Class 11

### A ray of light incident at an angle θ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5o and the prism is made of a material of refractive index 1.5, the angle of incidence isa) 7.5ob) 5oc) 15od) 2.5o

Answer: a) 7.5o The distance between the refracting surfaces is negligible with thin prisms, thus the prism angle (A) is very small. Because A = r1 + r2, if A is tiny, both r1 and r2 will be little...

### Mark (√) against the correct answer in the following: The range of , where is A. B. C. D.

Solution: Option(D) is correct. $f(x)=a x$, where $a>0$ Case 1 : When $x<0$, then ax lies between $(0,1)$ Case 2 : When $x \geq 0$, then $a x \geq 1$ Union of above two cases, gives us the...

### A gardener has a supply of fertilizers of the type 1 which consist of nitrogen and phosphoric acid, and of the type II which consist of nitrogen and phosphoric acid. After testing the soil condition, he finds that he needs at least of nitrogen and of phosphoric acid for his crop. If the type – I fertilizer costs 60 paise per kg and the type – II fertilizer costs 40 paise per kg, determine how many kilograms of each type of fertilizer should be used so that the nutrient requirement are met at a minimum cost. What is the minimum cost?

Let $x$ and $y$ be number of kilograms of fertilizer I and II, $\therefore$ According to the question, $0.10 x+0.05 y \geq 14,0.06 x+0.10 y \geq 14, x \geq 0, y \geq 0$ Minimize $Z=0.60 x+0.40 y$...

### A manufactures produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and for 3 minutes to manufacture the first type, while it takes 3 minutes on machine and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of and each. Assuming that the manufacture can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?

Let $x$ and $y$ be number of soaps be manufactured of $1^{\text {st }}$ and $2^{\text {nd }}$ type. $\therefore$ According to the question, $2 x+3 y \leq 480,3 x+5 y \leq 480, x \geq 0, y \geq 0$...

### A firm manufactures two types of products, and , and sells them at a profit of on type and B. Each product is processed on two machines, and . Type A requires one minute of processing time on and two minutes on Type requires one minute on and one minute on is available for not more than 6 hours 40 minutes while is available for at most 10 hours a day. Find how many products of each type the firm should produce each day in order to get maximum profit.

Let the firm manufacture $x$ number of Aand y number of $B$ products. $\therefore$ According to the question, $X+y \leq 400,2 x+y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=2 x+2 y$ The feasible...

### A dealer wishes to purchase a number of fans and sewing machines. He has only to invest and space and on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximize the profit?

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

### Question while B can stitch 10 shirts and 4 pairs of trousers per day. How many days should each of them work if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labor cost?

Let the total number of days tailor $A$ work be $x$ and tailor $B$ be $y$. $\therefore$ According to the question, $6 x+10 y \geq 60,4 x+4 y \geq 32, x \geq 0, y \geq 0$ Minimize $Z=300 x+400 y$ The...

### A manufacture produces nuts and bolts for industrial machinery. It takes 1 hour of work on machine and 3 hours on machine B to produces a packet of nuts while it takes 3 hours on machine and 1 hours on machine B to produce a packet of bolts. He earns a profit \mp17.50 per packet on nuts and \mp7 per packet on bolts. How many packets of each should be produced each day so as to maximize his profit if he operates, each machine for at the most 12 hours a day? Also find the maximum profit.

Let the number of packets of nuts and bolts be $x$ and y respectively. $\therefore$ According to the question, $x+3 y \leq 12,3 x+y \leq 12, x \geq 0, y \geq 0$ Maximize $Z=17.50 x+7 y$ The feasible...

### The most reactive amine towards dilute hydrochloric acid is ___________.

Solution: Option (ii) is the answer. Reason: The reactivity of amines is proportional to their basicity. If the R group is, the order of basicity is secondary amine ...

### A man has to purchase rice and wheat. A bag of rice and a bag of wheat cost \mp 180 and 120 respectively. He has storage capacity of 10 bags only. He earns a profit of and 78 per bag of rice and wheat respectively. How many bags of each must he buy to make maximum profit?

Let the number of wheat and rice bags be $x$ and $y$. $\therefore$ According to the question, $120 x+180 y \leq 1500, x+y \leq 10, x \geq 0, y \geq 0$ Maximize $Z=8 x+11 y$ The feasible region...

### A small firm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most 24. It takes 1 hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16 . If the profit on a necklace is and that on a bracelet is , how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Let the firm manufacture $x$ number of necklaces and y number of bracelets a day. $\therefore$ According to the question, $x+y \leq 24,0.5 x+y \leq 16 x \geq 1, y \geq 1$ Maximize $Z=100 x+300 y$...

### Maximize , subject to the constraints

The feasible region determined by the constraints $x \geq 0, y \geq 0, x+5 y \leq 200,2 x+3 y \leq 134$ is given by The corner points of feasible region are $A(10,38), B(0,40), C(0,0), D(67,0)$. The...

### Find the maximum value of , subject to the constraints. and

The feasible region determined by the constraints $x \geq 0, y \geq 0$, $x+y \geq 2,2 x+3 y \leq 6$ is given by The corner points of the feasible region is $A(0,2), B(2,0), C(3,0)$. The values of...

### Solve each of the following systems of simultaneous inequations: and

Consider the inequation $2 x+y>1:$ $\Rightarrow y>1-2 x$ Consider the equation $y=1-2 x$ Finding points on the coordinate axes: If $x=0$, the y value is 1 i.e, $y=1$ $\Rightarrow$ the point...

### Graph the solution sets of the following inequations:

Given $y-2 \leq 3 x$ $\Rightarrow y \leq 3 x+2$ Consider the equation $y=3 x+2$ Finding points on the coordinate axes: If $x=0$, the $y$ value is 2 i.e, $y=2$ $\Rightarrow$ the point on $Y$ axis is...

### Let and defined by and . Show that o .

Solution: We need to prove: $g$ o $f \neq f$ o $g$ Formula used: (i) f o $\mathrm{g}=\mathrm{f}(\mathrm{g}(\mathrm{x}))$ (ii) $g$ of $=g(f(x))$ Given that: (i) $f: R \rightarrow R: f(x)=x^{2}$ (ii)...

### For each of the following differential equations, find a particular solution satisfying the given condition: where and when

Solution: $\begin{array}{l} \cos \left(\frac{d y}{d x}\right)=a \\ \Rightarrow \frac{d y}{d x}=\cos ^{-1} a \\ \Rightarrow d y=\cos ^{-1} a d x \end{array}$ On integrating both sides we obtain:...

### Find the general solution of each of the following differential equations:

Solution: $\begin{array}{l} \frac{d y}{d x}=1+x+y+x y=1+y+x(1+y) \\ \Rightarrow \frac{d y}{d x}=(1+y)(1+x) \end{array}$ On rearranging the terms we obtain: $\Rightarrow \frac{d y}{1+y}=(1+x) d x$ On...

### Prove that

Solution: Left Hand Limit: $\lim _{x \rightarrow 3-} f(x)=\lim _{x \rightarrow 3-} \frac{x^{2}-x-6}{x-3}$ $=\lim _{\mathrm{x} \rightarrow 3-} \frac{(\mathrm{x}+2)(\mathrm{x}-3)}{\mathrm{x}-3}$ [By...

### The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...

### The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

Solution: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...

Solution: We need to find: $-x, y, z$, The given set of lines are : $\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 \\ \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\... read more Solution: We need to find:$-x, y, z$The given set of lines are : -$\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\...

### If , find . Using , solve the following system of equations:

Solution: It is given, $A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{array}\right]$ $\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj}(A)$...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x , y , z$ The given set of lines are : $\begin{array}{l} 4 x+3 y+2 z=60 \\ x+2 y+3 z=45 \\ 6 x+2 y+3 z=70 \end{array}$ Now converting the following equations in matrix...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y=3 \\ 2 x+3 y+4 z=17 \\ y+2 z=7 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-2 y+z=0 \\ y-z=2 \\ 2 x-3 z=10 \end{array}$ Now converting the following equations...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 5 x-y=-7 \\ 2 x+3 z=1 \\ 3 y-z=5 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y-2 z=3 \\ x+y=1 \\ x+z=-6 \end{array}$ Now converting the following equations in...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x+2 y+z=4 \\ -x+y+z=0 \\ x-3 y+z=4 \end{array}$ Now converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x+y-z=1 \\ x-y+z=2 \\ 3 x+y-2 z=-1 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x , y , z$ The given set of lines are : - $x + y - z = 1$ $\begin{array}{l} 3 x+y-2 z=3 \\ x-y-z=-1 \end{array}$ Now, converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x , y , z$ The given set of lines are : - $\begin{array}{l} 3 x-4 y+2 z=-1 \\ 2 x+3 y+5 z=7 \\ x+z=2 \end{array}$ Now, converting the following equations in matrix...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, z$ The given set of lines are : - $\begin{array}{l} 6 x-9 y-20 z=-4 \\ 4 x-15 y+10 z=-1 \\ 2 x-3 y-5 z=-1 \end{array}$ Now, converting the...

### Solve each of the following systems of equations using matrix method. ; : .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y+2 z=7 \\ 3 x+4 y-5 z=-5 \\ 2 x-y+3 z=12 \end{array}$ Now, converting the...

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 4 x-5 y-11 z=12 \\ x-3 y+z=1 \\ 2 x+3 y-7 z=2 \end{array}$ Now, converting the...
Solution: We need to find: - $x, y, z$ The given set of lines are : - $\begin{array}{l} 2 x+3 y+3 z=5 \\ x-2 y+z=-4 \\ 3 x-y-2 z=3 \end{array}$ Now, converting the following equations in matrix...