List of top Mathematics Foundation Questions asked in GATE Environmental Science and Engineering

Consider the equation for a curve, $y = f(x) = x^2 + x$. The area enclosed by the curve, the x-axis (y = 0 line); the vertical lines passing through x = 1 and x = 2 is ______(rounded off to 2 decimal places)

Consider the function f(x) = ln(sin(x)). Expand f(x+h) using Taylor's series. In this context, the correct statement(s) is/are

A population (with mean μ) follows normal distribution. Ten samples (N) are drawn at random with a mean value of "x" and standard deviation of "S". Following table provides the confidence limits, C(t) of the cumulative probability function for Student's t - distribution two-tailed test with degree of freedom, D.
Which one of the following expression is correct for testing the null hypothesis $H_o: μ = 0$ at 10% significance level?

D	C(t)
	0.9	0.95	0.975
9	1.38	1.83	2.26
10	1.37	1.81	2.23
11	1.36	1.80	2.20

Consider the equation - $\farc{dy}{dx}- x^2 + e^x = 0;$ with y =1 at x = 0.
The value of y at x = 1 is _________. (rounded off to 2 decimal places).
Take the value of e (base of natural logarithm) as 2.7.

Which one is the solution y(x) for the following ordinary differential equation and the specified boundary conditionmys?
\(\frac{d^2y}{dx^2} -3 \frac{dy}{dx}+2y = 2e ^{-x}; y(0) =2; (\frac{dy}{dx}_{x=0})=1\)

$f(x) = x^3 – 4.5x^2 - 12x$ has a local maximum at x = ___________(an integer value) in the range x = − 2 to + 2

If X is a vector, and A and B are linear operators; then the correct mathematical relationship(s) is/are

The eigen values of the matrix \(\begin{bmatrix}4 & 3 \\3 & 4 \end{bmatrix} \) are

Ten cards in a pack are numbered as 1, 2, 3, . . .10. The probability of drawing a card with an even number or a number which is a multiple of 5 from the pack is