Available Questions 607 found Page 30 of 31
Standalone Questions
#570
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If vector \(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\) and vector \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\), then which of the following is correct ?
(A) \(\vec{a} \parallel \vec{b}\)
(B) \(\vec{a} \perp \vec{b}\)
(C) \(|\vec{b}| > |\vec{a}|\)
(D) \(|\vec{a}| = |\vec{b}|\)
Key: B
Sol:
Sol:
**Correct Option if MCQ:** B
**Reasoning:**
* Calculate the dot product: \(\vec{a} \cdot \vec{b} = (3)(1) + (2)(-1) + (-1)(1) = 3 - 2 - 1 = 0\).
* Since the dot product is zero, the vectors are perpendicular.
* \(\vec{a} \perp \vec{b}\)
#569
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = \sqrt{37}\), \(|\vec{b}| = 3\) and \(|\vec{c}| = 4\), then the angle between \(\vec{b}\) and \(\vec{c}\) is
(A) \(\dfrac{\pi}{6}\)
(B) \(\dfrac{\pi}{4}\)
(C) \(\dfrac{\pi}{3}\)
(D) \(\dfrac{\pi}{2}\)
Key: C
Sol:
Sol:
#568
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The projection vector of vector \(\vec{a}\) on vector \(\vec{b}\) is
(A) \((\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^{2}})\vec{b}\)
(B) \(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\)
(C) \(\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}\)
(D) \((\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|^{2}})\vec{b}\)
Key: A
Sol:
Sol:
\((\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^{2}})\vec{b}\)
#567
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Let \(\vec{p}\) and \(\vec{q}\) be two unit vectors and \(\alpha\) be the angle between them. Then \((\vec{p}+\vec{q})\) will be a unit vector for what value of \(\alpha\)?
(A) \(\frac{\pi}{4}\)
(B) \(\frac{\pi}{3}\)
(C) \(\frac{\pi}{2}\)
(D) \(\frac{2\pi}{3}\)
Key: D
Sol:
Sol:
\(\frac{2\pi}{3}\)
#566
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If the sides AB and AC of \(\triangle ABC\) are represented by vectors \(\hat{j}+\hat{k}\) and \(3\hat{i}-\hat{j}+4\hat{k}\) respectively, then the length of the median through A on BC is:
(A) \(2\sqrt{2}\) units
(B) \(\sqrt{18}\) units
(C) \(\frac{\sqrt{34}}{2}\) units
(D) \(\frac{\sqrt{48}}{2}\) units
Key: C
Sol:
Sol:
The position vector of the midpoint \(D\) is the average of the position vectors of \(B\) and \(C\) relative to \(A\):\[\vec{AD} = \frac{\vec{AB} + \vec{AC}}{2}\]
\[\vec{AD} = \frac{3}{2}\hat{i} + \frac{5}{2}\hat{k}\]
The length of the median is the magnitude of the vector \(\vec{AD}\), denoted as \(|\vec{AD}|\):\[|\vec{AD}| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(0\right)^2 + \left(\frac{5}{2}\right)^2}\]
\[|\vec{AD}| = \frac{\sqrt{34}}{2}\]
#565
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Let \(\vec{a}\) be a position vector whose tip is the point \((2,-3)\). If \(\vec{AB}=\vec{a}\), where coordinates of A are \((-4, 5)\), then the coordinates of B are:
(A) \((-2,-2)\)
(B) \((2,-2)\)
(C) \((-2,2)\)
(D) \((2, 2)\)
Key:
Sol:
Sol:
#564
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
The respective values of \(|\vec{a}|\) and \(|\vec{b}|\), if given \((\vec{a}-\vec{b})\cdot(\vec{a}+\vec{b})=512\) and \(|\vec{a}|=3|\vec{b}|\), are:
(A) 48 and 16
(B) 3 and 1
(C) 24 and 8
(D) 6 and 2
Key:
Sol:
Sol:
#563
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Let \(|\vec{a}|=5\) and \(-2\le\lambda\le1\). Then, the range of \(|\lambda\vec{a}|\) is:
(A) [5, 10]
(B) [-2, 5]
(C) [2, 1]
(D) [-10, 5]
Key:
Sol:
Sol:
#562
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \(3\hat{i}+15\hat{j}+6\hat{k}\) and the other is along the vector \(2\hat{i}+10\hat{j}+\lambda\hat{k}\), then the value of \(\lambda\) is :
(A) 6
(B) 1
(C) \(\frac{1}{4}\)
(D) 4
Key:
Sol:
Sol:
#561
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\) for any two vectors, then vectors \(\vec{a}\) and \(\vec{b}\) are:
(A) orthogonal vectors
(B) parallel to each other
(C) unit vectors
(D) collinear vectors
Key:
Sol:
Sol:
#560
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Let \(\theta\) be the angle between two unit vectors \(\hat{a}\) and \(\hat{b}\) such that \(\sin\theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to:
(A) \(\pm\frac{3}{5}\)
(B) \(\pm\frac{3}{4}\)
(C) \(\pm\frac{4}{5}\)
(D) \(\pm\frac{4}{3}\)
Key:
Sol:
Sol:
#559
Mathematics
Vector Algebra
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The vector with terminal point \(A(2,-3,5)\) and initial point \(B(3, 4, 7)\) is:
(A) \(\hat{i}-\hat{j}+2\hat{k}\)
(B) \(\hat{i}+\hat{j}+2\hat{k}\)
(C) \(-\hat{i}-\hat{j}-2\hat{k}\)
(D) \(-\hat{i}+\hat{j}-2\hat{k}\)
Key:
Sol:
Sol:
#558
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
For any two vectors \(\vec{a}\) and \(\vec{b}\), which of the following statements is always true?
(A) \(\vec{a}.\vec{b}\ge
(B) \vec{a}
(C)
(D) \vec{b}
Key:
Sol:
Sol:
#557
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:
(A) \(2\hat{j}\)
(B) \(\hat{j}\)
(C) \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)
(D) \(\frac{\hat{i}+\hat{k}}{\sqrt{2}}\)
Key:
Sol:
Sol:
#556
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\), then \(\vec{a}\) and \(\vec{b}\):
(A) collinear vectors which are not parallel
(B) parallel vectors
(C) perpendicular vectors
(D) unit vectors
Key: C
Sol:
Sol:
#555
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If \(|\vec{a}|= 2\) and \(-3\le k\le2\), then \(|\vec{k}\vec{a}|\in\):
(A) [-6, 4]
(B) [0, 4]
(C) [4, 6]
(D) [0, 6]
Key:
Sol:
Sol:
#554
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If \(\vec{a}\) and \(\vec{b}\) are two vectors such that \(|\vec{a}|=1,|\vec{b}|=2~and\vec{a}\cdot\vec{b}=\sqrt{3}\) then the angle between \(2\vec{a}\) and \(-\vec{b}\) is:
(A) \(\frac{\pi}{6}\)
(B) \(\frac{\pi}{3}\)
(C) \(\frac{5\pi}{6}\)
(D) \(\frac{11\pi}{6}\)
Key:
Sol:
Sol:
#553
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of
(A) an equilateral triangle
(B) an obtuse-angled triangle
(C) an isosceles triangle
(D) a right-angled triangle
Key:
Sol:
Sol:
#552
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:
(A) \(a^{2}\)
(B) \(2a^{2}\)
(C) \(3a^{2}\)
(D) 0
Key:
Sol:
Sol:
#551
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
The position vectors of points P and Q are \(\vec{p}\) and \(\vec{q}\) respectively. The point R divides line segment PQ in the ratio 3:1 and S is the mid-point of line segment PR. The position vector of S is:
(A) \(\frac{\vec{p}+3\vec{q}}{4}\)
(B) \(\frac{\vec{p}+3\vec{q}}{8}\)
(C) \(\frac{5\vec{p}+3\vec{q}}{4}\)
(D) \(\frac{5\vec{p}+3\vec{q}}{8}\)
Key:
Sol:
Sol: