CBSE Class 12 Maths – MCQ and Online Tests – Unit 10 – Vector Algebra
Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.
These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.
CBSE Class 12 Maths – MCQ and Online Tests – Unit 10 – Vector Algebra
Question 1.
\(\vec{a}\). \(\vec{a}\) =
(a) 0
(b) 1
(c) |\(\vec{a}\)|²
(d) |\(\vec{a}\)|
Answer
Answer: (c) |\(\vec{a}\)|²
Question 2.
\(\vec{k}\) × \(\vec{j}\) =
(a) 0
(b) 1
(c) \(\vec{i}\)
(d) –\(\vec{i}\)
Answer
Answer: (d) –\(\vec{i}\)
Question 3.
The projection of the vector 2\(\hat{i}\) – \(\hat{j}\) + \(\hat{k}\) on the vector \(\hat{i}\) – 2\(\hat{j}\) + \(\hat{k}\) is
(a) \(\frac{4}{√6}\)
(b) \(\frac{5}{√6}\)
(c) \(\frac{4}{√3}\)
(d) \(\frac{7}{√6}\)
Answer
Answer: (b) \(\frac{5}{√6}\)
Question 4.
If \(\vec{a}\) = \(\vec{i}\) – \(\vec{j}\) + 2\(\vec{k}\) and b = 3\(\vec{i}\) + 2\(\vec{j}\) – \(\vec{k}\) then the value of (\(\vec{a}\) + 3\(\vec{b}\))(2\(\vec{a}\) – \(\vec{b}\))=.
(a) 15
(b) -15
(c) 18
(d) -18
Answer
Answer: (b) -15
Question 5.
If |\(\vec{a}\)|= \(\sqrt{26}\), |b| = 7 and |\(\vec{a}\) × \(\vec{b}\)| = 35, then \(\vec{a}\).\(\vec{b}\) =
(a) 8
(b) 7
(c) 9
(d) 12
Answer
Answer: (b) 7
Question 6.
\(\vec{i}\) – \(\vec{j}\) =
(a) 0
(b) 1
(c) \(\vec{k}\)
(d) –\(\vec{k}\)
Answer
Answer: (a) 0
Question 7.
The position vector of the point (1, 0, 2) is
(a) \(\vec{i}\) +\(\vec{j}\) + 2\(\vec{k}\)
(b) \(\vec{i}\) + 2\(\vec{j}\)
(c) \(\vec{2}\) + 3\(\vec{k}\)
(d) \(\vec{i}\) + 2\(\vec{K}\)
Answer
Answer: (d) \(\vec{i}\) + 2\(\vec{K}\)
Question 8.
If \(\vec{a}\) = 2\(\vec{i}\) – 3\(\vec{j}\) + 4\(\vec{k}\) and \(\vec{b}\) = \(\vec{i}\) + 2\(\vec{j}\) + \(\vec{k}\) then \(\vec{a}\) + \(\vec{b}\) =
(a) \(\vec{i}\) + \(\vec{j}\) + 3\(\vec{k}\)
(b) 3\(\vec{i}\) – \(\vec{j}\) + 5\(\vec{k}\)
(c) \(\vec{i}\) – \(\vec{j}\) – 3\(\vec{k}\)
(d) 2\(\vec{i}\) + \(\vec{j}\) + \(\vec{k}\)
Answer
Answer: (b) 3\(\vec{i}\) – \(\vec{j}\) + 5\(\vec{k}\)
Question 9.
If \(\vec{a}\) = \(\vec{i}\) + 2\(\vec{j}\) + 3\(\vec{k}\) and \(\vec{b}\) = 3\(\vec{i}\) + 2\(\vec{j}\) + \(\vec{k}\), then cos θ =
(a) \(\frac{6}{7}\)
(b) \(\frac{5}{7}\)
(c) \(\frac{4}{7}\)
(d) \(\frac{1}{2}\)
Answer
Answer: (b) \(\frac{5}{7}\)
Question 10.
The modulus of 7\(\vec{i}\) – 2\(\vec{J}\) + \(\vec{K}\)
(a) \(\sqrt{10}\)
(b) \(\sqrt{55}\)
(c) 3\(\sqrt{6}\)
(d) 6
Answer
Answer: (c) 3\(\sqrt{6}\)
Question 11.
If |\(\vec{a}\) + \(\vec{b}\)| = |\(\vec{a}\) – \(\vec{b}\)|, then
(a) \(\vec{a}\) || \(\vec{a}\)
(b) \(\vec{a}\) ⊥ \(\vec{b}\)
(c) |\(\vec{a}\)| = |\(\vec{b}\)|
(d) None of these
Answer
Answer: (b) \(\vec{a}\) ⊥ \(\vec{b}\)
Question 12.
The projection of the vector 2\(\hat{i}\) + 3\(\hat{j}\) – 6\(\hat{k}\) on the line joining the points (3, 4, 2) and (5, 6,3) is
(a) \(\frac{2}{3}\)
(b) \(\frac{4}{3}\)
(c) –\(\frac{4}{3}\)
(d) \(\frac{5}{3}\)
Answer
Answer: (b) \(\frac{4}{3}\)
Question 13.
If |\(\vec{a}\) × \(\vec{b}\)| – |\(\vec{a}\).\(\vec{b}\)|, then the angle between \(\vec{a}\) and \(\vec{b}\), is
(a) 0
(b) \(\frac{π}{2}\)
(c) \(\frac{π}{4}\)
(d) π
Answer
Answer: (c) \(\frac{π}{4}\)
Question 14.
The angle between two vector \(\vec{a}\) and \(\vec{b}\) with magnitude √3 and 4, respectively and \(\vec{a}\).\(\vec{b}\) = 2√3 is
(a) \(\frac{π}{6}\)
(b) \(\frac{π}{3}\)
(c) \(\frac{π}{2}\)
(d) \(\frac{5π}{2}\)
Answer
Answer: (b) \(\frac{π}{3}\)
Question 15.
The scalar product of 5\(\hat{i}\) + \(\hat{j}\) – 3\(\hat{k}\) and 3\(\hat{i}\) – 4\(\hat{j}\) + 7\(\hat{k}\) is
(a) 10
(b) -10
(c) 15
(d) -15
Answer
Answer: (b) -10
Question 16.
If \(\vec{a}\).\(\vec{b}\) = 0, then
(a) a ⊥ b
(b) \(\vec{a}\) || \(\vec{b}\)
(c) \(\vec{a}\) + \(\vec{b}\) = 0
(d) \(\vec{a}\) – \(\vec{b}\) = 0
Answer
Answer: (a) a ⊥ b
Question 17.
Unit vector perpendicular to each of the vector 3\(\hat{i}\) + \(\hat{j}\) + 2\(\hat{k}\) and 2\(\hat{i}\) – 2\(\hat{j}\) + 4\(\hat{k}\) is
(a) \(\frac{\hat{i}+\hat{j}+\hat{k}}{√3}\)
(b) \(\frac{\hat{i}-\hat{j}+\hat{k}}{√3}\)
(c) \(\frac{\hat{i}-\hat{j}-\hat{k}}{√3}\)
(d) \(\frac{\hat{i}+\hat{j}-\hat{k}}{√3}\)
Answer
Answer: (c) \(\frac{\hat{i}-\hat{j}-\hat{k}}{√3}\)
Question 18.
Which one of the following can be written for (\(\vec{a}\) – \(\vec{b}\)) × (\(\vec{a}\) + \(\vec{b}\))
(a) \(\vec{a}\) × \(\vec{b}\)
(b) 2\(\vec{a}\) × \(\vec{b}\)
(c) \(\vec{a}\)² – \(\vec{b}\)
(d) 2\(\vec{b}\) × \(\vec{b}\)
Answer
Answer: (b) 2\(\vec{a}\) × \(\vec{b}\)
Question 19.
If \(\vec{a}\) = 2\(\vec{i}\) – 5\(\vec{j}\) + k and \(\vec{b}\) = 4\(\vec{i}\) + 2\(\vec{j}\) + \(\vec{k}\) then \(\vec{a}\).\(\vec{b}\) =
(a) 0
(b) -1
(c) 1
(d) 2
Answer
Answer: (b) -1
Question 20.
If 2\(\vec{i}\) + \(\vec{j}\) + \(\vec{k}\), 6\(\vec{i}\) – \(\vec{j}\) + 2\(\vec{k}\) and 14\(\vec{i}\) – 5\(\vec{j}\) + 4\(\vec{k}\) be the position vector of the points A, B and C respectively, then
(a) The A, B and C are collinear
(b) A, B and C are not colinear
(c) \(\vec{AB}\) ⊥ \(\vec{BC}\)
(d) None of these
Answer
Answer: (a) The A, B and C are collinear
Question 21.
According to the associative lass of addition of addition of s ector
(\(\vec{a}\) + …….) + \(\vec{c}\) = …… + (\(\vec{b}\) + \(\vec{c}\))
(a) \(\vec{b}\), \(\vec{a}\)
(b) \(\vec{a}\), \(\vec{b}\)
(c) \(\vec{a}\), 0
(d) \(\vec{b}\), 0
Answer
Answer: (a) \(\vec{b}\), \(\vec{a}\)
Question 22.
The points with position vectors (2. 6), (1, 2) and (a, 10) are collinear if the of a is
(a) -8
(b) 4
(c) 3
(d) 12
Answer
Answer: (c) 3
Question 23.
|\(\vec{a}\) + \(\vec{b}\)| = |\(\vec{a}\) – \(\vec{b}\)| then the angle between \(\vec{a}\) and \(\vec{b}\)
(a) \(\frac{π}{2}\)
(b) 0
(c) \(\frac{π}{4}\)
(d) \(\frac{π}{6}\)
Answer
Answer: (a) \(\frac{π}{2}\)
Question 25.
If \(\vec{a}\) and \(\vec{b}\) are any two vector then (\(\vec{a}\) × \(\vec{b}\))² is equal to
(a) (\(\vec{a}\))²(\(\vec{b}\))² – (\(\vec{a}\).\(\vec{b}\))²
(b) (\(\vec{a}\))² (\(\vec{b}\))² + (\(\vec{a}\).\(\vec{b}\))²
(c) (\(\vec{a}\).\(\vec{b}\))²
(d) (\(\vec{a}\))²(\(\vec{b}\))²
Answer
Answer: (a) (\(\vec{a}\))²(\(\vec{b}\))² – (\(\vec{a}\).\(\vec{b}\))²
Question 26.
|\(\vec{a}\) × \(\vec{b}\)| = |\(\vec{a}\).\(\vec{b}\)| then the angle between \(\vec{a}\) and \(\vec{b}\)
(a) 0
(b) \(\frac{π}{2}\)
(c) \(\frac{π}{4}\)
(d) π
Answer
Answer: (a) 0
Question 27.
If ABCDEF is a regular hexagon then \(\vec{AB}\) + \(\vec{EB}\) + \(\vec{FC}\) equals
(a) zero
(b) 2\(\vec{AB}\)
(c) 4\(\vec{AB}\)
(d) 3\(\vec{AB}\)
Answer
Answer: (d) 3\(\vec{AB}\)
Question 28.
Which one of the following is the modulus of x\(\hat{i}\) + y\(\hat{j}\) + z\(\hat{k}\)?
(a) \(\sqrt{x^2+y^2+z^2}\)
(b) \(\frac{1}{\sqrt{x^2+y^2+z^2}}\)
(c) x² + y² + z²
(d) none of these
Answer
Answer: (a) \(\sqrt{x^2+y^2+z^2}\)
Question 29.
If C is the mid point of AB and P is any point outside AB then,
(a) \(\vec{PA}\) + \(\vec{PB}\) = 2\(\vec{PC}\)
(b) \(\vec{PA}\) + \(\vec{PB}\) = \(\vec{PC}\)
(c) \(\vec{PA}\) + \(\vec{PB}\) = 2\(\vec{PC}\) = 0
(d) None of these
Answer
Answer: (a) \(\vec{PA}\) + \(\vec{PB}\) = 2\(\vec{PC}\)
Question 30.
If \(\vec{OA}\) = 2\(\vec{i}\) – \(\vec{j}\) + \(\vec{k}\), \(\vec{OB}\) = \(\vec{i}\) – 3\(\vec{j}\) – 5\(\vec{k}\) then |\(\vec{OA}\) × \(\vec{OB}\)| =
(a) 8\(\vec{i}\) + 11\(\vec{j}\) – 5\(\vec{k}\)
(b) \(\sqrt{210}\)
(c) sin θ
(d) \(\sqrt{40}\)
Answer
Answer: (b) \(\sqrt{210}\)
Question 31.
The distance of the point (- 3, 4, 5) from the origin
(a) 50
(b) 5√2
(c) 6
(d) None of these
Answer
Answer: (b) 5√2
Question 32.
If |a| = |b| = |\(\vec{a}\) + \(\vec{b}\)| = 1 then |\(\vec{a}\) – \(\vec{b}\)| is equal to
(a) 1
(b) √3
(c) 0
(d) None of these
Answer
Answer: (b) √3
Question 33.
If \(\hat{a}\) and \(\hat{b}\) be two unit vectors and 0 is the angle between them, then |\(\hat{a}\) – \(\hat{b}\)| is equal to
(a) sin \(\frac{θ}{2}\)
(b) 2 sin \(\frac{θ}{2}\)
(c) cos \(\frac{θ}{2}\)
(d) 2 cos \(\frac{θ}{2}\)
Answer
Answer: (b) 2 sin \(\frac{θ}{2}\)
Question 34.
The angle between the vector 2\(\hat{i}\) + 3\(\hat{j}\) + \(\hat{k}\) and 2\(\hat{i}\) – \(\hat{j}\) – \(\hat{k}\) is
(a) \(\frac{π}{2}\)
(b) \(\frac{π}{4}\)
(c) \(\frac{π}{3}\)
(d) 0
Answer
Answer: (a) \(\frac{π}{2}\)
Question 34.
If O be the origin and \(\vec{OP}\) = 2\(\hat{i}\) + 3\(\hat{j}\) – 4\(\hat{k}\) and \(\vec{OQ}\) = 5\(\hat{i}\) + 4\(\hat{j}\) -3\(\hat{k}\), then \(\vec{PQ}\) is equal to
(a) 7\(\hat{i}\) + 7\(\hat{j}\) – 7\(\hat{k}\)
(b) -3\(\hat{i}\) + \(\hat{j}\) – \(\hat{k}\)
(c) -7\(\hat{i}\) – 7\(\hat{j}\) + 7\(\hat{k}\)
(d) 3\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)
Answer
Answer: (d) 3\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)
Question 35.
If \(\vec{a}\) = \(\hat{i}\) – \(\hat{j}\) + \(\hat{k}\), \(\vec{b}\) = \(\hat{i}\) + 2\(\hat{j}\) – \(\hat{k}\), \(\vec{c}\) = 3\(\hat{i}\) – p\(\hat{j}\) – 5\(\hat{k}\) are coplanar then P =
(a) 6
(b) -6
(c) 2
(d) -2
Answer
Answer: (a) 6
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