Mathematics-I Unit-wise Important Questions
Common to All Branches
Note: These are for your reference purpose and if you have time read all the units
UNIT I:
1. Define the convergence of an infinite series. Show that the nth term of a convergent series tends to zero. Is the converse true?
2.1 Test the convergence or divergence of the series:
• 1 + 1/2² + 2²/3³ + 3³/4⁴ + 4⁴/5⁵ + ...
• Σ(n=1 to ∞) 1/(n² + n)
• Σ(n=1 to ∞) 1/(n² + n)
2.2 State Maclaurin’s theorem with Lagrange’s form of remainder for f(x) = Cos x
3. State and prove comparison test. Test for convergence of the series 1/1×3 + 1/3×5 + 1/5×7 + ...
4.1 Verify Lagrange's mean value theorem for f(x) = (x-1)(x-2)(x-3) in [0, 4]
4.2 Find the Maclaurin’s series expansion of f(x)= sinhx
5. State Taylor's and Maclaurin's theorems. Verify Maclaurin's theorem for f(x) = (1-x)^(1/2) with Lagrange's form of remainder up to 3 terms in [0,1] at x = 1.
6. Using the integral test, discuss the convergence of the series Σ(n=1 to ∞) 1/(2n+3)
UNIT II:
1. Solve the Bernoulli equation:
• cosh x (dy/dx) + y sinh x = 2 cosh² x sinh x
• dy/dx + 2y tan x = y²
• dy/dx + 2y tan x = y²
2. Newton's Law of Cooling Problem:
A copper ball is heated to a temperature of 80°C at time t = 0, then it is placed in water maintained at 30°C. If at t = 3 minutes, the temperature of the ball is reduced to 50°C. Find the time at which the temperature of the ball is 40°C.
A copper ball is heated to a temperature of 80°C at time t = 0, then it is placed in water maintained at 30°C. If at t = 3 minutes, the temperature of the ball is reduced to 50°C. Find the time at which the temperature of the ball is 40°C.
3. Solve the exact differential equation:
• y(x⁴ + x² y² + xy) dx + x(x⁴ - x² y² + x²) dy = 0
• (2xy³eʸ + 2xy⁴) dx + (x²y²eʸ - x²y³ - 3x)dy = 0
• (2xy³eʸ + 2xy⁴) dx + (x²y²eʸ - x²y³ - 3x)dy = 0
4. Find the orthogonal trajectories of the family of curves:
• x² + y² = a²
• r² = a sin 2θ
• r² = a sin 2θ
5. Solve the linear differential equation:
• (1 - x²) dy/dx + 2xy = x√(1 - x²)
• dy/dx = (x² + y² + 1)/(2xy)
• dy/dx = (x² + y² + 1)/(2xy)
6. An RL circuit has an Emf given (in volts) by 4 sin t, a resistance of 100 ohms, an inductance of 4 henries with no initial current. Find the current at any time t.
UNIT III:
1. Find the Particular Integral of:
• (D² - 5D + 6) y = eˣ
• (D³ + 1)y = 3 + 5eˣ
• (D³ + 1)y = 3 + 5eˣ
2. Solve (D² - 1) y = cosh 2x
3. Solve (D² - 9) y = sin2x
4. Solve (D² + 5D + 6) y = eˣ + sin x
5. Solve (D² + 3D + 2) y = e⁻ˣ + cos x by the method of variation of parameters
6.Consider an electrical circuit containing an inductance L, Resistance R and capacitance C. Let q be the electrical charge on the capacitor plate and 'I' be the current in the circuit at any time t. There is applied E.M.F Esinωt in the circuit. Then find the charge on the capacitor.
UNIT IV:
1. Find ∂f/∂x, ∂f/∂y for f(x, y) = log√(x² + y²)
2. If u = x² tan⁻¹(y/x) - y² tan⁻¹(x/y), then show that ∂²u/∂x∂y = (x² - y²)/(x² + y²)
3. State and prove Euler's theorem for homogeneous functions. Find ∂u/∂x if u = sin(x² + y²), where a²x² + b²y² = c²
4. If x = r cos θ, y = r sin θ, find ∂(r,θ)/∂(x,y) and ∂(x,y)/∂(r,θ)
5. Determine whether the functions U = x/(y-z), V = y/(x-y), W = z/(z-x) are dependent. If dependent find the relationship between them.
6. Find the extreme values f(x, y) = x⁴ + y⁴ - 2x² + 4xy - 2y²
UNIT V: MULTIPLE INTEGRALS
1. Evaluate ∫∫ xy dx dy (with appropriate limits)
2. Evaluate ∫∫ (eˣ/y) dx dy (with appropriate limits)
3. By change of order of integration evaluate ∫₀² ∫₀^(x²) xy dy dx
4. Evaluate ∫₀ᵃ ∫₀^√(a²-y²) (x² + y²) dy dx by changing into polar coordinates
5. Find the area bounded by the lines x = 0, y = 1, x = 1 and y = 0
6. Find by using triple integrals, the volume of the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x/a + y/b + z/c = 1
KEY CONCEPTS Add Chesanu below:
UNIT I
• Series Convergence Tests:
• Mean Value Theorem Applications:
• Taylor/Maclaurin Series:
• Alternating Series:
UNIT II
• Equation Type Recognition:
• Newton's Law of Cooling:
• Integrating Factors:
• Orthogonal Trajectories:
UNIT III
• Method Selection:
• Particular Integral Forms:
• Operator Method:
• Circuit Applications:
UNIT IV
• Chain Rule Applications:
• Jacobian Computation:
• Euler's Theorem:
• Lagrange Multiplier Method:
UNIT V
• Integration Limits:
• Order Change:
• Polar Coordinates:
• Volume Applications: