1.
A complex number has:
Correct Answer
C. Both
Explanation
A complex number has both a real part and an imaginary part. The real part represents the horizontal component of the number, while the imaginary part represents the vertical component. Together, these two parts form a complex number in the form a + bi, where a is the real part and bi is the imaginary part. Therefore, the correct answer is "Both."
2.
Graphical representation of the complex numbers is called:
Correct Answer
A. Argands diagram
Explanation
An Argand diagram is a graphical representation of complex numbers. It consists of a coordinate plane where the real part of a complex number is represented on the x-axis and the imaginary part is represented on the y-axis. The diagram helps visualize complex numbers as points in a plane, making it easier to understand their properties and relationships. Polar form and non-polar form are not related to the graphical representation of complex numbers, so they are not the correct answers.
3.
Sin2x is:
Correct Answer
A. 2sinxcosx
Explanation
The correct answer is 2sinxcosx because the double angle identity for sine states that sin2x = 2sinxcosx. This means that the sine of twice an angle is equal to twice the sine of the angle multiplied by the cosine of the angle.
4.
Derivative of cos(x)
Correct Answer
A. -sin(x)
Explanation
The derivative of a cosine function is equal to the negative sine of the same angle. Therefore, the correct answer is -sin(x).
5.
Derivative of ln(x)
Correct Answer
A. 1/x
Explanation
The derivative of ln(x) is 1/x. This can be derived using the chain rule of differentiation. The derivative of ln(x) can be written as d/dx(ln(x)). Applying the chain rule, we can rewrite this as 1/x * d/dx(x). The derivative of x with respect to x is 1. Therefore, the derivative of ln(x) is 1/x.
6.
A derivative of tan(x)
Correct Answer
A. Sec^{2}(x)
Explanation
The derivative of tan(x) is sec^2(x). This is because the derivative of tan(x) can be found using the quotient rule, which states that the derivative of f(x)/g(x) is (g(x)f'(x) - f(x)g'(x))/[g(x)]^2. In this case, f(x) = sin(x) and g(x) = cos(x), so the derivative of tan(x) is (cos(x)*cos(x) - sin(x)*(-sin(x)))/[cos(x)]^2 = (cos^2(x) + sin^2(x))/[cos(x)]^2 = 1/[cos^2(x)] = sec^2(x). Therefore, the correct answer is sec^2(x).
7.
A derivative of e^{x }is
Correct Answer
C. E^{x}
Explanation
The correct answer is "ex" because a derivative of the function "ex" (which represents the exponential function) with respect to the variable x is simply "ex" itself.
8.
A derivative of 2x^{3} is
Correct Answer
B. 6x^{2}
Explanation
The correct answer is 6x2. To find the derivative of 2x3, we use the power rule of differentiation. The power rule states that when differentiating a term with a variable raised to a power, we bring down the power as a coefficient and decrease the power by 1. Applying this rule to 2x3, we bring down the power of 3 as a coefficient, resulting in 6x3. Then, we decrease the power by 1, giving us 6x2. Therefore, the derivative of 2x3 is 6x2.
9.
A derivative of cos x is
Correct Answer
A. -sin x
Explanation
The correct answer is "-sin x" because when we take the derivative of the cosine function, the result is the negative sine function. This can be derived using the chain rule and the derivative of the sine function. Therefore, the derivative of cos x is -sin x.
10.
A derivative of sec x is
Correct Answer
A. -sec x tan x
Explanation
The correct answer is -sec x tan x. The derivative of sec x can be found using the quotient rule. Applying the quotient rule, we differentiate the numerator (which is 1) and get 0, and then differentiate the denominator (which is cos x) and get -sin x. Dividing these results and simplifying, we get -sec x tan x.
11.
A matrix is said to be _______ if the number of rows is not equal to the number of columns of a matrix.
Correct Answer
A. Rectangular
Explanation
A matrix is said to be rectangular if the number of rows is not equal to the number of columns of a matrix. In a rectangular matrix, the number of rows and columns are different, meaning that the matrix does not have an equal number of rows and columns. This is in contrast to a square matrix, where the number of rows is equal to the number of columns. A diagonal matrix refers to a matrix where all the non-diagonal elements are zero. "None of these" is not the correct answer as it does not accurately describe a matrix with unequal rows and columns.
12.
A square matrix A is said to be ______ if |A| is equal to 0.
Correct Answer
A. Singular matrix
Explanation
A square matrix A is said to be singular if its determinant (|A|) is equal to 0. This means that the matrix does not have an inverse and its columns are linearly dependent. In other words, the matrix cannot be inverted and its rows or columns can be expressed as a linear combination of the other rows or columns. Therefore, if |A| is equal to 0, the matrix A is a singular matrix.
13.
A square matrix A is said to be ______ if |A| is not equal to 0.
Correct Answer
A. Non singular matrix
Explanation
A square matrix A is said to be non-singular if its determinant |A| is not equal to 0. This means that the matrix A has an inverse, which allows for the solution of linear equations involving A. In other words, if a square matrix A is non-singular, it is invertible and has a unique solution.
14.
A square matrix A is equal to [a_{ij}] is said to be ______ if [a_{ij}] = [a_{ji}]
Correct Answer
A. Symmetric
Explanation
A square matrix A is said to be symmetric if the elements of the matrix are equal to their corresponding elements in the transpose of the matrix. In other words, if A = [aij], then A is symmetric if aij = aji for all i and j. This means that the matrix is symmetric along its main diagonal.
15.
State the reflexive property.
Correct Answer
A. A = A
Explanation
The reflexive property states that any element or object is equal to itself. In this case, the statement "A = A" represents the reflexive property because it asserts that A is equal to itself.
16.
AA^{-1 }= ?
Correct Answer
A. I
Explanation
The given answer "I" is correct because when we subtract 1 from AA, it becomes I. In Roman numerals, A represents 1, so when we subtract 1 from A, it becomes I.
17.
A matrix is said to be ________ if all the non-diagonal elements are 0.
Correct Answer
A. Diagonal
Explanation
A matrix is said to be diagonal if all the non-diagonal elements are 0. In other words, a diagonal matrix is a square matrix where all the elements outside the main diagonal (the diagonal from the top left to the bottom right) are 0. This means that the only non-zero elements in a diagonal matrix are on the main diagonal.
18.
A derivative of cot x
Correct Answer
A. -cosec^{2} x
Explanation
The given expression is a derivative of cot x. The derivative of cot x can be found by using the quotient rule of differentiation. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) is given by (g'(x)h(x) - g(x)h'(x))/[h(x)]^2. In this case, g(x) = 1 and h(x) = tan x. Taking the derivatives of g(x) and h(x), we get g'(x) = 0 and h'(x) = sec^2 x. Plugging these values into the quotient rule formula, we get (-0 * tan x - 1 * sec^2 x)/[tan x]^2, which simplifies to -sec^2 x/[tan x]^2. Since sec^2 x = 1/[cos x]^2 and [tan x]^2 = [sin x]^2/[cos x]^2, we can further simplify the expression to -cosec^2 x. Therefore, the correct answer is -cosec^2 x.
19.
∫ sinx dx
Correct Answer
A. - cos x
Explanation
The integral of sin(x) with respect to x is equal to -cos(x). This is a well-known result in calculus. The antiderivative of sin(x) is -cos(x), meaning that when we take the derivative of -cos(x) with respect to x, we get sin(x). Therefore, the correct answer is -cos(x).
20.
∫ cosx dx
Correct Answer
A. Sin x
Explanation
The integral of cos(x) with respect to x is sin(x). This is a well-known result in calculus. The derivative of sin(x) is cos(x), so it follows that the integral of cos(x) is sin(x) plus a constant. Therefore, the correct answer is sin(x).
21.
∫ cosec^{2}x dx
Correct Answer
A. -cotx
Explanation
The integral of cosec^2x is equal to -cotx + C, where C is the constant of integration. This can be derived using the identity cosec^2x = 1 + cot^2x and integrating both sides. Therefore, the correct answer is -cotx.
22.
Sin^{2}x + cos^{2}x = ?
Correct Answer
A. 1
Explanation
The given expression is the sum of the squares of sine and cosine of 2x. According to the Pythagorean identity, sin^2(x) + cos^2(x) = 1. Therefore, sin^2(2x) + cos^2(2x) = 1. Hence, the correct answer is 1.
23.
An ___________ set of mn numbers arranged in a rectangular array of m rows and n columns and enclosed by a pair of brackets is called a matrix.
Correct Answer
A. Ordered
Explanation
A matrix is an ordered set of numbers arranged in a rectangular array of rows and columns. The numbers are enclosed by a pair of brackets. Therefore, the correct answer is "ordered" because the numbers in a matrix are arranged in a specific order.
24.
Which of these is not a matrix type?
Correct Answer
A. Rounded matrix
Explanation
A rounded matrix is not a recognized type of matrix. Matrices can be classified into different types based on their properties, such as square matrices, identity matrices, diagonal matrices, etc. However, a rounded matrix is not a standard classification. Therefore, the correct answer is "Rounded matrix."
25.
The determinant of a matrix is mentioned as?
Correct Answer
A. Mod
Explanation
The determinant of a matrix is mentioned as "Mod" because in mathematics, the determinant of a matrix is often denoted with the symbol "mod". The determinant is a scalar value that can be calculated for a square matrix and provides important information about the matrix, such as whether it is invertible or singular. The "mod" notation is commonly used to represent the determinant in mathematical equations and formulas.
26.
A matrix obtained by interchanging the rows and columns of a matrix A is called as?
Correct Answer
A. Transpose
Explanation
The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of matrix A.
27.
In a determinant, a ____________ of an element is a determinant by omitting rows and column in which that element is present.
Correct Answer
A. Minor
Explanation
In a determinant, a minor of an element is a determinant obtained by omitting the rows and columns in which that element is present. The minor is used to calculate the cofactor of an element, which is then used in various operations involving determinants. The cofactor matrix is an important tool in finding the inverse of a matrix and solving systems of linear equations.
28.
Which of these is a type of elementary transformation?
Correct Answer
C. Both
Explanation
Both row and column transformations are types of elementary transformations. Elementary transformations are operations performed on matrices that do not change the linear relationships between the rows or columns of the matrix. These transformations include multiplying a row or column by a non-zero scalar, adding a multiple of one row or column to another row or column, and interchanging two rows or columns. Both row and column transformations can be used to perform these elementary operations on a matrix.
29.
The number of non zero rows determines the __________ of a matrix.
Correct Answer
A. Rank
Explanation
The number of non-zero rows in a matrix determines its rank. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it represents the dimension of the vector space spanned by the rows or columns of the matrix. Therefore, the correct answer is "Rank".
30.
Standard form of complex number is z=x+iy.
Correct Answer
A. True
Explanation
The standard form of a complex number is indeed z=x+iy, where x represents the real part and iy represents the imaginary part. This form allows us to easily represent and perform operations on complex numbers. Therefore, the given answer, "True," is correct.
31.
A square matrix is said to be __________ if all the diagonal elements are unity and non-diagonal elements are zero.
Correct Answer
A. Unit
Explanation
A square matrix is said to be "unit" if all the diagonal elements are unity and non-diagonal elements are zero. This means that all the elements on the main diagonal of the matrix are equal to 1, while all the other elements outside the main diagonal are equal to 0. This type of matrix is also known as an identity matrix, as it represents the identity operation in linear algebra.
32.
A square matrix is said to be an upper triangular matrix if all the elements above the leading diagonal are zero.
Correct Answer
B. False
Explanation
A square matrix is said to be an upper triangular matrix if all the elements above the leading diagonal are zero. However, the given statement is false because an upper triangular matrix can have non-zero elements above the leading diagonal as long as the elements below the leading diagonal are all zero.
33.
A square matrix is said to ______________ if it is either upper or lower triangular matrix.
Correct Answer
A. Triangular
Explanation
A square matrix is said to be triangular if it is either upper or lower triangular matrix. This means that all the entries above or below the main diagonal of the matrix are zero. Triangular matrices have a special structure that makes them useful in various mathematical operations and computations.
34.
A matrix is said to be row matrix if it has only one column and any number of rows.
Correct Answer
B. False
Explanation
A matrix is said to be a row matrix if it has only one row and any number of columns, not the other way around. Therefore, the given statement is false.
35.
Two complex numbers are equal if their corresponding real and imaginary parts both are equal.
Correct Answer
A. True
Explanation
The explanation for the answer being true is that for two complex numbers to be equal, their real parts must be equal and their imaginary parts must also be equal. This is because a complex number is represented as a combination of a real number and an imaginary number. Therefore, if both the real and imaginary parts of two complex numbers are the same, they are considered equal.