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The interval, in which function y = x³ + 6x² + 6 is increasing is
To find the interval for which the function y = x³ + 6x² + 6 is increasing, we will look at its derivative. Compute the derivative of the function: y'(x) = d/dx(x³ + 6x² + 6) = 3x² + 12x Determine where the derivative is positive, so the function is increasing: y'(x) > 0 3x² + 12x > 0 Factor tRead more
To find the interval for which the function y = x³ + 6x² + 6 is increasing, we will look at its derivative.
Compute the derivative of the function:
y'(x) = d/dx(x³ + 6x² + 6) = 3x² + 12x
Determine where the derivative is positive, so the function is increasing:
y'(x) > 0
3x² + 12x > 0
Factor the expression:
3x(x + 4) > 0
This inequality holds when x 0. Thus, the function is increasing in the intervals (-∞, -4) and (0, ∞).
Conclusion:
The correct intervals where the function is increasing are (-∞, 0) U (4, ∞).
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If A is a matrix of order 3 x 3 such that A² = 4A – 3I, then A⁻¹ is
We are given a 3×3 matrix A that satisfies the equation: A² = 4A - 3I Step 1: Expressing A⁻¹ To find A⁻¹, we rearrange the given equation: A² - 4A + 3I = 0 Factoring, (A - I)(A - 3I) = 0 This implies that A satisfies the equation: (A - I)(A - 3I) = 0 Multiplying both sides by (A - I)⁻¹ (if it existsRead more
We are given a 3×3 matrix A that satisfies the equation:
A² = 4A – 3I
Step 1: Expressing A⁻¹
To find A⁻¹, we rearrange the given equation:
A² – 4A + 3I = 0
Factoring,
(A – I)(A – 3I) = 0
This implies that A satisfies the equation:
(A – I)(A – 3I) = 0
Multiplying both sides by (A – I)⁻¹ (if it exists),
A – 3I = (A – I)⁻¹ 0
Since A – I is invertible, we take its inverse on both sides,
A⁻¹ = 1/3 (4I – A)
Step 2: Selecting the Correct Option
Comparing with the given choices, the correct answer is: 1/3 (4I – A)
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If a matrix has 5 elements, write all possible orders it can have are
A matrix with 5 elements can have different possible orders, provided that the total number of elements is the product of its rows and columns. Let the number of rows be m and the number of columns be n, then: m × n = 5 The possible values of (m, n) that satisfy this equation are: - 1 × 5 (1 row andRead more
A matrix with 5 elements can have different possible orders, provided that the total number of elements is the product of its rows and columns.
Let the number of rows be m and the number of columns be n, then:
m × n = 5
The possible values of (m, n) that satisfy this equation are:
– 1 × 5 (1 row and 5 columns)
– 5 × 1 (5 rows and 1 column)
Since both 1 × 5 and 5 × 1 are possible, the correct answer is:
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If P and Q are two different matrices of order 3 x 4 and 4 x 3 respectively, then the order of matrix QP is
The order of a product of two matrices depends upon the order of the given matrices. Given: Matrix P has an order of 3 × 4 Matrix Q has an order of 4 × 3 For the matrix multiplication QP to be defined, the number of columns of Q must equal the number of rows of P. Since Q is 4 × 3 and P is 3 × 4, thRead more
The order of a product of two matrices depends upon the order of the given matrices. Given: Matrix P has an order of 3 × 4 Matrix Q has an order of 4 × 3 For the matrix multiplication QP to be defined, the number of columns of Q must equal the number of rows of P.
Since Q is 4 × 3 and P is 3 × 4, the multiplication QP is not possible
Since the question asks for the order of QP, but the multiplication is undefined, none of the given options are correct.
If the question meant PQ instead of QP, then:
– The order of PQ would be 3 × 3 since P is 3 × 4 and Q is 4 × 3.
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Suppose P and Q are two different matrices of order 3 x n and n x p, then the order of the matrix P x Q is
P is a 3 × n order matrix Q is an n × p order matrix For multiplication of matrices P × Q to be possible, the number of columns of P and the number of rows of Q must be equal. P has n columns. Q has n rows. Thus, multiplication is possible. The order of the resulting matrix will be determined by theRead more
P is a 3 × n order matrix
Q is an n × p order matrix
For multiplication of matrices P × Q to be possible, the number of columns of P and the number of rows of Q must be equal.
P has n columns.
Q has n rows.
Thus, multiplication is possible.
The order of the resulting matrix will be determined by the number of rows of P and the number of columns of Q, so the order of P × Q will be 3 × p.
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