IAS 2013 — CSAT Maths

Correct Option

Let the original speed of the train be $x$x km/hr.

• The distance traveled at the original speed is 63 km. The time taken for this segment is $\frac{63}{x}$x63​ hours.
• The train then travels 72 km at an increased speed of $(x + 6)$(x+6) km/hr. The time taken for this segment is $\frac{72}{x + 6}$x+672​ hours.
• The total journey time is given as 3 hours.

Formulating the equation based on the total time:

$$\frac{63}{x} + \frac{72}{x + 6} = 3$$x63​+x+672​=3

To solve for $x$x, multiply the entire equation by $x(x + 6)$x(x+6) to eliminate the denominators:

$$63(x + 6) + 72x = 3x(x + 6)$$63(x+6)+72x=3x(x+6)

Expand and simplify the equation:

$$63x + 378 + 72x = 3x^2 + 18x$$63x+378+72x=3x2+18x

$$135x + 378 = 3x^2 + 18x$$135x+378=3x2+18x

Rearrange the terms to form a quadratic equation:

$$3x^2 + 18x - 135x - 378 = 0$$3x2+18x−135x−378=0

$$3x^2 - 117x - 378 = 0$$3x2−117x−378=0

Divide the entire equation by 3 to simplify:

$$x^2 - 39x - 126 = 0$$x2−39x−126=0

Factor the quadratic equation:

$$x^2 - 42x + 3x - 126 = 0$$x2−42x+3x−126=0

$$x(x - 42) + 3(x - 42) = 0$$x(x−42)+3(x−42)=0

$$(x - 42)(x + 3) = 0$$(x−42)(x+3)=0

This yields two possible values for $x$x: $x = 42$x=42 or $x = -3$x=−3.

Since speed cannot be a negative value, $x = -3$x=−3 is rejected. Therefore, the original speed of the train is 42 km/hr.

Incorrect Options

Options 1 (24 km/hr), 2 (33 km/hr), and 4 (66 km/hr) are incorrect because substituting these values into the derived equation $$\frac{63}{x} + \frac{72}{x + 6} = 3$$x63​+x+672​=3 does not satisfy the condition that the total journey time is 3 hours.

• If $x = 24$x=24 km/hr: $$\frac{63}{24} + \frac{72}{24 + 6} = 2.625 + \frac{72}{30} = 2.625 + 2.4 = 5.025 \ne 3$$2463​+24+672​=2.625+3072​=2.625+2.4=5.025=3
• If $x = 33$x=33 km/hr: $$\frac{63}{33} + \frac{72}{33 + 6} \approx 1.909 + \frac{72}{39} \approx 1.909 + 1.846 \approx 3.755 \ne 3$$3363​+33+672​≈1.909+3972​≈1.909+1.846≈3.755=3
• If $x = 66$x=66 km/hr: $$\frac{63}{66} + \frac{72}{66 + 6} \approx 0.954 + \frac{72}{72} = 0.954 + 1 = 1.954 \ne 3$$6663​+66+672​≈0.954+7272​=0.954+1=1.954=3