Limits, Continuity & Derivatives · NDA previous year questions

Which one of the following functions is differentiable for all real values of x?

1. A.$\frac{x}{|x|}$
2. B.$x|x|$
3. C.$\frac{1}{|x|}$
4. D.$\frac{1}{x}$

What is the slope of the tangent to the curve $x=t^2+3t-8$, $y=2t^2-2t-5$ at $t=2$?

1. A.7/6
2. B.6/7
3. C.1
4. D.5/6

Which one of the following statements is correct?

1. A.$e^x$ is an increasing function
2. B.$e^x$ is a decreasing function
3. C.$e^x$ is neither increasing nor decreasing function
4. D.$e^x$ is a constant function

If $y=\cos t$ and $x=\sin t$, then what is $\frac{dy}{dx}$ equal to?

1. A.$xy$
2. B.$x/y$
3. C.$-y/x$
4. D.$-x/y$

Which one of the following statements is correct ?

1. A.The derivative of a function f(x) at a point will exist if there is one tangent to the curve y = f(x) at that point and the tangent is parallel to y-axis
2. B.The derivative of a function f(x) at a point will exist if there is one tangent to the curve y = f(x) at that point and the tangent must be parallel to x-axis
3. C.The derivative of a function f(x) at a point will exist if there is one and only one tangent to the curve y = f(x) at that point and the tangent is not parallel to y-axis
4. D.None of the above

What is ${\lim }_{x\to 0}{x}^2\sin \left(\frac{1}{x}\right)$ equal to ?

1. A.0
2. B.1
3. C.1/2
4. D.Limit does not exist

What is ${\lim }_{x\to -2}\left(\frac{x+2}{x^3+8}\right)$ equal to ?

1. A.1/4
2. B.-1/4
3. C.1/12
4. D.-1/12

What is the rate of change of $\sqrt{x^2+16}$ with respect to $x^2$ at $x=3$ ?

1. A.1/5
2. B.1/10
3. C.1/20
4. D.None of the above

Consider the following in respect of the function $f(x)=|x-3|$ : 1. f(x) is continuous at x = 3 2. f(x) is differentiable at x = 0. Which of the above statements is/are correct ?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the maximum value of ${\sin }^{2}x$?

1. A.-1
2. B.0
3. C.1
4. D.Infinity

The function $y=f(x)=mx+c$ has

1. A.maximum point but no minimum point
2. B.minimum point but no maximum point
3. C.both maximum and minimum points
4. D.neither maximum point nor minimum point

At an extreme point of a function $f(x)$, the tangent to the curve is

1. A.parallel to the x-axis
2. B.perpendicular to the x-axis
3. C.inclined at an angle $45^{\circ }$ to the x-axis
4. D.inclined at an angle $60^{\circ }$ to the x-axis

If $y=\left(1+x^{\frac{1}{4}}\right)\left(1+x^{\frac{1}{2}}\right)\left(1-x^{\frac{1}{4}}\right)$, then what is $\frac{dy}{dx}$ equal to?

1. A.1
2. B.$-1$
3. C.$x$
4. D.$x^{\frac{1}{2}}$

The point in the interval $(0,2\pi )$ where $f(x)=e^x\sin x$ has maximum slope is

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{2}$
3. C.$\pi$
4. D.$\frac{3\pi }{2}$

If $\tan \theta =\sqrt{m}$, where $m$ is a non-square natural number $m$, then $\sec 2\theta$ is

1. A.a negative number
2. B.a transcendental number
3. C.an irrational number
4. D.a rational number

What is the maximum value of $3\cos x+4\sin x+5$?

1. A.5
2. B.7
3. C.10
4. D.12

If $\sin \theta ={\cos }^2\theta$, then what is ${\cos }^2\theta (1+{\cos }^2\theta )$ equal to?

1. A.1
2. B.0
3. C.${\cos }^2\theta$
4. D.$2\sin \theta$

If the rate of change in volume of spherical soap bubble is uniform, then the rate of change of surface area varies as

1. A.square of radius
2. B.square root of radius
3. C.inversely proportional to radius
4. D.cube of the radius

What is $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}$ equal to?

1. A.$2\tan x$
2. B.$2\mathrm{cosec}x$
3. C.$2\cos x$
4. D.$2\sin x$

If $f(x)=x\ln x$, then $f(x)$ attains minimum value at which one of the following points?

1. A.$x=e^{-2}$
2. B.$x=e$
3. C.$x=e^{-1}$
4. D.$x=2e^{-1}$

If the roots of the equation $x^2-4x-{\log }_{3}N=0$ are real, then what is the minimum value of N?

1. A.$1/256$
2. B.$1/27$
3. C.$1/64$
4. D.$1/81$

1. A.1995
2. B.1996
3. C.1997
4. D.1998

1. A.1995
2. B.1996
3. C.1997
4. D.1998

1. A.$\frac{80}{3}\%$
2. B.$\frac{100}{3}\%$
3. C.$35\%$
4. D.$40\%$

NOTE: STUDY THE PIE CHART GIVEN BELOW AND ANSWER THE NEXT 04 (FOUR) QUESTIONS THAT FOLLOW: The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures: Agriculture (A), Industry (B), Education (C), Employment (D) and Miscellaneous (E). The total allocation is 36,000 (in crores of rupees). Which head is allocated maximum funds?

1. A.Agriculture
2. B.Industry
3. C.Employment
4. D.Miscellaneous