NCERT Exemplar

### (i) Consider a thin lens placed between a source (S) and an observer (O). Let the thickness of the lens vary as 2 0 () – α = b wb w, where b is the verticle distance from the pole. w0 is a constant. Using Fermat’s principle i.e. the time of transit for a ray between the source and observer is an extremum, find the condition that all paraxial rays starting from the source will converge at a point O on the axis. Find the focal length.(ii) A gravitational lens may be assumed to have a varying width of the form w(b) = k1 ln (k2/b) = k1 ln (k2/bmin). Show that an observer will see an image of a point object as a ring about the centre of the lens with an angular radius.β = √(n-1)k1 u/v / u + v

i) Time taken by the ray to travel from S to P1 is = t1 = √u2 + b2/c Time taken by the ray to travel from P1 to O is = t2 = v/c (1+ ½ b2/v2) Time taken to travel through the lens is =...

### Show that for a material with refractive index µ ≥ 2 , light incident at any angle shall be guided along a length perpendicular to the incident face. Answer:

Let the refractive index of the rectangular slab be μ ≥ √2. μ = 1/sin ic sin ic > 1/ μ cos r ≥ 1/ μ sin i/sin r = μ From Snell’s law Sin I = μ sin r i = 90o 1 + 1 ≤ μ2 2 ≤ μ2 Taking the square...

### A jar of height h is filled with a transparent liquid of refractive index µ. At the centre of the jar on the bottom surface is a dot. Find the minimum diameter of a disc, such that when placed on the top surface symmetrically about the centre, the dot is invisible.

tan ic d/2/h ic = d/2h d = 2h tan ic d = 2h ×1/√μ2 – 1

### A short object of length L is placed along the principal axis of a concave mirror away from focus. The object distance is u. If the mirror has a focal length f, what will be the length of the image? You may take L << |v-f|

The mirror formula is 1/v + 1/u = 1/f u is the object distance v is the image distance du = |u1 – u2| = L Differentiating on the both sides we get, dv/v2 = -du/u2 v/u = f/u-f du = L, therefore,...

### For a glass prism (µ = √3 ) the angle of minimum deviation is equal to the angle of the prism. Find the angle of the prism.

μ = sin[(A + δm)/2]/sin (A/2)

### An astronomical refractive telescope has an objective of focal length 20m and an eyepiece of focal length 2cm.(a) The length of the telescope tube is 20.02m. (b) The magnification is 1000. (c) The image formed is inverted. (d) An objective of a larger aperture will increase the brightness and reduce chromatic aberration of the image.

Answer: (a) The length of the telescope tube is 20.02m. (b) The magnification is 1000. (c) The image formed is inverted.                      ...

### A magnifying glass is used, as the object to be viewed can be brought closer to the eye than the normal near point. This results in(a) a larger angle to be subtended by the object at the eye and hence viewed in greater detail.(b) the formation of a virtual erect image.(c) increase in the field of view.(d) infinite magnification at the near point.

Answer: (a) a larger angle to be subtended by the object at the eye and hence viewed in greater detail. (b) the formation of a virtual erect image.              ...

### Between the primary and secondary rainbows, there is a dark band known as Alexandar’s dark band. This is because(a) light scattered into this region interfere destructively.(b) there is no light scattered into this region(c) light is absorbed in this region.(d) angle made at the eye by the scattered rays with respect to the incident light of the sun lies between approximately 42° and 50°.

Answer: (a) light scattered into this region interfere destructively. (d) angle made at the eye by the scattered rays with respect to the incident light of the sun lies between approximately 42° and...

### A rectangular block of glass ABCD has a refractive index 1.6. A pin is placed midway on the face AB. When observed from the face AD, the pin shall (a) appear to be near A.(b) appear to be near D.(c) appear to be at the centre of AD.(d) not be seen at all.

Answer: (a) appear to be near A. (d) not be seen at all. The pin will appear to be near A as long as the angle of incidence on AD of the ray emerging from the pin is smaller...

### Consider an extended object immersed in water contained in a plane trough. When seen from close to the edge of the trough the object looks distorted because(a) the apparent depth of the points close to the edge is nearer the surface of the water compared to the points away from the edge.(b) the angle subtended by the image of the object at the eye is smaller than the actual angle subtended by the object in the air.(c) some of the points of the object far away from the edge may not be visible because of total internal reflection.(d) water in a trough acts as a lens and magnifies the object.

Answer: (a) the apparent depth of the points close to the edge is nearer the surface of the water compared to the points away from the edge. (b) the angle subtended by the image of the object at the...

### A car is moving with at a constant speed of 60 km h–1 on a straight road. Looking at the rearview mirror, the driver finds that the car following him is at a distance of 100 m and is approaching with a speed of 5 km h –1. In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every 2 still the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct?(a) The speed of the car in the rear is 65 km h–1.(b) In the side mirror, the car in the rear would appear to approach with a speed of 5 km h–1 to the driver of the leading car.(c) In the rearview mirror the speed of the approaching car would appear to decrease as the distance between the cars decreases.(d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases.

Answer: (d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases.

### The optical density of turpentine is higher than that of water while its mass density is lower. The figure shows a layer of turpentine floating over water in a container. For which one of the four rays incident on turpentine in the figure, the path shown is correct?a) 1b) 2c) 3d) 4

Answer: b) 2 When light travels from (optically) rarer medium air to optically denser medium turpentine, it bends towards the normal, i.e., θ1 >...

### The direction of a ray of light incident on a concave mirror as shown by PQ while directions in which the ray would travel after reflection is shown by four rays marked 1, 2, 3, and 4. Which of the four rays correctly shows the direction of reflected ray?a) 1b) 2c) 3d) 4

Answer: b) 2 After reflection, the ray PQ of light that passes through focus F and strikes the concave mirror should become parallel to the primary...

### The phenomena involved in the reflection of radiowaves by ionosphere is similar toa) reflection of light by a plane mirrorb) total internal reflection of light in the air during a miragec) dispersion of light by water molecules during the formation of a rainbowd) scattering of light by the particles of air

Answer: b) total internal reflection of light in the air during a mirage The ionosphere, a layer of the atmosphere, reflects radio waves, allowing them to reach far-flung portions of the globe....

### The radius of curvature of the curved surface of a plano-convex lens is 20 cm. If the refractive index of the material of the lens be 1.5, it willa) act as a convex lens only for the objects that lie on its curved sideb) act as a concave lens only for the objects that lie on its curved sidec) act as a convex lens irrespective of the side on which the object liesd) act as a concave lens irrespective of the side on which the object lies

Answer: c) act as a convex lens irrespective of the side on which the object lies

### A passenger in an aeroplane shalla) never see a rainbowb) may see a primary and a secondary rainbow as concentric circlesc) may see a primary and a secondary rainbow as concentric arcsd) shall never see a secondary rainbow

Answer: b) may see a primary and a secondary rainbow as concentric circles As an aeroplane flies higher in the sky, passengers may notice a primary and secondary rainbow in the form of concentric...

### An object approaches a convergent lens from the left of the lens with a uniform speed 5 m/s and stops at the focus. The imagea) moves away from the lens with a uniform speed 5 m/sb) moves away from the lens with a uniform accelerationc) moves away from the lens with a non-uniform accelerationd) moves towards the lens with a non-uniform acceleration

Answer: c) moves away from the lens with a non-uniform acceleration In our case, the object approaches a convergent lens from the left at a uniform speed of 5 m/s, causing the image to travel away...

### A short pulse of white light is incident from air to a glass slab at normal incidence. After travelling through the slab, the first colour to emerge isa) blueb) greenc) violetd) red

Answer: d) Red The relation v = fλ describes the velocity of a wave. The frequency of light does not change when it travels from one medium to another. As a result, the bigger the wavelength, the...

### Equipotential surfaces a) are closer in regions of large electric fields compared to regions of lower electric fields b) will be more crowded near sharp edges of a conductor c) will be more crowded near regions of large charge densities d) will always be equally spaced

The correct answer is a) are closer in regions of large electric fields compared to regions of lower electric fields b) will be more crowded near sharp edges of a conductor c) will be more crowded...

### Consider a uniform electric field in the z direction. The potential is a constant a) in all space b) for any x for a given z c) for any y for a given z d) on the x-y plane for a given z

The correct answer is b) for any x for a given z c) for any y for a given z d) on the x-y plane for a given z

### A parallel plate capacitor is made of two dielectric blocks in series. One of the blocks has thickness d1 and dielectric constant k1 and the other has thickness d2 and dielectric constant k2 as shown in the figure. This arrangement can be thought of as a dielectric slab of thickness d = d1 + d2 and effective dielectric constant k. The k is a) k1d1 + k2d2/d1+d2 b) k1d1 + k2d2/k1 + k2 c) k1k2 (d1 + d2)/(k1d1 + k2d2) d) 2k1k2/k1 + k2

The correct answer is c) k1k2 (d1 + d2)/(k1d1 + k2d2)

### Equipotential at a great distance from a collection of charges whose total sum is not zero are approximately a) spheres b) planes c) paraboloids d) ellipsoids

The correct answer is a) spheres

### The electrostatic potential on the surface of a charged conducting sphere is 100V. Two statements are made in this regard: S1: At any point inside the sphere, the electric intensity is zero S2: At any point inside the sphere, the electrostatic potential is 100V Which of the following is a correct statement? a) S1 is true but S2 is false b) Both S1 and S2 are false c) S1 is true, S2 is also true, and S1 is the cause of S2 d) S1 is true, S2 is also true but the statements are independent

The correct answer is c) S1 is true, S2 is also true, and S1 is the cause of S2

### Figure shows some equipotential lines distributed in space. A charged object is moved from point A to point B. a) the work done in fig (i) is the greatest b) the work done in fig (ii) is least c) the work done is the same in fig (i), fig (ii), and fig (iii) d) the work done in fig (iii) is greater than fig (ii) but equal to that in fig (i)

The correct answer is c) the work done is the same in fig (i), fig (ii), and fig (iii)

### A capacitor of 4μF is connected as shown in the circuit. The internal resistance of the battery is 0.5Ω. The amount of charge on the capacitor plates will be a) 0 b) 4μC c) 16μC d) 8μC

The correct answer is d) 8μC

### A ball is dropped and its displacement vs time graph is as shown in the figure where displacement x is from the ground and all quantities are positive upwards. a) Plot qualitatively velocity vs time graph b) Plot qualitatively acceleration vs time graph

a) At t=0 and v=0 , v-t graph is: b) At x = 0, a-t graph is:

### A positively charged particle is released from rest in a uniform electric field. The electric potential energy of the charge a) remains a constant because the electric field is uniform b) increases because the charge moves along the electric field c) decreases because the charge moves along the electric field d) decreases because the charge moves opposite to the electric field

The correct answer is c) decreases because the charge moves along the electric field

### Graph the solution sets of the following inequations: Given $x+2 y>1$ $\begin{array}{l} \Rightarrow 2 y>1-x \\ \Rightarrow y>\frac{1}{2}-\frac{x}{2} \end{array}$ Consider the equation $y=\frac{1}{2}-\frac{x}{2}$ Finding points on the...

### 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 $y$...

### 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...

### 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 , 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 $2 x+y=6$...

### 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:...

### A die is tossed twice. A ‘success’ is getting an even number on a toss. Find the variance of the number of successes.

Let’s consider E to be the event of getting even number on tossing a die.

### Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard variation of the random variable X where X is the number of aces.

Let’s consider X to be the random variable such that X = $0,1,2$ Now, let E = the event of drawing an ace And, F = the event of drawing non – ace So,

### A bag contains coins. It is known that n of these coins has a head on both sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is , determine the value of n.

Given, n coins are two headed coins and the remaining $(n+1)$ coins are fair. Let  ${{E}_{1}}$ : the event that unfair coin is selected ${{E}_{2}}$ : the event that the fair coin is selected...

### There are two bags, one of which contains black and white balls while the other contains black and white balls. A die is thrown. If it shows up or , a ball is taken from the Ist bag; but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.

Let ${{E}_{1}}$ be the event of selecting Bag $1$ and ${{E}_{2}}$ be the event of selecting Bag $2$. Also, let ${{E}_{3}}$ be the event that black ball is selected Now,...

### A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.

Let E1 be the event that the letter comes from TATA NAGAR, E2 be the event that the letter comes from CALCUTTA And, E3 be the event that on the letter, two consecutive letters TA are visible Now,...

### Three bags contain a number of red and white balls as follows: Bag red balls, Bag red balls and white ball Bag white balls. The probability that bag i will be chosen and a ball is selected from it is . What is the probability that (i) a red ball will be selected? (ii) a white ball is selected?

Given: Bag $1:3$ red balls, Bag $2:2$ red balls and $1$ white ball Bag $3:3$  white balls Now, let E1, E2 and E3 be the events of choosing Bag $1$, Bag $2$ and Bag $3$ respectively and...

### 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...

### An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.

Let’s consider A to be the event of having m white and n black balls ${{E}_{1}}$ = First ball drawn of white colour ${{E}_{2}}$ = First ball drawn of black colour ${{E}_{3}}$  = Second ball...

### A box has blue and red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of second ball being blue?

Given that the box has $5$ blue and $4$ red balls. Let us consider ${{E}_{1}}$ be the event that first ball drawn is blue and ${{E}_{2}}$ be the event that first ball drawn is red. And, E is...

### Bag I contains black and white balls, Bag II contains black and white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

According to the question: Bag $1$ has $3$B, $2$W balls and Bag $2$ has $2$B, $4$W balls. Let ${{E}_{1}}$ = The event that bag $1$ is selected ${{E}_{2}}$ = The event that bag...

### A bag contains white and black balls. Another bag contains white and black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Let us consider  ${{W}_{1}}$ and ${{W}_{2}}$ to be two bags containing $\left( 4W,\text{ }5B \right)$and $\left( 9W,\text{ }7B \right)$balls respectively. Let us take ${{E}_{1}}$ be the...

### Suppose tickets are sold in a lottery each for Re . First prize is of Rs and the second prize is of Rs. . There are three third prizes of Rs. each. If you buy one ticket, what is your expectation.

Let’s take X to be the random variable where X = $0,500,2000$and $3000$

### Three dice are thrown at the same time. Find the probability of getting three two’s, if it is known that the sum of the numbers on the dice was six.

Given that the dice is thrown three times So, the sample space n(S) = ${{6}^{3}}~=\text{ }216$ Let E1 be the event when the sum of number on the dice was $6$ and ${{E}_{2}}$be the event when...

### Let E1 and E2 be two independent events such that and . Describe in words of the events whose probabilities are: Here, $p({{\mathbf{E}}_{\mathbf{1}}})\text{ }=~{{p}_{\mathbf{1}}}~$ and $\mathbf{P}({{\mathbf{E}}_{\mathbf{2}}})\text{ }=\text{ }{{\mathbf{p}}_{\mathbf{2}}}$ Now, its clearly seen that either...

### 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?

As per the solution of exercise 14, we have Maximize $Z\text{ }=\text{ }200x\text{ }+\text{ }120y$subject to constrains $3x\text{ }+\text{ }y\le 600$…. (i) $x\text{ }+\text{ }y\le 300$…. (ii)...

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

From the solution of exercise 13, we have The objective function for maximum profit $Z\text{ }=\text{ }100x\text{ }+\text{ }170y$ Subject to constraints, $x\text{ }+\text{ }4y\le 1800$…. (i)...

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

As per the solution of exercise 12, we have The objective function for minimum cost is $Z\text{ }=\text{ }400x\text{ }+\text{ }200y$ Subject to the constrains; $5x\text{ }+\text{ }2y\ge 30$….....

### Refer to Exercise . 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.

As per the solution of exercise $11$, we have Maximize $Z\text{ }=\text{ }50x\text{ }+\text{ }60y$subject to the constraints $20x\text{ }+\text{ }10y\le 200\text{ }2x\text{ }+\text{ }y\le 20$…...

### A man rides his motorcycle at the speed of km/hour. He has to spend Rs per km on petrol. If he rides it at a faster speed of km/hour, the petrol cost increases to Rs per km. He has at most Rs 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.

Let’s assume the man covers x km on his motorcycle at the speed of $50$km/hr and covers y km at the speed of $50$ km/hr and covers y km at the speed of $80$ km/hr. So, cost of petrol =...

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

In the given figure, it’s seen that the feasible region is ABCA. The corner points are $C\left( 0,\text{ }3 \right),\text{ }B\left( 0,\text{ }5 \right)$’and for A, we have to solve equations...

### Find the distance of a point (2, 4, –1) from the line (x+5)/1=(y+3)/4=(z-6)/-9

According to ques,

### If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane.

According to ques, Points are : $\left( 2,\text{ }\text{ }1,\text{ }\text{ }3 \right)\text{ }and\text{ }\left( 1,\text{ }\text{ }3,\text{ }3 \right)$ And, Direction ratios of the normal to the...

### Prove that the lines x = py + q, z = ry + s and x = p¢y + q¢, z = r¢y + s¢ are perpendicular if pp¢ + rr¢ + 1 = 0

According to ques,

### The position vector of the point which divides the join of points 2a-3b and a+b in the ratio 3 : 1 is (A) (3a-2b)/2 (B) (7a-8b)/4 (C) 3a/4 (D) 5a/4

CORRECT ANSWER: $\left( D \right)5a/4~$ ACCORDING TO QUES , Given ratio is 3:1

### The vector in the direction of the vector i-2j+k that has magnitude 9 is (A) i-2j+2k (B)(i-2j+2k)/3 (C)3(i-2j+2k) (D)9(i-2j+2k)

CORRECT ANSWER:  $\left( C \right)\text{ }3\left( i-2j+2k \right)$

### If a=i+j+k and b=j-k, find a vector c such that axc=b and a.c=3

according to ques,

### Show that area of the parallelogram whose diagonals are given by a and b is (axb)/2. Also find the area of the parallelogram whose diagonals are 2i-j+k and i+3j-k

according to ques,

### If a,b,c determine the vertices of a triangle, show that ½[bxc+cxa+axb] gives the vector area of the triangle. Hence deduce the condition that the three points a,b,c are collinear. Also find the unit vector normal to the plane of the triangle.

according to ques,

### Prove that in any triangle ABC, , where a, b, c are the magnitudes of the sides opposite to the vertices A, B, C, respectively.

according to ques,

### Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.

According to ques, proved.

### Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).

according to ques,

### If A, B, C, D are the points with position vectors i+j-k, 2i-j+3k, 2i-3k, 3i-2j+k respectively, find the projection of AB along CD.

According to ques,

### Find the sine of the angle between the vectors a=3i+j+2k and b=2i-2j+4k

according to ques,

### If a+b+c=0 , show that axb=bxc=cxa . Interpret the result geometrically?

according to ques,

### Find the angle between the vectors:2i-j+k and 3i+4j-k.

according to ques,

### Find a vector of magnitude 6, which is perpendicular to both the vectors 2i-j+2k and 4i-j+3k .

according to ques,

### A vector r has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of r , given that r makes an acute angle with x-axis.

According to ques,

### If y = e–x (A cos x + B sin x), then y is a solution of: (A) d^2y/dx^2+2dy/dx=0 (B)d^2y/dx^2-2dy/dx+2y=0 (C)d^2y/dx^2+2dy/dx=2y=0 (D)d^y/dx^2+2y=0

Correct option :(C). According to ques, $y~=~{{e}^{x}}~\left( A\text{ }cos~x~+\text{ }B\text{ }sin~x \right)$ Differentiating both sides with respect to x,

### The order and degree of the differential equation d^2y/dx^2=(dy/dx)^1/4+x^1/5=0 respectively, are (A) 2 and not defined (B) 2 and 2 (C) 2 and 3 (D) 3 and 3

Correct option : (A) 2 and not defined According to ques, Since the degree of dy/dx is in fraction its undefined and the degree is $2.$

### Choose the correct answer from the given four options: The degree of the differential equation [1+(dy/dx)^2]^3/2=d^2y/dx^2 is (A) 4 (B) 3/2 (C) not defined (D) 2

Correct option: D (2) According to ques, hence the answer is 2.

### Choose the correct answer from the given four options. The degree of the differential equation (d^2y/dx^2)^2+(dy/dx)^2=xsin(dy/dx)is: (A) 1 (B) 2 (C) 3 (D) not defined

Correct option: (D) not defined. As the value of sin (dy/dx) on expansion will be in increasing power of dy/dx,

### Solve : x dy/dx (log y-logx+1)

According to ques,

### Solve: dy/dx = cos(x + y) + sin (x + y). [Hint: Substitute x + y = z]

According to ques,

### Find the general solution of (1 + tan y) (dx – dy) + 2xdy = 0.

according to ques,

according to ques, concentric circles with centre $\left( 1,\text{ }2 \right)$ and with radius ‘r’ can be written as, \[{{\left( x\text{ }\text{ }1 \right)}^{2}}~+\text{ }{{\left( y\text{ }\text{...