Download App

Linear Programming — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSLinear Programming5 Marks
Question
Solve the following linear programming problem for minimisation by graphical method :
Objective function
$
\begin{aligned}Z = 5 x + y \\
constraints
3 x + 5 y & \geq 1 5 \\
5 x + 2 y & \leq 1 0 \\
x \geq 0 , y & \geq 0
\end{aligned}
$
✓

Answer

$
Z=5 x+y
$
Constraints are:
$
\begin{aligned}
3 x+5 y & \geq 15 \\
5 x+2 y & \leq 10 \\
x & \geq 0 \\
y & \geq 0
\end{aligned}
$
(i) Region of $3 x+5 y \geq 15$ :
Image
The line $3 x+5 y=15$ passes through the points $A (5,0)$ and $B (0,3)$. Its graph is AB .
Putting $x=0, y=0$ in $3 x+5 y \geq 15$, we get $0 \geq 15$ which is false.
i.e., this region contain AB and region above it.
(ii) Region of $5 x+2 y \leq 10$ :The line $5 x+2 y=10$ passes through the points P $(2,0)$ and $Q (0,5)$. Its graph is PQ .
Now putting $x=0, y=0$ is $5 x+2 y \leq 10$, we get $0 \leq 10$ which is true.
i.e., the region of $5 x+2 y \leq 10$ is the line PQ and below PQ towards the origin.
(iii) The area of $x \geq 0$ is on $y$-axis and to the right of $y$-axis.
(iv) The area of $y \geq 0$ is on $x$-axis and above the $x$-axis.
Thus, the feasible region of this problem is OBRP.
Corner PointCorresponding Value of Z = 5x + y
O(0, 0)0
P(2, 0)10
$R \left(\frac{20}{19}, \frac{45}{19}\right)$$\frac{ 1 4 5 }{ 1 9 }$ Maximum
B(0 , 3)3

Hence, at the corner point O (0, 0) value of Z = 0

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Linear Programming→Linear Programming — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

The adjacent sides of a parallelogram are represented by the vectors $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{b}}=-2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$. Find the unit vectors parallel to the diagonals of the parallelogram.
→
Differentiate the following functions with respect to x:
$\log(\text{x}+\sqrt{\text{x}^2+1})$
→
Show that the function $f: R \rightarrow\{x \in R:-1 < x<1\}$ defined by $f(x)=\frac{x}{1+|x|}, x \in R$ is one-one and onto function.
→
Find the variance of the distribution:
$\text{x}$ $0$ $1$ $2$ $3$ $4$ $5$
$\text{P}(\text{x})$ $\frac{1}{6}$ $\frac{5}{18}$ $\frac{2}{9}$ $\frac{1}{6}$ $\frac{1}{9}$ $\frac{1}{18}$
→
Find the equation of the tangent to the curve $\text{x}=\theta+\sin\theta,\text{y}+\cos\theta\text{ at }\theta=\frac{\pi}{4}.$
→
Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=\log(2+\text{x})-\frac{2\text{x}}{2+\text{x}},\text{x}\in\text{R}$
→
If $AD$ is the median of $\triangle\text{ABC},$ using vectors, prove that $\text{AB}^2+\text{AC}^2=2\big(\text{AD}^2+\text{CD}^2\big).$
→
Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}\text{ dx}$
→
Show that the lines $\frac{\text{x}+1}{-3}=\frac{\text{y}-3}{2}=\frac{\text{z}+2}{1}$ and $\frac{\text{x}}{1}=\frac{\text{y}-7}{-3}=\frac{\text{z}+7}{2}$ are coplanar. Also, find the equation of the plane containing them.
→
A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines, and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹ 7 profit and that of B at a profit of ₹ 4. Find the production level per day for maximum profit graphically.
→