If is the acute angle between two unit vectors a and b, then ()/2 = Option 1: 1/2 a + b Option 2: 1/2 a - b Option 3: 1/4 a + b Option 4: 1/4 a - b

Answer: 1/2 a + b. Solution: a+b, ^2 = 1 + 1 + 2 = 2(1+) = 4^2(/2). So, a+b, = 2(/2) (positive since acute). Hence (/2) = 1/2, a+b, .

Algebra

Vector Algebra

Medium

If $\theta$ is the acute angle between two unit vectors $\vec{a}$ and $\vec{b}$ , then $\cos\frac{\theta}{2} =$

$\frac{1}{2}|\vec{a} + \vec{b}|$

$\frac{1}{2}|\vec{a} - \vec{b}|$

$\frac{1}{4}|\vec{a} + \vec{b}|$

$\frac{1}{4}|\vec{a} - \vec{b}|$

✅ Correct Option: 1

$|\vec{a}+\vec{b}|^2 = 1 + 1 + 2\cos\theta = 2(1+\cos\theta) = 4\cos^2(\theta/2)$ . So $|\vec{a}+\vec{b}| = 2\cos(\theta/2)$ (positive since $\theta$ acute). Hence $\cos(\theta/2) = \frac{1}{2}|\vec{a}+\vec{b}|$ .

2025: 27 May Shift 1

Match List-I with List-II

List-I	List-II
(A) If vector $\vec{a}$ and $\vec{b}$ are such that $\vec{a} = \lambda \vec{b}$ and $\vert \vec{a}\vert = \vert \vec{b}\vert$ , then	(I) $\vec{a}$ and $\vec{b}$ are orthogonal
(B) Projection vector of $\vec{a}$ on $\vec{b}$	(II) $[0, 12]$
(C) $\vec{a}$ and $\vec{b}$ are non-zero vectors such that $\vert \vec{a} + \vec{b}\vert = \vert \vec{a} - \vec{b}\vert$ , then	(III) $\vec{a} = \pm \vec{b}$
(D) If $\vert \vec{a}\vert = 4, -3 \le \lambda \le 2$ , then the range of $\vert \lambda \vec{a}\vert$	(IV) $(\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2}) \vec{b}$

Choose the correct answer from the options given below:

2026: 25 May Shift 1

If $\lvert\vec{a}\rvert = 4, \lvert\vec{b}\rvert = 3$ and $\lvert\vec{a}+\vec{b}\rvert = 6$ , then $\lvert\vec{a}-\vec{b}\rvert =$

2025: 13 May Shift 1

Match List-I with List-II

List-I	List-II
(A) Angle between $\vec{i} - \vec{j}$ and $\vec{j} + \vec{k}$	(I) 0
(B) Angle between $2\vec{j} - \vec{k}$ and $\vec{j} + 2\vec{k}$	(II) $\frac{2\pi}{3}$
(C) Angle between $\vec{i} + 2\vec{j}$ and $5\vec{i} + 10\vec{j}$	(III) $\frac{\pi}{6}$
(D) Angle between $\sqrt{3}\vec{i} + \vec{j}$ and $\vec{i}$	(IV) $\frac{\pi}{2}$

Choose the correct answer from the options given below:

2025: 14 May Shift 2

Match List-I with List-II

Consider two vectors $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -3\hat{i} - 6\hat{j} + 3\hat{k}$ , then

List-I	List-II
(A) Angle between $\vec{a}$ and $\vec{b}$ is	(I) $\cos^{-1}\left(\frac{1}{\sqrt{6}}\right)$
(B) Angle between $\vec{a}$ and $x$ -axis is	(II) $\cos^{-1}\left(\frac{2}{\sqrt{6}}\right)$
(C) Angle between $\vec{b}$ and $x$ -axis is	(III) $\pi$
(D) Angle between $\vec{a}$ and $y$ -axis is	(IV) $\cos^{-1}\left(-\frac{1}{\sqrt{6}}\right)$

Choose the correct answer from the options given below:

2024: 16 May Shift 1

The angle between two lines whose direction ratios are propotional to $1,1,-2$ and $(\sqrt{3}-1),(-\sqrt{3}-1),-4$ is :

2026: 26 May Shift 2

Let $\vec{a},\vec{b},\vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=2, |\vec{c}|=1$ . If the projection of $\vec{b}$ along $\vec{c}$ is equal to the projection of $\vec{a}$ along $\vec{c}$ and $\vec{b} \perp \vec{a}$ , then the value of $|2\vec{a}-2\vec{b}+3\vec{c}|$ is equal to

If θ\thetaθ is the acute angle between two unit vectors a⃗\vec{a}a and b⃗\vec{b}b, then cos⁡θ2=\cos\frac{\theta}{2} =cos2θ​=

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CUET Mathematics Past Year Questions (Topic-Wise)

If $\theta$ is the acute angle between two unit vectors $\vec{a}$ and $\vec{b}$ , then $\cos\frac{\theta}{2} =$