Mathematics III Past Paper PDF

# MATHEMATICS - III ## Sequence and Series - Sequence and Series 2 - Multivariable Calculus (Differentiation) 28 - Multivariable Calculus (Integration) 57 - Ordinary Differential Equation 80 - Graph Theory 105 ## NOTE MAKAUT course structure and syllabus of 3rd Sem has been changed from 2019. Pr...

# MATHEMATICS - III ## Sequence and Series - Sequence and Series 2 - Multivariable Calculus (Differentiation) 28 - Multivariable Calculus (Integration) 57 - Ordinary Differential Equation 80 - Graph Theory 105 ## NOTE MAKAUT course structure and syllabus of 3rd Sem has been changed from 2019. Present syllabus of MATHEMATICS III [for CS & IT] has been designed and structured with selected topics from other MATHEMATICS of different years and branches. Taking special care of this matter we are providing the relevant MAKAUT university solutions as model questions & answers along with the complete solutions of new university papers, so that students can get an idea about university questions patterns. ## SEQUENCES AND SERIES ### Chapter at a Glance - A sequence *u* is said to be bounded above or bounded below according as the range set of *u* is bounded above or bounded below. The sequence *u* is said to be bounded if its range set is both bounded above and below. - A sequence *u* is said to be bounded above if there exist a real number U such that *u* ≤ *U*, *Vn* ∈ *N*. U is known as an upper bound of the sequences *u*. - A sequence *u* is said to be bounded below if there exist a real number L such that *u* ≥ *L*, *Vn* ∈ *N*. L is known as a lower bound of the sequence *u*. - A sequence *u* is said to be unbounded above or unbounded below if it is not bounded above or bounded below. - A sequence *u* is said to be bounded if real numbers *L,U* such that *L* ≤ *u* ≤ *U*, *Vn* ∈ *N*. - A sequence *u* is said to be Convergent if *lim* *u* is a finite quantity. - A sequence *u* is said to be Divergent if *lim* *u* = ±∞. - A sequence which is neither convergent nor divergent is known as oscillatory sequence. - If a series is convergent, then its nth term tends to zero as *n*→∞ i.e., *lim* *u* = 0. - If the nth term of a series does not tends to zero as *n* → ∞ i.e., if *lim* *u* ≠ 0, then the series is divergent. ### D'Alembert's Ratio Test: - **Statement:** Let ∑*u* be a series of positive terms and *lim* *u/u* = *l*. - Then ∑*u* converges if *l* < 1. - ∑*u* diverges if *l* > 1. - **No conclusion can be drawn if *l* = 1 i.e., the test fails for *l* = 1.** ### Cauchy's Root Test: - **Statement:** Let ∑*u* be a series of positive terms and *lim* √*u* = *l*. - Then ∑*u* converges if *l* < 1. - ∑*u* diverges if *l* > 1. - **No conclusion can be drawn if *l* = 1 i.e., the test fails for *l* = 1.** ### Alternating Series: - A series in which the terms are alternatively positive and negative is called an alternating series and the series is denoted by: ∑(-1)^n *u* = *u₁ - u₂ + u₃ - u₄......+u* - *u* ≥ 0, *n* is such that - *u*₁₊₁ ≤ *u* i.e., {*u*} is monotonic decreasing and - *u* → 0 as *n*→∞, *lim* *u* = 0 then the series is convergent. - **Note:** If *lim* *u* ≠ 0, the alternating series is Oscillatory. ### Absolutely convergent series: - An infinite series ∑*u* of real constants is said to be absolutely convergent if the series ∑|*u*| i.e., if |*u₁*| + |*u₂*| + |*u₃*| +...+|*u*| +... is convergent. - **Note:** A convergent series of real non-negative terms is absolutely convergent. ### Conditionally convergent series: - An infinite series ∑*u* of real constants is said to be conditionally convergent if the series ∑*u* is convergent but not absolutely convergent (i.e., ∑|*u*| is not convergent). ### Theorem: - If a series ∑ *u* of real constants is said to be absolutely convergent, then it is convergent i.e., absolute convergence of a series implies its convergence. - **Results:** For proving absolute convergence, we apply the known tests of positive series on ∑|*u*|. Thus, ∑|*u*| converges absolutely, if - *lim* √|*u*| < 1 (**Root Test**) - or, *lim* |*u*₁₊₁/u*| < 1 (**Ratio Test**) ### Very Short Answer Type Questions: 1. The series ∑ (-1)^n/√*n*₊₁ is convergent if: - a) p < 1 - b) p > 1 - c) p = 1 - d) none of these - **Answer:** (b) 2. On which region log(1+x) can be expanded in an infinite series? - **Answer:** -1 < x ≤ 1. 3. What is the nature of the series ∑ (-1)^n/√*n*√*n*₊₁ √*n*₊₂? - **Answer:** The series is convergent. 4. Determine whether the sequence *x*, where *x* = 1/(1-2) + 1/(2-3) + 1/(3-4) +....+1/(n-1) + 1*(n-1)(n+1) converges or not. - **Answer:** Given series is convergent. 5. Determine whether the series ∑ (sin^2(*n*))/(2*n* + 1) converges or not. - **Answer:** The series is divergent. 6. Give an example of a bounded sequence which is not convergent. - **Answer:** An example of a bounded sequence that is not convergent is the sequence {(-1)^n}. 7. The sequence {1/3, 1/3^2, 1/3^3, 1/3^4......}is: - a) monotonic increasing - b) oscillatory - c) monotonic decreasing - d) divergent - **Answer:** (d) 8. The Sequence {1, 1/3, 1/3^2, 1/3^3......}is: - a) divergent - b) oscillatory - c) convergent - d) none of these - **Answer:** (c) 9. ∑ (1/2*n* + 1) is: - a) convergent - b) divergent - c) neither convergent nor divergent - d) none of these - **Answer:** (b) 10. The series ∑ 1/(*n*(*n* + 1)) = 1/2 + 1/(2*3) + 1/(3*4) + 1/(4*5) +......is? - a) absolutely convergent - b) conditionally convergent - c) oscillatory - d) none of these - **Answer:** (b) 11. If f(x) = x sin x, -π ≤ x ≤ π, f(x) = ((a₀)/2) + ∑ (aₙ cos nx + bₙ sin nx), then the value of a₁ is - a) 2 - b) 0 - c) 4 - d) 1 - **Answer:** (c) 12. The period of cos 2x is: - a) 2π - b) π - c) 2 - d) 1 - **Answer:** (b) 13. The period of the function f(x) = sin x is: - a) 2π - b) π/2 - c) 3π - d) π - **Answer:** (c) 14. A function f(x) = x^2, -π ≤ x ≤ π is represented by a Fourier series ∑ (aₙ cos nx + bₙ sin nx). Then the value of b₁ is - a) 2π²/3 - b) 4(-1)^n - c) 0 - d) none of these - **Answer:** (c) 15. The least upper bound of the sequence {1, 1/2, 1/3, 1/4,..... 1/(*n* + 1)} is? - a) 0 - b) 1/2 - c) 1 - d) 2 - **Answer:** (c) 16. A function f(x), a<x<b, can be expanded in a Fourier series: - a) only if it is continuous everywhere - b) even if it is discontinuous at a finite number of points in (a, b) - c) even if it is unbounded in (a, b) - d) only if it is both continuous & bounded in (a, b) - **Answer:** (b) 17. The series ∑ (1/2*n* + 1) is - a) convergent - b) divergent - c) oscillatory - d) none of these - **Answer:** (a) ### Short Answer Type Questions: 1. Discuss the convergence of the series 1 + 2p/2! + 3p/3! + 4p/4! +...... 2. Test the series ∑(1/√*n*√*n*₊₁ √*n*₊₂) 3. Verify that e = 1+x+x²/2! + x³/3! + x⁴/4! + x⁵/5! + x⁶/6! +... 4. Give an example of a sequence which is bounded but not convergent. 5. Show that the sequence {*U*}, where *U* = 2(-1)^*n* does not converge. 6. Test the convergence of the series ∑ ((√*n* + 1) -√*n*)/√*n*√*n* + 1). 7. Test the convergence of the series ∑ (1/3^n + 1/4^n + 1/5^n +.....) 8. Test the nature of the series ∑ (3.5/1.2) + (3.5.7/1.2.3) + (3.5.7.9/1.2.3.4) +..... 9. For what values of x, the following series is convergent: x^2/(1.3) + x^4/(3.5) + x^6/(5.7) +..... 10. Find whether the following series is convergent: ∑ (2/3)^n + (2/4)^n + (2/5)^n +... 11. Discuss the convergence of the series ∑((1+1/√*n*) - 1)/√*n*. 12. Discuss the convergence of the series ∑cos(*n*π)/(*n*² + 1). Is it absolutely convergent? ### MULTIVARIABLE CALCULUS (DIFFERENTIATION) ## Chapter at a Glance - The ordered pair (x, y) of two numbers *x,y*∈R (set of real numbers) is a point in two dimensional space and the set of all such two-dimensional points forms a space which is known as two-dimensional Euclidean space and the space is denoted by R². Similarly, the ordered triple (x, y, z) of three numbers *x, y, z*∈R is a point in three-dimensional space and the set of all such three-dimensional points forms a space which is known as three dimensional Euclidean space and the space is denoted by R³ and so on. - Previously, we have studied functions of one variable, which we often wrote as *y* = *f(x)*, where *x* is the independent variable and *y* is the dependent variable. Here, we will extend our idea to functions of two variables, denoted as *z* = *f(x, y)*, where *x, y* are independent variables and *z* is the dependent variable. For example, *f(x, y) = x² + y²* is a function of two variables and *x, y* are real numbers, i.e., *(x,y)*∈R², *f* possesses a real value for all values of *(x, y)* i.e., for all *(x, y)*∈R², *f*∈R. ### Limit of a function: - A function *f(x, y)* has the limit *l* (as *(x, y)* → *(a, b)* if for every 𝛿 > 0, there is a 𝛿>0 such that |*f(x, y) - l*| < 𝛿, whenever 0 < (*x - a*)² + (*y - b*)² < 𝛿². This is denoted by *lim* _(x,y)→(a,b)_ *f(x,y)* = *l*.  ### Differentiation of Implicit Function: - Taking x as the only independent variable, it may be considered f(x, y) = C, in which both x and y are functions of x. - Now, differentiating both sides of the equation *f(x,y) = C* with respect to x, we get - *∂f/∂x* + *∂f/∂y*(*dy/dx*) = 0 - *∂f/∂x* + *∂f/∂y* (*dy/dx*) = 0 - *dy/dx* = -(*∂f/∂x) / (*∂f/∂y*). ### Jacobian: - Let u = u(x, y) and v = v(x,y). The Jacobian of *u,v* with respect to x, y is denoted by *J* and defined as: - *∂(u,v)/∂(x,y)* = *∂(u,v)/∂(x,y)* - *∂(u,v)/∂(x,y)* = *∂u/∂x* *∂v/∂y* - *∂u/∂y* *∂v/∂x*. ### Properties of Jacobian: - *∂(x,y)/∂(u,v)* = 1 / *∂(u,v)/∂(x,y)* i.e., *JJ' = 1*. - If u and v are functions of r,θ: then *∂(u,v)/∂(x,y)* = *∂(u,v)/∂(r,θ)* *∂(r,θ)/∂(x,y)*. ### Maxima & Minima of functions of several variables - **Stationary point or critical point:** The points satisfying the conditions *f*₁ = *f*₂ = 0 are called stationary points or, critical points. - **Extreme point:** The point at which the function attains maximum or minimum value is called the extreme point. - **Saddle point:** A saddle point is a stationary point but not an extreme point. In other words, at the saddle point the function has neither maxima nor minima. - **Necessary conditions:** The necessary conditions for f(x, y) to have an extreme value at (a, b) are *f*₁(*a,b*) = 0 and *f*₂(*a,b*) = 0. - **Sufficient Conditions:** Let *f*₁(*a,b*)=0 and *f*₂(*a,b*) = 0; and let *f(x,y)* have continuous partial derivatives up to the second order in the neighborhood of *(a,b)*, then if - A = *f*₁₁(*a,b*), B = *f*₁₂(*a,b*), C = *f*₂₂(*a,b*) - *f(a,b)* is maximum if *AC - B² > 0* and *A < 0* - *f(a, b)* is minimum if *AC - B² > 0* and *A > 0* - If *AC-B² < 0* then *f(a,b)* is not an extreme value, i.e., *(a, b)* is a saddle point. - If *AC-B² = 0*, then no conclusion can be made. ## **Very Short Answer Type Questions**: 1. *f(x,y) = (√*y* + √*x*)/(y + x)* is a homogeneous function of degree? - **Answer:** 1/2 2. If for any x, λx = 0, then λ is called as? - **Answer:** Irrotational vector field. 3. If *f(x,y) = x + y³, find the value of *f*₁(*0,0*)? - **Answer:** 0. 4. Is the function *f(x,y) = xy + x - y*, when *(x,y) ≠ (0,0)*, *f(x,y) = 0* when *(x,y) = (0,0)*, continuous at (1, 2)? - **Answer:** Yes. 5. Is the function *f(x, y) = (x² + y² + xy)*, when *(x,y) ≠ (2, 3)*, *f(x, y) = 10*, when *(x, y) = (2,3)*, continuous at (0,0)? - **Answer:** Yes. 6. Find the degree of the homogeneous function *f(x, y) = (√*x* + √*y*)/y*? - **Answer:** 0. 7. The *lim* _(x,y)→(0,0)_ *xy/(x^2 + y^2)* does not exist - **Answer:** True 8. If *x = r cosθ, y = r sinθ*, then d(x,y)/d(r,θ) is: - **Answer:** *r* 9. If *u = (x² + y²)/2*, find the value of 'n' so that *xu₁ + yu₂ = nu*. - **Answer:** 2. 10. *f(x,y) = (√*x* + √*x*)/(y + x)* is a homogeneous function of degree? - **Answer:** 0 11. If *u(x, y) = tan-¹* (*y/x)*, then the value of *x(∂u/∂x) + y(∂u/∂y)* is? - **Answer:** *u(x,y)*. ### MULTIVARIABLE CALCULUS (INTEGRATION) ## Chapter at a Glance ### Introduction: - In Calculus II, students defined the integral ∫*a*_*b* *f(x)* *dx* over a finite interval [*a,b*]. The function *f* was assumed to be continuous, or at least bounded, otherwise the integral was not guaranteed to exist. Assuming an antiderivative of *f* could be found, ∫*a*_*b* *f(x)* *dx* always existed and was a number. In this section, we investigate what happens when these conditions are not met. ### Definition (Improper Integral): - An integral is an improper integral if either the interval of integration is not finite (improper integral of type 1) or if the function to integrate is not continuous (not bounded) in the interval of integration (improper integral of type 2). ### Example: - ∫*0*_*∞* *e*^-*x* *dx* is an improper integral of type I since the upper limit of integration is infinite.   - ∫*0*_*1*  *(1/x)* *dx* is an improper integral of type 2 because *1/x* is not continuus at 0 - ∫*1*_*∞*  *(1/(*x*-1))* *dx* is an improper integral of types I since the upper limit of integration is infinite. It is also an improper integral of type 2 because *(1/(*x*-1))* is not continuous at 1 and 1 is in the interval of integration. - ∫*(-1)*_*1*  *(1/(*x*-1))* *dx* is an improper integral of type 2 because *(1/(*x*-1))* is not continuous at -1 and 1.  - ∫*0*_*π/2* *tan x* *dx* is an improper integral of type 2 because *tan x* is not continuous at π/2. ### Evaluation of Area of a plane region: - Let R be the region bounded by a closed curve C in XOY plane. Then the area of the region R = ∫∫*R* *dxdy*. - Similarly, the area lying in ZOX or in YOZ plane can be formulated. ### Evaluation of volume of a Solid region: - Volume of the cylindrical solid enclosed by the region R, lying on XOY plane and the surface *Z* = *f(x, y)* defined on R, where the generators are parallel to *z*-axis. - The volume of a solid region in three-dimensional space ∫∫∫*V* *dxdydz*. ### Evaluation of Volume and Surface of Revolution: - **When axis of revolution is X-axis**: - Let R be the region bounded by the curve *y* = *f(x)*, X-axis and the ordinates *x* = *a* and *x* = *b*. Then the formulae for finding the volume and surface area due to revolution of the region R about X-axis is   - Volume of revolution: *π∫*_*a*_*b* {*f(x)*}² *dx*    - Surface area of revolution: *2π∫*_*a*_*b* *f(x)*√(1 + {*f'(x)*}²) *dx* - **When axis of revolution is Y-axis**:    - Let R' be the region bounded by the curve *x* = *g(y)*, Y-axis and the straight lines *y* = *c* and *y* = *d*. Then the formulae for finding the volume and surface area due to revolution of the region R' about Y - axis is   - Volume of revolution: *π∫*_*c*_*d* {*g(y)*}² *dy*   - Surface area of revolution: *2π∫*_*c*_*d* *g(y)*√(1 + {*g'(y)*}²) *dy* ### Very Short Answer Type Questions: 1. What is the area of the region bounded by *x*-axis, *y* = *e*ˣ, *x* = 0, *x* = 1? - **Answer:** *e*⁻¹ 2. Find the value of ∫∫∫*V* *xyz*² *dxdydz* ,where *V* ={(x,y,z): -1 ≤ x ≤ -1, -2 ≤ y ≤ -2, -3 ≤ z ≤ -3}. - **Answer:** 0. 3. If ∫*C* *(ydx + xdy)* = *p* where *C* is given by *x* = cosθ, *y* = sinθ, 0 ≤ θ ≤ π, find the value of *p*. - **Answer:** *p* = 0. 4. If C is the circle *x² + y² = 4*, find the value of *∫*_*C* *xdx*. - **Answer:** 0. ### ORDINARY DIFFERENTIAL EQUATION (ODE) ## Chapter at a Glance ### Overview of Differential Equations - A differential equation is an equation, the unknown is a function, and both the function and its derivatives may appear in the equation. Differential equations are essential for a mathematical description of nature. They are at the core of many physical theories: Newton's and Lagrange equations for classical mechanics, Maxwell's equations for classical electromagnetism, Schrödinger's equation for quantum mechanics, and Einstein's equation for the general theory of gravitation, to mention a few of them. The following examples show how differential equations look like. #### Examples: - **(a)** Newton's second law of motion for a single particle. The unknown is the position in space of the particle, *x(t)*, at the time *t*. From a mathematical point of view, the unknown is a single variable vector-valued function in space. This function is usually written as *x* or *x:R→R³, * where the function domain is every *t∈R* and the function range is any point in space *x(t) ∈ R³*. The differential equation is - *d²x/dt² = f(t, x(t))*,  - where the positive constant *m* is the mass of the particle and *f: R x R³ → R³* is the force acting on the particle, which depends on the time and the position in space *x*. This is the well-known law of motion: mass times acceleration equals force. - **(b)** The time decay of a radioactive substance. The unknown is a scalar-valued function *u:R→R*, where *u(t)* is the concentration of the radioactive substance at the time *t*. The differential equation is   - *du/dt (t) = -ku(t)*,     - where *k* is a positive constant. The equation says the higher the material concentration, the faster it decays. - **(c)** The wave equation, which describes waves propagating in a media. An example is sound, where pressure waves propagate in the air. The unknown is a scalar-valued function of two variables *u: R x R→R*, where *u(t, x)* is a perturbation in the air density at the time *t*  and point *x = (x, y, z)* in space. (We used the same notation for vectors and points, although they are different types of objects). The equation is   - *∂²u(t,x)/∂t² = v² (∂²u(t,x)/∂x² + ∂²u(t,x)/∂y² + ∂²u(t,x)/∂z²)*,    - where *v* is a positive constant describing the wave speed and we have used the notation *∂* to mean partial derivative. - **(d)** The heat conduction equation, which describes the variation of temperature in a solid material. The unknown is a scalar-valued function *u*, where *u(t,x)* is the temperature at time *t* and the point *x = (x, y, z)* in the solid. The equation is   - *∂u(t,x)/∂t = v² *(∂²u(t,x)/∂x² + ∂²u(t,x)/∂y² + ∂²u(t,x)/∂z²)*.      - where *k* is a positive constant representing thermal properties of the material. - The equations in examples (a) and (b) are called Ordinary Differential Equations (ODE), since the unknown function depends on a single independent variable, t in these examples. The equations in examples (c) and (d) are called partial differential equations (PDE), since the unknown function depends on two or more independent variables, t, x, y, and z in these examples, and their partial derivatives appear in the equations. - The order of a differential equation is the highest derivative order that appears in the equation. Newton's equation in Example (a) is second order, the time decay equation in Example (b) is first order, the wave equation in Example (c) is a second derivative order in time and space variables and the heat equation in Example (d) is first order in time and second order in space variables. ### Linear Equations - A good start is a precise definition of the differential equations we are about to study in this Chapter. We use primes to denote derivatives,  - *dy(t)/dt = y'(t)*. - This is a compact notation and we use it when there is no risk of confusion. - **Definition 1.1:** A first order ordinary differential equation in the unknown y is   - *y'(t) = f(t, y(t))* - where *y: R→R* is the unknown function and *f: R x R → R* is a given function. The equation in (i) is called linear iff the function with values *f(t, y)* is linear on its second argument, that is, there exist functions *a, b: R→R* such that   - *y'(t) = a(t)y(t)+b(t), f(t,y) = a(t)y+b(t)* - A different sign convention for Eqn. (1.1.2) may be found in the literature. For example, Boyce-DiPrima [3] writes it as y' = -ay + b. The sign choice in front of function *a* is just a convention. Some people like the negative sign, because later on, when they write the equation as *y'+ay = b*, they get a plus sign on the left-hand side. In any case, we stick here to the convention *y'=ay+b*. - A linear first order equation has constant coefficients iff both functions *a* and *b* in Eqn. (ii) are constants. Otherwise, the equation has variable coefficients. #### Examples: - **(a)** An example of a first order linear ODE is the equation   - *y'(t) = 2y(t) + 3* - **(b)** Another example of a first order linear ODE is the equation  - *y'(t) = -2y/t+4t*. - In this case, the right-hand side is given by the function *f(t, y) = - 2y/t + 4t*, where *a(t) = -2/t* and *b(t) = 4t*. Since the coefficients are non-constant functions of *t*, this is a variable coefficients equation. - A function *y: DCR→R* is a solution of the differential equation in (i) iff the equation is satisfied for all values of the independent variable *t* in the domain *D*  of the function *y*. ### Linear Equations with Constant Coefficients - Constant coefficient equations are simpler to solve than variable coefficient ones. There are many ways to solve them. Integrating each side of the equation, however, does not work. For example, take the equation   -  *y' = 2y + 3*     - and integrate on both sides,    - ∫*y'(t)* *dt* = 2∫*y(t)* *dt* + 3*t* + *c*, *c* ∈ *R*. - The fundamental Theorem of Calculus implies *y(t) = ∫y'(t)* *dt*. Using this equality in the equation above we get   - *y(t) = 2∫y(t)* *dt* + 3*t* + *c*. - We conclude that integrating both sides of the differential equation is not enough to find the solution *y*. We still needs to find a primitive of *y*. Since we do not know *y*, we cannot find its primitive. The only thing we have done here is to rewrite the original differential equation as an integral equation. That is why *integrating both sides of a linear equation does not work*. - One needs a better idea to solve a linear differential equation. We describe here one possibility, the integrating factor method. Multiply the differential equation by a particularly chosen non-zero function, called the integrating factor. Choose the integrating factor having one important property. The whole equation is transformed into a total derivative of a function, called the potential function. Integrating the differential equation is now trivial, the result is the potential function equal to any constant. Any solution of the differential equation is obtained inverting the potential function. This whole idea is called the integrating factor method. - In the next Section we generalize this idea to find solutions to variable coefficients linear ODE and generalize this idea to certain non-linear differential equations, we are able to state in a theorem a precise formula for the solutions of constant coefficient linear equations. ### Theorem (Constant Coefficients): - The linear differential equation   -   *y'(t) = ay(t) + b*     - where *a ≠ 0, b* are constants, has infinitely many solutions labeled by *c* ∈ *R* as follows.   - *y(t) = ce*^(at/b)* ### Remarks: - **(a)** Why do we start the Proof of Theorem multiplying the equation by the function *e*^(at)? At first sight, it is not clear where this idea comes from. In Lemma we show that only functions proportional to the exponential *e*^(at)* have the property needed to be an integrating factor for any differential equation. In Lemma we multiply the differential equation by a function *μ(t) = ce*^(at)* and only then one finds that this function must be *μ(t) = ce*^(at)*. - **(b)** Since the function *μ* is used to multiply the original differential equation, we can freely choose the function with *c = 1*, as we did in the proof of Theorem. - **(c)** It is important we understand the origin of the integrating factor *e*^(at)* in order to extend results from constant coefficients equations to variable coefficients equations. ### Very Short Answer Type Questions: 1. The order and degree of the differential equation *d²y/dx²* = (dy/dx)^3 + *x*² are: - **Answer:** 2, 2 2. The general solution of *xp² - yp + 1 = 0* is: - **Answer:** *y = cx + 1/c* 3. If *iu + v = x, uv = y* then *∂(x,y)/∂(u,v)* is: - **Answer:** *u - v*. 4. If *f(x, y) = x² + 3xy² + y²*, then the value of *∂f/∂x* + *∂f/∂y* for *x+y* is: - **Answer:** 3*f* 5. The differential equation *(Ax + By)*dx + *(Cx + Dy)* dy = 0 is exact of: - **Answer:** *B = C*. 6. What is the general form of Clairaut's equation? - **Answer:** y = px + f(p), where *p = dy/dx*. 7. Find the value of *(D² + 4)^-1 (sin 2x)*. - **Answer:** -cos 2*x*. 8. Find the general and singular solution of *y = 4xp - 16y