List of top Mathematics Questions asked in CPET

For what value(s) of the constants \(a\), \(b\) and \(c\) is the quadrature formula \[\int_{-1}^1 f(x)\,dx \approx af(-1)+bf(0)+cf(1)\] exact for polynomials of degree up to 3?

What is the approximate value of the integral \(\int_0^1(x+x^2)\,dx\), when Simpson's 1/3-rd rule is applied with 2 sub-intervals?

Which one of the following statements about the central difference and averaging operators is correct?

Using Lagrange interpolation with the following function values, find the approximate value of \(f(2)\).

x	0	1	3
f(x)	1	2	4

Using Newton's divided difference method, the second divided difference for the function values f(1)=2, f(2)=3, f(4)=7 is approximately equal to ____.

Which of the following statements is/are true?
(I) Gauss-Seidel method for solving a linear system converges faster than the Gauss-Jacobi method.
(II) In the Gauss-Jordan method for solving a linear system, the coefficient matrix is transformed into an upper triangular matrix.

The solution of the system of ODEs: \(\frac{dx}{dt}=x+2y\), \(\frac{dy}{dt}=3x+2y\) with initial conditions x(0)=6 and y(0)=4 is ____.

Regula Falsi method is used to find a root of the equation \(f(x)=x^3-x-1\) by choosing the initial guesses \(x_0=1\) and \(x_1=2\). What would be the value of the next iteration (i.e. \(x_3\)) up to two decimal places?

Which of the following statements is/are true for the PDE: \((1+x^2)u_{xx}+(1+y^2)u_{yy}+xu_x+yu_y=0\)?
(I) It is classified as parabolic PDE.
(II) The canonical equation is \(u_{\xi\xi}+u_{\eta\eta}=0\).

In solving the heat equation using separation of variables, the eigenvalues correspond to ____.

Solution of the PDE \(u_x+yu_y=0\) with the initial condition \(u(0,y)=y^3\) is ____.

A string of length 1 meter is fixed at both ends and obeys the wave equation \(u_{tt}=4u_{xx}\) with initial conditions: \(u(x,0)=\sin(\pi x)\), \(u_t(x,0)=0\). Then its solution \(u(x,t)\) is ____.

If the roots of the characteristic equation of the Euler's ODE has a repeated root m, then what is the correct form of the general solution? (\(c_1,c_2\) are arbitrary constants.)

The general solution of the linear partial differential equation (PDE): \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=z\) is ____ (\(\phi\) is an arbitrary function).

The Bernoulli equation \(\frac{dy}{dx}+P(x)y=Q(x)y^3\) is best solved by making which of the following substitutions?

Which one of the following options is true for the initial value problem: \(\frac{dy}{dx}=2y^{1/3}\), \(y(0)=0\)?

The general solution of the ODE \((D^2+6D+9)y=\frac{e^{-3x}}{x^3}\), \(D=\frac{d}{dx}\) (\(c_1,c_2\) are arbitrary constants) is ____.

If the non-homogeneous term in an ordinary differential equation (ODE) is \(xe^{3x}\) and 3 is a root of the characteristic equation with multiplicity 2, then what is the form of the particular solution (i.e. \(y_p\))? (c is a constant.)

If \(u=x^3\) and \(v=y^2\) transforms the ODE: \(3x^5dx - y(y^2-x^3)dy=0\) into \(\frac{dy}{du}=\frac{\lambda u}{(u-v)}\), then what is the value of \(\lambda\)?

What does the differential equation \((2x+y+1)dx+(x+2y+1)dy=0\) represent?

Which one of the following statements is false?
(I) The linear transformation \(L:\mathbb{R}^2 \to \mathbb{R}^2\) that reflects a vector across the line \(y=x\) is both self-adjoint and normal.
(II) The linear transformation \(L:\mathbb{R}^2 \to \mathbb{R}^2\) which rotates a vector by an angle \(\pi/4\) in the anti-clockwise direction is both self-adjoint and normal.

The projection of the vector \(u=(2,-1,3) \in \mathbb{R}^3\) onto the vector \(v=(1,2,-1)\) of the vector space \(\mathbb{R}^3\) is ____.

If \(\lambda_1,\lambda_2,\lambda_3\) are the eigenvalues of the matrix \[\begin{pmatrix}-2&2&-3\\2&1&-6\\-1&-2&0\end{pmatrix}\] then the value of \(\lambda_1^2+\lambda_2^2+\lambda_3^2\) is ____.

If the eigenvalue \(\lambda\) of a matrix \(M\) has algebraic multiplicity 3 and geometric multiplicity 2, then which one of the following options is true?

Which one of the following matrices has \(p(x) = x^3 - 8x^2 + 5x + 7\) as the minimal polynomial?