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Option 2

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• Faculty Support #### If  for  then Option: 1 Option: 2 Option: 3 Option: 4 Use

Correct Option (3)

Option d

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• Faculty Support #### The following system of linear equations   has  Option: 1 infinitely many solutions, satisfying Option: 2 infinitely many solutions, satisfying Option: 3 no solution Option: 4 only the trivial solution.

System of Homogeneous linear equations - If ? ≠ 0, then x= 0, y = 0, z = 0 is the only solution of the above system. This solution is also known as a trivial solution.

If ? = 0, at least one of x, y and z are non-zero. This solution is called a non-trivial solution.

Explanation: using equation (ii) and (iii), we have  This is the condition for a system have Non-trivial solution.

-  so infinite non-trivial solution exist

now equation (1) + 3 equation (3)

10x - 20z = 0

x = 2z

Correct Option 2

#### Let If then :    Option: 1 Option: 2 Option: 3 Option: 4 Elementary row operations -

Elementary row operations

Row transformation: Following three types of operation (Transformation) on the rows of a given matrix are known as elementary row operation (transformation).

i) Interchange of ith row with jth row, this operation is denoted by ii) The multiplication of ith row by a constant k (k≠0) is denoted by iii) The addition of ith row to the elements of jth row multiplied by constant k (k≠0) is denoted by In the same way, three-column operations can also be defined too.

- Correct Option (3)

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• Faculty Support #### Let denote the greatest integer and Then the function, is discontinuous, when x is equal to:  Option: 1 Option: 2 Option: 3 Option: 4   Correct Option (1)

#### Let be such that the equation, has a repeated root , which is also a root of the equation, . If is the other root of this equation, then is equal to: Option: 1 Option: 2 Option: 3 Option: 4 Nature of Roots -

Let the quadratic equation is ax2 + bx + c = 0

D is the discriminant of the equation.

iii) if roots D = 0, then roots will be real and equal, then -

ax2 – 2bx + 5 = 0 having equal roots or and  Put in the second equation  Correct Option 2

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• Faculty Support #### In the expansion of , if is the least value of the term independent of  when  and is the least value of the term independent of  when , then the ratio is equal to :  Option: 1 Option: 2 Option: 3 Option: 4 General Term of Binomial Expansion

Term independent of x: It means term containing x0,

Now,

Correct option 1

#### Let and be differentiable functions on such that is the identity function. If for some then is equal to :    Option: 1 Option: 2 Option: 3 Option: 4 Rules of Differentiation (Chain Rule) -

Rules of Differentiation (Chain Rule)

Chain Rule or Derivation of Composite Function:

The chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Let f and g be functions. For all x in the domain of g for which g is differentiable at x and f is differentiable at g(x), the derivative of the composite function

h(x) = (f?g)(x) = f (g(x)) Is given by

h′(x) = f’(g(x))?g’(x)

Composites of Three or More Functions

For all values of x for which the function is differentiable, if k(x) = h(f(g(x)))Then, - Correct Option 4  