Question 8

Determine order and degree(if defined) of differential y^{'} + y =e^{x}

Answer

y^{'} + y =e^{x}

y^{'} + y - e^{x }=0

The highest order derivative present in the differential equation is y^{'}. Therefore, its order is one.

The given differential equation is a polynomial equation in y^{' }and the highest power raised to y^{' }is one. Hence, its degree is one.

- Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:-
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i)

*f*:**R → R**defined by*f(x)*= 3 – 4x(ii)

*f*:**R → R**defined by*f(x)*= 1 + x^{2 } - Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- Q:- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. - Q:-
Show that the Modulus Function

*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. - Q:-
Prove that the Greatest Integer Function

*f*: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. - Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

- Q:- Let R be the relation in the set N given by R = {(a, b): a = b − 2, b > 6}. Choose the correct answer.

(A) (2, 4) ∈ R

(B) (3, 8) ∈R

(C) (6, 8) ∈R

(D) (8, 7) ∈ R - Q:-
Determine order and degree(if defined) of differential equation y' + 5y = 0

- Q:- Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
- Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:- Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
- Q:-
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

- Q:- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- Q:-
Let

*f*: X → Y be an invertible function. Show that the inverse of*f*is f, i.e.,^{–1}*(f*=^{–1})^{–1}*f*. - Q:- \begin{align} \int\left(1-x\right).\sqrt {x}.dx\end{align}
- Q:-
If

*f(x)*=_{}, show that*fof*(x) = x, for all x ≠ 2/3. What is the inverse of*f*?

- NCERT Chapter

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