NIMCET 2025 — Mathematics PYQ

To find the obtuse angle between the two given lines, we first need to determine their slopes.

Step 1: Find the Slopes of the Lines

The standard slope-intercept form of a straight line is:

$$y=mx+c$$

where $m$ is the slope.

• First Line: $2y=x+1$

  Dividing both sides by 2 gives:

  $$y=\frac{1}{2}x+\frac{1}{2}$$

  So, the slope of the first line ($m_1$) is:

  $$m_1=\frac{1}{2}$$

• Second Line: $y=3x+2$

  This equation is already in slope-intercept form.

  So, the slope of the second line ($m_2$) is:

  $$m_2=3$$

Step 2: Apply the Angle Formula

The acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by the formula:

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$$

Substituting the values of $m_1$ and $m_2$:

$$\tan \theta =\left|\frac{3-\frac{1}{2}}{1+\left(\frac{1}{2}\right)(3)}\right|$$

Simplify the numerator and the denominator:

$$\tan \theta =\left|\frac{\frac{5}{2}}{1+\frac{3}{2}}\right|$$

$$\tan \theta =\left|\frac{\frac{5}{2}}{\frac{5}{2}}\right|$$

$$\tan \theta =|1|=1$$

Since $\tan \theta =1$, the acute angle is:

$$\theta =\frac{\pi }{4}(\text{or }45^{\circ })$$

Step 3: Calculate the Obtuse Angle

The question explicitly asks for the obtuse angle (an angle greater than $90^{\circ }$ or $\frac{\pi }{2}$).

Two intersecting lines form two supplementary angles ($\theta$ and $180^{\circ }-\theta$). Therefore, the obtuse angle is:

$$\text{Obtuse Angle}=\pi -\theta$$

$$\text{Obtuse Angle}=\pi -\frac{\pi }{4}=\frac{3\pi }{4}(\text{or }135^{\circ })$$

Correct Answer

The obtuse angle between the lines is Option B:

$$\frac{3\pi }{4}$$