Area and Volume — JEE Main Physics MCQs with Solutions
Free JEE Main Physics Area and Volume MCQs with step-by-step solutions (35 questions). Part of Vectors. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Area and Volume · medium · theory
The area of the parallelogram whose adjacent sides are $\vec A = \hat{i} + 2\hat{j}$ and $\vec B = 2\hat{i} + \hat{j}$ is
A. $3$ sq units ✓ Correct
B. $5$ sq units
C. $\sqrt{5}$ sq units
D. $6$ sq units
Solution: $\vec A\times\vec B = (1\cdot1 - 2\cdot2)\hat{k} = -3\hat{k}$, so the area $= |\vec A\times\vec B| = 3$ sq units.
Q2 — Area and Volume · easy · theory
The area of a parallelogram with sides of lengths $5$ and $4$ units inclined at $30^\circ$ is
A. $20$ sq units
B. $10$ sq units ✓ Correct
C. $5$ sq units
D. $10\sqrt{3}$ sq units
Solution: Area $= ab\sin\theta = 5\times4\times\sin 30^\circ = 10$ sq units.
Q3 — Area and Volume · easy · theory
The area of the triangle whose two sides are the vectors $3\hat{i}$ and $4\hat{j}$ is
A. $3$ sq units
B. $24$ sq units
C. $12$ sq units
D. $6$ sq units ✓ Correct
Solution: Area $= \tfrac{1}{2}|\vec A\times\vec B| = \tfrac{1}{2}(3)(4)\sin 90^\circ = 6$ sq units.
Q4 — Area and Volume · medium · numerical
The volume of the parallelepiped with edges $\hat{i}$, $\hat{j}$ and $\hat{k}$ is
A. $\sqrt{3}$
B. $1$ ✓ Correct
C. $3$
D. $0$
Solution: Volume $= \hat{i}\cdot(\hat{j}\times\hat{k}) = \hat{i}\cdot\hat{i} = 1$ (a unit cube).
Q5 — Area and Volume · medium · numerical
The volume of the parallelepiped with edges $2\hat{i}$, $3\hat{j}$ and $4\hat{k}$ is
A. $12$
B. $24$ ✓ Correct
C. $6$
D. $9$
Solution: The edges are mutually perpendicular: volume $= 2\times3\times4 = 24$ cubic units.
Q6 — Area and Volume · medium · theory
Three vectors $\vec A$, $\vec B$, $\vec C$ are coplanar. Their scalar triple product $\vec A\cdot(\vec B\times\vec C)$ is
A. Always positive
B. Zero ✓ Correct
C. $ABC$
D. Maximum
Solution: The scalar triple product measures the volume of the parallelepiped — for coplanar vectors that volume collapses to zero.
Q7 — Area and Volume · medium · numerical
The vectors $\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\hat{j}$ and $\hat{j}+\lambda\hat{k}$ are coplanar when $\lambda$ equals
A. $1$
B. $0$
C. $2$
D. $\frac{1}{2}$ ✓ Correct
Solution: Set the scalar triple product (determinant) to zero: $1(-\lambda - 0) - 1(\lambda - 0) + 1(1-0) = 1 - 2\lambda = 0 \Rightarrow \lambda = \tfrac{1}{2}$.
Q8 — Area and Volume · easy · theory
The diagonals of a parallelogram are $2\hat{i}$ and $4\hat{j}$. Its area is
A. $16$ sq units
B. $8$ sq units
C. $2$ sq units
D. $4$ sq units ✓ Correct
Solution: Area $= \tfrac{1}{2}|\vec d_1\times\vec d_2| = \tfrac{1}{2}(2)(4) = 4$ sq units.
Q9 — Area and Volume · medium · theory
The area of the triangle with vertices $O(0,0,0)$, $A(1,0,0)$ and $B(0,2,0)$ is
A. $2$ sq units
B. $1$ sq unit ✓ Correct
C. $\frac{1}{2}$ sq unit
D. $4$ sq units
Solution: $\vec{OA}\times\vec{OB} = \hat{i}\times 2\hat{j} = 2\hat{k}$; area $= \tfrac{1}{2}(2) = 1$ sq unit.
Q10 — Area and Volume · medium · theory
Geometrically, $|\vec A \times \vec B|$ represents
A. The area of the triangle with sides $\vec A$ and $\vec B$
B. The length of the diagonal
C. The volume of a parallelepiped
D. The area of the parallelogram with sides $\vec A$ and $\vec B$ ✓ Correct
Solution: $AB\sin\theta$ is exactly base × height of the parallelogram formed by the two vectors. (The triangle is half of it.)
Q11 — Area and Volume · medium · theory
The area of the triangle whose adjacent sides are $\vec A = 2\hat{i} + \hat{j}$ and $\vec B = \hat{i} + 3\hat{j}$ is
A. $\frac{7}{2}$ sq units
B. $5$ sq units
C. $\frac{5}{2}$ sq units ✓ Correct
D. $10$ sq units
Solution: $\vec A\times\vec B = (2\cdot3 - 1\cdot1)\hat{k} = 5\hat{k}$; triangle area $= \tfrac{1}{2}(5) = \tfrac{5}{2}$ sq units.
Q12 — Area and Volume · medium · theory
A unit vector normal to the plane containing $\hat{i}$ and $\hat{i}+\hat{j}$ is
A. $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
B. $\hat{k}$ ✓ Correct
C. $\hat{i}$
D. $\hat{j}$
Solution: $\hat{i}\times(\hat{i}+\hat{j}) = \hat{i}\times\hat{j} = \hat{k}$ — perpendicular to the plane of the two vectors.
Q13 — Area and Volume · medium · numerical
The volume of the parallelepiped with edges $\vec a = \hat{i}+\hat{j}$, $\vec b = \hat{j}+\hat{k}$ and $\vec c = \hat{i}+\hat{k}$ is
A. $3$
B. $1$
C. $2$ ✓ Correct
D. $0$
Solution: Triple product determinant: $1(1-0) - 1(0-1) + 0 = 1 + 1 = 2$ cubic units. (A classic JEE computation.)
Q14 — Area and Volume · easy · theory
A parallelogram has sides $3$ and $4$ units at right angles. Its area is
A. $12$ sq units ✓ Correct
B. $7$ sq units
C. $24$ sq units
D. $6$ sq units
Solution: Area $= ab\sin 90^\circ = 3\times4 = 12$ sq units (a rectangle).
Q15 — Area and Volume · easy · theory
Two sides of a triangle are equal vectors of magnitude $a$ with angle $\theta$ between them. The area of the triangle is
A. $a^2\sin\theta$
B. $\frac{1}{2}a^2\cos\theta$
C. $a^2(1-\cos\theta)$
D. $\frac{1}{2}a^2\sin\theta$ ✓ Correct
Solution: Area $= \tfrac{1}{2}|\vec A\times\vec B| = \tfrac{1}{2}a\cdot a\sin\theta$.
Q16 — Area and Volume · medium · theory
The scalar triple product $\vec A\cdot(\vec A\times\vec B)$ equals
A. $A^2B$
B. Zero ✓ Correct
C. The volume of a parallelepiped
D. $AB^2$
Solution: Two of the three vectors are identical, so the "parallelepiped" is flat — the triple product vanishes.
Q17 — Area and Volume · easy · theory
For fixed side lengths, the area of a parallelogram is maximum when the angle between the sides is
A. $60^\circ$
B. $180^\circ$
C. $90^\circ$ ✓ Correct
D. $45^\circ$
Solution: Area $= ab\sin\theta$ peaks at $\sin\theta = 1$, i.e. $\theta = 90^\circ$.
Q18 — Area and Volume · easy · theory
If $\vec A \times \vec B = 3\hat{k}$, the area of the triangle formed with $\vec A$ and $\vec B$ as adjacent sides is
A. $6$ sq units
B. $\frac{3}{2}$ sq units ✓ Correct
C. $3$ sq units
D. $\sqrt{3}$ sq units
Solution: Triangle area $= \tfrac{1}{2}|\vec A\times\vec B| = \tfrac{3}{2}$ sq units.
Q19 — Area and Volume · medium · theory
The scalar triple product $\vec a\cdot(\vec b\times\vec c)$ geometrically gives
A. The length of the longest edge
B. The area of a parallelogram
C. The total surface area
D. The volume of the parallelepiped with edges $\vec a, \vec b, \vec c$ ✓ Correct
Solution: $|\vec b\times\vec c|$ is the base area and the dot with $\vec a$ multiplies by the height component — base × height = volume.
Q20 — Area and Volume · medium · theory
The area of the parallelogram whose diagonals are $3\hat{i}+\hat{j}-2\hat{k}$ and $\hat{i}-3\hat{j}+4\hat{k}$ is
A. $5\sqrt{3}$ sq units ✓ Correct
B. $10\sqrt{3}$ sq units
C. $15$ sq units
D. $5$ sq units
Solution: $\vec d_1\times\vec d_2 = -2\hat{i} - 14\hat{j} - 10\hat{k}$, magnitude $\sqrt{4+196+100} = \sqrt{300} = 10\sqrt{3}$. Area $= \tfrac{1}{2}(10\sqrt{3}) = 5\sqrt{3}$ sq units. (A very well-known problem.)
Q21 — Area and Volume · easy · theory
A rhombus has diagonals of lengths $6$ and $8$ units. Its area is
A. $24$ sq units ✓ Correct
B. $14$ sq units
C. $12$ sq units
D. $48$ sq units
Solution: Area $= \tfrac{1}{2}d_1 d_2 = \tfrac{1}{2}(6)(8) = 24$ sq units.
Q22 — Area and Volume · medium · theory
The volume of a cube whose edges are represented by $a\hat{i}$, $a\hat{j}$ and $a\hat{k}$ is
A. $a^3$ ✓ Correct
B. $3a$
C. $a^2$
D. $\sqrt{3}a$
Solution: Triple product of mutually perpendicular equal edges: $a\cdot a\cdot a = a^3$.
Q23 — Area and Volume · medium · theory
Which of the following equals $\vec A\cdot(\vec B\times\vec C)$?
A. $\vec B\cdot(\vec A\times\vec C)$
B. $\vec A\cdot(\vec C\times\vec B)$
C. $\vec B\cdot(\vec C\times\vec A)$ ✓ Correct
D. $\vec C\cdot(\vec B\times\vec A)$
Solution: The scalar triple product is unchanged under cyclic rotation: $[\vec A\,\vec B\,\vec C] = [\vec B\,\vec C\,\vec A] = [\vec C\,\vec A\,\vec B]$. Swapping two vectors flips the sign.
Q24 — Area and Volume · medium · numerical
The value of $\hat{i}\cdot(\hat{k}\times\hat{j})$ is
A. $\hat{i}$
B. $-1$ ✓ Correct
C. $1$
D. $0$
Solution: $\hat{k}\times\hat{j} = -\hat{i}$ (anti-cyclic order), so the dot with $\hat{i}$ gives $-1$.
Q25 — Area and Volume · medium · theory
The area of the triangle with vertices $A(1,1,0)$, $B(2,3,0)$ and $C(3,1,0)$ is
A. $2$ sq units ✓ Correct
B. $\sqrt{2}$ sq units
C. $1$ sq unit
D. $4$ sq units
Solution: $\vec{AB} = (1,2,0)$, $\vec{AC} = (2,0,0)$; $\vec{AB}\times\vec{AC} = -4\hat{k}$. Area $= \tfrac{1}{2}(4) = 2$ sq units.
Q26 — Area and Volume · medium · theory
If $\vec B = 2\vec A$ (parallel vectors), the volume of the parallelepiped formed by $\vec A$, $\vec B$ and any third vector $\vec C$ is
A. Zero ✓ Correct
B. $2A^2C$
C. Twice the volume with $\vec A$ alone
D. Cannot be determined
Solution: Two parallel edges make the three vectors coplanar (linearly dependent), so the volume is zero.
Q27 — Area and Volume · easy · theory
A parallelogram has sides $6$ and $8$ units. Its maximum possible area is
A. $24$ sq units
B. $48$ sq units ✓ Correct
C. $96$ sq units
D. $14$ sq units
Solution: Maximum at $\theta = 90^\circ$: area $= 6\times8 = 48$ sq units.
Q28 — Area and Volume · medium · theory
A parallelepiped has volume $24$ cubic units and base area $8$ sq units. Its height is
A. $16$ units
B. $4$ units
C. $3$ units ✓ Correct
D. $2$ units
Solution: Volume $=$ base area $\times$ height $\Rightarrow h = \frac{24}{8} = 3$ units.
Q29 — Area and Volume · easy · theory
The direction of the area vector of a plane surface is
A. Perpendicular to the surface ✓ Correct
B. Parallel to the surface
C. Along the longest side
D. Along a diagonal
Solution: By convention (right-hand rule), an area vector points along the normal to the surface.
Q30 — Area and Volume · medium · numerical
The area of a parallelogram with adjacent sides 4 and 5 at 30° is:
A. 5
B. 17.3
C. 10 ✓ Correct
D. 20
Solution: Area = ab sinθ = 4×5×0.5 = 10.