Solution:
Vertical line test fails only for f₆ and therefore f₆(x) is not a function.
Question 2:
Choose the correct option.
○ f₁ and f₃ are one-one functions in the given domain.
○ f₂ and f₄ are one-one functions in the given domain.
○ f₃ and f₅ are one-one functions in the given domain.
○ f₅ is one-one function in the given domain.
Solution:
The functions f₂ and f₄ are strictly decreasing functions in the domain [-4, 4], therefore these are one-to-one functions.
Question 3:
Choose the correct option.
○ f₁ and f₅ are strictly increasing functions in the given domain.
○ f₂ and f₄ are strictly decreasing functions in the given domain.
○ f₄ and f₅ are strictly decreasing functions in the given domain.
○ f₅ is strictly increasing function in the given domain.
Solution:
From the given graph, f₂ and f₄ are strictly decreasing functions in the domain [-4, 4].
Questions 4 and 5 (Radioactive Decay Context)
Question 4:
Let N₀ be the number of atoms of a radioactive material at the initial stage (t = 0), and N(t) be the number of atoms at time t, given by N(t) = N₀e^{-λt}. If at time t₁, the number of atoms reduces to half of N₀, and at time t₂, the number of atoms reduces to one fourth of N₀, which equation is correct?
○ e^{t₁/t₂} = 2
○ e^{t₂/t₁} = 2
○ e^{t₂-t₁} = 2
○ e^{t₁-t₂} = 2
Solution:
e^{t₂-t₁} = 2
Question 5:
If N₁ is the number of atoms at time t = 1/λ, what is the ratio of N₀ to N₁?
○ 1 : e
○ e : 1
○ 1 : e^{-λ}
○ 1 : e^{λ}
Solution:
N₀ : N₁ = e : 1
Multiple Select Questions (MSQ)
Question 6:
Selvi deposits ₹P in bank A at 10% per year for 10 years, then withdraws and deposits in bank B at 12.5% per year for n years. M_A(x) is the amount after x years in bank A, M_B(y) is the amount after y years in bank B. Choose the correct options.
☐ M_A(x) is a one-one function of x, for x ∈ (0,10)
☐ M_B(y) is a one-one function of y
☐ M_A(12) = P × 1.1^{12}
☐ M_A(12) = 0
☐ M_A(x) is strictly increasing in x, for x ∈ (0,10)
☐ M_B(y) is decreasing in y
☐ M_B(n) = (P × 1.1^{10}) × (1.125)^n
☐ M_B(n) = (P × 1.1^n) × (1.125)^{10}
Solution:
Correct options:
M_A(x) is a one-one function of x, for x ∈ (0,10)
M_B(y) is a one-one function of y
M_A(12) = 0 (since withdrawn after 10 years)
M_A(x) is strictly increasing in x, for x ∈ (0,10)
M_B(n) = (P × 1.1^{10}) × (1.125)^n
Question 7:
There are two offers in a shop:
First offer: discount of M(n)% if n products are bought
Second offer: ₹1000 discount
Geeta shops for ₹15,000. f(n) and g(n) are total payable after first and second offers, respectively. T(n) is total payable after both offers.
☐ T(n) = (100 – M(n)) × 15000 is the total payable when first offer is applied after the second
☐ T(n) = (100 – M(n)) × 140 is the total payable when first offer is applied after the second
☐ T(n) = (100 – M(n)) × 150 – 1000 is the total payable when second offer is applied after the first
Solution:
Correct options:
f(n) = (100 – M(n)) × 150 and g(n) = 14000
T(n) = (100 – M(n)) × 140 (when first offer is applied after the second)
T(n) = (100 – M(n)) × 150 – 1000 (when second offer is applied after the first)
Question 8:
If M(n) = −n² + 18n − 72, n ∈ {6,7,8,9}, and Geeta can use offers in any sequence, choose the correct options.
☐ Total payable amount is same irrespective of the order in which the offers are applied.
☐ She should choose offer one and then offer two.
☐ She should choose offer two and then offer one.
☐ If she chooses offer one and then offer two, the minimum payable amount will be ₹12650.
Solution:
Correct options:
She should choose offer one and then offer two.
If she chooses offer one and then offer two, the minimum payable amount will be ₹12650.
Numerical Answer Type (NAT)
Question 9:
Given f(x) = (5x + 9)/2x, domain (−∞, m) ∪ (m, ∞); g(y) = √(y² – 9), domain ℝ \ (−n, n). If h(x) = f(g(x)), find m + n.
Solution:
m = 0, n = 3
Answer: 3
Question 10:
Range of f(x): (−∞, m) ∪ (m, ∞); range of g(x): [n, ∞). Find 2(m + n).
Solution:
m = 2.5, n = 0
Answer: 5
Question 11:
Domain of h(x) is (−∞, −3) ∪ (m, ∞). Find m.
Solution:Answer: 3
Question 12:
Domain of f⁻¹(x) is (−∞, m) ∪ (m, ∞). Find 2m.
Solution:
m = 2.5
Answer: 5
Question 13:
If f⁻¹(5) = 9/m, find m.
Solution:Answer: 5
Note:
All questions and solutions are as per the provided PDF1. For questions referencing graphs or tables, the description is summarized as given in the document.
For questions 14 and above, the PDF does not mention any further questions or solutions in the provided excerpt. The last question in the extracted content is Question 13.
