Let domain and codomain of a function y = f(x) are S and T respectively.
(A) There are infinitely many elements in domain and four elements in codomain.
⇒ There are infinitely many functions from S to T.
⇒ Option (A) is correct
(B) If number of elements in domain is greater than number of elements in codomain, then number of strictly increasing function is zero.
⇒ Option (B) is incorrect
(C) Maximum number of continuous functions = 4 × 4 × 4 = 64
(Every subset (0, 1),(1, 2),(3, 4) has four choices)
∵ 64<120⇒ option (C) is correct.
(D) For every point at which f(x) is continuous, f(x) = 0
⇒ Every continuous function from S to T is differentiable.
Option (D) is correct.
Q2: Let f : [0, 1] → [0, 1] be the function defined by . Consider the square region S = [0, 1] × [0, 1]. Let be called the green region and be called the red region. Let be the horizontal line drawn at a height ℎ ∈ [0, 1]. Then which of the following statements is(are) true?
(a) There exists an ℎ ∈ [1/4, 2/3] such that the area of the green region above the line L_{ℎ} equals the area of the green region below the line L_{ℎ}
(b) There exists an ℎ ∈ [1/4, 2/3] such that the area of the red region above the line L_{ℎ} equals the area of the red region below the line L_{ℎ}
(c) There exists an ℎ ∈ [1/4, 2/3] such that the area of the green region above the line L_{ℎ} equals the area of the red region below the line L_{ℎ}
(d) There exists an ℎ ∈ [1/4, 2/3] such that the area of the red region above the line L_{ℎ} equals the area of the green region below the line [JEE Advanced 2023 Paper 1]
Ans: (b), (c) & (d)
Given,
option (A) is incorrect
option (B) is correct.
⇒ h= 1/2 ⇒ option (C) is correct.
(D) ∵ Option (C) is correct ⇒ option (D) is also correct.
Q1: Let M denote the determinant of a square matrix M. Let be the function defined by
where
Let p(x) be a quadratic polynomial whose roots are the maximum and minimum values of the function g(θ), and p(2) =2 − √2. Then, which of the following is/are TRUE ?
(a)
(b)
(c)
(d) [JEE Advanced 2022 Paper 1]Ans: (a) & (c)
Given,
Here,
and
and
Also,
(skew symmetric)
For option (A) Correct.
For option (B) Incorrect.
For option (C) Correct.
For option (D) Incorrect.
Then the value of is ..........
Ans: 19
The given function f : [0, 1] → R be define by
So, f(x) + f(1 − x) = 1 .....(i)
=
= (19 times) {from Eq. (i)}
= 19.
Q2: Let the function be defined by Suppose the function f has a local minimum at θ precisely when , where . Then the value of is ............. [JEE Advanced 2020 Paper 2]
Ans: 0.5
The given function f : R → R be defined by
The local minimum of function 'f' occurs when
but
Where,
So, = 0.50
Q3: Let f : [0, 2] → R be the function defined by
If are such that , then the value of β  α is .......... [JEE Advanced 2020 Paper 1]
Ans: 1
The given function f : [0, 2] → R defined by
As, so,
Therefore the value of (β  α) = 1
Q3: For a polynomial g(x) with real coefficients, let m_{g} denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by
For a polynomial f, let f' and f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (m_{f'} + m_{f''}), where f ∈ S, is .............. [JEE Advanced 2020 Paper 1]
Ans: 5
Given set S of polynomials with real coefficients
and for a polynomial f ∈ S, Let
it have −1 and 1 as repeated roots twice, so graph of f(x) touches the Xaxis at x = −1 and x = 1, so f'(x) having at least three roots x = −1, 1 and α. Where α ∈ (−1, 1) and f''(x) having at least two roots in interval (−1, 1)
So, m_{f'} = 3 and m_{f''} = 2
∴ Minimum possible value of (m_{f'} + m_{f''}) = 5
Q4: If the function f : R → R is defined by f(x) = x (x − sin x), then which of the following statements is TRUE?
(a) f is oneone, but NOT onto
(b) f is onto, but NOT oneone
(c) f is BOTH oneone and onto
(d) f is NEITHER oneone NOR onto [JEE Advanced 2020 Paper 1]
Ans: (c)
The given function f : R → R is
The function 'f' is a odd and continuous function and as , so range is R, therefore, 'f' is a onto function.
⇒ f is strictly increasing function. ∀x ∈(0, ∞).
Similarly, for x < 0, −x + sin x > 0 and (− x) (1 − cos x) > 0, therefore, f′(x)> 0∀ x ∈(−∞, 0)
⇒ f is strictly increasing function, ∀x ∈ (0, ∞)
Therefore 'f' is a strictly increasing function for x ∈ R and it implies that f is oneone function.
Q1: Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of oneone functions from X to Y and β is the number of onto functions from Y to X, then the value of 1 / 5!(β − α) is .................. [JEE Advanced 2018 Paper 2]
Ans: 119
Given, X has exactly 5 elements and Y has exactly 7 elements.
∴ n(X) = 5
and n(Y) = 7
Now, number of oneone functions from X to Y is
Number of onto functions from Y to X is β
1, 1, 1, 1, 3 or 1, 1, 1, 2, 2
= 4 x 35  21= 140  21
= 119
Q2: Let and (Here, the inverse trigonometric function sin^{−1}^{ }x assumes values in [−π/2, π/2].).
Let f : E_{1} → R be the function defined by and g : E_{2} → R be the function defined by . [JEE Advanced 2018 Paper 2]
and
So,
The domain of f and g are
and Range of
Range of f is R − {0} or (−∞, 0) ∪ (0, ∞)
Range of g is
Now, P → 4, Q → 2, R → 1, S → 1
Hence, option (a) is correct answer.
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