Quiz 7: FE Civil Mathematics and Statistics Practice

Questions in this quiz (10)

1. The second moment of area about the $$y$$-axis is calculated using which of the following integrals?

   • $$\int ydA$$
   • $$\int x^2dA$$
   • $$\int y^2dA$$
   • $$\int xydA$$

2. Calculate the derivative of $$f(x)=\arctan(2x)$$ at $$x=1$$.

   • $$\frac{1}{5}$$
   • $$\frac{2}{5}$$
   • $$\frac{5}{2}$$
   • $$\frac{2}{3}$$

3. In hypothesis testing, the significance level $$\alpha$$ represents:

   • The probability that the null hypothesis is true
   • The probability of rejecting a true null hypothesis
   • The probability of accepting a false null hypothesis
   • The probability of making a Type $$II$$ error

4. Evaluate the line integral $$\int_C(x+2y)\,ds$$, where $$C$$ is the straight line segment from $$(0,0)$$ to $$(3,4)$$.

   • $$20.0$$
   • $$25.0$$
   • $$27.5$$
   • $$35.0$$

5. If $$E[X]=4$$ and $$E[Y]=7$$, calculate $$E[3X+2Y]$$.

   • $$20$$
   • $$22$$
   • $$26$$
   • $$28$$

6. Given $$f(x,y)=x^2y^3+3xy^2$$, calculate the mixed partial derivative $$\frac{\partial^2f}{\partial x\partial y}$$ at the point $$(1,1)$$. First differentiate with respect to $$y$$: $$\frac{\partial f}{\partial y}=3x^2y^2+6xy$$ Now differentiate with respect to $$x$$: $$\frac{\partial^2f}{\partial x\partial y}=6xy^2+6y$$ Evaluate at $$(1,1)$$: $$\frac{\partial^2f}{\partial x\partial y}(1,1)=6(1)(1)^2+6(1)$$ $$=6+6=12$$ Final answer: $$12$$

   • $$15$$
   • $$12$$
   • $$9$$
   • $$6$$

7. Find the absolute minimum value of $$f(x)=x^3-3x^2-9x+5$$ on the closed interval $$[0,4]$$.

   • $$-22$$
   • $$-15$$
   • $$-5$$
   • $$5$$

8. Two samples have the following statistics: Sample $$1$$ has $$n_1=12$$ and $$s_1^2=10$$; Sample $$2$$ has $$n_2=18$$ and $$s_2^2=14$$. Calculate the pooled variance.

   • $$12.43$$
   • $$12.82$$
   • $$13.25$$
   • $$13.60$$

9. Calculate the curl of the vector field $$F=\langle yz,zx,xy\rangle$$.

   • $$\langle0,0,0\rangle$$
   • $$\langle x,y,z\rangle$$
   • $$\langle2x,2y,2z\rangle$$
   • $$\langle y-z,z-x,x-y\rangle$$

10. A $$95\%$$ confidence interval for a population mean is correctly interpreted as:

    • There is a $$95%$$ probability that the true mean lies in this specific interval
    • $$95%$$ of all data points lie within this interval
    • If sampling were repeated many times, about $$95%$$ of the intervals would contain the true mean
    • The sample mean equals the population mean with $$95\%$$ probability