Limits, Continuity & Derivatives · NDA previous year questions

Where does the tangent to the curve $y=e^x$ at the point $(0,1)$ meet x-axis?

1. A.$(1,0)$
2. B.$(-1,0)$
3. C.$(2,0)$
4. D.$\left(-\frac{1}{2},0\right)$

Consider the following statements in respect of the function $f(x)=x+\frac{1}{x}$: 1. The local maximum value of f(x) is less than its local minimum value. 2. The local maximum value of f(x) occurs at $x=1$. 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 area of a rectangle that can be inscribed in a circle of radius 2 units?

1. A.4 square units
2. B.6 square units
3. C.8 square units
4. D.16 square units

What is $\int \frac{\mathrm{d}x}{x(x^2+1)}$ equal to?

1. A.$\frac{1}{2}\ln \left(\frac{x^2}{x^2+1}\right)+C$
2. B.$\ln \left(\frac{x^2}{x^2+1}\right)+C$
3. C.$\frac{3}{2}\ln \left(\frac{x^2}{x^2+1}\right)+C$
4. D.$\frac{1}{2}\ln \left(\frac{x^2+1}{x^2}\right)+C$

What is the derivative of $e^{e^x}$ with respect to $e^x$?

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

What is the condition that $f(x)=x^3+x^2+kx$ has no local extremum?

1. A.$4k<1$
2. B.$3k>1$
3. C.$3k<1$
4. D.$3k\le 1$

If $f(x)=2^x$, then what is ${\int }_2^{10}\frac{f^{\prime }(x)}{f(x)}\mathrm{d}x$ equal to?

1. A.$4\ln 2$
2. B.$\ln 4$
3. C.$\ln 5$
4. D.$8\ln 2$

If ${\int }_{-2}^{0}f(x)\mathrm{d}x=k$, then ${\int }_{-2}^0|f(x)|\mathrm{d}x$ is

1. A.less than k
2. B.greater than k
3. C.less than or equal to k
4. D.greater than or equal to k

If the function $f(x)=x^2-kx$ is monotonically increasing in the interval $(1,\infty )$, then which one of the following is correct?

1. A.$k<2$
2. B.$2<k<3$
3. C.$3<k<4$
4. D.$k>4$

The tangent to the curve $x^2=y$ at $(1,1)$ makes an angle $\theta$ with the positive direction of x-axis. Which one of the following is correct?

1. A.$\theta <\frac{\pi }{6}$
2. B.$\frac{\pi }{6}<\theta <\frac{\pi }{4}$
3. C.$\frac{\pi }{4}<\theta <\frac{\pi }{3}$
4. D.$\frac{\pi }{3}<\theta <\frac{\pi }{2}$

What is the derivative of $e^x$ with respect to $x^e$?

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

If a differentiable function f(x) satisfies ${\lim }_{x\to -1}\frac{f(x)+1}{x^2-1}=-\frac{3}{2}$ then what is ${\lim }_{x\to -1}f(x)$ equal to?

1. A.$-\frac{3}{2}$
2. B.-1
3. C.0
4. D.1

If the function $f(x)=\{\begin{array}{ll}5,& x=1\\ b-ax,& x>1\end{array}$ is continuous, then what is the value of $(a+b)$?

1. A.5
2. B.10
3. C.15
4. D.20

The curve $y=-x^3+3x^2+2x-27$ has the maximum slope at:

1. A.$x=-1$
2. B.$x=0$
3. C.$x=1$
4. D.$x=2$

A 24 cm long wire is bent to form a triangle with one of the angles as 60°. What is the altitude of the triangle having the greatest possible area?

1. A.4√3 cm.
2. B.2√3 cm.
3. C.6 cm.
4. D.3 cm.

If $f(x)=e^{|x|}$, then which one of the following is correct?

1. A.$f^{\prime }(0)=1$.
2. B.$f^{\prime }(0)=-1$.
3. C.$f^{\prime }(0)=0$.
4. D.$f^{\prime }(0)$ does not exist.

If x + y = 20 and P = xy, then what is the maximum value of P?

1. A.100
2. B.96
3. C.84
4. D.50

What is the derivative of sin(ln x) + cos(ln x) with respect to x at x = e?

1. A.$\frac{\cos 1-\sin 1}{e}$
2. B.$\frac{\sin 1-\cos 1}{e}$
3. C.$\frac{\cos 1+\sin 1}{e}$
4. D.1

If $x=e^t\cos t$ and $y=e^t\sin t$, then what is $\frac{\mathrm{d}x}{\mathrm{d}y}$ at $t=0$ equal to?

1. A.0
2. B.1
3. C.$2e$
4. D.-1

What is the maximum value of sin 2x ⋅ cos 2x?

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

What is the maximum number of points of intersection of 5 non-overlapping circles?

1. A.10
2. B.15
3. C.20
4. D.25

The radius of a circle is increasing at the rate of $0.7cm/sec$. What is the rate of increase of its circumference?

1. A.$4.4cm/sec$
2. B.$8.4cm/sec$
3. C.$8.8cm/sec$
4. D.$15.4cm/sec$

If ${\lim }_{x\to 1}\frac{x^4-1}{x-1}={\lim }_{x\to k}\frac{x^3-k^3}{x^2-k^2}$, where $k\ne 0$, then what is the value of $k$?

1. A.$\frac{2}{3}$
2. B.$\frac{4}{3}$
3. C.$\frac{8}{3}$
4. D.4

If $f(x)=\frac{\sin x}{x}$, where $x\in \mathbf{R}$, is to be continuous at $x=0$, then the value of the function at $x=0$

1. A.should be 0
2. B.should be 1
3. C.should be 2
4. D.cannot be determined

What is the minimum value of $|x-1|$, where $x\in \mathbf{R}$ ?

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