Q.1. The integral is equal to (where C is a constant of integration) (2020)
(1)
(2)
(3)
(4)
Ans. (1)
Let,
Let
So,
Q.2. If where C is a constant of integration, then the ordered pair (λ, f(θ)) is equal to (2020)
(1) (1,1 tan θ)
(2) ( 1,1 tan θ)
(3) (1,1 + tan θ)
(4) (1,1 + tan θ)
Ans. (3)
We have
Let tan θ = t ⇒sec^{2} θdθ = dt.
Therefore,
Hence,
Q.3. For x^{2} ≠ nπ + 1, n∈N (the set of natural numbers), the integral
(2019)
(where c is a constant of integration)
Ans. (3, 4)
Solution. Consider the given integral
Q.4. and f(0) = 0, then the value of f(1) is: (2019)
(3) 1/2
(4) 1/4
Ans. (4)
Solution.
f(x) =
Q.5. Let n ≥ 2 be a natural number and 0 < θ < π/2. Then is equal to (where C is a constant of integration (2019)
Ans. (1)
Solution.
Q.6. where C is a constant of integration, then f (x) is equal to: (2019)
(1)  2x^{3}  1
(2)  4x^{3}  1
(3) 2x^{3} + 1
(4) 4x^{3}+ 1
Ans. (2)
Solution.
Put 4x^{3} = θ
⇒ 12x^{2} dx = dθ
⇒
Then, by comparison
f(x) = 4x^{3}  1
Q.7. for a suitable chosen integer m and a function A (x), where C is a constant of integration, then (A(x))^{m} equals: (2019)
Ans. (1)
Solution.
Comparing both sides,
Q.8. where C is a constant of integration, then f(x) is equal to: (2019)
(1)
(2)
(3)
(4)
Ans. (4)
Solution.
Q.9. The integral is equal to:
(where C is a constant of integration) (2019)
Ans. (3)
Solution.
Q.10. The integral is equal to: (where C is a constant of integration) (2019)
Ans. (2)
Solution.
Q.11.
is equal to:
(where c is a constant of integration.) (2019)
(1) 2x + sinx + 2 sin2x + c
(2) x + 2 sinx + 2 sin2x + c
(3) x + 2 sinx + sin2x + c
(4) 2x + sinx + sin2x + c
Ans. (3)
Solution.
[ ∵ sin 2x = 2 sin x cos x and sin 3x = 3 sin x  4 sin^{3}x]
Q.12. where C is a constant of integration, then the function f(x) is equal to: (2019)
Ans. (4)
Solution.
Q.13. The integral ∫ sec^{2/3} x cosec^{4/3} xdx is equal to:
(1) 3 tan^{1/3} x + C
(3) 3 cot^{1/3} x + C
(4) 3 tan^{1/3} x + C
(Here C is a constant of integration) (2019)
Ans. (1)
Solution.
Q.14. If ∫e^{sec x} (sec x tan x f(x) + (sec x tan x + sec^{2} x)) dx = e^{sec}^{x} f(x) + C, then a possible choice of f(x) is: (2019)
Ans. (1)
Solution.
Q.15. where C is a constant of integration, then: (2019)
(1) A = 1/54 and f(x) = 3 (x  1)
(2) A = 1/81 and f(x) = 3 (x  1)
(3) A = 1/27 and f(x) = 9 (x  1)
(4) A = 1/54 and f(x) = 9 (x  1)^{2}
Ans. (1)
Solution.
Let (x  1)^{2} = 9 tan^{2} θ ....(1)
After differentiating equation ...(1), we get
2 (x  1) dx = 18 tan θ sec^{2}θ dθ
we get: A = 1/54 and f(x) = 3 (x  1)
Q.16. If where c is a constant of integration, then g(1) is equal to: (2019)
(1) 1
(2) 1
(3)
(4)
Ans. (3)
Solution.
Q.17. The integral is equal to: (2019)
(Here C is a constant of integration)
Ans. (3)
Solution.
Q.18. Let α ∈ (0, π/2) be fixed. If the integral A(x) cos2α+B(x) sin2α+C, where C is a constant of integration, then the functions A(x) and B(x) are respectively: (2019)
(1) x + α and log_{e}sin(x + α)
(2) x  α and log_{e}sin(x  α)
(3) x  α and log_{e} cos(x  α)
(4) x + α and log_{e} sin(x  a)
Ans. (2)
Solution.
Q.19. The integral is equal to: (2018)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.
Q.20. If f = 2x + 1, (x ∈ R − {1, −2}), then ∫ f(x)dx is equal to: (where C is a constant of integration) (2018)
(1) 12 log_{e} 1  x  3x + C
(2) – 12 log_{e} 1 – x + 3x + C
(3) – 12 log_{e} 1 – x  3x + C
(4) 12 log_{e} 1 – x + 3x + C
Ans. (3)
Solution.
= 3 {–4ℓn1–x – x + C = –12ℓn 1–x – 3x + C
Q.21. If f(x) = dt then: (2018)
(1) f''' (x)  f''(x) = cosx  2x sinx
(2) f'''(x) + f''(x)  f'(x) = cosx
(3) f'''(x) + f''(x) = sinx
(4) f'''(x) + f'(x) = cosx  2x sinx
Ans. (4)
Solution.
Q.22. If (C is a constant of integration), then the ordered pair (K, A) is equal to (2018)
(1) (2, 1)
(2) (2, 3)
(3) (–2, 1)
(4) (–2, 3)
Ans. (2)
Solution.
I =
=
Q.23. Let I_{n} = ∫tan^{n} xdx,(n > 1). If I_{4} +I_{6}= a tan^{5} x + bx^{5} + C, where C is a constant of integration, then the ordered pair (a, b) is equal to (2017)
(1) (1/5 , 0)
(2) (1/5 , 1)
(3) (1/5, 0)
(4) (1/5, 1)
Ans. (3)
Solution.
Let tanx = t
sec^{2}x dx = dt
Q.24. The integral is equal to:
(where C is a constant of integration) (2017)
(1)
(2)
(3)
(4)
Ans. (1)
Solution.
Q.25. If and then the ordered pair (A,B) is equal to :(where c is a constant of integration) (2017)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.
Q.26. The integral dx is equal to: (2016)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.
Dividing numerator and denominator by x^{15} we get,
Q.27. If , where k is a constant of integration, then A + B + C equals (2016)
(1) 15/5
(2) 21/5
(3) 7/10
(4) 27/10
Ans. (1)
Solution.
tan x = t
Q.28. The integral is equal to (where C is a constant of integration) (2016)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.
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