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The graph of the line $y=3x$ passing through $(0, 0)$ and $(2, 6)$ is given below.
(i) Identify the slope of the line from the graph and explain how you calculated it.
(ii) What is the y-intercept of the line? Explain its significance.
(iv) Without drawing a new graph, determine whether the point $(4, 10)$ lies on this line. Justify your answer.
Answer:
(i) Slope = $\frac{\text{Rise}}{\text{Run}} = \frac{6 - 0}{2 - 0} = \frac{6}{2} = 3$.
(ii) The line passes through $(0,0)$, so the y-intercept is 0. This signifies that at the starting point ($x=0$), the value of $y$ is also 0.
(iv) No. Substitute $x=4$ into the equation $y=3x$. $y = 3(4) = 12$. Since $12 \neq 10$, the point $(4, 10)$ does not lie on the line.
The formula to convert Celsius to Fahrenheit is $F = \frac{9}{5}C + 32$.
(ii) If the temperature is $30^\circ C$, what is the temperature in Fahrenheit?
(iii) If the temperature is $95^\circ F$, what is the temperature in Celsius?
(v) Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.
Answer:
(ii) $F = \frac{9}{5}(30) + 32 = 9(6) + 32 = 54 + 32 = \mathbf{86^\circ F}$.
(iii) $95 = \frac{9}{5}C + 32 \implies 63 = \frac{9}{5}C \implies C = \frac{63 \times 5}{9} = 7 \times 5 = \mathbf{35^\circ C}$.
(v) Yes. Set $F = C = x$.
$x = \frac{9}{5}x + 32 \implies x - \frac{9}{5}x = 32 \implies -\frac{4}{5}x = 32 \implies x = -\frac{32 \times 5}{4} = -4 \times 5 = \mathbf{-40}$. So, $-40^\circ C = -40^\circ F$.
Give a formula to find the x-intercept of the line $y = mx + c$.
Answer: The x-intercept occurs where the line crosses the x-axis, meaning $y = 0$.
Substitute $y=0$ into the equation: $0 = mx + c \implies mx = -c \implies x = -\frac{c}{m}$.
The formula for the x-intercept is $-\frac{c}{m}$ (assuming $m \neq 0$).
Plan A: ₹50 monthly fee + ₹0.20 per minute. Plan B: ₹30 monthly fee + ₹0.30 per minute.
Find the number of minutes at which both plans cost the same.
Answer: Let the number of minutes be $x$.
Cost of Plan A = $50 + 0.20x$
Cost of Plan B = $30 + 0.30x$
Equate them: $50 + 0.20x = 30 + 0.30x \implies 20 = 0.10x \implies x = \frac{20}{0.10} = 200$.
Both plans cost the same at 200 minutes. (Below 200 mins, Plan B is cheaper; above 200 mins, Plan A is cheaper).
For what values of $p$ does the pair of equations $4x + py + 8 = 0$ and $2x + 2y + 2 = 0$ have a unique solution?
Answer: For a unique solution, the ratio of coefficients must be $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.
$\frac{4}{2} \neq \frac{p}{2} \implies 2 \neq \frac{p}{2} \implies p \neq 4$.
The equations have a unique solution for all real values of $p$ except $p = 4$.
Find the value of 'k' for which the following system of equations represents a pair of coincident lines:
$x + 2y = 3$; $(k-1)x + (k+1)y = k+3$.
Answer: For coincident lines (infinitely many solutions), $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
$\frac{1}{k-1} = \frac{2}{k+1} = \frac{3}{k+3}$
Solve the first part: $\frac{1}{k-1} = \frac{2}{k+1} \implies k+1 = 2(k-1) \implies k+1 = 2k-2 \implies 3 = k$.
Verify with the third part: $\frac{2}{3+1} = \frac{2}{4} = \frac{1}{2}$ and $\frac{3}{3+3} = \frac{3}{6} = \frac{1}{2}$. It matches.
So, k = 3.
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