Question 1

What are the eigenvalues of the matrix [2, 1, 1; 1, 4, 1; 1, 1, 2]?

Accepted Answer

1, 2, 5. The eigenvalues of a matrix are the solutions to its characteristic polynomial, which in this case yields the values 1, 2, and 5. These values indicate the scaling factors for the eigenvectors associated with the matrix.

Question 2

A circle with center at (x,y) = (0.5, 0) and radius = 0.5 intersects with another circle with center at (x,y) = (1,1) and radius = 1 at two points. One of the points of intersection (x,y) is:

Accepted Answer

(0.2,0.4). The point (0.2, 0.4) lies on both circles, satisfying their equations, which confirms it as a point of intersection. The other options either do not satisfy the equations of both circles or lie outside their respective radii.

Question 3

Suppose λ is an eigenvalue of matrix A and x is the corresponding eigenvector. Let x also be an eigenvector of the matrix B = A − 2I, where I is the identity matrix. Then, the eigenvalue of B corresponding to the eigenvector x is equal to

Accepted Answer

λ − 2. If x is an eigenvector of A corresponding to eigenvalue λ, then Ax = λx. For the matrix B = A - 2I, we have Bx = (A - 2I)x = Ax - 2Ix = λx - 2x = (λ - 2)x. This shows that x is also an eigenvector of B with eigenvalue λ - 2.

Question 4

Let A = [[1, 1], [1, 3], [−2, −3]] and b = [b1, b2, b3]. For Ax = b to be solvable, which one of the following options is the correct condition on b1, b2, and b3:

Accepted Answer

3b1 + b2 + 2b3 = 0. The correct option is based on the requirement for the system of equations represented by Ax = b to have a solution. This condition arises from the need for the vector b to lie in the column space of matrix A, which is determined by the linear combinations of its columns. The equation 3b1 + b2 + 2b3 = 0 ensures that b can be expressed as a linear combination of the columns of A.

Question 5

If the quadrantal bearing of a line is N 30° W, then the whole circle bearing of the line is

Accepted Answer

330°. Quadrantal bearing N 30 W is measured 30 degrees toward west from north, so the whole circle bearing = 360 - 30 = 330 degrees. The stored 300 deg is incorrect.

Question 6

The matrix [2, -4; 4, -2] has

Accepted Answer

complex eigenvalues and eigenvectors. The matrix has a negative determinant and a non-zero trace, which indicates that its eigenvalues are complex. Consequently, the eigenvectors associated with these complex eigenvalues are also complex.

Question 7

The rank of the following matrix is
[1, 1, 0, -2; 2, 0, 2, 2; 4, 1, 3, 1]

Accepted Answer

2. With R2->R2-2R1 = [0,-2,2,6] and R3->R3-4R1 = [0,-3,3,9], we find R3' = 1.5*R2', so the third row is linearly dependent. Only two independent rows remain, so the rank is 2.

Question 8

Euclidean norm (length) of the vector [4 -2 -6]^T is

Accepted Answer

√56. The Euclidean norm is calculated as the square root of the sum of the squares of the vector's components. For the vector [4, -2, -6], this results in √(4² + (-2)² + (-6)²) = √(16 + 4 + 36) = √56.

Question 9

Consider the following equations of straight lines:
Line L1: 2x - 3y = 5
Line L2: 3x + 2y = 8
Line L3: 4x - 6y = 5
Line L4: 6x - 9y = 6
Which one among the following is the correct statement?

Accepted Answer

L4 is perpendicular to L2 and L4 is parallel to L3. Line L4 is perpendicular to L2 because the product of their slopes equals -1, indicating they intersect at a right angle. Additionally, L4 is parallel to L3 since both lines have the same slope, confirming they never intersect.

Question 10

P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct?

Accepted Answer

If P and Q are invertible, then [PQ]⁻¹ = Q⁻¹P⁻¹.. The correct option states that the inverse of the product of two invertible matrices P and Q is equal to the product of their inverses in reverse order, which is a fundamental property of matrix multiplication and inverses. This means that if both matrices are invertible, the relationship holds true as stated.

Question 11

Let y be a non-zero vector of size 2022 × 1. Which of the following statement(s) is/are TRUE?

Accepted Answer

yy^T is a symmetric matrix.. The matrix yy^T is formed by the outer product of the vector y with itself, which inherently results in a symmetric matrix since (yy^T)^T = yy^T.

Question 12

If the entries in each column of a square matrix M add up to 1, an eigenvalue of M is

Accepted Answer

1. An eigenvalue of 1 indicates that the matrix M preserves the sum of the entries in each column when multiplied by a vector of ones, reflecting the property that the column sums equal 1.

Question 13

Consider the following linear system.

x + 2y − 3z = a
2x + 3y + 3z = b
5x + 9y − 6z = c

This system is consistent if a, b and c satisfy the equation

Accepted Answer

3a + b − c = 0. The third equation is a linear combination of the first two: 3*(R1) + 1*(R2) gives (5,9,-6) on the left, so consistency requires c = 3a+b, i.e. 3a+b-c=0 (option 1), not 7a-b-c=0.

Question 14

For the matrix [A] given below, the transpose is _______.

[A] = [ [2 3 4], [1 4 5], [4 3 2] ]

Accepted Answer

[ [2 1 4], [3 4 3], [4 5 2] ]. The transpose of a matrix is obtained by swapping its rows and columns. In this case, the first row [2, 3, 4] becomes the first column, the second row [1, 4, 5] becomes the second column, and the third row [4, 3, 2] becomes the third column, resulting in the correct option.

Question 15

Pick the CORRECT eigen value(s) of the matrix [A] from the following choices.

[A] = [6 8
4 2]

Accepted Answer

10. For A = [[6,8],[4,2]], trace = 8 and det = 12 - 32 = -20, so lambda^2 - 8 lambda - 20 = 0 gives lambda = 10 and lambda = -2. The valid eigenvalues are indices 0 (10) and 2 (-2); the stored value 4 (index 1) is not an eigenvalue.

Question 16

Let p and q be two propositions. Consider the following two formulae in propositional logic.
S1: (¬p ∧ (p ∨ q)) → q
S2: q → (¬p ∧ (p ∨ q))
Which of the following is correct?

Accepted Answer

S1 is true and S2 is false.. S1 is true because if ¬p is true, then p is false, making the expression (p ∨ q) depend solely on q, which leads to q being true. S2 is false because it asserts that if q is true, then ¬p must also be true, which is not necessarily the case.

Question 17

A rectangular paper sheet of dimensions 54 cm × 4 cm is taken. The two longer edges of the sheet are joined together to create a cylindrical tube. A cube whose surface area is equal to the area of the sheet is also taken. Then, the ratio of the volume of the cylindrical tube to the volume

Accepted Answer

1/π. The volume of the cylindrical tube is calculated using its height and the radius derived from the circumference formed by the longer edge of the sheet. The cube's volume is derived from its surface area, which equals the area of the sheet. The ratio of these two volumes simplifies to 1/π, making option A the correct choice.

Question 18

The product of all eigenvalues of the matrix [1 2 3; 4 5 6; 7 8 9] is

Accepted Answer

0. The product of all eigenvalues of a matrix is equal to its determinant. Since the determinant of the given matrix is 0, it indicates that at least one eigenvalue is 0, making the product of all eigenvalues also 0.

Question 19

g(.) is a function from A to B, f(.) is a function from B to C, and their composition defined as f(g(.)) is a mapping from A to C.

If f(.) and f(g(.)) are onto (surjective) functions, which ONE of the following is TRUE about the function g(.)?

Accepted Answer

g(.) is not required to be a one-to-one or onto function.. The correctness of the option lies in the fact that the surjectivity of the composition f(g(.)) does not impose any restrictions on g(.) being onto or one-to-one. As long as f(.) maps the outputs of g(.) to all elements in C, g(.) can have any mapping characteristics.

Question 20

A = {0, 1, 2, 3,...} is the set of non-negative integers. Let F be the set of functions from A to itself. For any two functions, f1, f2 ∈ F, we define (f1 ⊙ f2)(n) = f1(n) + f2(n) for every number n in A. Which of the following is/are CORRECT about the mathematical structure (F, ⊙)?

Accepted Answer

(F, ⊙) is an Abelian monoid.. (F, ⊙) is an Abelian monoid because it satisfies the properties of closure, associativity, and the existence of an identity element (the zero function), and the operation ⊙ is commutative, meaning the order of functions does not affect the result.

Question 21

How many different non-isomorphic Abelian groups of order 4 are there?

Accepted Answer

2. There are two non-isomorphic Abelian groups of order 4: the cyclic group of order 4, denoted as Z₄, and the direct product of two cyclic groups of order 2, denoted as Z₂ × Z₂. These groups have different structures, which makes them non-isomorphic.

Question 22

Consider the set of (column) vectors defined by X = {x ∈ R³ | x1 + x2 + x3 = 0, where x^T = [x1,x2,x3]^T}. Which of the following is TRUE?

Accepted Answer

(A) {[1,-1,0]^T,[1,0,-1]^T} is a basis for the subspace X.. The set of vectors {[1,-1,0]^T,[1,0,-1]^T} is linearly independent and spans the subspace defined by the equation x1 + x2 + x3 = 0, making it a valid basis for X.

Question 23

If P,Q,R are Boolean variables, then (P + Q̅)(P̅Q + P.R)(P̅R + Q̅) simplifies to

Accepted Answer

P.Q̅. The expression simplifies to P.Q̅ because the terms combine in such a way that only the conjunction of P and the negation of Q remains, effectively eliminating other variables and combinations.

Question 24

How many of the following matrices have an eigenvalue 1? [ [1 0],[0 0] ], [ [0 1],[0 0] ], [ [1 -1],[1 1] ] and [ [-1 0],[1 -1] ]

Accepted Answer

one. Eigenvalues: [[1,0],[0,0]] -> 1,0 (has 1); [[0,1],[0,0]] -> 0,0 (no); [[1,-1],[1,1]] -> 1+/-i (no); [[-1,0],[1,-1]] -> -1,-1 (no). Only one matrix has eigenvalue 1.

Question 25

Statement for Linked Answer Questions 78 and 79:
Let xₙ denote the number of binary strings of length n that contain no consecutive 0s.
Which of the following recurrences does xₙ satisfy?

Accepted Answer

xₙ = xₙ₋₁ + xₙ₋₂. The recurrence relation xₙ = xₙ₋₁ + xₙ₋₂ correctly accounts for the construction of binary strings of length n without consecutive 0s. A valid string can either end with a 1 (which allows any valid string of length n-1 before it) or end with a 0 (which must be preceded by a 1, thus allowing any valid string of length n-2 before it).

Question 26

What is the possible number of reflexive relations on a set of 5 elements?

Accepted Answer

2²⁰. A reflexive relation must contain all 5 diagonal pairs, fixing them. The remaining n^2 - n = 25 - 5 = 20 ordered pairs may each be present or absent freely, giving 2^20 reflexive relations (index 2), not 2^10.

Question 27

Consider the set S = {1, ω, ω²}, where ω and ω² are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

Accepted Answer

a group. The set S = {1, ω, ω²} with multiplication satisfies the group properties: it is closed under multiplication, has an identity element (1), and every element has an inverse (1, ω, and ω² are all roots of unity). Thus, it forms a group.

Question 28

Statement for Linked Answer Questions 52 and 53:
For the grammar below, a partial LL(1) parsing table is also presented along with the grammar. Entries that need to be filled are indicated as E1, E2, and E3. ε is the empty string, indicates end of input, and | separates alternate right han

Accepted Answer

(A) FIRST(A) = {a, b, ε} = FIRST(B)
FOLLOW(A) = {a, b}
FOLLOW(B) = {a, b, }. The correct option is right because both A and B derive from S, which can produce ε, a and b, leading to the same FIRST set of {a, b, ε}. Additionally, the FOLLOW sets correctly reflect the symbols that can appear immediately after A and B in the derivations, including the end of input symbol for B.

Question 29

Statement for Linked Answer Questions 54 and 55:
A computer has a 256 KByte, 4-way set associative, write back data cache with block size of 32 Bytes. The processor sends 32 bit addresses to the cache controller. Each cache tag directory entry contains, in addition to address tag, 2 valid

Accepted Answer

(B) 14. To determine the number of bits in the tag field, we first calculate the total number of cache lines, which is derived from the cache size and block size. With a 256 KByte cache and a block size of 32 Bytes, there are 8192 blocks. Since it's a 4-way set associative cache, we have 2048 sets. The address consists of 32 bits, and we need to account for the index (11 bits for 2048 sets) and the block offset (5 bits for 32 Bytes), leaving us with 14 bits for the tag.

Question 30

Which one of the following expressions does NOT represent exclusive OR of x and y?

Accepted Answer

xy̅ + x̅y̅. The expression xy̅ + x̅y̅ represents a logical operation that outputs true when both inputs are either true or false, which is the definition of equivalence, not exclusive OR. Exclusive OR requires that only one of the inputs is true.

Question 31

If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is _____.

Accepted Answer

2. By the dimension formula dim(V1)+dim(V2)=dim(V1+V2)+dim(V1 cap V2). Since dim(V1+V2)<=6, we get dim(V1 cap V2)>=4+4-6=2. The smallest possible dimension is 2, not 1.

Question 32

Let aₙ be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for aₙ?

Accepted Answer

aₙ = aₙ₋₁ + aₙ₋₂. The correct option is right because an n-bit string that does not contain two consecutive 1s can be formed by either appending a '0' to an (n-1)-bit valid string or appending '10' to an (n-2)-bit valid string, leading to the recurrence relation aₙ = aₙ₋₁ + aₙ₋₂.

Question 33

In a quadratic function, the value of the product of the roots (α, β) is 4. Find the value of
(αⁿ + βⁿ)/(α^−n + β^−n)

Accepted Answer

4ⁿ. Since a^-n + b^-n = (b^n + a^n)/(ab)^n, the ratio (a^n+b^n)/(a^-n+b^-n) equals (ab)^n. With ab=4 this is 4^n, which is index 1, not the stored 4^(n-1).

Question 34

The minimum number of colours that is sufficient to vertex-colour any planar graph is

Accepted Answer

4. According to the Four Color Theorem, any planar graph can be colored using no more than four colors such that no two adjacent vertices share the same color, making four the minimum sufficient number for vertex-coloring any planar graph.

Question 35

Consider the systems, each consisting of m linear equations in n variables.
I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution
Which one of the following is CORRECT?

Accepted Answer

Only III is true. Statement III is correct because when the number of equations equals the number of variables, it is possible to have a consistent system that has a solution. Statement I is false as having fewer equations than variables does not guarantee a solution, and Statement II is also false since it is possible for an overdetermined system to still have a solution.

Question 36

Consider solving the following system of simultaneous equations using LU decomposition.

x1 + x2 - 2x3 = 4
x1 + 3x2 - x3 = 7
2x1 + x2 - 5x3 = 7

where L and U are denoted as

L = (L11 0 0
 L21 L22 0
 L31 L32 L33), U = (U11 U12 U13
 0 U22 U23
 0 0 U33)

Which one of the following is the cor

Accepted Answer

L32 = -1/2, U33 = -1/2, x1 = 0. Doolittle LU gives U22=2, U23=1, L32=(1-2)/2=-1/2, and U33=-5+4+1/2=-1/2. Forward solving Ly=b gives y=(4,3,1/2); back substitution gives x3=-1, x2=2, x1=0. So L32=-1/2, U33=-1/2, x1=0, index 3, not the stored option.

Question 37

Which of the following is/are the eigenvector(s) for the matrix given below?

[-9, -6, -2, -4; -8, -6, -3, -1; 20, 15, 8, 5; 32, 21, 7, 12]

Accepted Answer

[-1; 1; 0; 1]. The vector (-1, 1, 0, 1) satisfies the eigenvalue equation for the given matrix, indicating that it is a valid eigenvector. This means that when the matrix is multiplied by this vector, the result is a scalar multiple of the vector itself, confirming its eigenvector status.

Question 38

A series of natural numbers F1, F2, F3, F4, F5, F6, F7,... obeys Fₙ₊₁ = Fₙ + Fₙ₋₁ for all integers n ≥ 2. If F6 = 37, and F7 = 60, then what is F1 ?

Accepted Answer

4. The sequence follows the Fibonacci-like relation where each term is the sum of the two preceding terms. Given F6 = 37 and F7 = 60, we can work backwards to find F5 = F7 - F6 = 60 - 37 = 23, then F4 = F5 - F6 = 23 - 37 = -14, and continuing this process leads us to F1 = 4.

Question 39

Let
A = [1 2 3 4
4 1 2 3
3 4 1 2
2 3 4 1]
and
B = [3 4 1 2
4 1 2 3
1 2 3 4
2 3 4 1].
Let det(A) and det(B) denote the determinants of the matrices A and B, respectively.
Which one of the options given below is TRUE?

Accepted Answer

det(B) = -det(A). The correct option is true because matrix B can be obtained from matrix A by a series of row operations that include swapping rows, which changes the sign of the determinant. Therefore, the relationship between their determinants is that det(B) equals the negative of det(A).

Question 40

Geetha has a conjecture about integers, which is of the form ∀x (P(x) ⇒ ∃y Q(x,y)), where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha’s conjecture?

Accepted Answer

∀x∀y Q(x,y). The option ∀x∀y Q(x,y) implies that for every integer x, the statement Q holds for all integers y, which directly satisfies the condition of Geetha's conjecture that for each x where P(x) is true, there exists at least one y such that Q(x,y) is also true.

Question 41

Let f: A → B be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation ~ on the set A as a1 ~ a2 if f(a1) = f(a2), where a1, a2 ∈ A. Let E = {[x]: x ∈ A} be the set of all the equivalence classes under ~. Define a new mapping F: E → B as F([x]) =

Accepted Answer

F is a bijective function.. F is a bijective function because it is both onto and one-to-one. Since f is onto, every element in B has a pre-image in A, ensuring that F covers all of B. Additionally, because the equivalence relation groups elements of A that map to the same element in B, F is injective, as different equivalence classes map to different elements in B.

Question 42

Let X be a set and 2^X denote the powerset of X. Define a binary operation Δ on 2^X as follows: AΔB = (A - B) ∪ (B - A). Let H = (2^X, Δ). Which of the following statements about H is/are correct?

Accepted Answer

For every A ∈ 2^X, the inverse of A is A.. The operation Δ defined as AΔB = (A - B) ∪ (B - A) corresponds to the symmetric difference, which is associative and has an identity element (the empty set). The inverse of any set A under this operation is A itself, since AΔA results in the empty set, confirming that each element is its own inverse.

Question 43

If P e^x = Q e^(-x) for all real values of x, which one of the following statements is true?

Accepted Answer

P = Q = 0. The equation P e^x = Q e^(-x) must hold for all real x, which implies that both sides must equal zero for the equality to be true at all points, leading to the conclusion that P and Q must both be zero.

Question 44

Let p1 and p2 denote two arbitrary prime numbers. Which one of the following statements is correct for all values of p1 and p2?

Accepted Answer

p1p2 is not a prime number.. The product of two prime numbers, p1 and p2, is always composite (not prime) because it has at least three distinct positive divisors: 1, p1, and p2. Therefore, option B is correct.

Question 45

If A = [[1, 2], [2, -1]], then which ONE of the following is A⁸ ?

Accepted Answer

[[625, 0], [0, 625]]. For A=[[1,2],[2,-1]], A^2 = [[5,0],[0,5]] = 5I. Therefore A^8 = (A^2)^4 = 5^4 I = 625 I = [[625,0],[0,625]], not 25I.

Question 46

Consider a system of linear equations PX = Q where P ∈ R^(3×3) and Q ∈ R^(3×1). Suppose P has an LU decomposition, P = LU, where L = [ [1, 0, 0], [l21, 1, 0], [l31, l32, 1] ] and U = [ [u11, u12, u13], [0, u22, u23], [0, 0, u33] ]. Which of the following statement(s) is/are TRUE?

Accepted Answer

The system PX = Q can be solved by first solving LY = Q and then UX = Y.. This statement is correct because the LU decomposition allows us to break down the original system into two simpler systems: first, we solve for Y using the lower triangular matrix L, and then we use the upper triangular matrix U to find the solution X from Y.

Question 47

It is given that X1, X2, ···, XM are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors X1, X2, ···, XM, −X1, −X2, ···, −XM is

Accepted Answer

M. The dimension of the vector space spanned by the vectors X1, X2, ···, XM and their negatives is determined by the number of linearly independent vectors. Since the original M vectors are orthogonal and non-zero, they are linearly independent, and the negatives do not introduce any new dimensions, resulting in a total dimension of M.

Question 48

All the four entries of the 2×2 matrix P = [[p11, p12],[p21, p22]] are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?

Accepted Answer

p11p22 − p12p21 = 0. The determinant of a matrix is equal to the product of its eigenvalues. Since one of the eigenvalues is zero, the determinant must also be zero, which means that the expression p11p22 - p12p21, representing the determinant of the matrix, equals zero.

Question 49

The system of linear equations 4x + 2y = 7 and 2x + y = 6 has

Accepted Answer

no solution. The two equations represent parallel lines, which means they never intersect; therefore, there is no point that satisfies both equations simultaneously, resulting in no solution.

Question 50

Consider the matrix P = [[0, 1], [-2, -3]]. The value of e^P is

Accepted Answer

[[2e⁻¹ - e⁻², e⁻¹ - e⁻²], [-2e⁻¹ + 2e⁻², -e⁻¹ + 2e⁻²]]. Eigenvalues of P are -1 and -2. Writing e^P=c0 I+c1 P with e^-1=c0-c1 and e^-2=c0-2c1 gives c1=e^-1-e^-2, c0=2e^-1-e^-2. Then e^P=[[c0,c1],[-2c1,c0-3c1]]=[[2e^-1-e^-2, e^-1-e^-2],[-2e^-1+2e^-2, -e^-1+2e^-2]], which is option 3.

GATE Engineering Mathematics: Linear Algebra questions with solutions

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