Solution: Option(C) To find: Area of $A B C$ Given: $A(3,-2), B(k, 2)$ and $C(8,8)$ The formula used: $\Delta=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} &...

### Mark the tick against the correct answer in the following: If the points and are collinear then the value of is

### Mark the tick against the correct answer in the following: The vertices of a a are and The area of a is

A. units

B. 35 sq units

C. 32 sq units

D. units

Solution: Option(B) To find: Area of $A B C$ Given: $A(-2,4), B(2,-6)$ and $C(5,4)$ Formula used: $\Delta=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\...

### Mark the tick against the correct answer in the following: The solution set of the equation is

A.

B.

C.

D. None of these

Solution: Option(B) To find: Value of $x$ We have, $\left|\begin{array}{ccc}3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8\end{array}\right|=0$ Applying $\mathrm{R}_{1}...

### Mark the tick against the correct answer in the following: The solution set of the equation is

A.

B.

C.

D. None of these

Solution: Option(B) To find: Value of $x$ We have, $\left|\begin{array}{ccc}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0$ Applying...

### Mark the tick against the correct answer in the following: The solution set of the equation is

A.

B.

C.

D.

Solution: Option(A) To find: Value of $x$ We have, $\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ x-4 & 2 x-9 & 3 x-16 \\ x-8 & 2 x-27 & 3 x-64\end{array}\right|=0$ Applying...

### Mark the tick against the correct answer in the following: The solution set of the equation is

A.

B.

C.

D. none of these

Solution: Option(C) To find: Value of $x$ We have, $\left|\begin{array}{lll}\mathrm{x} & 3 & 7 \\ 2 & \mathrm{X} & 2 \\ 7 & 6 & \mathrm{x}\end{array}\right|=0$ Applying...

### Mark the tick against the correct answer in the following: If then

A. 0

B. 6

C.

D. 9

Solution: Option(C) To find: Value of $\mathrm{x}$ We have, $\left|\begin{array}{ccc}5 & 3 & -1 \\ -7 & x & 2 \\ 9 & 6 & -2\end{array}\right|=0$ Applying $R_{1} \rightarrow 2...

### Mark the tick against the correct answer in the following:

A.

B. 2

C.

D.

Solution: Option(A) To find: Value of $\left|\begin{array}{ccc}\mathrm{x}+1 & \mathrm{x}+2 & \mathrm{x}+4 \\ \mathrm{x}+3 & \mathrm{x}+5 & \mathrm{x}+8 \\ \mathrm{x}+7 &...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(C) To find: Value of $\left|\begin{array}{lll}a & 1 & b+c \\ b & 1 & c+a \\ c & 1 & a+b\end{array}\right|$ We have, $\left|\begin{array}{lll}a & 1 &...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. none of these

Solution: Option(A) To find: Value of $\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$ We have, $\left|\begin{array}{ccc}b+c & a...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. None of these

Solution: Option(A) To find: Value of $\left|\begin{array}{lll}\mathrm{bc} & \mathrm{b}+\mathrm{c} & 1 \\ \mathrm{ca} & \mathrm{a}+\mathrm{c} & 1 \\ \mathrm{ab} &...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. none of these

Solution: Option(C) To find: Value of $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$ We have, $\left|\begin{array}{ccc}1 & 1 & 1...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(C) To find: Value of $\left|\begin{array}{lll}b+c & a & b \\ c+a & c & a \\ a+b & b & c\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following:

A.

B.

C. 0

D. none of these

Solution: Option(B) To find: Value of $\left|\begin{array}{ccc}a & a+2 b & a+2 b+3 c \\ 3 a & 4 a+6 b & 5 a+7 b+9 c \\ 6 a & 9 a+12 b & 11 a+15 b+18 c\end{array}\right|$ We...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. none of these

Solution: Option(C) To find: Value of $\left|\begin{array}{ccc}a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following:

A. 0

B.

C.

D. none of these

Solution: Option( B) To find: Value of $\left|\begin{array}{ccc}x+y & x & x \\ 5 x+4 y & 4 x & 2 x \\ 10 x+8 y & 8 x & 3 x\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following: If be distinct positive real numbers then the value of is

A. positive

B. negative

C. a perfect square

D. 0

Solution: Option(B) To find: Nature of $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ We have, $\left|\begin{array}{lll}a & b & c \\...

### Mark the tick against the correct answer in the following:

A. 0

B. 1

C.

D. none of these

Solution: Option(A) To find: Value of $\left|\begin{array}{lll}\operatorname{sind} & \cos \alpha & \sin (\alpha+\bar{\delta}) \\ \sin \beta & \cos \beta & \sin (\beta+\bar{\delta})...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(C) To find: Value of $\left|\begin{array}{ccc}1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3}\end{array}\right|$ We have, $\left|\begin{array}{ccc}1 & 1...

### Mark the tick against the correct answer in the following:

A. 0

B. 1

C.

D. none of these

Solution: Option(B) To find: Value of $\left|\begin{array}{ccc}1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. 0

Solution: Option(D) To find: Value of $\left|\begin{array}{lll}a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c\end{array}\right|$ We have, $\left|\begin{array}{lll}a-b...

### Mark the tick against the correct answer in the following:

A. 2

B. 6

C. 24

D. 120

Solution: Option(24) To find: Value of $\left|\begin{array}{lll}1 ! & 2 ! & 3 ! \\ 2 ! & 3 ! & 4 ! \\ 3 ! & 4 ! & 5 !\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following:

A. 8

B.

C. 16

D. 142

Solution: Option(A) To find: Value of $\left|\begin{array}{lll}1^{2} & 2^{2} & 3^{2} \\ 2^{2} & 3^{2} & 4^{2} \\ 3^{2} & 4^{2} & 5^{2}\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following: If is a complex cube root of unity then the value of is

A. 2

B. 4

C. 0

D.

Solution: Option(0) To find: Value of $\left|\begin{array}{ccc}1 & \omega & 1+\omega \\ 1+\omega & 1 & \omega \\ \omega & 1+\omega & 1\end{array}\right|$ Formula used: (i)...

### Mark the tick against the correct answer in the following: If is a complex root of unity then

A. 1

B.

C. 0

D. none of these

Solution: Option(C) To find: Value of $\left|\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. none of these

Solution: Option(C) To find: Value of $\left|\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|$ Formula used: $\mathrm{i}^{2}=-1$ We have, $\left|\begin{array}{cc}a+i b...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(B) To find: Value of $\left|\begin{array}{cc}\sin 23^{\circ} & -\sin 7^{\circ} \\ \cos 23^{\circ} & \cos 7^{\circ}\end{array}\right|$ Formula used: (i) $\sin (A+B)=\sin A...

### Mark the tick against the correct answer in the following:

A. 1

B.

C.

D. none of these

Solution: Option(C) To find: Value of $\left|\begin{array}{ll}\cos 15^{\circ} & \sin 15^{\circ} \\ \sin 15^{\circ} & \cos 15^{\circ}\end{array}\right|$ Formula used: (i) $\cos (A+B)=\cos A...

### Mark the tick against the correct answer in the following:

A. 1

B. 0

C.

8D.

Solution: Option(B) To find: Value of $\left|\begin{array}{cc}\cos 70^{\circ} & \sin 20^{\circ} \\ \sin 70^{\circ} & \cos 20^{\circ}\end{array}\right|$ Formula used: (i) $\cos \theta=\sin...

### If the points and are collinear, prove that

Solution: Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3}...

### If and be three points such that area of a is 4 sq units, find the value of .

Solution: Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3}...

### Find the value of for which the area of a ABC having vertices and is 35 sq units.

Solution: Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3}...

### Find the value of for which the points and are collinear.

### Find the value of for which the points and are collinear.

### Find the value of for which the points and are collinear.

### Use determinants to show that the following points are collinear. and

### Use determinants to show that the following points are collinear. and

### Use determinants to show that the following points are collinear. and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the principal value of each of the following :

Solution: $\sin ^{-1}\left(\frac{-1}{2}\right)=-\sin ^{-1}\left(\frac{1}{2}\right)\left[\right.$ Formula: $\left.\sin ^{-1}(-x)=\sin ^{-1}(x)\right]$ $=-\frac{\pi}{6}$

### A die is thrown three times, find the probability that appears on the third toss if it is given that and appear respectively on first two tosses.

As per the given question, $n\;(A)\;=\;36$ And, let $B$ be the event of $6\;and\;5$ appearing respectively on first two tosses, if the die is tossed three times

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

Given $x+y \geq 4$ $\Rightarrow y \geq 4-x$ Consider the equation $y=4-x$. Finding points on the coordinate axes: If $x=0$, the $y$ value is 4 i.e, $y=4$ $\Rightarrow$ the point on the $Y$ axis is...

### Show that is continues at

Solution: L.H.L.: $\lim _{x \rightarrow 2-} f(x)=\lim _{x \rightarrow 2-} x^{2}$ $=4$ R.H.L.: $\lim _{\mathrm{x} \rightarrow 2^{*}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 2^{*}}...

(i)

(ii)

Solution: (i) We can write the given question as, $\sin ^{-1} \frac{1}{2}-2 \sin ^{-1} \frac{1}{\sqrt{2}}=\sin ^{-1} \frac{1}{2}-\sin ^{-1}\left(2 \times \frac{1}{\sqrt{2}}...

### Find the principal value of the following:

(i)

(ii)

Solution: (i) Suppose $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)=\mathrm{y}$ Therefore we can write the above equation as $\sin \mathrm{y}=\cos \frac{3 \pi}{4}=-\sin \left(\pi-\frac{3...

### Find the principal value of the following:

(i)

(ii)

Solution: (i) It is given that functions can be written as $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\frac{\sqrt{3}}{2 \sqrt{2}}-\frac{1}{2 \sqrt{2}}\right) $Taking $1 /...

### Find the principal value of the following:

(i)

(i i)

Solution: $(i)$ Let $\sin ^{-1}\left(\frac{-\sqrt{3}}{2}\right)=y$ Therefore $\sin y=\left(\frac{-\sqrt{3}}{2}\right)$ $\begin{array}{l} =-\sin \left(\frac{\pi}{3}\right) \\ =\sin...

### A manufacturer produces two Models of bikes – Model and Model . Model takes 6 man-hours to make per unit, while Model Y takes 10 man-hours per unit. There is a total of 450 man-hour available per week. Handling and Marketing costs are Rs 2000 and Rs 1000 per unit for Models and respectively. The total funds available for these purposes are Rs 80,000 per week. Profits per unit for Models and are Rs 1000 and Rs 500 , respectively. How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.

Solution: Let us take $\mathrm{x}$ an $\mathrm{y}$ to be the no. of models of bike produced by the manufacturer. From the question we have, Model $x$ takes 6 man-hours to make per unit Model $y$...

### Maximize subject to .

Solution: It is given that: $Z=x+y$ subject to constraints, $x+4 y \leq 8$ $2 x+3 y \leq 12,3 x+y \leq 9, x \geq 0, y \geq 0$ Now construct a constrain table for the above, we have Here, it can be...

### Refer to Exercise 15. Determine the maximum distance that the man can travel.

Solution: According to the solution of exercise 15, we have Maximize $Z=x+y$, subject to the constraints $2 x+3 y \leq 120 \ldots$ (i) $8 x+5 y \leq 400 \ldots$ (ii) $x \geq 0, y \geq 0$ Let's...

### Refer to Exercise 14. How many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit?

Solution: According to the solution of exercise 14, we have Maximize $Z=200 x+120 y$ subject to constrains $\begin{array}{l} 3 x+y \leq 600 \ldots \text { (i) } \\ x+y \leq 300 \ldots \text { (ii) }...

### Refer to Exercise 13. Solve the linear programming problem and determine the maximum profit to the manufacturer.

Solution: According to the solution of exercise 13, we have The objective function for maximum profit $\mathrm{Z}=100 \mathrm{x}+170 \mathrm{y}$ Subject to constraints, $x+4 y \leq 1800 \ldots(i)$...

### Refer to Exercise 12. What will be the minimum cost?

Solution: According to the solution of exercise 12, we have The objective function for minimum cost is $\mathrm{Z}=400 \mathrm{x}+200 \mathrm{y}$ Subject to the constrains; $5 x+2 y \geq 30 \ldots...

### Refer to Exercise 11. How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximize his profit? Determine the maximum profit.

Solution: According to the solution of exercise 11, we have Maximize $Z=50 x+60 y$ subject to the constraints $20 x+10 y \leq 2002 x+y \leq 20 \ldots$ (i) $10 x+20 y \leq 120 x+2 y \leq 12 \ldots$...

### A man rides his motorcycle at the speed of hour. He has to spend Rs 2 per km on petrol. If he rides it at a faster speed of hour, the petrol cost increases to Rs 3 per km. He has at most Rs 120 to spend on petrol and one hour’s time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.

Solution: Suppose the man covers $\mathrm{x} \mathrm{km}$ on his motorcycle at the speed of $50 \mathrm{~km} / \mathrm{hr}$ and covers $\mathrm{y} \mathrm{km}$ at the speed of $50 \mathrm{~km} /...

### A company manufactures two types of sweaters: type A and type B. It costs Rs 360 to make a type A sweater and Rs 120 to make a tvpe B sweater. The companv can make at most 300 sweaters and spend at most Rs 72000 a day. The number of sweaters of type B cannot exceed the number of sweaters of type A by more than 100 . The company makes a profit of Rs 200 for each sweater of type A and Rs 120 for every sweater of type B. Formulate this problem as a LPP to maximize the profit to the company.

Solution: Suppose $\mathrm{x}$ and $\mathrm{y}$ to be the number of sweaters of type $\mathrm{A}$ and type $\mathrm{B}$ respectively. The following constraints are: $360 x+120 y \leq 72000...

### A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of Type A screws requires 2 minutes on the threading machine and 3 minutes on the slotting machine. A box of type screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine. In a week, each machine is available for 60 hours. On selling these screws, the company gets a profit of Rs 100 per box on type A screws and Rs 170 per box on type B screws. Formulate this problem as a LPP given that the objective is to maximize profit.

Solution: Suppose that the company manufactures $\mathrm{x}$ boxes of type A screws and $y$ boxes of type B screws. The below table is constructed from the information provided:...

### A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs 400 and each small van is Rs 200 . Not more than Rs 3000 is to be spent on the iob and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.

Solution: Suppose $\mathrm{x}$ and $\mathrm{y}$ to be the number of large and small vans respectively. The below constrains table is constructed from the information provided:...

### A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs 50 and that on type B circuit is Rs 60 , formulate this problem as a LPP so that the manufacturer can maximize his profit.

Solution: Suppose $\mathrm{x}$ units of type A and $y$ units of type $\mathrm{B}$ electric circuits be produced by the manufacturer. The table is constructed from the information provided:...

### In Fig. 12.11, the feasible region (shaded) for a LPP is shown. Determine the maximum and minimum value of

Solution: It is seen from the given figure, that the corner points are as follows: $\mathrm{R}(7 / 2,3 / 4), \mathrm{Q}(3 / 2,15 / 4), \mathrm{P}(3 / 13,24 / 13)$ and $\mathrm{S}(18 / 7,2 / 7)$ On...

### The feasible region for a LPP is shown in Fig. 12.10. Evaluate at each of the corner points of this region. Find the minimum value of , if it exists.

Solution: It is given that: $Z=4 x+y$ In the figure given, $\mathrm{ABC}$ is the feasible region which is open unbounded. Here, we get $x+y=3\dots \dots(i)$ and $\quad x+2 y=4 \quad \ldots$ (ii) On...

### Refer to Exercise 7 above. Find the maximum value of .

Solution: It is clearly seen that the evaluating table for the value of $Z$, the maximum value of $Z$ is 47 at $(3,2)$

### The feasible region for a LPP is shown in Fig. 12.9. Find the minimum value of .

Solution: It is seen from the given figure, that the feasible region is $\mathrm{ABCA}$. Corner points are $\mathrm{C}(0,3), \mathrm{B}(0,5)$ and for A, we have to solve equations $x+3 y=9$ and...

### Feasible region (shaded) for a LPP is shown in Fig. 12.8. Maximize .

Solution: It is given that: $\mathrm{Z}=5 \mathrm{x}+7 \mathrm{y}$ and feasible region $\mathrm{OABC}$. Corner points of the feasible region are $\mathrm{O}(0,0), \mathrm{A}(7,0), \mathrm{B}(3,4)$...

### Determine the maximum value of if the feasible region (shaded) for a LPP is shown in Fig. 12.7.

Solution: OAED is the feasible region, as shown in the figure At $A, y=0$ in eq. $2 x+y=104$ we obtain, $\mathrm{x}=52$ This is a corner point $A=(52,0)$ At $D, x=0$ in eq. $x+2 y=76$ we obtain,...

### Minimize subject to the constraints: .

Solution: It is given that: $\mathrm{Z}=13 \mathrm{x}-15 \mathrm{y}$ and the constraints $\mathrm{x}+\mathrm{y} \leq 7$, $2 x-3 y+6 \geq 0, x \geq 0, y \geq 0$ Taking $x+y=7$, we have...

### Maximize the function , subject to the constraints:

Solution: It is given that: $\mathrm{Z}=\mathrm{Z}=11 \mathrm{x}+7 \mathrm{y}$ and the constraints $\mathrm{x}$ $\leq 3, y \leq 2, x \geq 0, y \geq 0$ Plotting all the constrain equations it can be...

### Maximize , subject to the constraints:

Solution: It is given that: $Z=3 x+4 y$ and the constraints $x+y \leq 1, x \geq 0$ $\mathrm{y} \geq 0$ Taking $x+y=1$, we have $$\begin{tabular}{|l|l|l|} \hline$x$ & 1 & 0 \\ \hline$y$ & 0 & 1 \\...

### Determine the maximum value of subject to the constraints:

Solution: It is given that: $\mathrm{Z}=11 \mathrm{x}+7 \mathrm{y}$ and the constraints $2 \mathrm{x}+\mathrm{y} \leq 6, \mathrm{x} \leq 2, \mathrm{x} \geq 0, \mathrm{y} \geq 0$ Let $2 x+y=6$...

### In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane

(a) (2, 3, -5) x + 2y – 2z = 9

(b) (-6, 0, 0) 2x – 3y + 6z – 2 = 0

Solution: (a) The length of perpendicular from the point $(2,3,-5)$ on the plane $x+2 y-2 z=9 \Rightarrow x+2 y-2 z-9=0$ is $\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$...

### In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 4x + 8y + z – 8 = 0 and y + z – 4 = 0

Solution: (a) $4 x+8 y+z-8=0$ and $y+z-4=0$ It is given that The eq. of the given planes are $4 x+8 y+z-8=0 \text { and } y+z-4=0$ It is known to us that, two planes are $\perp$ if the direction...

### In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

(b) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

Solution: (a) $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$ It is given that The eq. of the given planes are $2 x-2 y+4 z+5=0$ and $x-2 y+5=0$ It is known to us that, two planes are $\perp$ if the...

### Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Solution: Let's say that the eq. of the plane that passes through the two-given planes $x+y+z=1$ and $2 x+3 y+4 z=5$ is $\begin{array}{l} (x+y+z-1)+\lambda(2 x+3 y+4 z-5)=0 \\ (2 \lambda+1) x+(3...

### Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Solution: It is given that Eq. of the plane passes through the intersection of the plane is given by $(3 x-y+2 z-4)+\lambda(x+y+z-2)=0$ and the plane passes through the points $(2,2,1)$ Therefore,...

### Find the equation of the plane with intercept 3 on the -axis and parallel to ZOX plane.

Solution: It is known to us that the equation of the plane $\mathrm{ZOX}$ is $\mathrm{y}=0$ Therefore, the equation of plane parallel to $\mathrm{ZOX}$ is of the form, $\mathrm{y}=\mathrm{a}$ As the...

### Find the equations of the planes that passes through three points.

(a) (1, 1, –1), (6, 4, –5), (–4, –2, 3)

(b) (1, 1, 0), (1, 2, 1), (–2, 2, –1)

Solution: (a) It is given that, The points are $(1,1,-1),(6,4,-5),(-4,-2,3)$. Let, $\begin{array}{l} =\left|\begin{array}{ccc} 1 & 1 & -1 \\ 6 & 4 & -5 \\ -4 & -2 & 3...

### Find the Cartesian equation of the following planes:

(a)

(b)

Solution: (a) It is given that, The equation of the plane. Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\vec{r}=x...

### In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

(a) 2x + 3y – z = 5

(b) 5y + 8 = 0

Solution: (a) $2 x+3 y-z=5$ It is given that The eq. of the plane, $2 x+3 y-z=5 \ldots$. (1) The direction ratio of the normal $(2,3,-1)$ Using the formula,...

### Find the shortest distance between the lines whose vector equations are

Solution: Consider the given equations $\begin{array}{l} \Rightarrow \vec{r}=(1-t) \hat{i}+(t-2) \hat{j}+(3-2 t) \hat{k} \\ \vec{r}=\hat{i}-t \hat{i}+t \hat{j}-2 \hat{j}+3 \hat{k}-2 t \hat{k} \\...

### Find the shortest distance between the lines and

Solution: It is known to us that the shortest distance between two lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$ is given as:...

### Show that the lines and are perpendicular to each other.

Solution: The equations of the given lines are $\frac{\mathrm{x}-5}{7}=\frac{\mathrm{y}+2}{-5}=\frac{\mathrm{z}}{1}$ and $\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}}{3}$ Two lines...

### Find the values of p so that the lines and are at right angles.

Solution: The standard form of a pair of Cartesian lines is:...

### Find the vector and the Cartesian equations of the line that passes through the points (3, –2, –5), (3, –2, 6).

Solution: It is given that Let's calculate the vector form: The vector eq. of as line which passes through two points whose position vectors are $\vec{a}$ and $\vec{b}$ is...

### If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

Given, \[X\text{ }=\text{ }0,\text{ }1,\text{ }2,\text{ }3\] P(X = r) \[={{~}^{n}}{{C}_{r}}~{{p}^{r}}~{{q}^{n-r}}\] Where \[n\text{ }=\text{ }3,\text{ }p\text{ }=\text{ 1/2},\text{ }q\text{ }=\text{...

### The general solution of the differential equation

Therefore, the correct option is option(c).

### The general solution of a differential equation of the type is

SOLUTION: Therefore, yje correct option is option(c).

### The general solution of the differential equation is

Given question is Therefore, the correct option is OPTION(C)

### The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

### Find a particular solution of the differential equation, given that y = 0 when x = 0.

### Find a particular solution of the differential equation (x ≠ 0), given that y = 0 when x = π/2

### Solve the differential equation

### Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

### Solve the differential equation

### Find the particular solution of the differential equation

, given that y = 1 when x = 0.

### Find the equation of the curve passing through the point (0, π/4) whose differential equation is sin x cos y dx + cos x sin y dy = 0.

### Show that the general solution of the differential equation is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.

### Find the general solution of the differential equation

On integrating, we get, ⇒ sin-1x + sin-1y = C

### Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

(x -a)2 + (y –a)2 = a2 …………1 differentiating eq 1 with respect to x, we get, \[\begin{array}{*{35}{l}} 2\left( x-a \right)\text{ }+\text{ }2\left( y-a \right)\text{ }dy/dx~=\text{ }0 \\ \Rightarrow...

### Prove that

is the general solution of differential equation

where c is a parameter.

### Form the differential equation representing the family of curves given by (x – a)^2 + 2y^2 = a^2, where a is an arbitrary constant.

### verify that the given function (implicit or explicit) is a solution of the corresponding differential equation (iii) (iv)

### verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.(i) (ii)

### indicate its order and degree (if defined).

### indicate its order and degree (if defined). (i) (ii)

Therefore, its degree is three.

### The Integrating Factor of the differential equation is

Therefore, the correct option is OPTION(D)

### The Integrating Factor of the differential equation is A. e–x B. e–y C. 1/x D. x

Therefore, the correct option is OPTION(C).

### Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

Thus, equation (2) becomes: \[\begin{array}{*{35}{l}} 0\text{ }+\text{ }2\text{ }-\text{ }4\text{ }=\text{ }C\text{ }{{e}^{0}} \\ \Rightarrow ~C\text{ }=\text{ }-2 \\ \end{array}\] Substituting C...

### Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Since, the curve passes through origin. Thus, equation 2 becomes 1 = C Substituting C = 1 in equation 2, we get, \[x\text{ }+\text{ }y\text{ }+\text{ }1\text{ }=\text{ }{{e}^{x}}\] Therefore, the...

### find a particular solution satisfying the given condition:

### find a particular solution satisfying the given condition:

### find a particular solution satisfying the given condition:

### find the general solution:

⇒ x = 3y2 + Cy

### find the general solution:

$x=\frac{{{y}^{3}}}{3}+c$ $x=\frac{{{y}^{2}}}{3}+\frac{c}{y}$

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### find the general solution:

### If

is the angle between any two vectors

and

, then

when

is equal to (A)

(B)

(C)

(D)

### The value of

is (A)

(B)

(C)

(D)

It is given that, Hence the correct answer is C.

### Let

and

be two unit vectors and

is the angle between them. Then

is a unit vector if (A)

(B)

(C)

(D)

Here the correct answer is option d

### If

is the angle between two vectors

and

, Then

only when (A)

(B)

(C)

(D)

### Prove that

, if and only if

,

are perpendicular, given

.

It is given that Hence proved.

### If

,

,

. are mutually perpendicular vectors of equal magnitudes, show that the vector

is equally inclined to

,

and

.

let us assume,

### The scalar product of the vector

with a unit vector along the sum of vectors

and

is equal to one. Find the value of

.

Let us consider

### Let

,

and

. Find a vector

which is perpendicular to both

and

, and

Assume,

### Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are

,

,

.

Firstly, Let’s assume a vector to be equally inclined to axes OX, OY, and OZ at angle \[\alpha \]. Then, the direction cosines of the vector are \[\cos \alpha \],\[\cos \alpha \]and \[\cos \alpha...

### The two adjacent sides of a parallelogram are

and

. Find the unit vector parallel to its diagonal. Also, find its area.

we know that,

### Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are

and

externally in the ratio

. Also, show that P is the midpoint of the line segment RQ.

we know that,

### Show that the points A

, B

and C

are collinear, and find the ratio in which B divides AC.

Let us consider

### If

,

and

find a unit vector parallel to the vector

.

Let us consider

### Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

and

let us consider,

### Find the value of x for which

is a unit vector.

we know ,

### If

, then is it true that

? Justify your answer.

It is given that,

### A girl walks

km towards west, then she walks

km in a direction

east of north and stops. Determine the girl’s displacement from her initial point of departure.

It is given that, Let O and B be the initial and final positions of the girl respectively. Then, the girl’s position can be shown as:

### . Find the scalar components and magnitude of the vector joining the points P

and Q

.

let us consider,

### Write down a unit vector in XY-plane, making an angle of

with the positive direction of x-axis.

let us consider,

### Find the vector and the Cartesian equations of the lines that passes through the origin and (5, –2, 3).

Solution: Given: The origin $(0,0,0)$ and the point $(5,-2,3)$ It is known to us that The vector eq. of as line which passes through two points whose position vectors are $\vec{a}$ and $\vec{b}$ is...

### Find the Cartesian equation of the line which passes through the point and parallel to the line given by

Solution: It is given that The points $(-2,4,-5)$ It is known that Now, the Cartesian equation of a line through a point $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and having...

### Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

Solution: Given that, Line passes through the point $(1,2,3)$ and is parallel to the vector. It is known to us that Vector eq. of a line that passes through a given point whose position vector is...

### Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (–1, –2, 1), (1, 2, 5).

Solution: The points $(4,7,8),(2,3,4)$ and $(-1,-2,1),(1,2,5)$. Consider $A B$ be the line joining the points, $(4,7,8),(2,3,4)$ and $C D$ be the line through the points $(-1,-2$, 1), $(1,2,5)$. So...

### Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Solution: Given that The points $(1,-1,2),(3,4,-2)$ and $(0,3,2),(3,5,6)$. Let's consider $A B$ be the line joining the points, $(1,-1,2)$ and $(3,4,-2)$, and $C D$ be the line through the points...

### Which of the following is a homogeneous differential equation? A. (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0 B. (x y) dx – (x^3 + y^3) dy = 0 C. (x^3 + 2y^2) dx + 2xy dy = 0 D. y^2dx + (x^2 – x y – y^2) dy = 0

D. y2dx + (x2 – x y – y2) dy = 0

### A homogeneous differential equation of the from can be solved by making the substitution. (A) y = v x (B) v = y x (C) x = v y (D) x = v

(C) x = v y

### find the particular solution satisfying the given condition:

The required solution of the differential equation.

### find the particular solution satisfying the given condition:

### find the particular solution satisfying the given condition:

### find the particular solution satisfying the given condition:

### find the particular solution satisfying the given condition: (x + y) dy + (x – y) dx = 0; y = 1 when x = 1

### show that the given differential equation is homogeneous and solve each of them

### show that the given differential equation is homogeneous and solve each of them.

### show that the given differential equation is homogeneous and solve each of them.

### show that the given differential equation is homogeneous and solve each of them.

### Area of a rectangle having vertices A, B, C, and D with position vectors

and

respectively is

### show that the given differential equation is homogeneous and solve each of them.

### Let the vectors

and

be such that

and

, then

is a unit vector, if the angle between

and

is (A)

(B)

(C)

(D)

### Find the area of the parallelogram whose adjacent sides are determined by the vector .

and

Let us consider,

### show that the given differential equation is homogeneous and solve each of them.

### Find the area of the triangle with vertices A

, B

and C

.

We know that,

### show that the given differential equation is homogeneous and solve each of them.

x2 + y2 = Cx

### If either

or

then

. Is the converse true? Justify your answer with an example.

Firstly let us consider,

### Let the vectors

given as

,

,

. Then show that

It is given that,

### show that the given differential equation is homogeneous and solve each of them. (x – y) dy – (x + y) dx = 0

### Given that

and

. What can you conclude about the vectors

and

?

It is given that

### show that the given differential equation is homogeneous and solve each of them.

### Find

and

if

.

It is given that,

### show that the given differential equation is homogeneous and solve each of them.

### Show that

Firstly consider the LHS, We have,

### If a unit vector

makes an angles

with

with

and an acute angle

with

, then find

and hence, the compound of

Firstly,

### The general equation of the differential equation

The correct option is option(A) ex + e-y = C

### Find a unit vector perpendicular to each of the vector

and

, where

and

.

It is given that:

### In a culture, the bacteria count is 1, 00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2, 00,000, if the rate of growth of bacteria is proportional to the number present?

### Find

, if

and

It is given that: