Vectors and three-dimensional geometry together give about three questions in JEE Main. This set covers magnitude, dot and cross products, projections, the scalar triple product, direction cosines, distances, the equation of a plane and the shortest distance between skew lines, with worked solutions.
How to practise: read each question and tap the option you think is correct. You will see right away whether you got it, along with a short explanation, and your score is tracked at the bottom of the screen. This set has 15 questions.
Topics covered in this set
- Magnitude and dot product
- Angle between vectors and perpendicularity
- Cross product, area and projection
- Scalar triple product and coplanarity
- Direction cosines and distance in 3D
- Planes and shortest distance between lines
JEE Mathematics MCQs with answers and explanations
Q1. What is the magnitude of the vector a = i + 2j + 2k?
a) 5
b) √5
c) 3
d) 9
Q2. What is (2i − j + 3k) · (i + 4j + 2k)?
a) 4
b) 8
c) 0
d) 12
Q3. What is the angle between the vectors i + j and j + k?
a) 30°
b) 90°
c) 60°
d) 45°
Q4. What is (i + 2j) × (j + 3k)?
a) 6i − 3j − k
b) −6i + 3j − k
c) 6i + 3j + k
d) 6i − 3j + k
Q5. What is the area of the parallelogram with adjacent sides a = i + j and b = j + k?
a) 2
b) √2
c) 3
d) √3
Q6. For what value of λ are the vectors 2i + λj + k and i − 2j + 3k perpendicular?
a) 2
b) −5/2
c) 5/2
d) 5
Q7. What is the scalar projection of a = i + 2j + 3k on b = 2i − j + 2k?
a) 2/3
b) 3
c) 6
d) 2
Q8. What is the volume of the parallelepiped with edges i, 2j and i + j + 3k?
a) 3
b) 12
c) 6
d) 5
Q9. For what value of λ are the vectors i + j + k, i − j + 2k and 2i + λk coplanar?
a) 6
b) 2
c) 3
d) −3
Q10. A line has direction ratios 2, −1, 2. What are its direction cosines?
a) 2/9, −1/9, 2/9
b) 2, −1, 2
c) 2/3, −1/3, 2/3
d) 2/3, 1/3, 2/3
Q11. What is the distance between the points (1, 2, 3) and (4, 6, 15)?
a) √61
b) 12
c) 13
d) 169
Q12. What is the distance of the point (2, 1, −1) from the plane x − 2y + 2z = 5?
a) 7
b) 5/3
c) 7/3
d) 1
Q13. What is the angle between the planes 2x − y + z = 6 and x + y + 2z = 7?
a) 90°
b) 45°
c) 30°
d) 60°
Q14. Which plane cuts intercepts 2, 3 and 6 on the x, y and z axes?
a) 3x + 2y + z = 6
b) 6x + 3y + 2z = 6
c) 2x + 3y + 6z = 6
d) x + y + z = 11
Q15. What is the shortest distance between the lines r = t i and r = (2j + 3k) + s(j + k)?
a) 1
b) √2
c) 1/√2
d) 3
Quick revision notes
- a·b = |a||b| cos θ; a·b = 0 means perpendicular.
- |a × b| is the area of the parallelogram on a and b.
- [a b c] = volume of the parallelepiped; zero means coplanar.
- Distance of (x₁, y₁, z₁) from ax + by + cz = d is |ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²).
- Shortest distance between skew lines = |(a₂ − a₁)·(d₁ × d₂)|/|d₁ × d₂|.
Frequently asked questions
Are these JEE previous year questions?
No. They are original practice questions at the JEE Main level.
How do I find the angle between two planes?
Find the angle between their normal vectors, which are the coefficients of x, y and z. Take the acute angle.
What is the difference between direction ratios and direction cosines?
Direction ratios are any numbers proportional to the direction. Direction cosines are those numbers divided by their magnitude, so their squares add to 1.
Comments
Post a Comment