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Z = 30x1 + 30x2, subject to x1 ≥ 0, x2 ≥ 0, x1 + 3x2 ≤ 6, 4x1 + 8x2 ≥ 16, x1 + x2 ≤ 4. The minimum value of Z occurs at
  • a)
    (3, 0)
  • b)
    (2, 1)
  • c)
    (0, 2)
  • d)
    (4, 0)
Correct answer is option 'C'. Can you explain this answer?

Tanuja Kapoor answered
Corner Points are (0, 2), (2, 1), (4, 0) and (3, 1).
The objective function for (x, y) is 30x1 + 30x2.
After putting the corner points in the objective function, we get the minimum value of Z.
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If matrix A = [2 3 5], then the value of A.A' is:
  • a)
    38
  • b)
    26
  • c)
    39
  • d)
    28
Correct answer is option 'A'. Can you explain this answer?

Sreemoyee Choudhury answered
Understanding Matrix Multiplication
Matrix multiplication involves taking the dot product of rows and columns. In this case, we have a 1x3 matrix A = [2, 3, 5]. To find A.A, we will multiply matrix A by itself.
Calculating A.A
1. Matrix A:
A = [2, 3, 5]
2. Transpose of A:
A^T =
[
2
3
5
]
3. Multiplication:
A.A = A * A^T = [2, 3, 5] * [2, 3, 5]
4. Dot Product Calculation:
The product is calculated as follows:
- For the first element: 2*2 + 3*3 + 5*5
- Calculation:
- 2*2 = 4
- 3*3 = 9
- 5*5 = 25
5. Summing the Products:
- 4 + 9 + 25 = 38
Conclusion
Thus, the value of A.A is 38, which corresponds to option 'A'.
Final Verification
- Steps Recap:
- Identify the elements of matrix A.
- Multiply A with its transpose.
- Sum the results of the products to get the final value.
By following these steps, we confirm that the correct answer is indeed 38.

Consider the function 3x4 +20x3 −36x2 +44 in the interval [−5,10] , function is maximum at
  • a)
    1
  • b)
    0
  • c)
    2
  • d)
    3
Correct answer is option 'B'. Can you explain this answer?

Vaishnavi Dasgupta answered
The function you provided, 3x^4 - 20x^3, is a polynomial function of degree 4. It is a fourth-degree polynomial function.

Polynomial functions are functions that are expressed as the sum of terms, where each term consists of a coefficient multiplied by a variable raised to a non-negative integer power. In this case, the function has two terms: 3x^4 and -20x^3.

The highest power of x in the function is 4, so the degree of the polynomial is 4. The degree of a polynomial is the highest power of the variable in the function.

Polynomial functions of degree 4 are known as quartic functions. They can have various shapes and characteristics depending on the coefficients of the terms. The specific shape and behavior of the function can be determined by analyzing its graph or by using calculus techniques such as finding its critical points, intervals of increase/decrease, and concavity.

Without further information about the context or the purpose of the function, it is not possible to provide more specific details about its properties.

Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R is
  • a)
    Non commutative relation
  • b)
    A universal relation
  • c)
    An equivalence relation
  • d)
    An empty relation
Correct answer is option 'C'. Can you explain this answer?

Jatin Dasgupta answered
Equivalence Relation Explanation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Let's see how relation R in set T satisfies these properties:

Reflexive:
- For any triangle T1, T1 is congruent to itself (by definition of congruence).
- Therefore, R contains pairs like (T1, T1) for all T1 in T, making it reflexive.

Symmetric:
- If triangle T1 is congruent to triangle T2, then triangle T2 is also congruent to triangle T1 (by the properties of congruence).
- This means that if (T1, T2) is in R, then (T2, T1) must also be in R, satisfying symmetry.

Transitive:
- If triangle T1 is congruent to triangle T2, and triangle T2 is congruent to triangle T3, then triangle T1 is congruent to triangle T3 (by the transitive property of congruence).
- This implies that if (T1, T2) and (T2, T3) are in R, then (T1, T3) must also be in R, meeting the transitive criterion.
Since relation R in set T satisfies the properties of reflexivity, symmetry, and transitivity, it is an equivalence relation.

What is the domain of the cot–1 x ?
  • a)
    [∞,-∞]
  • b)
    (-∞, ∞)
  • c)
    (–∞, –1] ∪ [1, ∞)
  • d)
    [–1,1]
Correct answer is option 'B'. Can you explain this answer?

Vivek Rana answered
The cot function is periodic so to calculate its inverse function we need to make the function bijective. For that we have to consider an interval in which all values of the function exist and do not repeat. Now for the inverse of a function the domain becomes range and the range becomes domain. Thus the range of cot function, that is, (-∞, ∞) becomes the domain of inverse function.

The function y = 5x2 – 32x has a local minimum in the interval (0,10).
  • a)
    x = 1
  • b)
    x = 2
  • c)
    x = 3.2
  • d)
    No local minimum
Correct answer is option 'C'. Can you explain this answer?

Vaishnavi Dasgupta answered
The function y = 5x² represents a quadratic function. The variable x represents the input or independent variable, and the variable y represents the output or dependent variable. This function describes a parabola that opens upward and has a vertical stretch factor of 5. The exponent of 2 indicates that the variable x is squared, meaning that the function is quadratic.

If a matrix has 4 rows and 3 columns then how many elements will be there in this matrix?
  • a)
    4
  • b)
    3
  • c)
    1
  • d)
    12
Correct answer is option 'D'. Can you explain this answer?

Malavika Shah answered
To find the number of elements in a matrix, we multiply the number of rows by the number of columns.

Given that the matrix has 4 rows and 3 columns, we can calculate the number of elements as follows:

Number of elements = Number of rows × Number of columns

Number of elements = 4 × 3

Number of elements = 12

Therefore, the matrix will have 12 elements.

Explanation:
The elements of a matrix are the individual values that make up the matrix. In this case, the matrix has 4 rows and 3 columns, which means it will have 4 × 3 = 12 elements.

Each element in the matrix is identified by its position in the matrix, which is given by the row number and the column number. For example, the element in the first row and second column is referred to as A[1][2].

In general, the elements of a matrix are denoted as A[i][j], where i represents the row number and j represents the column number. In this case, the matrix has 4 rows, so i can take values from 1 to 4. Similarly, the matrix has 3 columns, so j can take values from 1 to 3.

Therefore, the matrix will have a total of 4 × 3 = 12 elements, each identified by its unique row and column number.

For a square matrix A = [aij] the quantity calculated for any element aij in A as the determinant of thesquare sub-matrix of order (n – 1) obtained by leaving the ith row and jth column of A is known as:
  • a)
    Cofactor
  • b)
    Minor
  • c)
    Coefficient
  • d)
    Elements
Correct answer is option 'B'. Can you explain this answer?

Roshni Desai answered
Understanding Minors in Matrices
In the context of a square matrix A = [aij], the term "minor" is crucial for various calculations in linear algebra, particularly in finding determinants and inverses.
What is a Minor?
- A minor of an element aij in matrix A is defined as the determinant of the square sub-matrix formed by deleting the ith row and jth column from A.
- This sub-matrix has an order of (n - 1), where n is the order of the original matrix A.
Significance of Minors
- Minors play a vital role in calculating the determinant of larger matrices.
- The concept of minors is foundational in the computation of cofactors and adjugates, which are used in finding the matrix inverse.
Example for Clarity
- Consider a 3x3 matrix A:
| a11 a12 a13 |
| a21 a22 a23 |
| a31 a32 a33 |
- To find the minor M11 of the element a11, you would take the determinant of the matrix formed by removing the first row and the first column:
| a22 a23 |
| a32 a33 |
- The determinant of this 2x2 matrix is the minor M11.
Conclusion
Thus, the correct answer to the question is option 'B' - Minor, as it refers specifically to the determinant of the sub-matrix formed by removing the respective row and column associated with an element in the matrix. Understanding minors is essential for mastering the concepts of determinants and matrix theory in linear algebra.

Let R be the relation in the set {5, 6, 7, 8} given by R = {(5, 6), (6, 6), (5, 5), (8, 8), (5, 7), (7, 7), (7, 6)}.
Choose the correct answer:
  • a)
    R is reflexive and symmetric but not transitive
  • b)
    R is reflexive and transitive but not symmetric
  • c)
    R is symmetric and transitive but not reflexive
  • d)
    R is an equivalence relation
Correct answer is option 'B'. Can you explain this answer?

Jyoti Sengupta answered
Let R be the relation in the set {1, 2,3, 4} is given by:
R = {(5,6), (6,6), (5,5), (8,8), (5,7), (7,7), (7,6)}
(a) (5,5), (6,6), (7,7), (8,8) ∈ R Therefore, R is reflexive.
(b) (5,6) ∈ R but (6,5) ∈ R. Therefore, R is not symmetric.
(c) If (5, 7) ∈ R and (7, 6) ∈ R then (5, 6) ∈ R. Therefore, R is transitive.

If A is a square matrix of order 3, such that A(adj A) = 10I, then |adj A| is equal to
  • a)
    1
  • b)
    10
  • c)
    100
  • d)
    101
Correct answer is option 'C'. Can you explain this answer?

Ameya Pillai answered
Given: A is a square matrix of order 3, such that A(adj A) = 10I

To find: |adj A|

Solution:

1. Using the property of adjoint of a matrix

We know that, adj(A) = (Cofactor of A)T

where Cofactor of A is the matrix obtained by taking the determinant of each minor of A and multiplying it by (-1)^(i+j), where i and j are the row and column indices of the element.

So, A(adj A) = A((Cofactor of A)T) = (ACofactor of A)T

2. Using the given condition

We are given that A(adj A) = 10I

Substituting this in the above equation, we get:

(ACofactor of A)T = 10I

Taking determinant on both sides, we get:

|ACofactor of A|T = 10^3

|ACofactor of A| = 10^3 (since determinant of a matrix is equal to the determinant of its transpose)

3. Using the property of determinant

We know that, |AB| = |A||B|

Substituting A = adj(A), we get:

|adj(A)Cofactor of A| = |adj(A)||Cofactor of A|

Since adj(A)Cofactor of A = |A|I (where I is the identity matrix), we get:

|A||Cofactor of A| = |adj(A)||Cofactor of A|

|A| = |adj(A)|

Substituting this in the previous equation, we get:

|adj(A)| |Cofactor of A| = 10^3

|adj(A)| = (10^3)/|Cofactor of A|

4. Finding the value of |Cofactor of A|

Since A is a square matrix of order 3, its adjoint matrix adj(A) is of order 3. Therefore, the Cofactor of A will be a matrix of order 3 as well.

Using the formula for finding the Cofactor of a matrix, we get:

Cofactor of A = (−1)^{i+j} M_{ij}

where M_{ij} is the determinant of the matrix obtained by deleting the i-th row and j-th column of A.

So, we need to find the determinant of 9 matrices (3x3) to find the Cofactor of A. However, we can simplify this process by using the property of symmetry of the Cofactor matrix.

We know that the Cofactor matrix is symmetric, i.e., Cofactor of A = (Cofactor of A)T

Therefore, we can find the determinant of only 4 matrices and use them to find the determinant of the remaining 5 matrices.

The 4 matrices are:

M_{11} = det\begin{pmatrix}a_{22} & a_{23}\\a_{32} & a_{33}\end{pmatrix}

M_{22} = det\begin{pmatrix}a_{11} & a_{13}\\a_{31} & a_{33}\end{pmatrix}

M_{33} = det\begin{pmatrix}a_{11} & a_{12}\\a_{21} &

Choose the correct option for the given matrix.
  • a)
    Lower triangular matrix
  • b)
    Upper triangular matrix
  • c)
    Diagonal matrix
  • d)
    Unit matrix
Correct answer is option 'A'. Can you explain this answer?

Riya Banerjee answered
A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal is zero.
Hence, given matrix is lower triangular matrix.

What is the principal value branch of cot–1x ?
  • a)
    (–1, 1)
  • b)
    [–1, 1]
  • c)
    (0, π)
  • d)
    [0, π]
Correct answer is option 'C'. Can you explain this answer?

Suresh Iyer answered
The cotangent function is periodic so to calculate its inverse function we need to make the function bijective. For that we have to consider an interval in which all values of the function exist and do not repeat. For cotangent function this interval is considered as (0, π).
Thus when we take the inverse of the function the domain becomes range and the range becomes domain. Hence the principal value branch is the range of cot–1 x that is (0, π).

if  then A2−5A + 7I2 is equal to :
  • a)
    0
  • b)
    1
  • c)
    -1
  • d)
    2
Correct answer is option 'A'. Can you explain this answer?

Neha Sharma answered
Since, all the elements of matrix are zero. So, given matrix is null/zero matrix.

The matrix 
  • a)
    identity matrix
  • b)
    symmetric matrix
  • c)
    skew-symmetric matrix
  • d)
    None of these
Correct answer is option 'B'. Can you explain this answer?

Suresh Iyer answered

So, the given matrix is a symmetric matrix. [Since, in a square matrix A, if A’ = A, then A is called symmetric matrix.]

The principal value of 
  • a)
    3π/4
  • b)
    π/4
  • c)
    π/2
  • d)
    π/6
Correct answer is option 'D'. Can you explain this answer?

Hansa Sharma answered
The principal value of  means that we need to find an angle in the principal branch of the function where the cosine function is equal to √3/2 . Hence the required value
is π/6

The function f(x) = e|x| is
  • a)
    continuous everywhere but not differentiable at x = 0
  • b)
    continuous and differentiable everywhere
  • c)
    not continuous at x = 0
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?

Vivek Rana answered
Given that,
f (x) = e|x|
The functions ex and |x| are continuous functions for all real value of x. Since ex is differentiable everywhere but |x| is non-differentiable at x = 0.
Thus, the given functions f(x) = e|x| is continuous everywhere but not differentiable at x = 0.

Evaluate the determinant of the matrix 
  • a)
    sin θ
  • b)
    cos θ
  • c)
    1
  • d)
    0
Correct answer is option 'C'. Can you explain this answer?

Shalini Patel answered
The determinant of a square matrix of order 2 is given by the difference of the product of diagonal elements and the product of the offdiagonal elements.

if  then the value of x is
  • a)
    3
  • b)
    0
  • c)
    – 1
  • d)
    1
Correct answer is option 'C'. Can you explain this answer?

Rajesh Gupta answered

On expanding along R1
2(x – 9x) – 3(x – 4x)+ 2(9x – 4x) + 3 = 0
2(–8x) – 3(–3x) + 2(5x) + 3 = 0
– 16x + 9x + 10x + 3 = 0
3x + 3 = 0
3x = –3
x = - 3/3
x = –1

All the trigonometric functions have inverse functions irrespective of the domain.
  • a)
    True
  • b)
    False
  • c)
    True but for only sine, cos and tan
  • d)
    None of the above
Correct answer is option 'B'. Can you explain this answer?

Suresh Iyer answered
For the inverse of a function to exist the function should be bijective which none of the trigonometric function is as they are periodic functions.

Which of the following functions is decreasing on 
  • a)
    sin 2x
  • b)
    tan x
  • c)
    cos x
  • d)
    cos 3x
Correct answer is option 'C'. Can you explain this answer?

Jyoti Sengupta answered
In the given interval 
f(x) = cos x
On differentiating with respect to x, we get f´(x) = – sin x
which gives

Hence, f(x) = cosx is decreasing in 

Which of the given values of x and y make the following pair of matrices equal
  • a)
    x=-1/3 , y = 7
  • b)
    Not possible to find
  • c)
    y = 7, x = -2/3
  • d)
    x = -1/3, y = -2/3
Correct answer is option 'B'. Can you explain this answer?

Gaurav Kumar answered
It is given that

Equating the corresponding elements, we get
3x + 7 = 0

We find that on comparing the corresponding elements of the two matrices, we get two different values of x, which is not possible.
Hence, it is not possible to find the values of x and y for which the given matrices are equal.

If A and B are symmetric matrices of same order, then AB – BA is a:
  • a)
    Skew-symmetric matrix
  • b)
    Symmetric matrix
  • c)
    Zero matrix
  • d)
    Identity matrix
Correct answer is option 'A'. Can you explain this answer?

Shalini Patel answered
A and B are symmetric matrices.
⇒ A = A’ and B = B’
Now, (AB – BA)’ = (AB)’ – (BA)’ ...(i)
⇒ (AB – BA)’=B’A’ – A’B’ [By reversal law]
⇒ (AB – BA)’ = BA – AB [From Eq. (i)]
⇒ (AB – BA)’ = –(AB – BA)
⇒ (AB – BA) is a skew-symmetric matrix.

If A = {1, 2}, B = {3,4, 5} and f = {(1, 3), (2, 5)} is a function from A to B, then f(x) is
  • a)
    onto
  • b)
    bijective
  • c)
    one-one
  • d)
    None of these
Correct answer is option 'C'. Can you explain this answer?

Krishna Iyer answered
Given, A = {1, 2}, B = {3,4, 5} and f : A → B is defined as f = {(1, 3), (2, 5)} i.e., f(1) = 3, f(2) = 5.
We can see that the images of distinct elements of A under f are distinct. So, f is one-one.

The matrix 
  • a)
    diagonal matrix
  • b)
    symmetric matrix
  • c)
    skew symmetric matrix
  • d)
    scalar matrix
Correct answer is option 'C'. Can you explain this answer?

Shalini Patel answered
We know that, in a square matrix, if bij = 0 when i ≠ j then it is said to be a diagonal matrix. Here, b12, b13…. ≠ 0
so the given matrix is not a diagonal matrix.
Now,

So, the given matrix is a skew-symmetric matrix, since we know that in a square matrix B, if B’ = − B, then it is called skew-symmetric matrix.

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