If a.b = a.c and axb = axc, a≠0, then Option 1: b+c=0 Option 2: b-c=0 Option 3: b+c=a Option 4: a+b=0

Answer: b-c=0. Solution: The first condition gives a*( b- c)=0, so b- c is perpendicular to a. The second gives ax( b- c)=0, so it is parallel to a. Both can hold only if b- c=0.

Algebra

Vector Algebra

Easy

core

If $\vec{a}.\vec{b} = \vec{a}.\vec{c}$ and $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$ , $\vec{a}\neq\vec{0}$ , then

$\vec{b}+\vec{c}=\vec{0}$

$\vec{b}-\vec{c}=\vec{0}$

$\vec{b}+\vec{c}=\vec{a}$

$\vec{a}+\vec{b}=\vec{0}$

✅ Correct Option: 2

The first condition gives $\vec a\cdot(\vec b-\vec c)=0$ , so $\vec b-\vec c$ is perpendicular to $\vec a$ . The second gives $\vec a\times(\vec b-\vec c)=\vec0$ , so it is parallel to $\vec a$ . Both can hold only if $\vec b-\vec c=\vec0$ .

2026: 21 May Shift 1

Let $\vec{a}$ be any vector such that $\vec{a}.\hat{i} = \vec{a}.(\hat{i}+\hat{j}) = \vec{a}.(\hat{i}+\hat{j}+\hat{k}) = 1$ , then $\vec{a}$ is

2026: 19 May Shift 1

The position vector of a point R which divides the line joining the points A(-2, 1, 3) and B(3, 5, -2) internally in the ratio 2:1, is

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:

2025: 3 June Shift 2

Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$ . Then match List-I with List-II

List-I	List-II
(A) $\sin \theta$	(I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert \vert \vec{b}\vert }$
(B) $\cos \theta$	(II) $\vert \vec{a} \times \vec{b}\vert$
(C) Area of the parallelogram with adjacent sides represented by $\vec{a}$ and $\vec{b}$	(III) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert }$
(D) Projection of $\vec{a}$ on $\vec{b}$	(IV) $\dfrac{\vert \vec{a} \times \vec{b}\vert }{\vert \vec{a}\vert \vert \vec{b}\vert }$

Choose the correct answer from the options given below:

2025: 3 June Shift 1

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$ , $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$ , is

2026: 19 May Shift 2

Vectors $\vec{a}, \vec{b}, \vec{c}$ are such that $\vec{a} + \vec{b} + \vec{c} = 0$ and $|\vec{a}| = |\vec{b}| = |\vec{c}|$ then, the angle between any two of them, is

If a⃗.b⃗=a⃗.c⃗\vec{a}.\vec{b} = \vec{a}.\vec{c}a.b=a.c and a⃗×b⃗=a⃗×c⃗\vec{a}\times\vec{b} = \vec{a}\times\vec{c}a×b=a×c, a⃗≠0⃗\vec{a}\neq\vec{0}a=0, then

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CUET Mathematics Past Year Questions (Topic-Wise)

If $\vec{a}.\vec{b} = \vec{a}.\vec{c}$ and $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$ , $\vec{a}\neq\vec{0}$ , then