Karnataka 2nd PUC Mathematics Vector Algebra MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: VECTOR ALGEBRA

Identifying Scalars and Vectors:
Tell the difference between scalars and vector
Types of Vectors:
Identifying zero, unit, position, coplanar, and collinear vectors.
Magnitude and Direction:
Calculate the length of a vector and its direction cosines or ratios
Addition and Subtraction:
Add and subtract vectors using the triangle law, parallelogram law, and
component methods.
Multiplication by a scalar:
Multiply a vector by a real number to change its size or reverse its direction.
Dot Product:
Compute the scalar product of two vectors to find angles and check for
perpendicularity. Get the projection of a vector in the direction of another vector.
Cross Product:
Compute the vector product in 3D space to find a vector perpendicular to two given
vectors.
Calculating area of triangle and parallelogram using cross product
IMPORTANT FORMULAE
⃗⃗
𝒂
 Unit vector for a vector 𝑎⃗ is given by 𝒂
̂ = |𝒂⃗⃗|

 Magnitude of a vector 𝑎⃗ = 𝑎𝑖̂ + 𝑏𝑗̂ + 𝑐𝑘̂ is given by |𝒂
⃗⃗| = √𝒂𝟐 + 𝒃𝟐 + 𝒄𝟐
𝒂 𝒃 𝒄
 Direction cosines of a vector 𝑎⃗ = 𝑎𝑖̂ + 𝑏𝑗̂ + 𝑐𝑘̂ are 𝒍 = ,𝒎 = ,𝒏 =
√𝒂𝟐 +𝒃𝟐 +𝒄 𝟐 √𝒂𝟐 +𝒃𝟐 +𝒄
𝟐 √𝒂𝟐 +𝒃𝟐 +𝒄𝟐

 Vector ⃗⃗⃗⃗⃗⃗
𝑃𝑄 joining two points 𝑃 (𝑥1 , 𝑦1 , 𝑧1 ) and 𝑄(𝑥2 , 𝑦2 , 𝑧2 ) is given by

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⃗⃗⃗⃗⃗⃗⃗ ̂
𝑷𝑸 = (𝒙𝟐 − 𝒙𝟏 )𝒊̂ + (𝒚𝟐 − 𝒚𝟏 )𝒋̂ + (𝒛𝟐 − 𝒛𝟏 )𝒌
 Section formula: The position vector of a point R dividing the line segment joining the
points P and Q whose position vectors are 𝑎⃗ and 𝑏⃗⃗
⃗⃗+𝒏𝒂
𝒎𝒃 ⃗⃗
 In the ratio m:n internally, is given by 𝒎+𝒏
⃗⃗−𝒏𝒂
𝒎𝒃 ⃗⃗
 In the ratio m:n externally, is given by 𝒎−𝒏

 Scalar(dot) product of two vectors 𝑎⃗ and 𝑏⃗⃗ having an angle 𝜃 between them is defined as
⃗⃗ = |𝒂
⃗⃗ . 𝒃
𝒂 ⃗⃗|𝒄𝒐𝒔𝜽
⃗⃗||𝒃

 For vectors 𝑎⃗ = 𝑎1 𝑖̂ + 𝑏1 𝑗̂ + 𝑐1 𝑘̂ and 𝑏⃗⃗ = 𝑎2 𝑖̂ + 𝑏2 𝑗̂ + 𝑐2 𝑘̂ dot product is given by
⃗⃗ = 𝒂𝟏 𝒂𝟐 + 𝒃𝟏 𝒃𝟐 + 𝒄𝟏 𝒄𝟐
⃗⃗ . 𝒃
𝒂
⃗⃗ . 𝒃
𝒂 ⃗⃗ ⃗⃗ . 𝒃
𝒂 ⃗⃗
 Projection of 𝑎⃗ along 𝑏⃗⃗ is ⃗⃗ and the projection vector of 𝑎⃗ along 𝑏⃗⃗ is ( ⃗⃗ ) ⃗𝒃⃗
|𝒃| |𝒃|

 Vector (cross) product of two vectors 𝑎⃗ and 𝑏⃗⃗ having an angle 𝜃 between them is defined
as 𝒂 ⃗⃗ = |𝒂
⃗⃗ 𝐗 𝒃 ⃗⃗|𝒔𝒊𝒏𝜽 𝒏
⃗⃗||𝒃 ̂ where 𝑛̂ is a unit vector perpendicular to the plane containing

𝑎⃗ and 𝑏⃗⃗ and 𝑎⃗ , 𝑏⃗⃗ 𝑎𝑛𝑑 𝑛̂ form a right handed system
 For vectors 𝑎⃗ = 𝑎1 𝑖̂ + 𝑏1 𝑗̂ + 𝑐1 𝑘̂ and 𝑏⃗⃗ = 𝑎2 𝑖̂ + 𝑏2 𝑗̂ + 𝑐2 𝑘̂ dot product is given by
𝒊̂ 𝒋̂ ̂
𝒌
𝒂 ⃗⃗
⃗⃗ 𝐗 𝒃 = |𝒂𝟏 𝒃𝟏 𝒄𝟏 |
𝒂𝟐 𝒃𝟐 𝒄𝟐
𝟏
 ⃗⃗ 𝐗 ⃗𝒃⃗|
If 𝑎⃗ and 𝑏⃗⃗ represent the adjacent sides of a triangle then its area is given by 𝟐 |𝒂

Vector addition, magnitude and section formula
1. Which of the following is a vector (Easy)
(1) time
(2) volume
(3) force
(4) speed
2. In the given triangle ABC which of the following is not true (Easy)
⃗⃗⃗⃗⃗⃗ + 𝐵𝐶
(1) 𝐴𝐵 ⃗⃗⃗⃗⃗⃗ − 𝐶𝐴
⃗⃗⃗⃗⃗⃗ = ⃗0⃗

(2) ⃗⃗⃗⃗⃗⃗
𝐴𝐵 + 𝐵𝐶⃗⃗⃗⃗⃗⃗ − ⃗⃗⃗⃗⃗⃗
𝐴𝐶 = ⃗0⃗
(3) ⃗⃗⃗⃗⃗⃗
𝐴𝐵 + 𝐵𝐶⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗
𝐶𝐴 = ⃗0⃗
(4) ⃗⃗⃗⃗⃗⃗
𝐴𝐵 − 𝐶𝐵⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗
𝐶𝐴 = ⃗⃗0

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⃗⃗ and ⃗𝒃⃗ are two collinear vectors, then which of the following is incorrect (Easy)
3. If 𝒂
(1) 𝑏⃗⃗ = 𝜆𝑎⃗, for some 𝜆
(2) 𝑎⃗ = ±𝑏⃗⃗
(3) The respective components of 𝑎⃗ and 𝑏⃗⃗ are not proportional
(4) Both the vectors 𝑎⃗ and 𝑏⃗⃗ have same direction, but different magnitudes
4. If 𝒂
⃗⃗ is a nonzero vector of magnitude 𝒂 and 𝝀 a nonzero scalar, then 𝝀𝒂
⃗⃗ is unit vector
if (Easy)
(1) 𝜆 = 1
(2) 𝜆 = −1
(3) 𝑎 = |𝜆|
1
(4) 𝑎 = |𝜆|

5. Area of a rectangle having vertices A, B, C and D with position vectors
𝟏
̂ , 𝒊̂ + 𝟏 𝒋̂ + 𝟒𝒌
– 𝒊̂ + 𝟐 𝒋̂ + 𝟒𝒌 ̂ , 𝒊̂ − 𝟏 𝒋̂ + 𝟒𝒌
̂ and −𝒊̂ − 𝟏 𝒋̂ + 𝟒𝒌
̂ , respectively is (Average)
𝟐 𝟐 𝟐
1
(1) 2

(2) 1
(3) 2
(4) 4
̂ is (Easy)
6. A vector of magnitude 5 units and in the direction of the vector 𝟐𝒊̂ − 𝟔𝒋̂ + 𝟑𝒌
5
(1) 7 (2𝑖̂ − 6𝑗̂ + 3𝑘̂)
5
(2) 7 (−2𝑖̂ + 6𝑗̂ − 3𝑘̂)
1
(3) 7 (2𝑖̂ − 6𝑗̂ + 3𝑘̂)
1
(4) 7 (−2𝑖̂ + 6𝑗̂ − 3𝑘̂)

7. The unit vector in the direction of the vector 𝒂 ̂ is
⃗⃗ = 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌 (Easy)
1 2 2
(1) 9 𝑖̂ − 9 𝑗̂ + 9 𝑘̂
1 2 2
(2) − 3 𝑖̂ + 3 𝑗̂ − 3 𝑘̂
1 2 2
(3) − 9 𝑖̂ + 9 𝑗̂ − 9 𝑘̂
1 2 2
(4) 3 𝑖̂ − 3 𝑗̂ + 3 𝑘̂

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8. If 𝒂
⃗⃗ is a non zero vector of magnitude 𝒂 and λ is a non zero scalar, then λ𝒂
⃗⃗ is a unit
vector if (Easy)
(1) 𝜆 = |𝑎⃗|
1
(2) 𝜆 = |𝑎⃗⃗|
(3) 𝜆 = 1
(4) 𝜆 = 𝑎
⃗⃗⃗⃗⃗⃗⃗
9. If the vertices of a triangle are 𝑨(𝟏, 𝟐, 𝟑), 𝑩(−𝟏, 𝟎, 𝟐) and 𝑪(𝟐, 𝟏, −𝟏) then the vector 𝑨𝑩
(Easy)
(1) −2𝑖̂ − 2𝑗̂ − 𝑘̂
(2) 𝑖̂ − 𝑗̂ + 3𝑘̂
(3) 3𝑖̂ − 𝑗̂ − 2𝑘̂
(4) 2𝑖̂ + 3𝑗̂ + 𝑘̂
10. The position vector of a point C which divides the line segment joining the points A
⃗⃗ and ⃗𝒃⃗ internally in the ratio 1:2 is
and B whose position vectors are 𝒂 (Average)
⃗⃗
𝑎⃗⃗+2𝑏
(1) 3
⃗⃗
⃗⃗⃗⃗⃗⃗+𝑏
2𝑎
(2) 3
⃗⃗
𝑎⃗⃗−𝑏
(3) 3
⃗⃗
𝑎⃗⃗+𝑏
(4) 3
̂ are
11. The direction cosines of the vector 𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌 (Easy)
(1) 1,2,3
1 2 3
(2) , ,
√14 √14 √14
1 2 3
(3) 14 , 14 , 14
1 2 3
(4) , ,
√12 √12 √12

̂ are mutually perpendicular unit vectors, then the value of 𝒊̂. (𝒋̂ 𝐗 𝒌
12. If 𝒊̂, 𝒋̂ , 𝒌 ̂ ) is (Easy)
(1) 0
(2) 1
(3) -1
(4) 3

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13. The sum of two unit vectors is a unit vector, then the magnitude of their difference
is (Easy)
(1) 0
(2) 1
(3) √2
(4) √3
⃗⃗ 𝐚𝐧𝐝 ⃗𝒃⃗ are two collinear vectors, then
14. If 𝒂 (Easy)
(1) 𝑎⃗X𝑏⃗⃗ = 0
(2) 𝑎⃗ = λ𝑏⃗⃗ for some scalar λ
(3) 𝑎⃗. 𝑏⃗⃗ = |𝑎⃗| |𝑏⃗⃗|
(4) All of the above
15. If 𝒂 ̂ , ⃗𝒃⃗ = 𝜶𝒊̂ + 𝜷𝒋̂ + 𝟐𝒌
⃗⃗ = 𝟐𝒊̂ + 𝟐𝒋̂ − 𝒌 ̂ and |𝒂
⃗⃗ + ⃗𝒃⃗| = |𝒂
⃗⃗ − ⃗𝒃⃗| , then 𝟐α+2β is equal to
(KCET 2026) (Average)
(1) 2
(2) 4
(3) 1
(4) 0
̂ , 2𝒊̂ + 𝒋̂ − 𝒌
16. Value of λ for coplanar vectors 2𝒊̂ − 𝟑𝒋̂ + 𝟒𝒌 ̂ and λ𝒊̂ − 𝒋̂ + 𝟐𝒌
̂
(KCET 2026) (Average)
(1) 5
(2) -5
(3) 6
(4) -6
̂ and 𝒊̂ + 𝟑𝒋̂ + 𝟓𝒌
17. The two vectors 𝒊̂ + 𝒋̂ + 𝒌 ̂ represent the two sides 𝑨𝑩 ⃗⃗⃗⃗⃗⃗ respectively
⃗⃗⃗⃗⃗⃗⃗ and 𝑨𝑪
of ∆𝑨𝑩𝑪. The length f the median through 𝑨 is (KCET 2020) (Difficult)
(1) √14
√14
(2) 2

(3) 14
(4) 7

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̂ , 𝒊̂ + 𝒃𝒋̂ + 𝒌
18. If the vectors 𝒂𝒊̂ + 𝒋̂ + 𝒌 ̂ and 𝒊̂ + 𝒋̂ + 𝒄𝒌
̂ are coplanar (𝒂 ≠ 𝒃 ≠ 𝒄 ≠ 𝟏), then the
value of 𝒂𝒃𝒄 − (𝒂 + 𝒃 + 𝒄) = (KCET 2018) (Difficult)
(1) 2
(2) -2
(3) 0
(4) -1
Scalar (dot) product of vectors
19. If θ is the angle between two vectors 𝒂 ⃗⃗, then 𝒂
⃗⃗ and 𝒃 ⃗⃗ ≥ 𝟎 only when
⃗⃗. 𝒃 (Average)
𝜋
(1) 0 < 𝜃 < 2
𝜋
(2) 0 ≤ 𝜃 ≤ 2

(3) 0 < 𝜃 < 𝜋
(4) 0 ≤ 𝜃 ≤ 𝜋
⃗⃗ and ⃗𝒃⃗ be two unit vectors and θ is the angle between them. Then 𝒂
20. Let 𝒂 ⃗⃗ + ⃗𝒃⃗ is a unit
vector if (Average)
𝜋
(1) 𝜃 = 4
𝜋
(2) 𝜃 = 3
𝜋
(3) 𝜃 = 2
2𝜋
(4) 𝜃 =
3

21. If θ is the angle between two vectors 𝒂 ⃗⃗, then |𝒂
⃗⃗ and 𝒃 ⃗⃗| = |𝒂
⃗⃗. 𝒃 ⃗⃗| when θ is equal to
⃗⃗𝑿𝒃
(Easy)
(1) 0
𝜋
(2) 4
𝜋
(3) 2

(4) 𝜋
22. The projection of the vector 𝒂 ̂ on the vector 𝒃
⃗⃗ = 𝒊̂ − 𝟐𝒋̂ + 𝒌 ⃗⃗ = 𝟐𝒊̂ + 𝒋̂ + 𝟐𝒌
̂ is
(Easy)
1
(1) 3
2
(2) 3
1
(3) 2

(4) 2

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⃗⃗ + ⃗𝒃⃗| = |𝒂
23. If |𝒂 ⃗⃗ − ⃗𝒃⃗|, then the angle between 𝒂
⃗⃗ and ⃗𝒃⃗ is (Average)
(1) 0
𝜋
(2) 4
𝜋
(3) 2

(4) 𝜋
24. The vectors of 𝒂 ̂ and ⃗𝒃⃗ = 𝝀𝒊̂ − 𝒋̂ − 𝒌
⃗⃗ = 𝒊̂ + 𝟑𝒋̂ + 𝟓𝒌 ̂ are perpendicular if λ is equal to
(Average)
(1) -2
(2) 2
(3) -8
(4) 8
⃗⃗ and ⃗𝒃⃗ is 3 and |𝒂
25. The scalar product of two vectors 𝒂 ⃗⃗| = 𝟒 then the
⃗⃗ 𝐗𝒃

angle between 𝒂 ⃗⃗ is
⃗⃗ and 𝒃 (Difficult)
𝜋
(1) 3
𝜋
(2) 4
4
(3) tan−1 (3)
3
(4) tan−1 (4)
26. The vectors 𝒂 ̂ and ⃗𝒃⃗ = 𝒊̂ + 𝟐𝒋̂ − 𝟐𝒌
⃗⃗ = 𝟐𝒊̂ + 𝟑𝒋̂ + 𝟒𝒌 ̂ are (Easy)
(1) Perpendicular
(2) Parallel
(3) Collinear
(4) Neither parallel nor perpendicular
27. If 𝒂 ̂ and ⃗𝒃⃗ = 𝟔𝒊̂ − 𝟑𝒋̂ + 𝟐𝒌
⃗⃗ = 𝟐𝒊̂ + 𝟐𝒋̂ − 𝒌 ̂ , then 𝒂
⃗⃗ . ⃗𝒃⃗ is (Easy)
(1) 4
(2) 10
(3) 6
(4) -6
⃗⃗ . ⃗𝒃⃗ = 𝟎 then
28. If 𝒂 (Easy)
(1) 𝑎⃗ and 𝑏⃗⃗ are parallel
(2) 𝑎⃗ and 𝑏⃗⃗ are perpendicular
(3) 𝑎⃗ or 𝑏⃗⃗ is a zero vector
(4) Both B and C are possible
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29. For any vector 𝒂
⃗⃗ , the value of (𝒂
⃗⃗. 𝒊̂)𝟐 + (𝒂
⃗⃗. 𝒋̂)𝟐 + (𝒂 ̂ )𝟐 is
⃗⃗. 𝒌 (Average)
(1) |𝑎⃗|2
(2) 2|𝑎⃗|2
(3) 3|𝑎⃗|2
(4) 0
30. If 𝒂
⃗⃗ is any non-zero vector, then of (𝒂
⃗⃗. 𝒊̂)𝒊̂ + (𝒂
⃗⃗. 𝒋̂)𝒋̂ + (𝒂 ̂ )𝒌
⃗⃗. 𝒌 ̂ is equal to (Easy)
(1) 0
(2) 𝑎⃗
(3) 3𝑎⃗
(4) |𝑎⃗|2
31. The value f λ for which the vectors 𝒂 ̂ and ⃗𝒃⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌
⃗⃗ = 𝟐𝒊̂ + 𝝀𝒋̂ + 𝒌 ̂ are orthogonal is
(KCET2026) (Average)
5
(1) 2
5
(2) − 2
3
(3) 2
3
(4) − 2
32. If 𝒂 ̂,𝒃
⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝒌 ⃗⃗ = 𝒊̂ − 𝒋̂ + 𝟒𝒌
̂ and 𝒄 ̂ are such that 𝒂
⃗⃗ = 𝒊̂ + 𝒋̂ + 𝒌 ⃗⃗ is perpendicular to
⃗⃗ + 𝛌 𝒃
⃗𝒄⃗, then the value of λ is (KCET2025) (Difficult)
(1) 1
(2) -1
(3) 2
(4) -2
⃗⃗ 𝐚𝐧𝐝 ⃗𝒃⃗ are unit vectors, the angle between 𝒂
33. If 𝒂 ⃗⃗ 𝐚𝐧𝐝 ⃗𝒃⃗for √𝟑𝒂
⃗⃗ − ⃗𝒃⃗ to be a unit vector is
(KCET2025) (Difficult)
(1) 30o
(2) 45o
(3) 60o
(4) 90o
𝟐 𝟐
34. If |𝒂 ⃗⃗| +|𝒂
⃗⃗𝐗𝒃 ⃗⃗. ⃗𝒃⃗| =144 and |𝒂 ⃗⃗| is equal to
⃗⃗|=4, then |𝒃 (KCET2023) (Difficult)
(1) 3
(2) 4
(3) 8
(4) 12

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35. A vector 𝒂
⃗⃗ makes equal acute angles with the coordinate axes. Then the projection of
⃗⃗ = 𝟓𝒊̂ + 𝟕𝒋̂ − 𝒌
the vector 𝒃 ̂ on 𝒂
⃗⃗ is (KCET2023) (Average)
11
(1) 3
11
(2)
√3
3
(3) 5√3

(4) 11
36. If 𝒂 ⃗⃗ = 𝟎 and 𝒂
⃗⃗⃗⃗. 𝒃 ⃗⃗ makes an angle of 𝟔𝟎𝐨 with 𝒂
⃗⃗⃗⃗ + 𝒃 ⃗⃗⃗⃗ then (KCET2023) (Difficult)
⃗⃗⃗⃗| = 2|𝑏⃗⃗|
(1) |𝑎

⃗⃗⃗⃗| = |𝑏⃗⃗|
(2) 2|𝑎

(3) |𝑏⃗⃗| = √3|𝑎
⃗⃗⃗⃗|

⃗⃗⃗⃗| = √3|𝑏⃗⃗|
(4) |𝑎
𝜽
37. If 𝒂 ⃗⃗ are unit vectors and θ is the angle between 𝒂
⃗⃗⃗⃗ and 𝒃 ⃗⃗, then 𝒔𝒊𝒏 ( ) is
⃗⃗⃗⃗ and 𝒃
𝟐

(KCET2020) (Difficult)
(1) |𝑎 ⃗⃗⃗⃗|
⃗⃗⃗⃗ − 𝑏

⃗⃗⃗⃗ + ⃗⃗⃗⃗
(2) |𝑎 𝑏|
|𝑎 ⃗⃗⃗⃗|
⃗⃗⃗⃗+𝑏
(3) 2

|𝑎 ⃗⃗⃗⃗|
⃗⃗⃗⃗−𝑏
(4) 2

𝟐 𝟐
38. If |𝒂 ⃗⃗⃗⃗| = 𝟒 then √|𝒂
⃗⃗⃗⃗| = 𝟏𝟔 and |𝒃 ⃗⃗⃗⃗| + |𝒂
⃗⃗⃗⃗𝐗𝒃 ⃗⃗⃗⃗. ⃗⃗⃗⃗
𝒃| (KCET2019) (Average)

(1) 8
(2) 64
(3) 4
(4) 16
𝟐𝝅
39. If the angle between 𝒂 ⃗⃗ is
⃗⃗⃗⃗ and 𝒃 and the projection of 𝒂 ⃗⃗ is -2,
⃗⃗⃗⃗ in the direction of 𝒃
𝟑

then |𝒂
⃗⃗⃗⃗| = (KCET2019) (Average)
(1) 3
(2) 1
(3) 4
(4) 2

2026-27 MATHEMATICS CET MATERIAL Page 9 of 14

⃗⃗⃗⃗ is perpendicular to ⃗𝒃⃗ and (𝒂
40. If 𝒂 ⃗⃗⃗⃗ + ⃗𝒃⃗) is perpendicular to ⃗⃗⃗⃗), then m=
⃗⃗⃗⃗ + 𝐦𝒃
(𝒂
(KCET2013) (Average)
(1) -1
(2) 1
⃗⃗⃗⃗|2
|𝑎
(3) − 2
⃗⃗⃗⃗|
|𝑏
2
⃗⃗⃗⃗|
|𝑏
(4) − |𝑎⃗⃗⃗⃗|2

⃗⃗⃗⃗ and ⃗⃗⃗⃗
41. If 𝒂 𝒃 are mutually perpendicular unit vectors then (𝟑𝒂 ⃗⃗⃗⃗). (𝟓𝒂
⃗⃗⃗⃗ + 𝟐𝒃 ⃗⃗⃗⃗) =
⃗⃗⃗⃗ − 𝟔𝒃
(KCET2018) (Easy)
(1) 5
(2) 3
(3) 6
(4) 12
Vector (cross) product of vectors

⃗⃗ and ⃗𝒃⃗ be such that |𝒂
42. Let the vectors 𝒂 ⃗⃗| = √𝟐 , then 𝒂
⃗⃗| = 𝟑 and |𝒃 ⃗⃗ X ⃗𝒃⃗ is a unit vector ,
𝟑

if the angle between 𝒂 ⃗⃗ is
⃗⃗ and 𝒃 (Average)
𝜋
(1) 6
𝜋
(2)
4
𝜋
(3) 3
𝜋
(4) 2
̂ ) + 𝒋̂. (𝒊̂𝑿𝒌
43. The value of 𝒊̂. (𝒋̂𝑿𝒌 ̂) + 𝒌
̂ . (𝒊̂𝑿𝒋̂) is (Average)
(1) 0
(2) -1
(3) 1
(4) 3
̂ , 𝒊̂ + 𝟐𝒋̂ − 𝟑𝒌
44. If the vectors 𝟐𝒊̂ − 𝒋̂ + 𝒌 ̂ and 3𝒊̂ + 𝝀𝒋̂ + 𝟓𝒌
̂ are coplanar, then the value of λ is
(Average)
(1) -4
(2) 4
(3) -2
(4) 2

2026-27 MATHEMATICS CET MATERIAL Page 10 of 14

45. The magnitude of the vector product of 𝒂 ̂ and ⃗𝒃⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝒌
⃗⃗ = 𝟐𝒊̂ + 𝒋̂ − 𝒌 ̂ is (Easy)

(1) √26
(2) √27
(3) √30
(4) √35
46. Area of the triangle with adjacent sides determined by the vectors 𝒂 ̂ and
⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝒌
⃗⃗ = 𝟐𝒊̂ + 𝒋̂ + 𝒌
𝒃 ̂ is (Easy)
√11
(1) 2

√6
(2) 2

(3) √6
(4) √11
̂ + (𝒋̂𝐗𝒌
47. The value of (𝒊̂𝐗𝒋̂). 𝒌 ̂ ). 𝒊̂ is (Average)
(1) 0
(2) 1
(3) 2
(4) -1
⃗⃗ = ⃗𝟎⃗ and ⃗𝒂⃗ . ⃗𝒃⃗ = 𝟎 then
48. If ⃗𝒂⃗ 𝐗𝒃 (Average)
(1) 𝑎⃗ and 𝑏⃗⃗ are both zero vectors
(2) 𝑎⃗ or 𝑏⃗⃗ is a zero vector
(3) 𝑎⃗ and 𝑏⃗⃗ are mutually perpendicular vectors
(4) 𝑎⃗ and 𝑏⃗⃗ are both collinear vectors
49. If |𝒂 ⃗⃗| = 𝟓 and |𝒂
⃗⃗| = 𝟐, |𝒃 ⃗⃗| = 𝟖, then |𝒂
⃗⃗𝐗𝒃 ⃗⃗. ⃗𝒃⃗| is equal to (Average)
(1) 6
(2) 8
(3) 10
(4) 12
̂ is
50. The unit vector perpendicular to both vectors 𝒊̂ + 𝒋̂ and 𝒋̂ + 𝒌 (Average)
̂
𝑖̂−𝑗̂ +𝑘
(1)
√3
̂
𝑖̂+𝑗̂ +𝑘
(2)
√3
̂
𝑖̂+𝑗̂ −𝑘
(3)
√3
̂
𝑖̂−𝑗̂ −𝑘
(4)
√3
2026-27 MATHEMATICS CET MATERIAL Page 11 of 14

51. If 𝒂 ̂ , ⃗𝒃⃗ = 𝒋̂ − 𝒌
⃗⃗ = 𝒊̂ + 𝒋̂ + 𝒌 ̂ and 𝒂 ⃗⃗ = ⃗𝒃⃗ , 𝒂
⃗⃗𝐗𝒄 ⃗⃗ = 𝟑, then 𝒄
⃗⃗. 𝒄 ⃗⃗ is equal to (KCET2026) (Difficult)
1
(1) 3 (5𝑖̂ + 2𝑗̂ + 2𝑘̂)
1
(2) 3 (2𝑖̂ + 5𝑗̂ + 2𝑘̂)
1
(3) 3 (5𝑖̂ + 2𝑗̂ − 𝑘̂)

(4) 𝑖̂ + 𝑗̂ + 𝑘̂
52. If |𝒂 ⃗⃗| = 𝟐 and 𝒂
⃗⃗| = 𝟏𝟎, |𝒃 ⃗⃗⃗⃗. ⃗𝒃⃗ = 𝟏𝟐 , then the value of |𝒂 ⃗⃗| is
⃗⃗⃗⃗𝐗𝒃 (KCET2025) (Difficult)
(1) 8
(2) 12
(3) 16
(4) 4
53. If the area of the parallelogram with 𝒂 ⃗⃗ as two adjacent sides is 15sq. units, then
⃗⃗ 𝐚𝐧𝐝 𝒃
the area of the parallelogram having 𝟑𝒂 ⃗⃗ and 𝒂
⃗⃗ + 𝟐𝒃 ⃗⃗ as two adjacent sides in sq.
⃗⃗ + 𝟑𝒃
unit is (KCET2025) (Average)
(1) 45
(2) 75
(3) 105
(4) 120
54. If 𝒂 ⃗⃗ + 𝟑𝒄
⃗⃗ + 𝟐𝒃 ⃗⃗ = 𝟎 and (𝒂 ⃗⃗) + (𝒃
⃗⃗𝐗𝒃 ⃗⃗𝐗𝒄
⃗⃗) + (𝒄
⃗⃗𝐗𝒂 ⃗⃗𝐗𝒄
⃗⃗) = 𝝀(𝒃 ⃗⃗) then value of λ is equal to
(KCET2023) (Average)
(1) 3
(2) 4
(3) 6
(4) 0
55. A unit vector perpendicular to the plane containing the vectors
̂ and −𝟐𝒊̂ + 𝒋̂ + 𝟑𝒌
𝒊̂ + 𝟐𝒋̂ + 𝒌 ̂ is (KCET2019) (Easy)
̂
𝑖̂+𝑗̂ −𝑘
(1)
√3
̂
−𝑖̂−𝑗̂ −𝑘
(2)
√3
̂
𝑖̂+𝑗̂ +𝑘
(3)
√3
̂
−𝑖̂+𝑗̂ −𝑘
(4)
√3

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⃗⃗⃗⃗ is perpendicular to both ⃗𝒃⃗ and 𝒄
56. If 𝒂 ⃗⃗ , then (KCET2013) (Easy)
(1) 𝑎 ⃗⃗⃗⃗X𝑐⃗⃗⃗) = ⃗0⃗
⃗⃗⃗⃗. (𝑏

(2) 𝑎 ⃗⃗⃗⃗X𝑐⃗⃗⃗) = ⃗0⃗
⃗⃗⃗⃗X(𝑏

(3) 𝑎 ⃗⃗⃗⃗ + 𝑐⃗⃗⃗) = ⃗0⃗
⃗⃗⃗⃗X(𝑏

(4) 𝑎 ⃗⃗⃗⃗ + 𝑐⃗⃗⃗) = ⃗0⃗
⃗⃗⃗⃗ + (𝑏
𝟐 𝟐
⃗⃗⃗⃗| + |𝒂
⃗⃗⃗⃗𝐗𝒃
57. |𝒂 ⃗⃗⃗⃗| = 𝟏𝟒𝟒 and |𝒂
⃗⃗⃗⃗. 𝒃 ⃗⃗⃗⃗| is
⃗⃗⃗⃗| = 𝟒 , then the value of |𝒃 (KCET2018) (Average)
(1) 1
(2) 2
(3) 3
(4) 4
̂ , −𝟐𝒊̂ + 𝟑𝒋̂ − 𝒌
58. The area of the triangle having vertices 𝒊̂ − 𝟐𝒋̂ + 𝟑𝒌 ̂ and 𝟒𝒊̂ − 𝟕𝒋̂ + 𝟕𝒌
̂ is
(KCET2018) (Easy)
(1) 36 sq units
(2) 0 sq units
(3) 39 sq units
(4) 11 sq units
59. If |𝒂 ⃗⃗⃗⃗| = 𝟐√𝟑, find the area of parallelogram whose adjacent sides are
⃗⃗⃗⃗𝐗𝒃

⃗⃗⃗⃗ + ⃗⃗⃗⃗
𝒂 ⃗⃗⃗⃗ − ⃗⃗⃗⃗
𝒃 and 𝒂 𝒃 (Avearge)
(1) 2√3
(2) 4√3
(3) 6√3
(4) 8√3
60. If |𝒂 ⃗⃗⃗⃗| = 𝟑 and 𝒂
⃗⃗⃗⃗| = 𝟓 , |𝒃 ⃗⃗⃗⃗ = 𝟑𝒊̂ + 𝟒𝒋̂ find 𝒂
⃗⃗⃗⃗𝑿𝒃 ⃗⃗⃗⃗. ⃗⃗⃗⃗
𝒃 (Difficult)

(1) 5√2
(2) 8
(3) 10√2
(4) 12

2026-27 MATHEMATICS CET MATERIAL Page 13 of 14

VECTOR ALGEBRA
1 2 3 4 5 6 7 8 9 10
3 1 3 4 3 1 4 2 1 2
11 12 13 14 15 16 17 18 19 20
2 2 4 4 1 3 1 2 2 4
21 22 23 24 25 26 27 28 29 30
2 2 3 4 3 1 1 2 1 2
31 32 33 34 35 36 37 38 39 40
2 2 1 1 2 3 4 2 3 3
41 42 43 44 45 46 47 48 49 50
2 2 3 1 2 1 3 2 1 1
51 52 53 54 55 56 57 58 59 60
1 3 3 3 4 1 3 2 2 3

2026-27 MATHEMATICS CET MATERIAL Page 14 of 14