Linear Programming

1. A.
   unbounded in $1^{st}$ quadrant
2. B.
   bounded in $1^{st}$ quadrant
3. C.
   unbounded in $2^{nd}$ quadrant
4. D.
   bounded in $2^{nd}$ quadrant

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1. A.
   $p = 2q$
2. B.
   $p = 3q$
3. C.
   $q = 3p$
4. D.
   $q = 2p$

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1. A.
   200
2. B.
   300
3. C.
   240
4. D.
   120

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1. A.
   a feasible function
2. B.
   an objective function
3. C.
   an optimal function
4. D.
   a constraint

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1. A.
   $8$
2. B.
   $29$
3. C.
   $35$
4. D.
   $43$

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1. A.
   $x + y \leq 5$, $x + 3y \leq 9$
2. B.
   $x + y \leq 5$, $x + 3y \geq 9$
3. C.
   $x + y \geq 5$, $x + 3y \leq 9$
4. D.
   $x + y \geq 5$, $3x + y \leq 9$

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1. A.
   $0$
2. B.
   $1$
3. C.
   $2$
4. D.
   Any natural number

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   Z is minimum at $S\left(\dfrac{18}{7},\ \dfrac{2}{7}\right)$
2. B.
   Z is maximum at $R\left(\dfrac{7}{2},\ \dfrac{3}{4}\right)$
3. C.
   (Value of Z at P) > (Value of Z at Q)
4. D.
   (Value of Z at Q) < (Value of Z at R)

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1. A.
   Z is maximum at (2, 72), minimum at (15, 20)
2. B.
   Z is maximum at (15, 20) minimum at (40, 15)
3. C.
   Z is maximum at (40, 15), minimum at (15, 20)
4. D.
   Z is maximum at (40, 15), minimum at (2, 72)

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1. A.
   lies in the shaded unbounded region.
2. B.
   lies in $\triangle$ AOB.
3. C.
   does not exist.
4. D.
   lies in the combined region of $\triangle$ AOB and unbounded shaded region.

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1. A.
   ABC
2. B.
   AOEC
3. C.
   CED
4. D.
   Open unbounded region BCD

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1. A.
   It will only have a maximum value.
2. B.
   It will only have a minimum value.
3. C.
   It will have both maximum and minimum values.
4. D.
   It will have neither maximum nor minimum value.

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1. A.
   The objective function maximizes the difference of the profit earned from products X and Y.
2. B.
   The objective function measures the total production of products X and Y.
3. C.
   The objective function maximizes the combined profit earned from selling X and Y.
4. D.
   The objective function ensures the company produces more of product X than product Y.

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1. A.
   a = b
2. B.
   a = 3b
3. C.
   b = 6a
4. D.
   3a = 2b

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true but Reason (R) is false.
4. D.
   Assertion (A) is false but Reason (R) is true.

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   logarithmic function
2. B.
   linear function
3. C.
   quadratic function
4. D.
   exponential function

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1. A.
   an unbounded region
2. B.
   an optimal region
3. C.
   a bounded region
4. D.
   a feasible region

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1. A.
   feasible solutions
2. B.
   constraints
3. C.
   optimal solutions
4. D.
   infeasible solutions

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1. A.
   50
2. B.
   110
3. C.
   120
4. D.
   170

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1. A.
   0
2. B.
   1
3. C.
   2
4. D.
   3

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1. A.
   $x + 2y \le 76$, $2x + y \ge 104$, $x, y \ge 0$
2. B.
   $x + 2y \le 76$, $2x + y \le 104$, $x, y \ge 0$
3. C.
   $x + 2y \ge 76$, $2x + y \le 104$, $x, y \ge 0$
4. D.
   $x + 2y \ge 76$, $2x + y \ge 104$, $x, y \ge 0$

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1. A.
   Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   whole xy-plane except the points lying on the line $3x + 5y = 7$.
2. B.
   whole xy-plane along with the points lying on the line $3x + 5y = 7$.
3. C.
   open half plane containing the origin except the points of line $3x + 5y = 7$.
4. D.
   open half plane not containing the origin.

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1. A.
   $(-2, 4)$
2. B.
   $(3, 2)$
3. C.
   $(-5, 6)$
4. D.
   $(4, 2)$

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1. A.
   $a = 9, b = 1$
2. B.
   $a = 5, b = 2$
3. C.
   $a = 3, b = 5$
4. D.
   $a = 5, b = 3$

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1. A.
   1
2. B.
   2
3. C.
   3
4. D.
   infinite

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1. A.
   z is maximum at (2, 72), minimum at (15, 20)
2. B.
   z is maximum at (15, 20), minimum at (40, 15)
3. C.
   z is maximum at (40, 15), minimum at (15, 20)
4. D.
   z is maximum at (40, 15), minimum at (2, 72)

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1. A.
   2
2. B.
   3
3. C.
   4
4. D.
   5

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1. A.
   $x + 2y \geq 4$, $x + y \leq 3$, $x \geq 0$, $y \geq 0$
2. B.
   $x + 2y \leq 4$, $x + y \leq 3$, $x \geq 0$, $y \geq 0$
3. C.
   $x + 2y \geq 4$, $x + y \geq 3$, $x \geq 0$, $y \geq 0$
4. D.
   $x + 2y \geq 4$, $x + y \geq 3$, $x \leq 0$, $y \leq 0$

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1. A.
   40
2. B.
   96
3. C.
   120
4. D.
   136

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