MTH 165 LPU Question Paper: Complete Guide with Questions, Answers, and Exam Preparation Strategy

Mathematics forms the backbone of engineering and science education, and at Lovely Professional University (LPU), MTH 165 is one of the most important foundational courses that students encounter in their first or second semester. The subject covers a wide array of mathematical concepts ranging from calculus and differential equations to linear algebra and complex numbers. For students who are searching for the mth 165 lpu question paper, this comprehensive guide is designed to help you understand the pattern of the examination, prepare effectively, and build a strong conceptual foundation. This blog covers more than just questions and answers — it walks you through the complete framework of what to expect, how to study, and what kinds of problems are most likely to appear in your examination.

The mth 165 lpu question paper typically follows a structured format that tests both theoretical understanding and practical problem-solving ability. Many students feel overwhelmed when they first look at the syllabus, but with the right approach and the right study material, cracking this paper becomes entirely achievable. In this blog, we have compiled an extensive set of questions and detailed answers across all major topics covered in MTH 165, along with tips, strategies, and topic-wise breakdowns to ensure you are fully prepared when exam day arrives.

Understanding the MTH 165 Course at LPU

MTH 165 at Lovely Professional University is a core mathematics course that is mandatory for most engineering and technology programs. The course is designed to develop analytical thinking and mathematical rigor in students who will go on to apply these concepts in fields like computer science, electrical engineering, mechanical engineering, and more.

The course is typically divided into five major units, each carrying equal or proportional weightage in the final examination. These units generally cover the following broad areas:

• Differential Calculus — including limits, continuity, differentiability, and applications of derivatives
• Integral Calculus — definite and indefinite integrals, techniques of integration, and applications
• Ordinary Differential Equations — first-order and higher-order equations, methods of solving them
• Linear Algebra — matrices, determinants, rank, eigenvalues, and eigenvectors
• Complex Numbers and Series — complex algebra, series expansions, convergence tests

Understanding this structure is crucial because when you obtain the mth 165 lpu question paper from previous years, you will notice that questions are generally distributed across these five areas. The paper usually has multiple sections — short-answer questions worth 2 to 5 marks each, followed by long-answer questions worth 8 to 10 marks each. Having clarity on this format helps you allocate your preparation time wisely.

Detailed Topic-Wise Questions and Answers

Unit 1: Differential Calculus

Differential calculus is the study of rates of change and slopes of curves. It is one of the most heavily tested areas in the mth 165 lpu question paper. Below are detailed questions along with complete explanations and solutions.

Q1. Define the concept of a limit and explain what it means for lim(x→a) f(x) = L.

A limit describes the value that a function approaches as the input approaches a specific point. Formally, lim(x→a) f(x) = L means that for every epsilon > 0, there exists a delta > 0 such that whenever 0 < |x – a| < delta, we have |f(x) – L| < epsilon. In simpler terms, as x gets arbitrarily close to ‘a’, f(x) gets arbitrarily close to L. It is important to note that the function need not be defined at x = a for the limit to exist. For example, lim(x→2) (x² – 4)/(x – 2) = lim(x→2)(x + 2) = 4, even though the original expression is undefined at x = 2.

Q2. State and prove the chain rule of differentiation.

The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). In other words, the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Proof sketch: Let u = g(x), so y = f(u). By definition of the derivative: dy/dx = lim(Δx→0) [f(g(x+Δx)) – f(g(x))]/Δx. We multiply and divide by Δu = g(x+Δx) – g(x), which gives dy/dx = lim(Δu→0)[f(u+Δu)-f(u)]/Δu * lim(Δx→0)[g(x+Δx)-g(x)]/Δx = f'(u) * g'(x) = f'(g(x)) * g'(x). Example: If y = sin(x²), then dy/dx = cos(x²) * 2x.

Q3. Find the absolute maximum and minimum values of f(x) = x³ – 3x² + 1 on the interval [-1, 3].

Step 1: Find critical points by setting f'(x) = 0. f'(x) = 3x² – 6x = 3x(x-2) = 0, so x = 0 and x = 2. Step 2: Evaluate f(x) at critical points and endpoints. f(-1) = (-1)³ – 3(-1)² + 1 = -1 – 3 + 1 = -3. f(0) = 0 – 0 + 1 = 1. f(2) = 8 – 12 + 1 = -3. f(3) = 27 – 27 + 1 = 1. Step 3: Compare: Absolute maximum = 1 (at x = 0 and x = 3). Absolute minimum = -3 (at x = -1 and x = 2).

Q4. Apply L’Hopital’s rule to evaluate lim(x→0) (sin x)/x.

When we substitute x = 0 directly into (sin x)/x, we get 0/0, which is an indeterminate form. L’Hopital’s rule states that if lim(x→a) f(x)/g(x) is of the form 0/0 or infinity/infinity, then lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x), provided the latter limit exists. Differentiating numerator and denominator separately: d/dx(sin x) = cos x and d/dx(x) = 1. Therefore, lim(x→0) (sin x)/x = lim(x→0) (cos x)/1 = cos(0)/1 = 1. This is a fundamental limit in calculus with wide applications in engineering and physics.

Q5. What is Rolle’s theorem and what conditions must be satisfied for it to apply?

Rolle’s theorem states that if a function f(x) satisfies three conditions: (1) f is continuous on the closed interval [a, b], (2) f is differentiable on the open interval (a, b), and (3) f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0. Geometrically, this means that if a smooth curve starts and ends at the same height, there must be at least one point where the tangent to the curve is horizontal. Example: f(x) = x² – 4x + 3 on [1, 3]. f(1) = 0 = f(3), f is continuous and differentiable everywhere. By Rolle’s theorem, there exists c where f'(c) = 2c – 4 = 0, giving c = 2, which lies in (1, 3). Verified.

Q6. Differentiate y = x^x using logarithmic differentiation.

When the variable appears both in the base and the exponent, logarithmic differentiation is required. Step 1: Take the natural logarithm of both sides: ln y = ln(x^x) = x * ln x. Step 2: Differentiate both sides with respect to x: (1/y)(dy/dx) = x*(1/x) + ln x * 1 = 1 + ln x. Step 3: Multiply both sides by y: dy/dx = y(1 + ln x) = x^x(1 + ln x). This technique is extremely useful for products and powers involving variables and is commonly tested in the mth 165 lpu question paper.

Unit 2: Integral Calculus

Integral calculus deals with the accumulation of quantities and the areas under and between curves. It is the inverse process of differentiation and is equally important in the examination.

Q7. Evaluate the integral of (1/(x² – 5x + 6)) with respect to x using partial fractions.

Step 1: Factor the denominator: x² – 5x + 6 = (x-2)(x-3). Step 2: Express using partial fractions: 1/[(x-2)(x-3)] = A/(x-2) + B/(x-3). Multiply through by (x-2)(x-3): 1 = A(x-3) + B(x-2). Setting x = 3: 1 = B(1), so B = 1. Setting x = 2: 1 = A(-1), so A = -1. Step 3: Integrate: Integral of [-1/(x-2) + 1/(x-3)] dx = -ln|x-2| + ln|x-3| + C = ln|(x-3)/(x-2)| + C.

Q8. Evaluate the definite integral of x*sin(x) from 0 to pi using integration by parts.

Integration by parts formula: Integral of u dv = uv – Integral of v du. Let u = x (so du = dx) and dv = sin(x) dx (so v = -cos x). Integral of x*sin(x) dx = x(-cos x) – Integral of (-cos x) dx = -x*cos x + sin x + C. Now evaluate from 0 to pi: [-x*cos x + sin x] from 0 to pi = [(-pi)(cos pi) + sin pi] – [0 + sin 0] = (-pi)(-1) + 0 – 0 = pi. Therefore, the definite integral equals pi.

Q9. State and explain the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus has two parts. Part 1 (differentiation of integral): If F(x) = Integral from a to x of f(t) dt, and f is continuous on [a, b], then F'(x) = f(x) for all x in [a, b]. This tells us that differentiation and integration are inverse processes. Part 2 (evaluation of definite integrals): If f is continuous on [a, b] and F is any antiderivative of f, then Integral from a to b of f(x) dx = F(b) – F(a). This is the most practical result in calculus as it gives us a method to evaluate definite integrals without computing limits of Riemann sums. Example: Integral from 1 to 3 of 2x dx = [x²] from 1 to 3 = 9 – 1 = 8.

Q10. Find the area bounded by the curves y = x² and y = 2x.

Step 1: Find intersection points. Set x² = 2x: x² – 2x = 0, x(x-2) = 0, so x = 0 and x = 2. Step 2: Determine which curve is on top between 0 and 2. At x = 1: y = 2(1) = 2 (line) and y = 1² = 1 (parabola). So the line y = 2x is above the parabola y = x². Step 3: Compute the area: Area = Integral from 0 to 2 of (2x – x²) dx = [x² – x³/3] from 0 to 2 = (4 – 8/3) – 0 = 12/3 – 8/3 = 4/3 square units.

Q11. Evaluate Integral of sqrt(1 – x²) dx using trigonometric substitution.

Let x = sin(theta), so dx = cos(theta) d(theta) and sqrt(1-x²) = sqrt(1-sin²theta) = cos(theta). Integral of sqrt(1-x²) dx = Integral of cos(theta) * cos(theta) d(theta) = Integral of cos²(theta) d(theta). Using the identity cos²(theta) = (1 + cos 2theta)/2: = Integral of (1 + cos 2theta)/2 d(theta) = theta/2 + sin(2theta)/4 + C. Converting back: theta = arcsin(x) and sin(2theta) = 2 sin(theta)cos(theta) = 2x*sqrt(1-x²). Final answer: (arcsin x)/2 + x*sqrt(1-x²)/2 + C.

Unit 3: Ordinary Differential Equations

Differential equations describe how quantities change and are fundamental to modeling physical, biological, and engineering systems. This unit is one of the most mathematically demanding parts of the mth 165 lpu question paper.

Q12. Solve the first-order linear ODE: dy/dx + 2y = 4x.

This is a linear ODE of the form dy/dx + P(x)y = Q(x), where P(x) = 2 and Q(x) = 4x. Step 1: Find the integrating factor (IF): IF = e^(Integral of P dx) = e^(2x). Step 2: Multiply both sides by the IF: e^(2x) dy/dx + 2e^(2x) y = 4x*e^(2x). The left side is d/dx[y*e^(2x)]. Step 3: Integrate both sides: y*e^(2x) = Integral of 4x*e^(2x) dx. Using integration by parts: Integral of 4x*e^(2x) dx = 4[x*e^(2x)/2 – e^(2x)/4] + C = 2x*e^(2x) – e^(2x) + C. Step 4: Divide by e^(2x): y = 2x – 1 + Ce^(-2x).

Q13. Solve the separable differential equation: dy/dx = (y² + 1)/(xy).

Rearranging: y/(y²+1) dy = (1/x) dx. Step 1: Separate variables: [y/(y²+1)] dy = (1/x) dx. Step 2: Integrate both sides. Left side: Let u = y²+1, then du = 2y dy, so Integral of y/(y²+1) dy = (1/2) ln(y²+1). Right side: Integral of (1/x) dx = ln|x|. Step 3: Combine: (1/2) ln(y²+1) = ln|x| + C1. Multiply by 2: ln(y²+1) = 2 ln|x| + 2C1 = ln(x²) + C2. Exponentiate: y²+1 = Ax² where A = e^(C2). Final answer: y² = Ax² – 1, where A is an arbitrary constant.

Q14. Solve the second-order linear homogeneous ODE: y” – 5y’ + 6y = 0.

For a second-order linear homogeneous ODE with constant coefficients, we assume a solution of the form y = e^(rx). Substituting: r²e^(rx) – 5re^(rx) + 6e^(rx) = 0. Dividing by e^(rx) gives the characteristic equation: r² – 5r + 6 = 0. Factoring: (r-2)(r-3) = 0, so r = 2 or r = 3. Since we have two distinct real roots, the general solution is: y = C1*e^(2x) + C2*e^(3x), where C1 and C2 are arbitrary constants determined by initial conditions.

Q15. Use the method of undetermined coefficients to solve: y” + y = cos(x).

Step 1: Solve the homogeneous part y” + y = 0. Characteristic equation: r² + 1 = 0, so r = ±i. Homogeneous solution: y_h = C1*cos(x) + C2*sin(x). Step 2: Since cos(x) is part of the homogeneous solution, we multiply the particular solution guess by x: y_p = x[A*cos(x) + B*sin(x)]. Step 3: Compute y_p” and substitute into the equation. y_p’ = A*cos(x) + B*sin(x) + x[-A*sin(x) + B*cos(x)]. y_p” = -2A*sin(x) + 2B*cos(x) – x[A*cos(x) + B*sin(x)]. Substituting into y” + y: -2A*sin(x) + 2B*cos(x) = cos(x). So A = 0, B = 1/2. Particular solution: y_p = (x/2)*sin(x). General solution: y = C1*cos(x) + C2*sin(x) + (x/2)*sin(x).

Q16. What is an exact differential equation? Give an example and solve it.

A differential equation M(x,y)dx + N(x,y)dy = 0 is said to be exact if partial M/partial y = partial N/partial x. The solution is found by computing a potential function F(x,y) such that dF = M dx + N dy, making F(x,y) = C the implicit general solution. Example: (2xy + y²)dx + (x² + 2xy)dy = 0. Here M = 2xy + y² and N = x² + 2xy. Check: partial M/partial y = 2x + 2y = partial N/partial x. So it is exact. F = Integral of M dx = x²y + xy² + g(y). Differentiate with respect to y: x² + 2xy + g'(y) = N = x² + 2xy. So g'(y) = 0, meaning g(y) = constant. Solution: x²y + xy² = C.

Unit 4: Linear Algebra

Linear algebra deals with systems of linear equations, matrices, determinants, and vector spaces. These concepts are widely used in computer graphics, data science, machine learning, and physics.

Q17. Find the rank of the matrix A = [[1,2,3],[4,5,6],[7,8,9]].

To find the rank, we perform row reduction to echelon form. Starting matrix: R1=[1,2,3], R2=[4,5,6], R3=[7,8,9]. Operation R2 = R2 – 4*R1: R2 = [4-4, 5-8, 6-12] = [0,-3,-6]. Operation R3 = R3 – 7*R1: R3 = [7-7, 8-14, 9-21] = [0,-6,-12]. Operation R3 = R3 – 2*R2: R3 = [0-0, -6+6, -12+12] = [0,0,0]. The echelon form has two non-zero rows, so the rank of A = 2. Notice that the third row became all zeros, indicating that the rows are linearly dependent (R3 = 2*R2 is a relationship that can be derived).

Q18. Explain eigenvalues and eigenvectors. Find them for the matrix A = [[4,1],[2,3]].

An eigenvector of a matrix A is a non-zero vector v such that Av = lambda*v, where lambda is a scalar called the eigenvalue. Eigenvalues represent scaling factors and eigenvectors represent directions that are only scaled (not rotated) by the transformation. To find eigenvalues, solve det(A – lambda*I) = 0. A – lambda*I = [[4-lambda, 1],[2, 3-lambda]]. Determinant: (4-lambda)(3-lambda) – 2 = lambda² – 7*lambda + 12 – 2 = lambda² – 7*lambda + 10 = 0. Roots: lambda = (7 ± sqrt(49-40))/2 = (7±3)/2. So lambda1 = 5, lambda2 = 2. For lambda1 = 5: (A-5I)v = 0 gives [[-1,1],[2,-2]]v = 0, so v1 = v2, eigenvector = [1,1]^T. For lambda2 = 2: [[2,1],[2,1]]v = 0 gives 2v1 + v2 = 0, eigenvector = [1,-2]^T.

Q19. Solve the system of linear equations using Cramer’s Rule: 2x + y = 5, x – y = 1.

Cramer’s Rule uses determinants to solve systems of equations. The system in matrix form is AX = B where A = [[2,1],[1,-1]], X = [x,y]^T, B = [5,1]^T. Step 1: Compute det(A) = (2)(-1) – (1)(1) = -2 – 1 = -3. Step 2: For x, replace first column with B: D_x = det([[5,1],[1,-1]]) = (5)(-1)-(1)(1) = -5-1 = -6. So x = D_x/det(A) = -6/-3 = 2. Step 3: For y, replace second column with B: D_y = det([[2,5],[1,1]]) = (2)(1)-(5)(1) = 2-5 = -3. So y = D_y/det(A) = -3/-3 = 1. Solution: x = 2, y = 1. Verification: 2(2)+1 = 5 and 2-1 = 1. Correct.

Q20. Define linear independence of vectors. Are the vectors v1 = [1,2,3], v2 = [0,1,4], v3 = [5,6,0] linearly independent?

Vectors v1, v2, …, vn are linearly independent if the only solution to c1*v1 + c2*v2 + … + cn*vn = 0 is c1 = c2 = … = cn = 0. To test the given vectors, we check if the determinant of the matrix formed by placing them as rows (or columns) is non-zero. Matrix = [[1,2,3],[0,1,4],[5,6,0]]. det = 1*(1*0 – 4*6) – 2*(0*0 – 4*5) + 3*(0*6 – 1*5) = 1*(0-24) – 2*(0-20) + 3*(0-5) = -24 + 40 – 15 = 1. Since the determinant is 1 (non-zero), the vectors are linearly independent. This means no one of them can be expressed as a linear combination of the others.

Q21. Find the inverse of the matrix B = [[2,1],[5,3]].

For a 2×2 matrix [[a,b],[c,d]], the inverse is (1/det) * [[d,-b],[-c,a]], provided det is not zero. Step 1: det(B) = 2*3 – 1*5 = 6 – 5 = 1. Step 2: Adjugate matrix = [[3,-1],[-5,2]]. Step 3: B inverse = (1/1) * [[3,-1],[-5,2]] = [[3,-1],[-5,2]]. Verification: B * B_inverse = [[2,1],[5,3]] * [[3,-1],[-5,2]] = [[2*3+1*(-5), 2*(-1)+1*2],[5*3+3*(-5), 5*(-1)+3*2]] = [[6-5, -2+2],[15-15,-5+6]] = [[1,0],[0,1]] = I. Confirmed.

Unit 5: Complex Numbers and Infinite Series

Complex numbers extend the real number line into a two-dimensional plane, enabling solutions to equations that have no real solutions. Series theory explores the convergence of infinite sums and expansions, which are critical in applied mathematics.

Q22. Express the complex number z = 3 + 4i in polar form and find its modulus and argument.

For a complex number z = a + bi, the polar form is z = r(cos theta + i sin theta) = r*e^(i*theta), where r = |z| = sqrt(a² + b²) is the modulus and theta = arctan(b/a) is the argument. For z = 3 + 4i: Modulus r = sqrt(3² + 4²) = sqrt(9+16) = sqrt(25) = 5. Argument theta = arctan(4/3) ≈ 53.13° = 0.9273 radians. Polar form: z = 5(cos 53.13° + i sin 53.13°) = 5e^(0.9273i). Euler’s formula e^(i*theta) = cos(theta) + i sin(theta) connects trigonometry and complex exponentials.

Q23. Use De Moivre’s theorem to find (1 + i)^8.

De Moivre’s theorem states that (r(cos theta + i sin theta))^n = r^n(cos n*theta + i sin n*theta). Step 1: Convert 1+i to polar form. Modulus = sqrt(1²+1²) = sqrt(2). Argument = arctan(1/1) = pi/4. So 1+i = sqrt(2)(cos(pi/4) + i sin(pi/4)) = sqrt(2)*e^(i*pi/4). Step 2: Apply De Moivre: (1+i)^8 = (sqrt(2))^8 * (cos(8*pi/4) + i sin(8*pi/4)) = 16 * (cos(2*pi) + i sin(2*pi)) = 16 * (1 + 0) = 16. So (1+i)^8 = 16, which is a real number.

Q24. State the ratio test for convergence of an infinite series and apply it to the series sum of n!/n^n.

The Ratio Test states: For a series sum of a_n, compute L = lim(n→∞) |a_(n+1)/a_n|. If L < 1, the series converges absolutely. If L > 1 (or L = infinity), the series diverges. If L = 1, the test is inconclusive. Applying to sum of a_n = n!/n^n: a_(n+1)/a_n = [(n+1)!/(n+1)^(n+1)] / [n!/n^n] = [(n+1)! * n^n] / [n! * (n+1)^(n+1)] = (n+1) * n^n / (n+1)^(n+1) = n^n/(n+1)^n = (n/(n+1))^n = (1 – 1/(n+1))^n. As n→∞, this approaches e^(-1) = 1/e ≈ 0.368. Since L = 1/e < 1, the series converges.

Q25. Write the Taylor series expansion of e^x about x = 0 (Maclaurin series).

The Maclaurin series is the Taylor series centered at x = 0. For f(x) = e^x, we compute all derivatives: f(0) = 1, f'(0) = 1, f”(0) = 1, and so on for all orders (since the derivative of e^x is e^x). The Maclaurin series is: e^x = f(0) + f'(0)*x + f”(0)*x²/2! + f”'(0)*x³/3! + … = 1 + x + x²/2! + x³/3! + x⁴/4! + … = sum from n=0 to infinity of x^n/n!. This series converges for all real values of x (the radius of convergence is infinity). Important special cases: e^1 = 1 + 1 + 1/2 + 1/6 + 1/24 + … ≈ 2.71828.

Common Mistakes Students Make in MTH 165 Examinations

After analyzing multiple iterations of the mth 165 lpu question paper and talking to students who have appeared in the examination, certain recurring mistakes stand out. Being aware of these pitfalls can significantly improve your score.

1. Skipping the verification step in differential equations. Always substitute your answer back into the original equation to confirm it is correct.
2. Forgetting constants of integration. Every indefinite integral must include ‘+C’ and every ODE solution must include arbitrary constants.
3. Confusing similar-looking formulas. For example, the integrals of 1/(1+x²) and 1/sqrt(1-x²) are arctan(x) and arcsin(x) respectively — they are easy to mix up under exam pressure.
4. Not checking conditions before applying theorems. Rolle’s theorem and the Mean Value Theorem require specific conditions (continuity, differentiability) that must be verified.
5. Arithmetic errors in matrix operations. A single sign error in a determinant calculation can invalidate an entire eigenvalue problem.
6. Ignoring the case of repeated or complex roots in ODEs. Each type of root (distinct real, repeated, complex) has a different form of general solution.
7. Not simplifying answers fully. Examiners expect answers in simplest form, and leaving an answer as ln(e^x) instead of x, for example, costs marks.

Smart Study Strategy for MTH 165 at LPU

Knowing the material is only half the battle. How you organize your preparation has an enormous impact on your performance. Here is a structured 30-day study plan that covers everything tested in the mth 165 lpu question paper.

Week 1: Foundation Building

Spend the first week revisiting core concepts from 11th and 12th standard mathematics. Limits, basic differentiation, and integration form the bedrock of everything else in MTH 165. Do not rush past this phase. A weak foundation in limits will make differential equations nearly impossible to grasp.

• Days 1-2: Revise limits, continuity, and the epsilon-delta definition
• Days 3-4: Practice differentiation — all rules including product, quotient, chain, and implicit
• Days 5-7: Work on applications of derivatives — maxima/minima, tangent and normal lines, curve sketching

Week 2: Integration and ODEs

Integration is essentially a table of patterns. The more integrals you practice, the better you become at recognizing which technique to apply. Similarly, ODEs require both formula recall and procedural accuracy.

• Days 8-9: Master integration techniques — substitution, parts, partial fractions, trigonometric substitution
• Days 10-11: Learn definite integral applications — area, volume, arc length
• Days 12-14: First-order ODEs — separable, linear, exact, and homogeneous

Week 3: Higher-Level Topics

• Days 15-16: Second-order ODEs — characteristic equations, particular integrals, method of undetermined coefficients
• Days 17-18: Linear algebra — matrix operations, rank, determinants
• Days 19-21: Eigenvalues, eigenvectors, and their geometric interpretations

Week 4: Complex Numbers, Series, and Revision

• Days 22-23: Complex numbers in rectangular, polar, and exponential form
• Days 24-25: Power series, convergence tests, Taylor and Maclaurin expansions
• Days 26-28: Solve 3-4 complete previous year mth 165 lpu question paper sets under timed conditions
• Days 29-30: Review weak areas, focus on high-weightage topics, and get adequate rest

Additional Practice Questions for Exam Readiness

The following questions are additional practice problems modeled on the style and difficulty of the mth 165 lpu question paper. Attempt these independently before checking the hints provided.

Q26. Find dy/dx if x³ + y³ = 6xy using implicit differentiation.

Differentiate both sides with respect to x: 3x² + 3y²(dy/dx) = 6y + 6x(dy/dx). Rearrange: 3y²(dy/dx) – 6x(dy/dx) = 6y – 3x². Factor out dy/dx: dy/dx(3y² – 6x) = 6y – 3x². Divide: dy/dx = (6y – 3x²)/(3y² – 6x) = (2y – x²)/(y² – 2x). This curve is the folium of Descartes, a famous curve in mathematics.

Q27. Evaluate the double integral of xy over the region R = {(x,y): 0 ≤ x ≤ 2, 0 ≤ y ≤ x}.

Double integral = Integral from x=0 to 2 [Integral from y=0 to x of xy dy] dx. Inner integral: Integral from 0 to x of xy dy = x[y²/2] from 0 to x = x * x²/2 = x³/2. Outer integral: Integral from 0 to 2 of x³/2 dx = (1/2)[x⁴/4] from 0 to 2 = (1/2)(16/4) = (1/2)(4) = 2. The double integral equals 2.

Q28. Verify that u = e^x*cos(y) satisfies Laplace’s equation d²u/dx² + d²u/dy² = 0.

Laplace’s equation is a fundamental PDE in physics and engineering. Compute partial derivatives: du/dx = e^x * cos(y). d²u/dx² = e^x * cos(y). du/dy = e^x * (-sin(y)) = -e^x * sin(y). d²u/dy² = -e^x * cos(y). Sum: d²u/dx² + d²u/dy² = e^x * cos(y) + (-e^x * cos(y)) = 0. Therefore, u = e^x * cos(y) is a harmonic function and satisfies Laplace’s equation. Such functions appear frequently in fluid flow and electrostatics problems.

Q29. Find the Maclaurin series of sin(x) and determine its radius of convergence.

For f(x) = sin(x): f(0) = 0, f'(0) = cos(0) = 1, f”(0) = -sin(0) = 0, f”'(0) = -cos(0) = -1, f””(0) = sin(0) = 0, and the pattern repeats with period 4. The Maclaurin series is: sin(x) = x – x³/3! + x⁵/5! – x⁷/7! + … = sum from n=0 to infinity of (-1)^n * x^(2n+1)/(2n+1)!. To find radius of convergence, apply the ratio test to the absolute terms: lim |a_(n+1)/a_n| = lim x^(2n+3)/((2n+3)!) * (2n+1)!/x^(2n+1) = lim x²/((2n+3)(2n+2)) = 0 for any finite x. Since L = 0 < 1 for all x, the radius of convergence is infinity (converges for all real x).

Q30. Solve the Bernoulli equation: dy/dx + y/x = y² * ln(x).

A Bernoulli equation has the form dy/dx + P(x)y = Q(x)y^n. Here P(x) = 1/x, Q(x) = ln(x), n = 2. Step 1: Divide by y²: y^(-2) dy/dx + y^(-1)/x = ln(x). Step 2: Let v = y^(-1), so dv/dx = -y^(-2) dy/dx, meaning y^(-2) dy/dx = -dv/dx. Step 3: Substitute: -dv/dx + v/x = ln(x), or dv/dx – v/x = -ln(x). This is now a linear ODE. Integrating factor: e^(-Integral(1/x)dx) = e^(-ln x) = 1/x. Multiply: (1/x)dv/dx – v/x² = -ln(x)/x, which is d/dx(v/x) = -ln(x)/x. Integrating: v/x = -Integral(ln(x)/x)dx = -(ln x)²/2 + C. Thus y^(-1)/x = -(ln x)²/2 + C, giving 1/y = x(C – (ln x)²/2).

How to Effectively Use Previous Year Question Papers

One of the most valuable resources available to any student is the collection of previous year question papers. The mth 165 lpu question paper from past semesters reveals patterns, repeated question types, and the level of difficulty expected by the examination board.

Here is how to use these papers most effectively:

• First Attempt: Solve the paper without any reference material under strict time conditions (typically 3 hours). This simulates the real examination environment and exposes your genuine weaknesses.
• Gap Analysis: After finishing, compare your answers with solutions and categorize every mistake. Are they conceptual errors, calculation mistakes, or time management failures? Each type requires a different corrective strategy.
• Topic Mapping: Identify which units appear most frequently across multiple year papers. If linear algebra eigenvalue problems appear in 4 out of 5 years, that is a high-priority topic deserving extra attention.
• Pattern Recognition: Notice the structure of long-answer questions. They often have two or three parts. Part (a) might be straightforward, while part (b) requires deeper application. Practice both.
• Formula Consolidation: After solving the paper, create a personalized formula sheet listing every formula you needed. Review this sheet daily in the final week before exams.

Conclusion

Mathematics is a subject that rewards consistent practice and honest engagement with difficult problems. The mth 165 lpu question paper covers a rich and interconnected set of topics that build upon each other in meaningful ways. Calculus provides the language for differential equations; linear algebra provides the tools for solving systems that appear in engineering; complex numbers and series bridge abstract mathematics with real-world applications in signal processing and physics.

This blog has walked you through more than 30 detailed questions and answers spanning all five major units of the course. Beyond the questions themselves, we have covered study strategies, common mistakes, and how to use past papers effectively. If you approach your preparation with discipline and genuine curiosity, the mth 165 lpu question paper will hold no surprises.

Remember that the goal of mathematics education is not merely to pass an examination but to develop a way of thinking — precise, logical, systematic, and creative. Every problem you solve strengthens that thinking. Good luck with your preparation, and may your examinations reflect the hours of effort you invest.