MCQ Engineering Mathematics

MCQ Engineering Mathematics

Topics: Linear Algebra · Calculus · Differential Equations · Complex Variables · Probability & Statistics · Numerical Methods

Q1. The rank of a matrix is equal to:

Number of zero rows Order of the largest non-singular submatrix Number of columns Determinant of the matrix

Q2. A system of linear equations Ax = b has a unique solution when:

A is singular det(A) = 0 A is non-singular b = 0

Q3. Eigenvalues of a real symmetric matrix are always:

Complex Real Purely imaginary Zero

Q4. If λ is an eigenvalue of A, then eigenvalue of A⁻¹ is:

λ 1/λ −λ λ²

Q5. The trace of a square matrix equals:

Product of eigenvalues Sum of eigenvalues Determinant Rank

Q6. If limₓ→a f(x) exists and equals f(a), then f is:

Differentiable at a Continuous at a Discontinuous at a Bounded only

Q7. Rolle's theorem requires that f is continuous on [a,b], differentiable on (a,b) and:

f(a) = f(b) f′(a) = 0 f(a) ≠ f(b) f is constant

Q8. The gradient of a scalar field φ points in the direction of:

Minimum rate of change Maximum rate of change Zero change Curl

Q9. Divergence of a vector field F measures:

Rotation Source/sink strength (flux density) Path length Potential

Q10. Stokes' theorem relates:

Line integral of F to surface integral of curl F Volume integral to surface integral of F Line integral to gradient None

Q11. Green's theorem in the plane is a special case of:

Gauss theorem Stokes theorem Divergence theorem only Mean value theorem

Q12. A critical point of f(x,y) where fₓₓ fᵧᵧ − (fₓᵧ)² > 0 and fₓₓ > 0 is a:

Local maximum Local minimum Saddle point Inflection

Q13. The integrating factor of dy/dx + P(x)y = Q(x) is:

e^∫P dx e^∫Q dx ∫P dx 1/P

Q14. The characteristic equation of y'' + ay' + by = 0 with roots m₁, m₂ distinct real gives solution:

c₁e^{m₁x} + c₂e^{m₂x} e^{αx}(c₁ cos βx + c₂ sin βx) (c₁+c₂x)e^{mx} Polynomial

Q15. Cauchy–Euler equation has the form:

a x²y'' + b xy' + cy = 0 y'' + y = 0 y' + P y = Q y'' = f(x)

Q16. Laplace transform of f'(t) (under standard conditions) is:

s F(s) − f(0) F(s)/s s F(s) F'(s)

Q17. One-dimensional heat equation is:

u_t = c² u_xx u_tt = c² u_xx u_xx + u_yy = 0 u_t + u_x = 0

Q18. A function f(z) is analytic in a domain if it is:

Continuous only Differentiable at every point of the domain Bounded Polynomial

Q19. Cauchy's integral theorem states that if f is analytic inside and on a simple closed curve C:

∮_C f dz = 0 ∮_C f dz = 2πi ∮_C f dz = f(0) None

Q20. Laurent series is used when the function has:

No singularity A singularity inside the annulus of expansion Only Taylor expansion Real coefficients only

Q21. If A and B are independent, P(A∩B) equals:

P(A)+P(B) P(A)P(B) P(A|B) 1

Q22. Mean of a Poisson random variable with parameter λ is:

λ √λ λ² 1/λ

Q23. For a standard normal distribution, mean and variance are:

0 and 1 1 and 0 0 and 0 1 and 1

Q24. Binomial distribution Bin(n,p) has variance:

np np(1−p) n(1−p) p(1−p)

Q25. Conditional probability P(A|B) is:

P(A∩B)/P(B) P(A)/P(B) P(A∪B) P(B|A)

Q26. Newton–Raphson iteration for finding root of f(x)=0 is:

x_{n+1} = x_n − f(x_n)/f'(x_n) x_{n+1} = x_n + f(x_n) x_{n+1} = f(x_n) Bisection only

Q27. Trapezoidal rule for ∫_a^b f(x) dx with one interval approximates the integral by:

(b−a)[f(a)+f(b)]/2 (b−a)f((a+b)/2) Simpson only None

Q28. Simpson's 1/3 rule requires the number of subintervals to be:

Odd Even Multiple of 3 Any

Q29. Euler's method for y' = f(x,y) advances by:

y_{n+1} = y_n + h f(x_n,y_n) y_{n+1} = y_n + h Runge–Kutta only None

Q30. The order of the classical 4th-order Runge–Kutta method is:

1 2 3 4

Engineering Mathematics – Theory & Formulae

📖 Linear Algebra

Systems of linear equations

$$ A\mathbf{x} = \mathbf{b} $$

• Unique solution ⇔ \(A\) non-singular (invertible), \(\det A \ne 0\).
• Infinite / no solution when \(A\) is singular; consistency checked via rank(\([A|b]\)) vs rank(\(A\)).

Eigenvalues / eigenvectors

$$ A\mathbf{v} = \lambda \mathbf{v}, \quad \det(A - \lambda I) = 0 $$

Trace = \(\sum \lambda_i\), Determinant = \(\prod \lambda_i\). Real symmetric matrices have real eigenvalues and orthogonal eigenvectors.

📖 Calculus

Mean value theorems

• Rolle: \(f(a)=f(b)\) ⇒ \(f'(c)=0\) for some \(c\in(a,b)\).
• Lagrange MVT: \(f(b)-f(a)=f'(c)(b-a)\).

Gradient, divergence, curl

$$ \nabla \phi = \frac{\partial\phi}{\partial x}\mathbf{i}+\frac{\partial\phi}{\partial y}\mathbf{j}+\frac{\partial\phi}{\partial z}\mathbf{k} $$

$$ \nabla\cdot\mathbf{F} = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z},\quad \nabla\times\mathbf{F} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \partial_x&\partial_y&\partial_z\\ F_x&F_y&F_z\end{vmatrix} $$

Directional derivative of \(\phi\) in direction \(\mathbf{u}\) (unit): \(\nabla\phi\cdot\mathbf{u}\).

Integral theorems

$$ \oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla\times\mathbf{F})\cdot d\mathbf{S} \quad\text{(Stokes)} $$

$$ \iiint_V (\nabla\cdot\mathbf{F})\,dV = \iint_S \mathbf{F}\cdot d\mathbf{S} \quad\text{(Gauss / divergence)} $$

$$ \oint_C (P\,dx+Q\,dy) = \iint_D (Q_x - P_y)\,dA \quad\text{(Green)} $$

Maxima / minima (two variables): critical points from \(f_x=f_y=0\); \(D=f_{xx}f_{yy}-(f_{xy})^2\); \(D>0,f_{xx}>0\) min; \(D>0,f_{xx}<0\) max; \(D<0\) saddle.

📖 Differential Equations

First-order linear

$$ \frac{dy}{dx}+P(x)y=Q(x),\quad \text{IF}=e^{\int P\,dx},\quad y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C $$

Constant-coefficient linear homogeneous: assume \(y=e^{mx}\); characteristic polynomial; general solution from roots (real distinct, repeated, or complex).

Cauchy–Euler: \(a x^2 y''+b x y'+c y=0\); trial \(y=x^m\).

Laplace transforms

$$ \mathcal{L}\{f\}(s)=\int_0^\infty e^{-st}f(t)\,dt,\quad \mathcal{L}\{f'\}=sF(s)-f(0),\quad \mathcal{L}\{f''\}=s^2F(s)-sf(0)-f'(0) $$

PDEs

• Heat: \(u_t = c^2 u_{xx}\)
• Wave: \(u_{tt} = c^2 u_{xx}\)
• Laplace: \(u_{xx}+u_{yy}=0\)

Solved by separation of variables with appropriate ICs/BCs.

📖 Complex Variables

\(f(z)\) analytic ⇔ Cauchy–Riemann equations hold and partials are continuous (sufficient conditions).

Cauchy integral theorem: \(\oint_C f\,dz=0\) if \(f\) analytic inside/on simple closed \(C\).

Cauchy integral formula: \(f(a)=\frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz\).

Taylor series about \(z_0\) in a disk of analyticity; Laurent series in an annulus, may contain negative powers (principal part) when \(z_0\) is an isolated singularity.

📖 Probability & Statistics

$$ P(A|B)=\frac{P(A\cap B)}{P(B)},\quad P(A\cup B)=P(A)+P(B)-P(A\cap B) $$

Independent ⇔ \(P(A\cap B)=P(A)P(B)\).

Distributions

• Binomial(\(n,p\)): \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\); mean \(np\), var \(np(1-p)\).
• Poisson(\(\lambda\)): \(P(X=k)=e^{-\lambda}\lambda^k/k!\); mean = var = \(\lambda\).
• Normal(\(\mu,\sigma^2\)); standard normal \(\mu=0,\sigma=1\).

Sample mean, median, mode; standard deviation \(\sigma=\sqrt{\mathrm{Var}}\).

📖 Numerical Methods

Nonlinear equations

$$ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\quad\text{(Newton–Raphson)} $$

Bisection requires a sign-changing interval; converges linearly.

Numerical integration

$$ \int_a^b f\approx\frac{h}{2}\big(f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\big)\quad\text{(Trapezoidal, }h=(b-a)/n\text{)} $$

$$ \int_a^b f\approx\frac{h}{3}\big(f(x_0)+4f(x_1)+2f(x_2)+\cdots+f(x_n)\big)\quad\text{(Simpson 1/3, }n\text{ even)} $$

ODE initial-value problems

$$ y_{n+1}=y_n+h f(x_n,y_n)\quad\text{(Euler)} $$

Improved Euler / Heun and classical RK4 are multi-stage higher-order single-step methods; multistep methods (Adams–Bashforth/Moulton) use several past points.

Yogendra Gopal Borse

Civil Engineer | Assistant Engineer Grade-I, Maharashtra PWD

B.Tech (Civil) from VJTI Mumbai. Experienced in bridge design, road works, estimation, project monitoring and digital engineering tools. Creator of YogiPWD – practical technical resources for civil engineers.

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