1
Consider the following statements
1. The function f(x) = x2 + 2 cos x is increasing in the interval (0, $$\pi$$).
2. The function $$f(x) = \ln (\sqrt {1 + {x^2}} - x)$$ is decreasing in the interval ($$-$$ $$\infty$$, $$\infty$$)
Which of the above statements is/are correct?
2
Consider the following statements
1. f(x) = ln x is an increasing function on (0, $$\infty$$).
2. f(x) = ex $$-$$ x(ln x) is an increasing function on (1, $$\infty$$).
Which of the above statements is/are correct?
3
At what value of x does the function attain maximum value?
4
The maximum value of the function is
5
The function f(x) is an increasing function in the interval
6
The function f(x) is a decreasing function in the interval
7
Which one of the following statements is correct in respect of the function f(x) = x3 sin x?
8
Which of the following statements is/are correct?
I. f(x) is increasing in the interval {$$-$$1, 2}.
II. f(x) is decreasing in the interval {2, 3}.
Select the correct answer using the code given below.
9
Which of the following statements are correct?
I. f(x) is continuous at x = 2.
II. f(x) attains greatest value at x = 2.
III. f(x) is differentiable at x = 2.
Select the correct answer using the code given below.
10
Let $$f(x) = \left\{ {\matrix{
{{{{e^x} - 1} \over x},} & {x > 0} \cr
{0,} & {x = 0} \cr
} } \right.$$ be a real valued function. Which one of the following statements is correct?
11
Which one of the following statements is correct?
12
Which one of the following statements is correct?
13
Which one of the following is correct in respect of the function
$$f(x) = x(x - 1)(x + 1)$$ ?
14
The maximum value of $${{\ln x} \over x}$$ is
15
A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is
16
What is the length of the longest interval in which the function $$f(x) = 3\sin x - 4{\sin ^3}x$$ is increasing?
17
Let $$f(x) = x + {1 \over x}$$, where x$$\in$$ (0, 1). Then which one of the following is correct?
18
Let the slope of the curve y = cos$$-$$1 (sin x) be tan$$\theta$$. Then, the value of $$\theta$$ in the interval (0, $$\pi$$) is
19
Which one of the following is correct in respect of the function $$f(x) = x\sin x + \cos x + {1 \over 2}{\cos ^2}x$$ ?
20
In which one of the following intervals is the function $$f(x) = {x^2} - 5x + 6$$ decreasing?
21
A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector?
22
What is the minimum value of $${[x(x - 1) + 1]^{{1 \over 3}}}$$, where 0 $$\le$$ x $$\le$$ 1 ?
23
If the area of the trapezium is the maximum possible, then what is $$\alpha$$ equal to?
24
If the area of the trapezium is maximum, what is the length of the fourth side?
25
What is the maximum area of the trapezium?
26
What is the slope of the curve at the point of intersection P?
27
How much angle does the tangent at P make with y-axis?
28
What is the equation of tangent to the curve at P?
29
If $$f(x) = {{{x^3}} \over 3} - {{5{x^2}} \over 2} + 6x + 7$$ increases in the interval T and decreases in the interval S, then which one of the following is correct?
30
In which one of the following intervals is the function increasing?
31
In which one of the following intervals is the function decreasing?
32
A given quantity of metal is to be case into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is
33
The minimum distance from the point (4, 2) to y2 = 8x is equal to
34
What is the minimum value of a2x + b2y where xy = c2 ?
40
What is the maximum value of y?
NDA Mathematics 3 September 2023
41
What is the maximum value of xy ?
NDA Mathematics 3 September 2023
43
What is the derivative of sin2 x with respect to cos2x ?
NDA Mathematics 16 April 2023
44
Under which one of the following conditions does the function f(x) = (p sec x)2 + (q cosec x)2 attain minimum value ?
NDA Mathematics 4 September 2022
45
Where does the function f(x) = $\displaystyle\sum_{j=1}^7$(x − j)2 attain its minimum value ?
NDA Mathematics 4 September 2022
46
How many extreme values does sin4x + 2x, where $0 < x < \frac{\pi}{2} $ have ?
NDA Mathematics 10 April 2022
47
If the derivative of the function$f(x) =\frac{m}{x} +2nx + 1$ vanishes at x = 2, then what is the value of m + 8n ?
NDA Mathematics 10 April 2022
48
If xy = 4225 where x, y are natural numbers, then what is the minimum value of x + y ?
NDA Mathematics 10 April 2022
50
The curve y = -x3 + 3x2 + 2x - 27 has the maximum slope at:
NDA Mathematics 18 April 2021
51
If x + y = 20 and P = xy, then what is the maximum value of P?
NDA Mathematics 18 April 2021
52
What is the maximum value of sin 2x ⋅ cos 2x?
NDA Mathematics 18 April 2021