1
JEE Main 2023 (Online) 11th April Evening Shift
+4
-1
Out of Syllabus

If four distinct points with position vectors $$\vec{a}, \vec{b}, \vec{c}$$ and $$\vec{d}$$ are coplanar, then $$[\vec{a} \,\,\vec{b} \,\,\vec{c}]$$ is equal to :

A
$$[\vec{d} \,\,\,\,\,\vec{b} \,\,\,\,\,\vec{a}]+[\vec{a} \,\,\,\,\,\vec{c} \,\,\,\,\,\vec{d}]+[\vec{d} \,\,\,\,\,\vec{b} \,\,\,\,\,\vec{c}]$$
B
$$[\vec{b} \,\,\,\,\,\vec{c} \,\,\,\,\,\vec{d}]+[\vec{d} \,\,\,\,\,\vec{a} \,\,\,\,\,\vec{c}]+[\vec{d} \,\,\,\,\,\vec{b} \,\,\,\,\,\vec{a}]$$
C
$$[\vec{a} \,\,\,\,\,\vec{d} \,\,\,\,\,\vec{b}]+[\vec{d} \,\,\,\,\,\vec{c} \,\,\,\,\,\vec{a}]+[\vec{d} \,\,\,\,\,\vec{b} \,\,\,\,\,\vec{c}]$$
D
$$[\vec{d} \,\,\,\,\,\vec{c} \,\,\,\,\,\vec{a}]+[\vec{b} \,\,\,\,\,\vec{d} \,\,\,\,\,\vec{a}]+[\vec{c} \,\,\,\,\,\vec{d} \,\,\,\,\,\vec{b}]$$
2
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1

For any vector $$\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$$, with $$10\left|a_{i}\right|<1, i=1,2,3$$, consider the following statements :

(A): $$\max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\} \leq|\vec{a}|$$

(B) : $$|\vec{a}| \leq 3 \max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\}$$

A
Only (B) is true
B
Only (A) is true
C
Neither (A) nor (B) is true
D
Both (A) and (B) are true
3
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1

Let $$\vec{a}$$ be a non-zero vector parallel to the line of intersection of the two planes described by $$\hat{i}+\hat{j}, \hat{i}+\hat{k}$$ and $$\hat{i}-\hat{j}, \hat{j}-\hat{k}$$. If $$\theta$$ is the angle between the vector $$\vec{a}$$ and the vector $$\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}$$ and $$\vec{a} \cdot \vec{b}=6$$, then the ordered pair $$(\theta,|\vec{a} \times \vec{b}|)$$ is equal to :

A
$$\left(\frac{\pi}{3}, 3 \sqrt{6}\right)$$
B
$$\left(\frac{\pi}{3}, 6\right)$$
C
$$\left(\frac{\pi}{4}, 3 \sqrt{6}\right)$$
D
$$\left(\frac{\pi}{4}, 6\right)$$
4
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1

Let $$\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$$ and $$\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d}=12$$. Then $$(-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})$$ is equal to :

A
24
B
42
C
44
D
48
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