ is less than the value at $P(0,40)$.
(D) The value of $Z$ at $Q(30,20)$ is less than the value at $R(40,0)$.

Showing the 12 most recent of 14 on this concept.

1. GUJCET 2025Set 031 markMCQ
   Q.The coordinates of the corner points of the bounded feasible region are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(60, 0)$. The maximum of the objective function $z = 40x + 30y$ is _____. (A) 2000 (B) 3400 (C) 2400 (D) 3000

   ›Reveal solutionSolution

   For a bounded LP region the optimum sits at a corner; test all corners.

   With $z=40x+30y$:

   • $(0,0)\to0$
   • $(0,40)\to1200$
   • $(20,40)\to800+1200=2000$ …

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2. GUJCET 2023Set 091 markMCQ
   Q.The coordinates of the corner points of the bounded feasible region are $(0, 10)$, $(5, 5)$, $(15, 15)$ and $(0, 20)$. The maximum of the objective function $Z = 10x + 20y$ is : (A) $600$ (B) $550$ (C) $400$ (D) $450$

   ›Reveal solutionSolution

   [!TLDR]

   $[L]=M^1L^2T^{-2}A^{-2}$.

   Concept

   Self inductance relates to stored magnetic energy $U=\tfrac12 L I^2$ (NCERT/GSEB electromagnetic induction), so $L=\dfrac{2U}{I^2}$.

   Solution

   Energy has dimensions $[ML^2T^{-2}]$ and current $[A]$: …

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3. GUJCET 2019Set 171 markMCQ
   Q.The coordinates of the corner points of the bounded feasible region are $(10, 0)$, $(2, 4)$, $(1, 5)$ and $(0, 8)$. The maximum of objective function $z = 60x + 10y$ is $\underline{\hspace{2cm}}$. (A) $800$ (B) $600$ (C) $700$ (D) $110$

   ›Reveal solutionSolution

   Checking corner points, $z=60x+10y$ is largest at $(10,0)$ giving $600$.

   Concept: The maximum of a linear objective on a bounded feasible region occurs at a corner.

   • $(10,0):600$ …

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4. GUJCET 2022Set 081 markMCQ
   Q.Corner points of the feasible region of objective function $Z=3x+9y$ of a linear programming problem are $(0,10)$, $(5,5)$, $(15,15)$ and $(0,20)$. Minimum value of $Z$ is ______. (A) $70$ (B) $90$ (C) $50$ (D) $60$

   ›Reveal solutionSolution

   In LP the optimum of a linear objective occurs at a corner point — just test them all.

   Concept. Evaluate $Z=3x+9y$ at every corner and pick the smallest.

   Solution.

   • $(0,10):\;0+90=90$
   • $(5,5):\;15+45=60$ …

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5. GUJCET 2026Set x1 markMCQ
   Q.The coordinates of the corner points of the bounded feasible region are $(0,10), (5,5), (15,15), (0,20)$. The minimum of the objective function $z = 3x + 9y$ is ______ (A) 180 (B) 30 (C) 90 (D) 60

   ›Reveal solutionSolution

   [!TLDR] The minimum of $z=3x+9y$ over the corner points is $60$ at $(5,5)$.

   Concept

   In linear programming, a linear objective attains its optimum over a bounded feasible region at a corner (vertex). So evaluate $z$ at every corner point and choose the smallest for the minimum.

   Solution …

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6. GUJCET 2024Set 131 markMCQ
   Q.The coordinates of the corner points of the bounded feasible region are $(0, 6), (3, 3), (9, 9), (0, 12)$. The maximum of the objective function $z = 6x + 12y$ is : (A) $166$ (B) $152$ (C) $144$ (D) $162$

   ›Reveal solutionSolution

   Evaluate $z=6x+12y$ at every corner point; the maximum is the answer.

   Steps.

   • $(0,6):\ 0+72 = 72$
   • $(3,3):\ 18+36 = 54$
   • $(9,9):\ 54+108 = 162$ …

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7. GSEB Higher Secondary Certificate (HSC) Examination 2026Set ANNUAL1 markMCQ
   Q.For a linear programming problem, the objective function $Z=510x+675y$ has the corner points of the bounded feasible region $(0,0)$, $(300,0)$, $(180,120)$ and $(0,240)$. The maximum value of $Z$ is ____.

   (a) 1,72,800

   (b) 1,62,000

   (c) 1,53,000

   (d) 1,70,000

   ›Reveal solutionSolution

   Evaluate $Z$ at each corner point and pick the largest.

   $Z=510x+675y$: $(0,0)\to0$; $(300,0)\to153000$; $(180,120)\to91800+81000=172800$; $(0,240)\to162000$.

   …

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8. GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQ
   Q.In a linear programming problem, $Z = px + qy$, $p, q > 0$. At the corner points $(0,10)$ and $(5,5)$ of the bounded feasible region, the values of $Z$ are $90$ and $60$ respectively. Then the relation between $p$ and $q$ = ____.

   (a) $p = 3q$

   (b) $q = 2p$

   (c) $q = 3p$

   (d) $p = 2q$

   ›Reveal solutionSolution

   Plug both corner points into $Z=px+qy$ to get two equations in $p,q$, then solve.

   At $(0,10)$: $10q=90 \Rightarrow q=9$.

   …

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9. GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQ
   Q.The corner points of the feasible region of the objective function $Z = -50x + 20y$ are $(0,5)$, $(0,3)$, $(1,0)$ and $(6,0)$. Then the minimum value of $Z$ = ____.

   (a) $-100$

   (b) $-300$

   (c) $-200$

   (d) $-500$

   ›Reveal solutionSolution

   Evaluate $Z$ at every corner point of the bounded feasible region; the minimum is the smallest value.

   $Z(0,5)=0+100=100$

   $Z(0,3)=0+60=60$

   $Z(1,0)=-50+0=-50$

   $Z(6,0)=-300+0=-300$

   …

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10. GSEB Higher Secondary Certificate (HSC) Examination 2024Set ANNUAL1 markMCQ
    Q.For linear programming problem, the objective function is $Z = 3x + 2y$. If the corner points of the bounded feasible region are $(12,0), (4,2), (1,5)$ and $(0,10)$, then the maximum value of $Z$ is ______.

    (a) $36$

    (b) $46$

    (c) $13$

    (d) $56$

    ›Reveal solutionSolution

    In an LP problem the optimum of a linear objective function always occurs at a corner point of the feasible region.

    Evaluate $Z=3x+2y$ at each corner point:

    $(12,0): Z=36$

    $(4,2): Z=12+4=16$

    $(1,5): Z=3+10=13$ …

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11. GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
    Q.The corner points of the feasible region determined by the following system of linear inequalities $2x + y \leq 10$, $x + 3y \leq 15$, $x, y \geq 0$ are $(0, 0)$, $(5, 0)$, $(3, 4)$ and $(0, 5)$. Let $Z = qx + py$ where $p, q > 0$, condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3, 4)$ and $(0, 5)$ is ___.

    (a) $q = p$

    (b) $q = 2p$

    (c) $q = 3p$

    (d) $p = 3q$

    ›Reveal solutionSolution

    If the maximum of $Z$ occurs at two adjacent corner points, $Z$ is equal there.

    $Z = qx+py$. Set $Z(3,4) = Z(0,5)$:

    …

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12. GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
    Q.The corner points of feasible region determined by the system of linear constraints are $(2, 72)$, $(15, 20)$, $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. Minimum of $Z$ occurs at ___.

    (a) $(2, 72)$

    (b) $(15, 20)$

    (c) $(40, 15)$

    (d) $(0, 0)$

    ›Reveal solutionSolution

    Evaluate the objective at each corner point and pick the smallest.

    $Z(2,72) = 12+216 = 228.$

    $Z(15,20) = 90+60 = 150.$

    $Z(40,15) = 240+45 = 285.$

    …

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