UG Mathematics Booster Test 3 - Equality of Matrices -

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UG Mathematics Booster Test 3 - Equality of Matrices

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Mathematics Booster Test 3 - Equality of Matrices

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QUESTION 1 OF 20

Matrix \(A\) represents the 5-day moving average of 3 stocks ⇒ order \(5\times 3\).
Matrix \(B\) represents the 3-day moving average of 5 stocks ⇒ order \(3\times 5\).
For \(A=B\), what must be the order of \(B\)?

Answer Review

1 Matrix \(A\) represents the 5-day moving average of 3 stocks ⇒ order \(5\times 3\).
Matrix \(B\) represents the 3-day moving average of 5 stocks ⇒ order \(3\times 5\).
For \(A=B\), what must be the order of \(B\)?

A \(3\times 5\)

B \(5\times 5\)

C \(5\times 3\)

D \(3\times 3\)

💡 Short Explanation

Equal matrices must have identical orders. Matrix \(A\) is \(5\times 3\). Therefore \(B\) must also be \(5\times 3\).

✅ Correct Answer Explanation

Matrix equality requires two conditions: same order and equal corresponding elements. Since matrix \(A\) has order \(5\times 3\), matrix \(B\) must also contain 5 rows and 3 columns. Therefore Option C is correct. Options A, B, and D fail because their dimensions differ.

❓ Why Other Options Are Incorrect

Option A → Rows and columns are interchanged, so order differs.
Option B → Number of columns does not match matrix \(A\).
Option D → Both rows and columns differ from matrix \(A\).

🎯 Strategy

Used: Elimination

Application:

Compare matrix dimensions directly.

Final Logic:

Equal matrices always have identical row–column structure.

🧠 Memory Trick

"Equal matrix = equal size"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Matrices

2 Let
A=[█(3@4)],B=[█(x@y)]
If A=B, find the magnitude of the vector.

A 3

B 4

C 25

D 5

💡 Short Explanation

Equality gives \(x=3, y=4\). Magnitude formula uses Pythagoras theorem. Result equals 5.

✅ Correct Answer Explanation

Since \(A=B\), \(x=3,y=4\) Vector magnitude: \(\sqrt{x^{2}+y^{2}}\) \(=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5\) Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Uses only one component.
Option B → Ignores complete magnitude formula.
Option C → Gives squared magnitude instead of magnitude.

🎯 Strategy

Used: Substitution

Application:

Use corresponding entries and apply vector magnitude formula.

Final Logic:

\(3\)-\(4\)-\(5\) triangle gives magnitude 5.

🧠 Memory Trick

"3-4-5 triangle"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Column Matrices

3 If corresponding entries give triangle vertices (0,0),(1,0),(0,2), find the area.

A 1.0 sq units

B 2.5 sq units

C 5.0 sq units

D 4.0 sq units

💡 Short Explanation

Base length equals 1. Height equals 2. Triangle area becomes 1 square unit.

✅ Correct Answer Explanation

Using triangle area formula: \(Area=\frac{1}{2}\times base\times height\) Base \(=1\), height \(=2\). \(Area=\frac{1}{2}(1)(2)=1\) Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect multiplication.
Option C → Far larger than actual geometric area.
Option D → Uses rectangle area instead of triangle area.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Identify geometric dimensions correctly.

Final Logic:

Half of base × height gives area.

🧠 Memory Trick

"Triangle = half rectangle"

📚 Class XI – NCERT Mathematics – Chapter: Coordinate Geometry / Matrices – Topic: Area of Triangle

4 If
[■(m+1&2@3&n-1)]=[■(4&2@3&5)]
find m,n.

A \(m=2,n=6\)

B \(m=3,n=6\)

C \(m=4,n=5\)

D \(m=1,n=4\)

💡 Short Explanation

Compare corresponding entries. \(m+1=4\) and \(n-1=5\). Hence \(m=3, n=6\).

✅ Correct Answer Explanation

From matrix equality: \(m+1=4\Rightarrow m=3n-1=5\Rightarrow n=6\) Thus the correct pair is \(\left(3,\ 6\right)\). The provided option label should be Option B.

❓ Why Other Options Are Incorrect

Option A → Incorrect value of \(m\).
Option C → Values do not satisfy equations.
Option D → Both variables incorrect.

🎯 Strategy

Used: Substitution

Application:

Equate matching matrix entries.

Final Logic:

Equal positions produce linear equations.

🧠 Memory Trick

"Box equals box"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Solving Variables Using Equality

5 Let
A=[■(a&b)],B=[■(1&1)]
where a,b∈{0,1}.
Find the probability that A=B.
Total outcomes = 4 → only (1ⓜ,1)works

A \(1/4\)

B \(1/2\)

C \(3/4\)

D \(1/8\)

💡 Short Explanation

Four equally likely outcomes exist. Only one equals matrix \(B\). Probability equals \(1/4\).

✅ Correct Answer Explanation

Possible ordered pairs: \(\left(0,0),(0,1),(1,0),(1,1\right)\) Only \(\left(1,\ 1\right)\)satisfies matrix equality. Hence: \(P(A=B)=\frac{1}{4}\) Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Counts two favorable cases incorrectly.
Option C → Exceeds actual probability.
Option D → Assumes eight total outcomes.

🎯 Strategy

Used: Option Grouping

Application:

List all possible ordered pairs.

Final Logic:

One favorable outcome among four total outcomes.

🧠 Memory Trick

"Only all ones work"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Corresponding Elements

6 Let
\(A=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right],B=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right],C=\left[\begin{pmatrix}1 & 2\\ 4 & 3\end{pmatrix}\right]\)

List I	List II
1. \(A=B\)	a. True
2. \(A=C\)	b. False
3. \(A-B=O\)	c. True
4. \(A-C=O\)	d. False

A 1-a, 2-a, 3-c, 4-d

B 1-a, 2-b, 3-c, 4-d

C 1-b, 2-b, 3-c, 4-c

D 1-a, 2-b, 3-d, 4-c

💡 Short Explanation

Equal matrices have identical corresponding elements.
Subtracting equal matrices gives the zero matrix.
Even one unequal corresponding element makes two matrices unequal.

✅ Correct Answer Explanation

Compare the matrices element by element.

[

A=\begin{bmatrix}1&2\3&4\end{bmatrix},

\qquad

B=\begin{bmatrix}1&2\3&4\end{bmatrix}

]

All corresponding elements of A and B are equal. Therefore,

A = B

Subtracting equal matrices gives

[

A-B=\begin{bmatrix}0&0\0&0\end{bmatrix}=O

]

Now compare A and C.

[

C=\begin{bmatrix}1&2\4&3\end{bmatrix}

]

The corresponding elements differ:

(a_{21}): 3 ≠ 4
(a_{22}): 4 ≠ 3

Hence,

A ≠ C

and

[

A-C=

\begin{bmatrix}

0&0\

\end{bmatrix}

\neq O

]

Therefore, the correct matching is:

List I List II

1. A = B a. True

2. A = C b. False

3. A − B = O c. True

4. A − C = O d. False

Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect because A ≠ C.
Option C → Incorrect because A = B is true and A − C ≠ O.
Option D → Incorrect because A − B = O is true, whereas A − C ≠ O.

🎯 Strategy

Elimination

Application:

Compare the corresponding entries of the matrices. If all entries are equal, the matrices are equal; otherwise, they are unequal. Then verify the subtraction.

Final Logic:

Equal matrices have identical corresponding elements and their difference is the zero matrix.

🧠 Memory Trick

"Same entries → Same matrix → Zero difference."

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality and Zero Matrix

7 Which pairs can never be equal due to different orders?
(I) \(2\times 3\) and \(3\times 2\) matrices
(II) Column matrix with 4 elements and row matrix with 4 elements
(III) Two square matrices of order 3

A I and II only

B I and III only

C II and III only

D I, II, and III

💡 Short Explanation

Equal matrices need same order. Cases I and II differ structurally. Case III may still be equal.

✅ Correct Answer Explanation

(I) Orders differ. (II) Column and row matrices have different dimensions. (III) Two \(3\times 3\) matrices can be equal if entries match. Thus only I and II can never be equal. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Includes a possible equality case.
Option C → Excludes statement I incorrectly.
Option D → Statement III is not always false.

🎯 Strategy

Used: Elimination

Application:

Check dimensional compatibility.

Final Logic:

Different orders prevent equality.

🧠 Memory Trick

"Different shape → never equal"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Same Order Requirement

8 Let
\(A=\left[\begin{pmatrix}x & 2\\ 3 & y\end{pmatrix}\right],B=\left[\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}\right]\)
Which statement is incorrect?

A If \(x=1\) and \(y=4\), matrices are equal

B If \(x\neq 1\), matrices are unequal regardless of \(y\)

C Equal entries alone don't guarantee full matrix equality

D Equal determinants imply matrices are equal

💡 Short Explanation

Equal determinants do not ensure equality. Corresponding entries must match. Matrix equality is stricter.

✅ Correct Answer Explanation

Matrices are equal only when every corresponding element matches. Equal determinants merely indicate equal scalar values, not identical matrices. Therefore Option D is incorrect. Options A, B, and C correctly describe matrix equality conditions.

❓ Why Other Options Are Incorrect

Option A → Correct equality condition.
Option B → One unequal entry breaks equality.
Option C → Full entry-wise equality is necessary.

🎯 Strategy

Used: Extreme Word Filter

Application:

Check whether the condition guarantees equality universally.

Final Logic:

Equal determinants cannot ensure equal matrices.

🧠 Memory Trick

"Same determinant ≠ same matrix"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Matrices

9 If
\(\left[\begin{pmatrix}x & 2\\ 3 & y\end{pmatrix}\right]=\left[\begin{pmatrix}1 & 2\\ 3 & 1\end{pmatrix}\right]\)
and \(x>y>0\), find \(x-y\).

A 5

B 7

C 0

D 3

💡 Short Explanation

Equality gives \(x=1, y=1\). Difference equals zero. Given inequality condition is inconsistent.

✅ Correct Answer Explanation

From equality: \(x=1,y=1\) Thus: \(x-y=0\) Although the condition \(x>y\) is inconsistent with equality, the matrix comparison still yields difference \(0\). Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Arithmetic mismatch.
Option B → Impossible from obtained values.
Option D → Difference is not three.

🎯 Strategy

Used: Substitution

Application:

Directly compare corresponding entries.

Final Logic:

Equal entries give equal variable values.

🧠 Memory Trick

"Equal boxes → equal numbers"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Variable Determination Using Equality

10 The system
\(y=x^{2}andy=2x+1\)
represents a parabola and a line.
How many real intersection points exist?

A 0

B 1

C 2

D 3

💡 Short Explanation

Equate both equations. Quadratic equation gives two roots. Therefore two intersections exist.

✅ Correct Answer Explanation

Equating: \(x^{2}=2x+1\) \(x^{2}-2x-1=0\) Discriminant: \((-2)^{2}-4(1)(-1)=8>0\) Hence there are two distinct real roots, so the parabola and line intersect at two points. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Discriminant is positive, not negative.
Option B → Roots are distinct, not repeated.
Option D → Quadratic and line intersect at maximum two points.

🎯 Strategy

Used: Substitution

Application:

Set equations equal and analyze roots.

Final Logic:

Positive discriminant gives two real intersections.

🧠 Memory Trick

"Quadratic + line → at most 2 cuts"

📚 Class XI – NCERT Mathematics – Chapter: Quadratic Equations / Matrices – Topic: Graphical Intersection of Curves

11 Assertion (A): The matrix equality
\(\left[\begin{pmatrix}x+1 & y\\ 2 & 3\end{pmatrix}\right]=\left[\begin{pmatrix}4 & 5\\ 2 & 3\end{pmatrix}\right]\)
gives a unique solution for \(x\) and \(y\).
Reason (R): A system of linear equations has infinitely many solutions if the corresponding lines are coincident.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Matrix equality gives two equations. Unique values of \(x\) and \(y\) result. Reason is unrelated here.

✅ Correct Answer Explanation

From matrix equality: \(x+1=4,y=5\) Thus: \(x=3,y=5\) A unique solution exists, so Assertion is true. The Reason statement is generally true in coordinate geometry, but it does not explain this matrix situation. Therefore Option C should logically be correct, not D.

❓ Why Other Options Are Incorrect

Option A → Assertion is clearly true.
Option C → Reason is true but unrelated to matrix equality explanation.
Option D → Assertion is not false.

🎯 Strategy

Used: Elimination

Application:

Check assertion and reason independently.

Final Logic:

Unique values arise from direct element comparison.

🧠 Memory Trick

"Equal entries → fixed values"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Matrices and Variable Comparison

12 If
\(x=2,y=4,z=1,w=0.5(w>0)\)
arrange in ascending order.

A \(w,z,x,y\)

B \(w,x,z,y\)

C \(z,w,y,x\)

D \(x,y,w,z\)

💡 Short Explanation

Compare numerical values directly. Smallest is \(0.5\). Largest is \(4\).

✅ Correct Answer Explanation

Values are: \(w=0.5,z=1,x=2,y=4\) Ascending order means smallest to largest: \(0.5<1<2<4\) Thus the correct arrangement is: \(w,z,x,y\) Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Places 2 before 1 incorrectly.
Option C → Places 1 before 0.5 incorrectly.
Option D → Completely reverses ascending order.

🎯 Strategy

Used: Option Grouping

Application:

Arrange values numerically from least to greatest.

Final Logic:

Ascending means increasing numerical order.

🧠 Memory Trick

"Half, one, two, four"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Variable Comparison from Equality

13 If
\(\left[\begin{pmatrix}C & 20\\ 30 & 40\end{pmatrix}\right]=\left[\begin{pmatrix}30 & 20\\ 30 & 40\end{pmatrix}\right]\)
find \(C\).

A 20

B 30

C 40

D 50

💡 Short Explanation

Equal matrices have equal entries. Compare first corresponding elements. Hence \(C=30\).

✅ Correct Answer Explanation

Matrix equality requires corresponding entries to match: \(C=30\) All remaining entries are already equal. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Does not satisfy equality.
Option C → First entries differ.
Option D → Matrix equality fails.

🎯 Strategy

Used: Substitution

Application:

Compare entries in identical positions.

Final Logic:

Equal positions imply equal values.

🧠 Memory Trick

"Top-left equals top-left"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Corresponding Elements

14 Match the Following

List I	List II
1. Step 1	a. Solve simultaneous equations
2. Step 2	b. Validate dimensions are equal
3. Step 3	c. Substitute back to check
4. Step 4	d. Equate elements

A 1-b, 2-a, 3-d, 4-c

B 1-d, 2-b, 3-a, 4-c

C 1-b, 2-d, 3-a, 4-c

D 1-a, 2-d, 3-c, 4-b

💡 Short Explanation

First verify matrix order. Then equate entries. Solve and finally verify.

✅ Correct Answer Explanation

Correct procedure: 1. Validate equal dimensions. 2. Equate corresponding elements. 3. Solve resulting equations. 4. Substitute answers back for checking. Thus mapping becomes: \(1-b, 2-d, 3-a, 4-c\) Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Solving cannot occur before equations form.
Option B → Step order is inconsistent.
Option D → Verification cannot precede solving.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Arrange steps logically from identification to verification.

Final Logic:

Equality checking always begins with same order.

🧠 Memory Trick

"Check → Compare → Solve → Verify"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Solving Unknowns Using Equality

15 A 3D coordinate is given by
\(\left[\begin{aligned}x\\ y\\ z\end{aligned}\right]=\left[\begin{aligned}0\\ 0\\ 0\end{aligned}\right]\)
This implies \(x=0, y=0, z=0\).
The coefficient matrix of the system is the identity matrix:
\(I=\left[\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}\right]\)
Find \(det⁡(I)\).

A 1

B 3

C -2

D 10

💡 Short Explanation

Identity matrix has determinant 1. Product of diagonal ones remains 1. Therefore determinant equals 1.

✅ Correct Answer Explanation

For any identity matrix: \(det⁡(I)=1\) Since all diagonal entries are 1 and off-diagonal entries are 0, determinant equals 1. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Determinant is not sum of diagonals.
Option C → Determinant cannot be negative here.
Option D → Arbitrary incorrect value.

🎯 Strategy

Used: Elimination

Application:

Recall standard determinant property of identity matrices.

Final Logic:

Identity matrices always have determinant 1.

🧠 Memory Trick

"I for Identity → determinant 1"

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Determinant of Identity Matrix

16 By simplifying the matrix relation, the parametric equations are obtained as:
\(x=2cos⁡t,y=3sin⁡t\)
which represent an ellipse.
Find the area enclosed by this curve.

A \(4\pi\)

B \(5\pi\)

C \(6\pi\)

D \(12\pi\)

💡 Short Explanation

Semi-axes are 2 and 3. Ellipse area equals \(\pi ab\). Final area becomes \(6\pi\).

✅ Correct Answer Explanation

The ellipse has semi-major axis \(a=2\) and semi-minor axis \(b=3\). Ellipse area formula: \(Area=\pi ab\) \(=\pi (2)(3)=6\pi\) Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Uses incorrect axis product.
Option B → Arithmetic error.
Option D → Double the actual area.

🎯 Strategy

Used: Substitution

Application:

Identify ellipse parameters from parametric equations.

Final Logic:

Multiply semi-axes and \(\pi\).

🧠 Memory Trick

"Ellipse → \(\pi ab\)"

📚 Class XI – NCERT Mathematics – Chapter: Conic Sections – Topic: Ellipse Area Concept

17 Solve the matrix equation
\(\left[\begin{pmatrix}x+2 & 0\\ 0 & x+2\end{pmatrix}\right]=\left[\begin{pmatrix}19 & 0\\ 0 & 19\end{pmatrix}\right]\)
given the boundary condition \(x>0\).
Find \(x\).

A 15

B 17

C 12

D 20

💡 Short Explanation

Compare diagonal entries directly. Equation becomes \(x+2=19\). Hence \(x=17\).

✅ Correct Answer Explanation

From equality of corresponding entries: \(x+2=19x=17\) The condition \(x>0\) is satisfied. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Gives 17 only after adding 2.
Option C → Does not satisfy equality.
Option D → Produces incorrect matrix entries.

🎯 Strategy

Used: Substitution

Application:

Use equality of corresponding diagonal elements.

Final Logic:

Equal entries form simple linear equations.

🧠 Memory Trick

"19 minus 2"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Solving Matrix Equations

18 Let
\(A=\left[\begin{pmatrix}u & v\end{pmatrix}\right],B=\left[\begin{pmatrix}2 & 2\end{pmatrix}\right]\)
where \(u,v\in \{1,2,3,4\}\).
Find the probability that \(A=B\).

A \(1/4\)

B \(1/2\)

C \(1/3\)

D \(1/8\)

💡 Short Explanation

Both entries must equal 2. Total outcomes are 16. Only one favorable outcome exists.

✅ Correct Answer Explanation

Possible ordered pairs: \(4\times 4=16\) Only \(\left(2,\ 2\right)\)satisfies equality. Thus: \(P(A=B)=\frac{1}{16}\) The provided answer \(1/4\) is incorrect.

❓ Why Other Options Are Incorrect

Option A → Considers only one variable.
Option B → Probability too large.
Option C → Does not match total outcomes.

🎯 Strategy

Used: Option Grouping

Application:

Count total and favorable ordered pairs.

Final Logic:

One successful pair among sixteen possibilities.

🧠 Memory Trick

"Two exact matches needed"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Corresponding Entries

19 Given the constraint:
\(x+y=4\)
which point lies exactly on the boundary?

A \(\left(3,\ 2\right)\)

B \(\left(2,\ 2\right)\)

C \(\left(4,\ 0\right)\)

D \(\left(0,\ 4\right)\)

💡 Short Explanation

Boundary points satisfy equation exactly. Substitute coordinates into relation. \(\left(2,\ 2\right)\)satisfies \(x+y=4\).

✅ Correct Answer Explanation

Testing options: \(2+2=4\) Thus \(\left(2,\ 2\right)\)lies on the line. Also \(\left(4,\ 0\right)\)and \(\left(0,\ 4\right)\)satisfy the equation, so the question has multiple valid answers. Option B is one correct choice.

❓ Why Other Options Are Incorrect

Option A → \(3+2=5\), not 4.
Option C → Actually satisfies the equation.
Option D → Also satisfies the equation.

🎯 Strategy

Used: Substitution

Application:

Substitute coordinates into equation.

Final Logic:

Boundary points make the equation true.

🧠 Memory Trick

"Add coordinates to test"

📚 Class XI – NCERT Mathematics – Chapter: Linear Inequalities / Matrices – Topic: Boundary of Linear Equation

20 Assertion (A): The matrix equation
\(\left[\begin{pmatrix}x & 2\end{pmatrix}\right]=\left[\begin{pmatrix}3 & 2\end{pmatrix}\right]\)
gives a real solution \(x=3\).
Reason (R): In matrix equality, only the first elements need to be equal.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Equality gives \(x=3\). All corresponding entries must match. Reason statement is false.

✅ Correct Answer Explanation

From matrix equality: \(x=3\) Hence Assertion is true. The Reason is false because matrix equality requires every corresponding element to match, not only the first entry. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Assertion is true.
Option C → Reason is false.
Option D → Assertion is not false.

🎯 Strategy

Used: Elimination

Application:

Evaluate assertion and reason separately.

Final Logic:

Matrix equality compares all entries.

🧠 Memory Trick

"All boxes must match"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Equality of Matrices

UG Mathematics Booster Test 3 - Equality of Matrices

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