If the angle between the vectors a=2^2 i+4 j+ k and b=7 i-2 j+ k is obtuse, then Option 1: (0,∞) Option 2: (0,1/2) Option 3: (-1,1) Option 4: (-∞,0) (1/2,∞)

Answer: (0,1/2). Solution: An obtuse angle requires a* b < 0. a* b = 14^2 - 8 + = 14^2 - 7 = 7(2-1). This is negative exactly when 0 < < 1/2.

Algebra

Vector Algebra

Medium

core

If the angle between the vectors $\vec a=2\lambda^2\hat i+4\lambda\hat j+\hat k$ and $\vec b=7\hat i-2\hat j+\lambda\hat k$ is obtuse, then

$\lambda \in (0,\infty)$

$\lambda \in \left(0,\dfrac{1}{2}\right)$

$\lambda \in (-1,1)$

$\lambda \in (-\infty,0) \cup \left(\dfrac{1}{2},\infty\right)$

✅ Correct Option: 2

An obtuse angle requires $\vec a\cdot\vec b < 0$ .

$\vec a\cdot\vec b = 14\lambda^2 - 8\lambda + \lambda = 14\lambda^2 - 7\lambda = 7\lambda(2\lambda-1)$ .

This is negative exactly when $0 < \lambda < \dfrac{1}{2}$ .

2025: 22 May Shift 2

If $\vec{a} = 2\hat{i} + m\hat{j} - n\hat{k}$ and $\vec{b} = l\hat{i} - 3\hat{j} + 4\hat{k}$ such that $\vec{a} = 2\vec{b}$ then the value of $14{l} + m + n$ is:

2026: 18 May Shift 2

There are two unit vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}+\vec{b}|=\sqrt{3}$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ , then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\vert \vec{a} - \vec{b} \vert$	I.	$0$
B.	$\cos\theta$	II.	$1$
C.	$\theta$	III.	$\dfrac{1}{2}$
D.	$\sin 3\theta$	IV.	$\dfrac{\pi}{3}$

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:

2026: 14 May Shift 2

If $\vec a, \vec b, \vec c$ are mutually perpendicular vectors of equal magnitude, the angle between $\vec a+\vec b+\vec c$ and $\vec a$ is

2025: 14 May Shift 1

If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$ , $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

2025: 3 June Shift 1

The projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$ is

If the angle between the vectors a⃗=2λ2i^+4λj^+k^\vec a=2\lambda^2\hat i+4\lambda\hat j+\hat ka=2λ2i^+4λj^​+k^ and b⃗=7i^−2j^+λk^\vec b=7\hat i-2\hat j+\lambda\hat kb=7i^−2j^​+λk^ is obtuse, then

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If the angle between the vectors $\vec a=2\lambda^2\hat i+4\lambda\hat j+\hat k$ and $\vec b=7\hat i-2\hat j+\lambda\hat k$ is obtuse, then