UG Mathematics Booster Test 2 - Matrix Fundamentals -

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UG Mathematics Booster Test 2 - Matrix Fundamentals

⏱️ 30 Minutes

❓ 20 Questions

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

UG Mathematics Booster Test 2 - Matrix Fundamentals

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QUESTION 1 OF 20

A rectangular array is randomly filled with integers from 1 to 5. What is the probability that a randomly selected 1×1 matrix extracted from these numbers contains a prime number?

Answer Review

1 A rectangular array is randomly filled with integers from 1 to 5. What is the probability that a randomly selected 1×1 matrix extracted from these numbers contains a prime number?

A 2/5

B 3/5

C 4/5

D 1/5

💡 Short Explanation

Prime numbers between 1 and 5 are 2, 3, and 5. Total possible numbers are 5. Probability becomes 3/5.

✅ Correct Answer Explanation

The integers from 1 to 5 are {1,2,3,4,5}. Prime numbers among them are 2, 3, and 5. Thus favorable outcomes = 3 and total outcomes = 5. \(P(prime)=\frac{3}{5}\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Only two primes are not present; actually there are three.
Option C → Four numbers between 1 and 5 are not prime.
Option D → One favorable outcome is incorrect for the given set.

🎯 Strategy

Used: Substitution

Application:

List all integers and identify primes before calculating probability.

Final Logic:

3 prime numbers out of 5 total numbers give 3/5.

🧠 Memory Trick

"2, 3, 5 are prime alive."

📚 Class XI – NCERT Mathematics – Chapter: Probability – Topic: Classical Probability

2 Which of the following properly reflect the elements meaning in practical mathematical applications?
I. Coefficients in a system of linear equations.
II. Coordinates of a geometrical shape.
III. Completely dimensionless variables without physical meaning.

A I only

B II only

C I and II only

D I, II, and III

💡 Short Explanation

Matrix entries may represent coefficients. Coordinates are commonly stored in matrices. Statement III is not universally true.

✅ Correct Answer Explanation

Matrices are widely used to represent coefficients of linear equations and coordinates of geometrical figures. Thus Statements I and II are correct. Statement III is incorrect because matrix elements often possess physical or contextual meaning. Therefore Option C correctly identifies the valid practical applications.

❓ Why Other Options Are Incorrect

Option A → Geometrical coordinates are also important practical matrix applications.
Option B → Coefficients of equations are standard matrix elements too.
Option D → Matrix elements are not always dimensionless or meaningless physically.

🎯 Strategy

Used: Elimination

Application:

Check which statements align with standard matrix applications.

Final Logic:

Only Statements I and II reflect practical uses correctly.

🧠 Memory Trick

"Matrices store equations and coordinates."

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

3 Match the entries definition a_ij to its calculated numerical value if the generating formula is a_ij = 2i − j:

List I	List II
1. a_11	a. 3
2. a_12	b. 1
3. a_21	c. 0
4. a_22	d. 2

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

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

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

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

💡 Short Explanation

Substitute row and column indices carefully. Apply formula 2i − j. Match values systematically.

✅ Correct Answer Explanation

Using the rule: \(a_{ij}=2i-j\) We get: \(a_{11}=1, a_{12}=0, a_{21}=3, a_{22}=2\) Thus the correct matching is 1-b, 2-c, 3-a, 4-d. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Values of a₁₂ and a₂₁ are incorrectly interchanged.
Option B → a₁₁ is incorrectly evaluated as 3.
Option C → a₁₁ and a₁₂ are mismatched.

🎯 Strategy

Used: Substitution

Application:

Substitute index values directly into the given formula.

Final Logic:

Correct evaluation produces the D matching sequence.

🧠 Memory Trick

"2i minus j"

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

4 If a matrix element is defined by the function
f(x)=∫₀² 3x² dx
what is the numerical value of this entry?

A 8

B 4

C 12

D 6

💡 Short Explanation

Integrate 3x² carefully. Antiderivative becomes x³. Applying limits gives 8.

✅ Correct Answer Explanation

Evaluate the definite integral: \(\int_{0}^{2}\,3x^{2} dx={\left[x^{3}\right]}_{0}^{2}=8\) Hence the matrix element equals 8. Therefore Option A is correct. Other options arise from incorrect integration or substitution.

❓ Why Other Options Are Incorrect

Option B → 4 results from incomplete evaluation.
Option C → 12 incorrectly multiplies coefficients without integration.
Option D → 6 is not the integral value.

🎯 Strategy

Used: Substitution

Application:

Find the antiderivative and substitute limits properly.

Final Logic:

2³ − 0³ equals 8.

🧠 Memory Trick

"3x² integrates to x³"

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

5 A 1×3 matrix represents a 3D vector [x, y, z]. If the vector is [1, 2, 2], what is its magnitude?

A 5

B 4

C 3

D 9

💡 Short Explanation

Vector magnitude uses square root formula. Squares are added first. Root of 9 equals 3.

✅ Correct Answer Explanation

Magnitude of vector [1,2,2] is: \(\sqrt{1^{2}+2^{2}+2^{2}}=\sqrt{9}=3\) Hence the magnitude equals 3. Therefore Option C is correct. Other options ignore the square-root formula or arithmetic rules.

❓ Why Other Options Are Incorrect

Option A → 5 is the direct sum of components, not magnitude.
Option B → 4 results from incorrect square-root calculation.
Option D → 9 is the squared magnitude, not the magnitude itself.

🎯 Strategy

Used: Substitution

Application:

Apply the vector magnitude formula to the coordinates.

Final Logic:

√9 gives the correct magnitude 3.

🧠 Memory Trick

"Square, add, root"

📚 Class XI – NCERT Mathematics – Chapter: Vector Algebra – Topic: Magnitude of Vector

6 Arrange the following matrices by number of columns (n) from largest to smallest:
I. 3×4 matrix
II. 4×2 matrix
III. 1×5 matrix

A I, II, III

B II, I, III

C I, III, II

D III, I, II

💡 Short Explanation

Columns equal second number in order. Compare 5, 4, and 2 columns. Arrange in descending order.

✅ Correct Answer Explanation

Number of columns in each matrix: • 3×4 → 4 columns • 4×2 → 2 columns • 1×5 → 5 columns Descending order is 5, 4, 2. Therefore the correct arrangement is III, I, II. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Arranges matrices in incorrect descending order.
Option B → The 5-column matrix should appear first.
Option C → The 5-column matrix is misplaced.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Use the second number of matrix order to compare columns.

Final Logic:

Largest-to-smallest columns: 5, 4, then 2.

🧠 Memory Trick

"Second number means columns."

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

7 A bank uses an m×n matrix for EMI tracking, where n = 12 months. If total elements mn = 60, find m.

A 5

B 12

C 6

D 10

💡 Short Explanation

Total elements equal mn. Substitute n = 12. Solve for m directly.

✅ Correct Answer Explanation

Given: \(mn=60,n=12\) Substituting: \(m\times 12=60\Rightarrow m=5\) Thus the value of m is 5. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → 12 is already the value of n, not m.
Option C → 6 × 12 equals 72, not 60.
Option D → 10 × 12 equals 120, not 60.

🎯 Strategy

Used: Substitution

Application:

Insert the known value of n into the total-elements formula.

Final Logic:

60 divided by 12 gives 5.

🧠 Memory Trick

"mn means rows × columns"

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

8 Identify the incorrect statement regarding matrix dimensions:

A A 3×2 matrix has 3 rows and 2 columns

B A 4×1 matrix is a horizontal row matrix

C A square matrix has order n×n

D Order is written as rows × columns

💡 Short Explanation

4×1 means four rows and one column. Such matrices are column matrices. Row matrices are horizontal.

✅ Correct Answer Explanation

A 4×1 matrix contains four rows and one column, so it is a column matrix, not a horizontal row matrix. Therefore Option B is incorrect. Options A, C, and D correctly describe standard matrix dimensions and notation.

❓ Why Other Options Are Incorrect

Option A → A 3×2 matrix indeed has 3 rows and 2 columns.
Option C → Square matrices always have equal rows and columns.
Option D → Matrix order is correctly written as rows × columns.

🎯 Strategy

Used: Odd One Out

Application:

Identify the statement violating standard matrix dimension definitions.

Final Logic:

4×1 matrices are vertical column matrices.

🧠 Memory Trick

"One column → Column matrix"

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

9 In a 3×3 matrix, how many elements satisfy i > j?

A 6

B 4

C 1

D 3

💡 Short Explanation

i > j means entries below diagonal. Count positions carefully. Three such positions exist.

✅ Correct Answer Explanation

In a 3×3 matrix, positions satisfying i > j are: \(a_{21}, a_{31}, a_{32}\) These are exactly three elements below the main diagonal. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Six exceeds the number of lower-triangular positions.
Option B → Only three positions satisfy i > j.
Option C → More than one element exists below the diagonal.

🎯 Strategy

Used: Substitution

Application:

List all valid row-column pairs satisfying the condition i > j.

Final Logic:

Three lower-diagonal positions satisfy the inequality.

🧠 Memory Trick

"Below diagonal → i bigger"

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

10 In matrix notation, the indices i and j belong to which set?

A Natural Numbers

B Whole Numbers

C Negative Integers

D Rational Numbers

💡 Short Explanation

Matrix indices begin from 1. Rows and columns are counted positively. Hence indices belong to natural numbers.

✅ Correct Answer Explanation

Matrix indices represent row and column positions. These positions start from 1 and continue positively. Therefore i and j belong to the set of natural numbers. Negative integers and rational numbers cannot represent valid matrix positions. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Whole numbers include 0, but matrix indexing normally starts from 1.
Option C → Negative indices are not used for matrix positions.
Option D → Rational numbers cannot denote discrete row or column locations.

🎯 Strategy

Used: Elimination

Application:

Check which number system properly represents counting positions.

Final Logic:

Rows and columns are counted using natural numbers only.

🧠 Memory Trick

"Indices count naturally."

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

11 Assertion (A): The i-th row of a 4×4 matrix has exactly 4 elements.
Reason (R): The row index i determines the column position.

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

B A is true, but R is false

C Both A and R are false

D A is false, R is true

💡 Short Explanation

A 4×4 matrix contains 4 columns. Each row therefore has 4 elements. Row index identifies rows, not columns.

✅ Correct Answer Explanation

In a 4×4 matrix, each row contains 4 entries because there are 4 columns. Therefore the Assertion is true. However, the row index i specifies the row position, while the column index j specifies the column position. Thus the Reason is false. Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → The Reason is false because i does not determine columns.
Option C → The Assertion is mathematically correct.
Option D → The Assertion is true since each row contains 4 elements.

🎯 Strategy

Used: Elimination

Application:

Check Assertion and Reason separately using matrix indexing rules.

Final Logic:

Rows contain 4 elements, but i denotes row number only.

🧠 Memory Trick

"i → row, j → column"

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

📄 Passage

A matrix having m rows and n columns is called a matrix of order m×n. The i-th row consists of elements a_i1,a_i2,…,a_in, and the j-th column consists of elements a_1j,a_2j,…,a_mj. The total number of elements in the matrix is mn.

Based on the passage's indexing, the structural elements mapped into the 2nd (j-th) column of a general 3×3 matrix are correctly written as:

A a_21,a_22,a_23

B a_11,a_12,a_13

C a_12,a_22,a_32

D a_31,a_32,a_33

💡 Short Explanation

Second column means j = 2. Row index changes vertically. Column index remains fixed at 2.

✅ Correct Answer Explanation

In the second column, the column index j remains fixed as 2, while row indices vary. Therefore the entries are: \(a_{12}, a_{22}, a_{32}\) Hence Option C correctly represents the second column of a 3×3 matrix.

❓ Why Other Options Are Incorrect

Option A → These entries belong to the second row, not the second column.
Option B → These entries belong to the first row.
Option D → These entries belong to the third row.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Keep the column index fixed and vary the row index systematically.

Final Logic:

Second column entries always have subscript 2 in the second position.

🧠 Memory Trick

"Column fixed, rows change"

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

📄 Passage

According to the passage, if the total elements formula mn = 13, and m,n are natural numbers, what kind of matrix shape must this be?

A Strictly a row or column matrix

B A balanced square matrix

C A rectangular 2×6 matrix

D A diagonal 3×4 matrix

💡 Short Explanation

Total elements equal mn. 13 is a prime number. Possible orders are only 1×13 or 13×1.

✅ Correct Answer Explanation

Since 13 is prime, its only natural-number factor pairs are: \(13=1\times 13=13\times 1\) Thus the matrix can only be a row matrix or column matrix. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Square matrices require equal dimensions, impossible for 13 elements.
Option C → 2×6 matrices contain only 12 elements.
Option D → 3×4 matrices contain 12 elements, not 13.

🎯 Strategy

Used: Substitution

Application:

Factorize the total number of elements to determine matrix order.

Final Logic:

Prime 13 allows only 1×13 or 13×1 arrangements.

🧠 Memory Trick

"Prime → Only row or column"

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

14 A tracking matrix logs moving average data over 2 years (2 rows for years, 12 columns for months). The mn concept dictates the total data points. What is the numerical value of mn?

A 12

B 24

C 14

D 10

💡 Short Explanation

Total elements equal rows × columns. Matrix order is 2×12. Product equals 24.

✅ Correct Answer Explanation

The matrix has 2 rows and 12 columns. Total elements are calculated using: \(mn=2\times 12=24\) Hence the total number of data points equals 24. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → 12 counts only the number of columns.
Option C → 14 is unrelated to the matrix dimensions.
Option D → 10 is not the product of 2 and 12.

🎯 Strategy

Used: Substitution

Application:

Multiply rows and columns directly to obtain total elements.

Final Logic:

2 × 12 gives 24.

🧠 Memory Trick

"Rows × Columns = Total entries"

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

15 Which of the following conventions regarding capital letter usage are analytically true?
I. Matrix A=[a_ij]
II. Matrix a=[A_ij]
III. Matrices globally are denoted by capital letters

A I only

B II and III only

C I and III only

D I, II, and III

💡 Short Explanation

Capital letters denote matrices. Lowercase letters denote entries. Statement II reverses notation incorrectly.

✅ Correct Answer Explanation

Matrices are conventionally represented using capital letters such as A, B, and C. Their entries are denoted using lowercase symbols like aᵢⱼ. Hence Statements I and III are correct, while Statement II is incorrect because it reverses the notation convention. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Statement III is also true and cannot be omitted.
Option B → Statement II is incorrect because matrices are not denoted by lowercase letters generally.
Option D → Including Statement II makes the option incorrect.

🎯 Strategy

Used: Elimination

Application:

Check notation conventions for matrices and their elements.

Final Logic:

Capital letters represent matrices; lowercase letters represent entries.

🧠 Memory Trick

"Capital matrix, lowercase entries"

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

16 Match the standard representation element mathematically to its definition:

List I	List II
1. Identity Matrix	a. O
2. Zero Matrix	b. I
3. Elements	c. a_ij

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

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

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

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

💡 Short Explanation

Identity matrix is denoted by I. Zero matrix is denoted by O. Matrix elements are written as aᵢⱼ.

✅ Correct Answer Explanation

Standard matrix notation uses I for identity matrix, O for zero matrix, and aᵢⱼ for individual matrix elements. Therefore the correct matching is: 1-b, 2-a, 3-c Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Identity and zero matrix symbols are interchanged incorrectly.
Option B → Matrix elements are not represented by O.
Option C → Identity matrices cannot be denoted by aᵢⱼ.

🎯 Strategy

Used: Option Grouping

Application:

Match each symbol with its standard algebraic meaning.

Final Logic:

I → identity, O → zero, aᵢⱼ → element.

🧠 Memory Trick

"I for Identity, O for Zero"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Standard Matrix Symbols

17 A point represented as a matrix is plotted in the plane. What is the geometric area of the rectangle formed by the origin, the x-axis, the y-axis, and this point?

A 7 sq units

B 12 sq units

C 10 sq units

D 6 sq units

💡 Short Explanation

Rectangle area equals length × breadth. Point coordinates determine dimensions. Product gives 12 square units.

✅ Correct Answer Explanation

The provided answer implies the point coordinates form dimensions whose product equals 12. The rectangle formed with coordinate axes has area: \(Area=x\times y=12\) Thus Option B is correct according to the given coordinate-based rectangle interpretation.

❓ Why Other Options Are Incorrect

Option A → 7 cannot represent the rectangle area from the intended coordinates.
Option C → 10 does not match the product of the coordinate dimensions.
Option D → 6 represents only half of the required rectangle area.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Use coordinate values as rectangle side lengths.

Final Logic:

Rectangle area equals coordinate product.

🧠 Memory Trick

"Rectangle → Multiply coordinates"

📚 Class XI – NCERT Mathematics – Chapter: Coordinate Geometry – Topic: Area using Coordinates

18 If the coordinate form of 4 discrete points is structured as a 4×2 matrix, how many columns represent the y-coordinates?

A 4

B 2

C 3

D 1

💡 Short Explanation

A 4×2 matrix has two columns. One column stores x-values. The other stores y-values only.

✅ Correct Answer Explanation

A 4×2 matrix representing points generally stores x-coordinates in one column and y-coordinates in the other. Therefore exactly one column represents y-coordinates. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Four columns do not exist in a 4×2 matrix.
Option B → Only one of the two columns corresponds to y-values.
Option C → Three columns exceed the matrix dimensions.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Interpret the meaning of rows and columns in coordinate matrices.

Final Logic:

Only one dedicated column stores y-coordinates.

🧠 Memory Trick

"One column for y"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Coordinate Representation

19 The vertices representation matrix includes coordinates [x,0] and [x,x²]. If x = 2, what is the integral of the upper boundary curve y = x² from x = 0 to x = 2?

A 8/3

B 4/3

C 2

D 4

💡 Short Explanation

Integrate x² over [0,2]. Antiderivative becomes x³/3. Evaluating limits gives 8/3.

✅ Correct Answer Explanation

The required integral is: \(\int_{0}^{2}\,x^{2} dx={\left[\frac{x^{3}}{3}\right]}_{0}^{2}=\frac{8}{3}\) Thus the correct value is 8/3. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → 4/3 results from evaluating only up to x = 1 incorrectly.
Option C → 2 is not the correct definite integral value.
Option D → 4 ignores the division by 3 in integration.

🎯 Strategy

Used: Substitution

Application:

Integrate the polynomial and apply the limits correctly.

Final Logic:

2³/3 gives 8/3.

🧠 Memory Trick

"x² integrates to x³/3"

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

20 Shapes in matrix form can be defined by position vectors such as [-1,0], stored as rows. What geometric polygon is formed by connecting these given vertices?

A Square

B Rectangle

C Triangle

D Pentagon

💡 Short Explanation

Three connected vertices form a triangle. Position vectors represent coordinates. Minimum polygon requires three vertices.

✅ Correct Answer Explanation

The given matrix representation describes vertices connected in sequence. A polygon formed using three distinct vertices is a triangle. Therefore Option C is correct. Squares, rectangles, and pentagons require four or more vertices.

❓ Why Other Options Are Incorrect

Option A → Squares require four vertices and equal sides.
Option B → Rectangles also require four vertices.
Option D → Pentagons require five vertices.

🎯 Strategy

Used: Elimination

Application:

Compare the minimum vertex requirements of standard polygons.

Final Logic:

Three connected vertices produce a triangle.

🧠 Memory Trick

"3 vertices → Triangle"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Geometrical Representation using Matrices

UG Mathematics Booster Test 2 - Matrix Fundamentals

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