Online Test: Application of Calculus (Commerce) | Class 12 Mathematics

Question 1 of 10 easy

Marginal cost is:

$C(x)/x$

$dC/dx$

$C(x+1)-C(x-1)$

$C'(x)/x$

Explanation: Marginal cost MC $=dC/dx$ — the rate of change of total cost with respect to output.

Question 2 of 10 easy

If $C(x)=x^2+5x+10$, then marginal cost at $x=5$ is:

$15$

$60$

$10$

$5$

Explanation: $MC=C'(x)=2x+5$. At $x=5$: $MC=10+5=15$.

Question 3 of 10 easy

Profit is maximised when:

$MR>MC$

$MR=MC$ and $P''<0$

$MC=AC$

$P'(x)>0$

Explanation: Profit $P(x)$ is maximum when $P'(x)=MR-MC=0$ (critical point) and $P''(x)<0$ (second derivative test confirms maximum).

Question 4 of 10 medium

If demand $x=200-4p$, the elasticity at $p=25$ is:

$1$

$2$

$0.5$

$1/2$

Explanation: $dx/dp=-4$. At $p=25$: $x=200-100=100$. $E_d=-(25/100)(-4)=1$.

Question 5 of 10 medium

If $C(x)=100+5x+x^2$, average cost is minimum at $x=$

$10$

$5$

$100$

$50$

Explanation: $AC=100/x+5+x$. $d(AC)/dx=-100/x^2+1=0\Rightarrow x^2=100\Rightarrow x=10$.

Question 6 of 10 easy

Total revenue is $R=20x-x^2$. Marginal revenue = 0 when $x=$

$10$

$20$

$5$

$2$

Explanation: $MR=dR/dx=20-2x=0\Rightarrow x=10$.

Question 7 of 10 easy

Demand is inelastic means:

$E_d>1$

$E_d<1$

$E_d=1$

$E_d=0$

Explanation: Inelastic demand: $E_d<1$ — proportional change in quantity demanded is less than proportional change in price.

Question 8 of 10 easy

If $R(x)=10x$ and $C(x)=x^2+2x+5$, profit-maximising output is:

$4$

$5$

$3$

$8$

Explanation: $P=R-C=-x^2+8x-5$. $P'=-2x+8=0\Rightarrow x=4$.

Question 9 of 10 medium

Relationship between AC and MC:

AC is always greater than MC

When MC < AC, AC is decreasing; when MC > AC, AC is increasing

AC = MC always

MC = AC/x

Explanation: When MC < AC, each additional unit costs less than average, pulling AC down. When MC > AC, each additional unit costs more than average, pushing AC up. MC = AC at the minimum of AC.

Question 10 of 10 easy

Break-even point is where:

$MR=MC$

$P(x)=0$, i.e. $R(x)=C(x)$

$AC$ is minimum

$MC=0$

Explanation: Break-even: total revenue equals total cost, so profit = 0.

Online Test — Application of Calculus (Commerce)