Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

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Important Questions on Continuity and Differentiability

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

A

be any

3 \times 3

invertible matrix. Then which one of the following is not always true?

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

A, B, C

be three

4 \times 4

matrices such that

det A = 5, det B = - 3

, and

det C = \frac{1}{2} .

Then the

det (2 A B^{- 1} C^{3} B^{T})

is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

A = [\begin{matrix} - 1 & 2 \\ 2 & - 1 \end{matrix}]

and

B = [\begin{matrix} 3 \\ 1 \end{matrix}], A X = B

, then

X

is equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Using elementary row transformations, find the inverse of the matrix $A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ - 2 & - 4 & - 5 \end{matrix}]$ .

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}]$ , find $A^{- 1}$ . Use it to solve the system of equations

$2 x - 3 y + 5 z = 11$

$3 x + 2 y - 4 z = - 5$

$x + y - 2 z = - 3$

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Find the inverse of the matrix using elementary operations.

$A = [\begin{array}{r} 1 & 2 & - 2 \\ - 1 & 3 & 0 \\ 0 & - 2 & 1 \end{array}]$ .

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

A

and

B

are two non-zero square matrices of the same order then

A B = O

Implies both A and B must be singular.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

f (x) = \log_{e} (s i n x), (0 < x < π)

and

g (x) = \sin^{- 1} (e^{- x}), (x \geq 0) .

If

α

is a positive real number such that

a = {(f o g)}^{'} (α)

and

b = (f o g) (α),

then

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

A = [\begin{matrix} 2 & 0 & 0 \\ - 1 & 2 & 3 \\ 3 & 3 & 5 \end{matrix}]

then find

A (adj A)

.

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let

A

be a non-singular square matrix of order

3 \times 3 .

Then

|a d j A|

is equal to:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Variations caused by mutation, as proposed by Hugo de Vries, are:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

A

is a

3 \times 3

matrix such that

|5 a d j A| = 5

, then

|A|

is equal to

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Find the order and the degree of the differential equation:

x^{2} \frac{d^{2} y}{d x^{2}} = {\{1 + {(\frac{d y}{d x})}^{2}\}}^{4}

.

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

B = [\begin{matrix} 5 & 2 α & 1 \\ 0 & 2 & 1 \\ α & 3 & - 1 \end{matrix}]

is the inverse of a

3 \times 3

matix

A,

then the sum of all values of

α

for which

d e t (A) + 1 = 0,

is:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

The order and the degree of the differential equation

{(\frac{d^{3} y}{d x^{3}})}^{\frac{1}{6}} - {(\frac{d y}{d x})}^{\frac{1}{3}} = 0

are respectively

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

\log (x^{2} + y^{2}) = 2 \tan^{- 1} (\frac{y}{x})

, show that

\frac{d y}{d x} = \frac{x + y}{x - y}

.

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Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Differentiate

\tan^{- 1} (\frac{1 + \cos x}{\sin x})

with respect to

x

.[Write your answer as decimal only]

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

A = [\begin{matrix} 2 & 3 \\ 1 & - 4 \end{matrix}]

and

B = [\begin{matrix} 1 & - 2 \\ - 1 & 3 \end{matrix}],

then verify that

(AB)^{- 1} = B^{- 1} A^{- 1} .

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Given

A = [\begin{matrix} 2 & - 3 \\ - 4 & 7 \end{matrix}]

, compute

A^{- 1}

and show that

2 A^{- 1} = 9 I - A

.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If

P = [\begin{matrix} 1 & α & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{matrix}]

is the adjoint of a

3 \times 3

matrix

A

and

|A| = 4

, then

α

is equal to