CUET PG 2022 — Mathematics PYQ

• $\vec{a}$ and $\vec{b}$ are unit vectors, which means:

  $$|\vec{a}|=1,|\vec{b}|=1$$

• The vectors $(\vec{a}+2\vec{b})$ and $(5\vec{a}-4\vec{b})$ are perpendicular.

• For two perpendicular vectors, their dot product is zero:

  $$(\vec{a}+2\vec{b})\cdot (5\vec{a}-4\vec{b})=0$$

Step 2: Expand the Dot Product

Using the distributive property of the dot product:

Since $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$, we can group the terms:

Step 3: Substitute the Magnitudes

Substitute $|\vec{a}|=1$ and $|\vec{b}|=1$ into the equation:

Solve for the dot product:

Step 4: Find the Angle ($\theta$)

The formula for the dot product is $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$.

Substitute the known values:

We know that $\cos \theta =\frac{1}{2}$ when: