CUET UG 2025 — Mathematics Algebra

2025mcqmedium

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \vec{c} = 3\hat{i} + \hat{j}$ be three vectors. If $(\vec{a} + \lambda\vec{b})$ is perpendicular to $\vec{c}$ , then the value of $\lambda$ is

Official previous-year question

Held on 3 Jun 2025 · Verified 13 Jul 2026.

Options

A
3
B
5
C
2
D
-3

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Solution

$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \quad \vec{c} = 3\hat{i} + \hat{j}$

Since $(\vec{a} + \lambda\vec{b}) \perp \vec{c}$ , their dot product equals zero:

$(\vec{a} + \lambda\vec{b}) \cdot \vec{c} = 0$

$\vec{a} + \lambda\vec{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(-\hat{i} + 2\hat{j} + \hat{k})$

$= (1 - \lambda)\hat{i} + (2 + 2\lambda)\hat{j} + (3 + \lambda)\hat{k}$

Taking the dot product with $\vec{c} = 3\hat{i} + \hat{j} + 0\hat{k}$ :

$(\vec{a} + \lambda\vec{b}) \cdot \vec{c} = 3(1 - \lambda) + 1(2 + 2\lambda) + 0(3 + \lambda)$

$= 3 - 3\lambda + 2 + 2\lambda$

$= 5 - \lambda$

$5 - \lambda = 0$

$\lambda = 5$

Therefore, $\boxed{\lambda = 5}$

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Solution

$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \quad \vec{c} = 3\hat{i} + \hat{j}$

Since $(\vec{a} + \lambda\vec{b}) \perp \vec{c}$ , their dot product equals zero:

$(\vec{a} + \lambda\vec{b}) \cdot \vec{c} = 0$

$\vec{a} + \lambda\vec{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(-\hat{i} + 2\hat{j} + \hat{k})$

$= (1 - \lambda)\hat{i} + (2 + 2\lambda)\hat{j} + (3 + \lambda)\hat{k}$

Taking the dot product with $\vec{c} = 3\hat{i} + \hat{j} + 0\hat{k}$ :

$(\vec{a} + \lambda\vec{b}) \cdot \vec{c} = 3(1 - \lambda) + 1(2 + 2\lambda) + 0(3 + \lambda)$

$= 3 - 3\lambda + 2 + 2\lambda$

$= 5 - \lambda$

$5 - \lambda = 0$

$\lambda = 5$

Therefore, $\boxed{\lambda = 5}$

CUET UG 2025 — Mathematics Algebra

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Solution

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CUET UG 2025 — Mathematics Algebra

Options

Did you get this right?

Solution

Your note

CUET UG Algebra in other years

More CUET UG Algebra questions

Other Mathematics topics for CUET UG

Calculus

Applied-Mathematics

Statistics & Applications

Geometry

Arithmetic

Coordinate Geometry

Keep practicing Algebra