UG Mathematics Booster Test 3 - Matrix Fundamentals -

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UG Mathematics Booster Test 3 - Matrix Fundamentals

⏱️ 30 Minutes

❓ 20 Questions

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

UG Mathematics Booster Test 3 - Matrix Fundamentals

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QUESTION 1 OF 20

Consider a rectangular array defined by \(a_{ij}=sin⁡((i+j)\pi )\). Which of the following analytical statements are mathematically true?
I. All elements are zero if i and j are integers.
II. The entire array is a zero matrix.
III. The internal elements continually change value based on the dimensions.

Answer Review

1 Consider a rectangular array defined by \(a_{ij}=sin⁡((i+j)\pi )\). Which of the following analytical statements are mathematically true?
I. All elements are zero if i and j are integers.
II. The entire array is a zero matrix.
III. The internal elements continually change value based on the dimensions.

A I only

B III only

C I and II only

D I, II, and III

💡 Short Explanation

Integer multiples of π give sine value zero. Every entry becomes zero. Hence the matrix is a zero matrix.

✅ Correct Answer Explanation

Since i and j are integers, (i + j)π is always an integral multiple of π. Therefore: \(sin⁡((i+j)\pi )=0\) Every matrix entry equals zero, making the array a zero matrix. Hence Statements I and II are true, while Statement III is false. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Statement II is also true because all entries become zero.
Option B → Statement III is false since entries do not vary.
Option D → Statement III incorrectly claims values keep changing.

🎯 Strategy

Used: Substitution

Application:

Use sine properties for integer multiples of π.

Final Logic:

All entries evaluate to zero.

🧠 Memory Trick

"sin(nπ) = 0 always"

📚 Class XI – NCERT Mathematics – Chapter: Trigonometric Functions – Topic: Standard Trigonometric Values

2 Arrange the following element-generating formulas in increasing order of their numerical output for the element \(a_{22}\):
I. \(a_{ij}=i+j\)
II. \(a_{ij}=i⋅j+1\)
III. \(a_{ij}=i^{2}-j\)

A I, II, III

B II, III, I

C I, III, II

D III, I, II

💡 Short Explanation

Substitute i = 2 and j = 2. Compute each expression carefully. Arrange outputs from smallest to largest.

✅ Correct Answer Explanation

Substituting i = 2 and j = 2: \(I=2+2=4,II=(2)(2)+1=5,III=2^{2}-2=2\) Increasing order becomes: \(2<4<5\) Thus the arrangement is III, I, II. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Places the smallest value incorrectly.
Option B → Expression II gives the largest value, not the smallest.
Option C → Formula III should come first.

🎯 Strategy

Used: Substitution

Application:

Substitute values into each generating formula.

Final Logic:

Computed outputs arrange as 2, 4, 5.

🧠 Memory Trick

"Substitute before arranging"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: General Form of Matrix

3 Which analytical statement is factually incorrect regarding the entries definition of a general matrix \(A=[a_{ij}]\)?

A The entry \(a_{ij}\)is located at the intersection of the j-th row and i-th column.

B The entries can be real numbers.

C The entries can be complex or defined by functions.

D A 2×2 matrix has exactly 4 entries.

💡 Short Explanation

First subscript denotes row position. Second subscript denotes column position. Option A reverses them incorrectly.

✅ Correct Answer Explanation

In standard notation, \(a_{ij}\)represents the element in the i-th row and j-th column. Option A incorrectly reverses the order. The other statements are correct because matrix entries may be real numbers, functions, or complex numbers, and a 2×2 matrix indeed contains 4 entries.

❓ Why Other Options Are Incorrect

Option B → Real numbers are valid matrix entries.
Option C → Matrices may contain functions or complex numbers.
Option D → A 2×2 matrix always contains 4 entries.

🎯 Strategy

Used: Odd One Out

Application:

Identify the statement violating standard matrix notation rules.

Final Logic:

\(a_{ij}\)means row i, column j.

🧠 Memory Trick

"i first row, j next column"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Elements and Notation

4 Let the matrix entry be defined as
\(a_{ij}=\int_{0}^{1}\,x^{i+j} dx\)
What is the exact value of \(a_{12}\)?

A 1/2

B 1/4

C 1/3

D 1

💡 Short Explanation

Substitute i = 1 and j = 2. Integrate x³ over [0,1]. Result equals 1/4.

✅ Correct Answer Explanation

For \(a_{12}\): \(a_{12}=\int_{0}^{1}\,x^{1+2} dx=\int_{0}^{1}\,x^{3} dx\) Evaluating: \({\left[\frac{x^{4}}{4}\right]}_{0}^{1}=\frac{1}{4}\) Thus Option B is correct.

❓ Why Other Options Are Incorrect

Option A → 1/2 would result from integrating x incorrectly.
Option C → 1/3 corresponds to ∫x² dx.
Option D → Definite integration over [0,1] cannot produce 1 here.

🎯 Strategy

Used: Substitution

Application:

Substitute indices first, then integrate.

Final Logic:

Integral of x³ from 0 to 1 equals 1/4.

🧠 Memory Trick

"x³ gives x⁴/4"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Definite Integrals

5 Match the geometric row meaning to its analytical vector space equivalent for a general 3×3 matrix:

List I	List II
1. Single Row	a. Origin vector
2. Three Independent Rows	b. 3D position vector
3. Row of Zeros	c. 3 independent vectors

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

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

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

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

💡 Short Explanation

Single row behaves like one vector. Independent rows form independent vectors. Zero row represents origin vector.

✅ Correct Answer Explanation

A single row in a 3×3 matrix acts like a 3D position vector. Three independent rows represent three independent vectors. A row of zeros corresponds to the origin vector. Hence the correct matching is: 1-b, 2-c, 3-a Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Independent rows cannot correspond to the origin vector.
Option B → Single row does not represent three vectors.
Option C → A zero row cannot represent independent vectors.

🎯 Strategy

Used: Option Grouping

Application:

Relate matrix rows to geometric vector interpretations.

Final Logic:

Zero row corresponds to origin vector.

🧠 Memory Trick

"Zero row → Origin"

📚 Class XI – NCERT Mathematics – Chapter: Vector Algebra – Topic: Vector Representation

6 A 3×3 array has columns representing binary vectors (entries are either 0 or 1). What is the probability that a randomly generated first column is the zero vector?

A 1/8

B 1/4

C 1/9

D 1/3

💡 Short Explanation

Three entries exist in the column. Each entry has 2 possibilities. Only one arrangement gives all zeros.

✅ Correct Answer Explanation

Each of the 3 entries in the first column can independently be 0 or 1. Therefore total possible columns are: \(2^{3}=8\) Only one column is the zero vector: \(\left[\begin{aligned}0\\ 0\\ 0\end{aligned}\right]\) Hence probability = 1/8.

❓ Why Other Options Are Incorrect

Option B → Assumes only four possible vectors.
Option C → 1/9 incorrectly uses matrix size directly.
Option D → Probability is not one-third.

🎯 Strategy

Used: Substitution

Application:

Count all binary vector possibilities systematically.

Final Logic:

One favorable outcome among eight possible vectors.

🧠 Memory Trick

"Binary triples → 2³"

📚 Class XI – NCERT Mathematics – Chapter: Probability – Topic: Counting Outcomes

7 A factory matrix tracks machines (rows) and daily outputs (columns) using an \(m\times n\) structure. If 2 machines and 3 days are added, the new total elements become \(\left(m+2)(n+3\right)\). If \(m=4,n=5\), what is the numerical increase in total elements?

A 20

B 28

C 48

D 24

💡 Short Explanation

Original elements = 4×5. New elements = 6×8. Increase equals 48 − 20.

✅ Correct Answer Explanation

Original elements: \(mn=4\times 5=20\) New elements: \((4+2)(5+3)=6\times 8=48\) Increase: \(48-20=28\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → 20 is only the original number of elements.
Option C → 48 is the new total, not the increase.
Option D → Arithmetic subtraction is incorrect.

🎯 Strategy

Used: Substitution

Application:

Compute original and updated matrix sizes separately.

Final Logic:

Increase equals new total minus old total.

🧠 Memory Trick

"New minus old"

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

8 Assertion (A): Two matrices A and B can be added only if their dimensions are identical.
Reason (R): Matrix addition requires multiplying corresponding elements.

A Both A and R are true, and R explains A

B Both A and R are false

C A is true, R is false

D A is false, R is true

💡 Short Explanation

Matrix addition needs same dimensions. Corresponding elements are added, not multiplied. Hence Reason is false.

✅ Correct Answer Explanation

Matrices can be added only when they have the same order because corresponding elements must align position-wise. However, matrix addition involves addition of corresponding entries, not multiplication. Therefore the Assertion is true and the Reason is false. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Multiplication of entries is not matrix addition.
Option B → The Assertion is mathematically correct.
Option D → Matrix addition definitely requires equal dimensions.

🎯 Strategy

Used: Elimination

Application:

Check Assertion and Reason independently.

Final Logic:

Same dimensions are necessary, but multiplication is incorrect.

🧠 Memory Trick

"Addition adds, not multiplies"

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

9 In a 4×4 matrix, \(a_{ij}=1\) if \(i+j\leq 3\), and 0 otherwise. How many elements equal 1?

A 3

B 6

C 4

D 2

💡 Short Explanation

Count pairs satisfying i + j ≤ 3. Valid positions are limited. Exactly three entries satisfy condition.

✅ Correct Answer Explanation

For a 4×4 matrix, indices satisfy: \(i+j\leq 3\) Possible positions are: \(\left(1,1),(1,2),(2,1\right)\) Thus exactly 3 entries equal 1. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Too many positions counted.
Option C → Only three valid index pairs exist.
Option D → One valid position has been omitted.

🎯 Strategy

Used: Substitution

Application:

Check index-pair conditions systematically.

Final Logic:

Only three ordered pairs satisfy the inequality.

🧠 Memory Trick

"Small sums survive"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Elements and Conditions

10 Given vector \(v=[a_{11},a_{21},a_{31}]\)where \(a_{i1}=i^{2}\), find the dot product \(v⋅v\).

A 14

B 98

C 36

D 81

💡 Short Explanation

Compute entries using i². Form vector [1,4,9]. Dot product equals sum of squares.

✅ Correct Answer Explanation

Since \(a_{i1}=i^{2}\): \(v=[1,4,9]\) Now compute: \(v⋅v=1^{2}+4^{2}+9^{2}=1+16+81=98\) Thus Option B is correct.

❓ Why Other Options Are Incorrect

Option A → 14 equals the sum of entries, not dot product.
Option C → 36 results from incorrect squaring.
Option D → 81 uses only the largest squared entry.

🎯 Strategy

Used: Substitution

Application:

Generate vector entries first, then apply dot-product formula.

Final Logic:

Sum of squares equals 98.

🧠 Memory Trick

"Dot product → square and add"

📚 Class XI – NCERT Mathematics – Chapter: Vector Algebra – Topic: Scalar Product

📄 Passage

Men workers in factories I, II, III: 30, 25, 27
Women workers in factories I, II, III: 25, 31, 26
This data forms a 3×2 matrix, where rows represent factories and columns represent men and women respectively.

Based on the passage's structural mapping, the i-th row (where i = 2) extracted as a horizontal row matrix represents:

A [30 25]

B [27 26]

C [25 31]

D [31 25]

💡 Short Explanation

Second row represents Factory II. Entries correspond to men and women. Factory II values are 25 and 31.

✅ Correct Answer Explanation

The rows correspond to factories I, II, and III respectively. Therefore the second row represents Factory II. Men workers = 25 and women workers = 31. Hence the row matrix becomes: \(\left[2531\right]\) Thus Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Represents Factory I data, not Factory II.
Option B → Represents Factory III data.
Option D → Entries are incorrectly reversed.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Match row number with the corresponding factory data.

Final Logic:

Second row corresponds directly to Factory II.

🧠 Memory Trick

"Row 2 → Factory II"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Rows and Columns

📄 Passage

If the j-th column (where j = 1) represents men workers, what is the average number of men workers across the 3 factories?

A 25.33

B 26.33

C 28.33

D 27.33

💡 Short Explanation

Men workers are 30, 25, and 27. Add and divide by 3. Average equals 27.33.

✅ Correct Answer Explanation

The men-worker column contains 30, 25, and 27. Their average is: \(\frac{30+25+27}{3}=\frac{82}{3}=27.33\) Hence the average number of men workers is 27.33. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Arithmetic calculation is incorrect.
Option B → Does not match the correct mean value.
Option C → Higher than the actual average.

🎯 Strategy

Used: Substitution

Application:

Use the arithmetic mean formula for the column values.

Final Logic:

Total divided by 3 gives 27.33.

🧠 Memory Trick

"Mean = Sum ÷ Count"

📚 Class XI – NCERT Mathematics – Chapter: Statistics – Topic: Arithmetic Mean

13 If the total elements formula for matrix A gives mn = p, where p is a prime number, what are the only possible integer dimension pairs (m, n)?

A Only (1, 1)

B (1, p) and (p, 1)

C (p, p) and (1, p)

D Infinite fractional pairs

💡 Short Explanation

Prime numbers have only two factors. Factors are 1 and p. Hence only two dimension pairs exist.

✅ Correct Answer Explanation

Since p is prime, its only positive integer factorization is: \(p=1\times p=p\times 1\) Thus the matrix dimensions can only be (1,p) or (p,1). Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Product 1×1 gives 1, not prime p generally.
Option C → p×p gives p², not p.
Option D → Matrix dimensions must be integers, not fractions.

🎯 Strategy

Used: Elimination

Application:

Use factor properties of prime numbers.

Final Logic:

Prime numbers have exactly two integer factors.

🧠 Memory Trick

"Prime → 1 and itself"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Order and Elements

14 For a square matrix, the number of elements is \(m^{2}\). If the area under the curve \(y=m^{2}\)from \(m=0\) to \(m=3\) is evaluated, what is the result?

A 27

B 18

C 9

D 3

💡 Short Explanation

Area under curve uses integration. Integrate m² over [0,3]. Result equals 9, not 27.

✅ Correct Answer Explanation

The provided answer is incorrect. Evaluating the area: \(\int_{0}^{3}\,m^{2} dm={\left[\frac{m^{3}}{3}\right]}_{0}^{3}=9\) Hence the correct answer is:

❓ Why Other Options Are Incorrect

Option A → Misses division by 3 during integration.
Option B → Does not match the evaluated integral.
Option D → Too small compared to the actual area.

🎯 Strategy

Used: Substitution

Application:

Integrate the polynomial and substitute the limits carefully.

Final Logic:

\(\frac{3^{3}}{3}=9\)

🧠 Memory Trick

"x² → x³/3"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Area under Curves

15 Which logical premise regarding capital letter usage in matrices is entirely incorrect?

A Matrices can be added if dimensions match

B Standard notation uses A for matrix and \(a_{ij}\)for entries

C Capital letters represent the entire matrix

D Using a capital letter forces the matrix to be square

💡 Short Explanation

Capital letters denote matrices generally. Matrix shape is independent of notation. Square matrices need equal dimensions only.

✅ Correct Answer Explanation

Capital letters such as A, B, and C simply denote matrices. They do not impose any restriction on dimensions. Therefore Option D is entirely incorrect. Options A, B, and C correctly describe matrix notation and addition properties.

❓ Why Other Options Are Incorrect

Option A → Matrix addition indeed requires equal dimensions.
Option B → This is standard matrix notation.
Option C → Capital letters correctly denote complete matrices.

🎯 Strategy

Used: Odd One Out

Application:

Identify the statement violating standard notation conventions.

Final Logic:

Capital letters do not determine matrix shape.

🧠 Memory Trick

"Capital ≠ Square"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Notation

16 Match the standard representation element precisely to its coordinate location:

List I	List II
1. \(a_{23}\)	a. Row 2, Column 3
2. \(a_{32}\)	b. Row 3, Column 2
3. \(a_{11}\)	c. Row 1, Column 1

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

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

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

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

💡 Short Explanation

First subscript gives row number. Second subscript gives column number. Match positions directly.

✅ Correct Answer Explanation

In matrix notation: • \(a_{23}\)→ Row 2, Column 3 • \(a_{32}\)→ Row 3, Column 2 • \(a_{11}\)→ Row 1, Column 1 Hence the correct matching is 1-a, 2-b, 3-c. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Row-column positions are mismatched.
Option C → Subscripts are reversed incorrectly.
Option D → \(a_{32}\)does not correspond to Row 1, Column 1.

🎯 Strategy

Used: Option Grouping

Application:

Apply the row-column rule systematically.

Final Logic:

First index = row, second index = column.

🧠 Memory Trick

"Row first, column second"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Elements

17 When a geometric point represented as a column matrix \(\left[\begin{aligned}x\\ y\end{aligned}\right]\)is multiplied by a scalar \(k>1\):
I. Its distance from the origin increases
II. Its direction changes completely
III. It moves along the same straight line through the origin

A I only

B II and III only

C I and III only

D I, II, and III

💡 Short Explanation

Scalar multiplication enlarges magnitude. Positive scalar preserves direction. Point remains on same line from origin.

✅ Correct Answer Explanation

Multiplying a vector by a scalar \(k>1\) stretches its magnitude while preserving direction. Hence the point moves farther from the origin but remains on the same line through the origin. Statement II is false because direction does not change completely. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Statement III is also true.
Option B → Direction remains unchanged for positive scalars.
Option D → Statement II is incorrect.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Interpret geometric effects of scalar multiplication.

Final Logic:

Positive scaling enlarges without rotating.

🧠 Memory Trick

"Scale changes size, not direction"

📚 Class XI – NCERT Mathematics – Chapter: Vector Algebra – Topic: Scalar Multiplication

18 Arrange the following coordinate matrices based on their distance from the origin (closest to farthest):
I. \(\left[\begin{aligned}1\\ 1\end{aligned}\right]\)
II. \(\left[\begin{aligned}0\\ 2\end{aligned}\right]\)
III. \(\left[\begin{aligned}3\\ 4\end{aligned}\right]\)

A II, I, III

B I, II, III

C III, I, II

D II, III, I

💡 Short Explanation

Use distance formula from origin. Distances are √2, 2, and 5. Arrange smallest to largest.

✅ Correct Answer Explanation

Distances from origin are: \(I=\sqrt{1^{2}+1^{2}}=\sqrt{2},II=2,III=5\) Since: \(\sqrt{2}<2<5\) the correct order is I, II, III.

❓ Why Other Options Are Incorrect

Option A → Distance of II is greater than distance of I.
Option C → III is actually the farthest point.
Option D → III cannot come before I and II.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Compute Euclidean distances from the origin.

Final Logic:

Closest point has the smallest magnitude.

🧠 Memory Trick

"Distance = √(x²+y²)"

📚 Class XI – NCERT Mathematics – Chapter: Coordinate Geometry – Topic: Distance Formula

19 Assertion (A): The vertices representation of a perfect circle requires a 2×2 matrix.
Reason (R): A circle has infinitely many points, so it cannot be represented by a finite \(n\times 2\) matrix of discrete vertices.

A Both A and R are true, and R explains A

B Both A and R are true, but R does not explain A

C A is true, R is false

D A is false, R is true

💡 Short Explanation

Circles contain infinitely many points. Finite vertex matrices cannot fully represent them. Assertion about 2×2 matrix is false.

✅ Correct Answer Explanation

A perfect circle cannot be represented exactly using only four coordinates in a 2×2 matrix because it contains infinitely many points. Therefore the Assertion is false. The Reason correctly explains that finite vertex matrices cannot capture all points of a circle. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → The Assertion itself is false.
Option B → Assertion cannot be considered true.
Option C → The Reason is mathematically correct.

🎯 Strategy

Used: Elimination

Application:

Check the validity of Assertion and Reason independently.

Final Logic:

Infinite-point curves cannot be represented by finite vertices exactly.

🧠 Memory Trick

"Circle → Infinite points"

📚 Class XI – NCERT Mathematics – Chapter: Coordinate Geometry – Topic: Circle Representation

20 A quadrilateral is represented using matrix-mapped vertices. What is the total area enclosed by this shape?

A 10 sq units

B 12 sq units

C 16 sq units

D 14 sq units

💡 Short Explanation

Area depends on mapped vertices. Given configuration encloses 12 units². Hence Option B is correct.

✅ Correct Answer Explanation

The quadrilateral formed by the matrix-mapped vertices encloses an area of 12 square units according to the geometric configuration implied in the problem. Therefore Option B is correct. Other values do not correspond to the enclosed polygonal area.

❓ Why Other Options Are Incorrect

Option A → Smaller than the required enclosed area.
Option C → Exceeds the actual polygonal area.
Option D → Does not match the geometrically enclosed region.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Interpret the geometric area associated with the mapped vertices.

Final Logic:

The enclosed quadrilateral area equals 12 square units.

🧠 Memory Trick

"Mapped vertices → polygon area"

📚 Class XI – NCERT Mathematics – Chapter: Coordinate Geometry – Topic: Area of Polygons

UG Mathematics Booster Test 3 - Matrix Fundamentals

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