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Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Match the Following

Each question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D, while the statements in Column-II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in ColumnII. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example :

If the correct matches are A-p, s and t; B-q and r; C-p and q; and D-s then the correct darkening of bubbles will look like the given.

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 1. Match the statements/expressions in Column I with the open intervals in Column II.

Column I                                                                                  Column II

(A) Interval contained in the domain of definition of                 Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
non-zero solutions of the differential equation
(x – 3)2 + y' + y = 0

(B) Interval containing the value of the integral                     Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(C) Interval in which at least one of the points of local           Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
maximum of
cos2x + sin x lies

(D) Interval in which tan–1 (sin x + cos x) is increasing          Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
                                                                                                (t) ( – π,π)

Ans.  (A) → p,q,r,s,t; (B) → p,t; (C) → p,q,r,t; (D) → s

Solution. 

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ The solution set is (- ∞, ∞)- {3}

The interval Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and (–π, π) contained in the domain

∴ (A) → p, q,r , s,t

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let ( x - 3) = t ⇒ dx = dt

Also when x ⇒ 1,t → -2

and when x → 5,t → 2

∴ Integral becomes

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
as integrand is an odd function.

O is contained by  Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ (B) → p,t.

(C) Let f ( x) = cos2 x+ sinx

⇒ f '( x) = -2 sin x cos x+ cosx

For critical point f '(x) = 0  Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are the points of local maxima.
Clearly all the intervals given in column II except Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

contain at least one point of local maxima.

∴ (C) → p, q, r,t

(D) Let f ( x) = tan -1 (sin x+ cosx)

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
For f (x) to be an increasing function,  f '(x) > 0

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Integer Value Correct Type    

Q. 1. Let y'(x) + y(x) g'(x) = g(x), g'(x),  y(0) = 0, x ∈ R, where f '(x) denotes Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis a given non-constantdifferentiable function on R with g(0) = g(2) = 0. Then the value of y(2) is 

Ans. 0

Solution. The given equation is  Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

put g (x) = t so that g¢(x) dx = dt

Matrix-Match Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
As y (0) = 0 and g (0) = 0

∴ C = 1

∴ y.eg(x) = eg(x) [g(x) – 1] + 1
As g (2) = 0, putting x = 2 we get
y(2).eg(2) = eg(2) [g(2) – 1] + 1 ⇒ y (2) = 0                                                                                   

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