M.C.Q (1 Marks)

MCQ 511 Mark

The solution set of the constraints $x+2 y \leq 2000, x+y \leq 1500, y \leq 600$ and $x \geq 0$ does not include the point

A
$(1000,0)$
B
$(0,500)$
C
$(2,0)$
✓
$(2000,0)$

Answer

Correct option: D.

$(2000,0)$

d
$(1000,0)$

$x+2 y=1000 \leq 2000$

$x+y=1000 \leq 15000$

$y=0 \leq 600$

$x \geq 0$

This is true. so, this point include in the region.

$(0,500)$

$x+2 y=1000 \leq 2000$

$x+y=500 \leq 15000$

$y=0 \leq 600$

$x=500 \geq 0$

This point also include in the region.

$(2,0)$

$x+2 y=2 \leq 2000$

$x+y=2 \leq 15000$

$y=0 \leq 600$

$x=2 \geq 0$

This point also include in the region.

For $(2000,0) \quad x+2 y=2000 \leq 2000$

$x+y=2000\,<\,1500$

which is not true.

$\therefore$ The point $(2000,0)$ is not in the region.

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MCQ 521 Mark

The corner points of the bounded feasible region are $(0,1),(0,7),(2,7),(6,3)(6,0)(1,0)$ For the objective function $\mathrm{Z}=3 x-y$

$(i)$ At which point, $Z$ is minimum?

$(ii)$ At which point, $Z$ is maximum ?

$(iii)$ The maximum value of $\mathrm{Z}$ is $\ldots \ldots \ldots$

$(iv)$ The minimum value of $\mathrm{Z}$ is $\ldots \ldots \ldots$

A
$(i) (2,7) \,\, (ii) (6,3) \,\, (iii) 20 \,\, (iv) -1$
✓
$(i) (0,7) \,\, (ii) (6,0) \,\, (iii) 18 \,\, (iv) -7$
C
$(i) (0,1)\,\, (ii) (6,3) \,\, (iii) 18 \,\, (iv) -1$
D
$(i) (0,7) \,\, (ii) (6,0) \,\, (iii) 15 \,\, (iv) -7$

Answer

Correct option: B.

$(i) (0,7) \,\, (ii) (6,0) \,\, (iii) 18 \,\, (iv) -7$

b

Corner point	Corresponding value of $Z =3 x-y$
$(0,1)$	$Z =0-1=-1$
$(0,7)$	$Z =0-7=-7 \text { Minimum value }$
$(2,7)$	$Z =3(2)-7=-1$
$(6,3)$	$Z =3(6)-3=15$
$(6,0)$	$Z =3(6)-0=18 \text { Maximum value }$
$(1,0)$	$Z =3(1)-0=3$

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MCQ 531 Mark

The corner points of the bounded feasible region are $(0,0),(2,0),(4,2),(2,4)$ and $\left(0, \frac{10}{3}\right)$

Then for the objective function $z=-x+2 y$

$(i)$ Maximum value of $z$ has at $\ldots \ldots \ldots . . .$

$(ii)$ Minimum value of $z$ has at $\ldots \ldots \ldots . . .$

$(iii)$ The maximum value of $z$ is $\ldots \ldots \ldots . . .$

$(iv)$ The minimum value of $z$ is $\ldots \ldots \ldots . . .$

A
$(i)\,\, (2,4)\,\,( ii )(0,0)\,\,( iii ) 6 \,\,( iv ) 0$
B
$(i) (0, \frac{10}{3} ) \,\, (ii) (4,2) \,\, ( iii ) 6 \,\, ( iv ) 0$
C
$(i) (2,4) \,\, ( ii)(2,0) \,\, ( iii ) 6 \,\,(iv)-2$
✓
$(i) (0, \frac{10}{3}) \,\, (ii) (2,0) \,\,(iii) \frac{20}{3} \,\, (\mathrm{iv})-2$

Answer

Correct option: D.

$(i) (0, \frac{10}{3}) \,\, (ii) (2,0) \,\,(iii) \frac{20}{3} \,\, (\mathrm{iv})-2$

d
$(\mathrm{i})\left(0, \frac{10}{3}\right)\, (ii)\, (2,0) \,(iii)\,\frac{20}{3}(\mathrm{iv})-2$

Corner points	Corresponding value of $z=x+2 y$
$(0,0)$	$0$
$(2,0)$	$-2$
$(4,2)$	$0$
$(2,4)$	$6$
$(0, \frac{10}{3})$	$\frac{20}{3}$

$\Rightarrow$ The minimum value of $z$ is $-2$ at $(2,0)$

$\Rightarrow$ The maximum value of $z$ is $\frac{20}{3}$ at $\left(0, \frac{10}{3}\right)$

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MCQ 541 Mark

The corner points of the bounded feasible region are $(60,0),(120,0),(60,40),(40,20)$ and $(20,30)$. For the objective function $z=5 x+10 y \ldots$

$(i)$ Maximum value of $z$.

$(ii)$ Minimum value of $z$.

$(iii)$ Maximum value of $z$ has at

$(iv)$ Minimum value of $z$ has at

A
$700,600,(60,40),(120,0)$
B
$600,400,(120,0),(40,20)$
C
$600,300,(120,0),(60,0)$
✓
$700,300,(60,40),(60,0)$

Answer

Correct option: D.

$700,300,(60,40),(60,0)$

d

Corner point	Corrsponding value of $z=5 x+10 y$
$(60,0)$	$z=5(60)+10(0)=$	$300$
$(120,0)$	$z=5(120)+10(0)=$	$600$
$(60,40)$	$z=5(60)+10(40)=$	$700$
$(40,20)$	$z=5(40)+10(20)=$	$400$
$(20,30)$	$z=5(20)+10(30)=$	$400$

$\therefore$ The maximum value of $z$ is $700$ at $(60,40).$

The minimum value of $z$ is $300$ at $(60,0).$

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MCQ 551 Mark

Corner points of the feasible region for an $\operatorname{LPP}$ are $(0,2),(3,0),(6,0),(6,8)$ and $(0,5)$ Let $F=4 x+6 y$ be the objective function. The Minimum value of $F$ occurs at $....$

A
only $(0,2)$
B
only $(3,0)$
C
the mid-point of the line segment joining the points $(0,2)$ and $(3,0)$ only
✓
any point on the line segment joining the points $(0,2)$ and $(3,0)$

Answer

Correct option: D.

any point on the line segment joining the points $(0,2)$ and $(3,0)$

d

Corner point	Objective function $z=4 x+6 y$
$(0,2)$	$z=4(0)+6(2)=12($ minimum value $)$
$(3,0)$	$z=4(3)+6(0)=12$ (minimum value)
$(6,0)$	$z=4(6)+6(0)=24$
$(6,8)$	$z=4(6)+6(8)=72($ maximum value $)$
$(0,5)$	$z=4(0)+6(5)=30$

Objective function is minimum of $(0,2)$ and $(3,0)$

$\therefore$ Hence objective function is minimum on line segment joining points $(0,2)$ and $(0,3)$

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Mathematics M.C.Q (1 Marks)