Q.1. The number of real roots of the equation is (2020)
(1) 1
(2) 3
(3) 2
(4) 4
Ans. (1)
The given equation is
Let e^{x} = t, so
Dividing both sides of the equation by t^{2},we get
Again,
Let
Then, y^{2} +y −3 =0 ⇒ (y − 2)(y +3) = 0
⇒ y = 2, 3
Now, and (not possible)
Hence, e^{x} = 1 ⇒ x = 0
Q.2. If x = 2 sinθ  sin2θ and y = 2cosθ  cos2θ, θ∈[0,2π], then at θ = π is (2020)
(1) 3/4
(2)
(3) 3/2
(4)
Ans. (Bonus)
We have,
(1)
(2)
From Eqs. (1) and (2), we have
Now,
Q.3. If x = 3 tan t and y = 3 sec t, then the value of at t = π/4, is: (2019)
(4) 1/6
Ans. (2)
Solution.
Q.4. Let f : R → R be a function such that f(x) = x^{3} + x^{2}f'(1) + xf"(2) + f"'(3), x∈R Then f(2) equals: (2019)
(1) 4
(2) 30
(3) 2
(4) 8
Ans. (3)
Solution.
Q.5. If x log_{e} (log_{e} x)  x^{2} + y^{2} = 4(y > 0), then dy/dx at x = e is equal to: (2019)
Ans. (2)
Solution. Consider the equation,
x log_{e} (log_{e} x)  x^{2} + y^{2} = 4
Differentiate both sides w.r.t. x,
Q.6. For x > 1, if (2x)^{2y} = 4e^{2x2y}, then (1 + log_{e} 2x)^{2} dy/dx equal to: (2019)
(2) log_{e} 2x
(4) x log_{e} 2x
Ans. (1)
Solution.
(2x)^{2y} = 4e^{2x2y}
⇒
Q.7. Let f be a differentiable function such that f(1) = 2 and f' (x) = f(x) for all x ∈ R. If h (x) = f(f(x)), then h' (1) is equal to: (2019)
(1) 2e^{2}
(2) 4e
(3) 2e
(4) 4e^{2}
Ans. (2)
Solution.
Since, f'(x) = f(x)
Since, the given condition
f(1) = 2
From eq^{n} (1)
f(x) = e^{x+c} = e^{c}e^{x}
Then, f(1) = e^{c}.e^{1}
⇒ 2 = e^{c}.e
⇒ 2/e = e^{c}
Then, from eq^{n} (1)
⇒
Now
⇒
Q.8. is equal to: (2019)
Ans. (Bouns)
Solution.
Q.9. If f(1) = 1, f'(1) = 3, then the derivative of f(f(f(x))) + (f(x))^{2} at x = 1 is: (2019)
(1) 33
(2) 12
(3) 15
(4) 9
Ans. (1)
Solution.
Q.10. If e^{y} + xy = e, the ordered pair at x = 0 is equal to: (2019)
Ans. (2)
Solution.
Given, e^{y} +xy = e ...(i)
Putting x = 0 in (i), ⇒ e^{y} = e ⇒ y = 1
On differentiating (i) w. r. to x
...(ii)
Putting y = 1 and x= 0 in (ii),
On differentiating (ii) w. r. to x,
Q.11. The derivative of , with respect to x/2, where (2019)
(1) 1
(2) 2/3
(3) 1/2
(4) 2
Ans. (4)
Solution.
Q.12. If then dy/dx is equal to: (2018)
(1) y/x
(1)  y/x
(1)  x/y
(1) x/y
Ans: (2)
Solution:
Q.13. If x^{2} + y^{2} + sin y = 4, then the value of d^{2}y/dx^{2} at the point (2, 0) is: (2017)
(1) 34
(2) 2
(3) 4
(4) 32
Ans: (1)
Solution:
Q.14. Let f be a polynomial function such that f(3x) = f'(x). f"(x), for all x ∈ R. Then: (2017)
(1) f(2) + f'(2) = 28
(2) f"(2)  f'(2) = 0
(3) f(2)  f'(2) + f"(2) = 10
(4) f"(2)  f(2) = 4
Ans. (2)
Solution.
Q.15. For x ∈ R, x ≠ 0, x ≠ 1, let f_{0}(x)= 1/1x and f_{n + 1}(x) = f_{0} (f(_{n}(X)), n = 0, 1, 2, ....... Then the value of is equal to: (2016)
(1) 4/3
(2) 1/3
(3) 5/3
(4) 8/3
Ans. (3)
Solution.
We have
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129 videos359 docs306 tests
