Limits, Continuity & Derivatives · NDA previous year questions

What is the derivative of ${\sin }^{2}x$ with respect to ${\cos }^{2}x$ ?

1. A.-1
2. B.1
3. C.$\sin 2x$
4. D.$\cos 2x$

What is the derivative of cosec(x°) ?

1. A.$-\mathrm{cosec}(x^{\circ })\cot (x^{\circ })$
2. B.$-\frac{\pi }{180}\mathrm{cosec}(x^{\circ })\cot (x^{\circ })$
3. C.$\frac{\pi }{180}\mathrm{cosec}(x^{\circ })\cot (x^{\circ })$
4. D.$-\frac{\pi }{180}\mathrm{cosec}(x)\cot (x)$

What is ${\lim }_{x\to 5}\frac{5-x}{|x-5|}$ equal to ?

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

What is ${\lim }_{x\to 1}\frac{x^9-1}{x^3-1}$ equal to ?

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

What is the ratio of the first term of A to that of B ?

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

1. A.11/29
2. B.22/49
3. C.33/59
4. D.44/69

If d is the common difference of A, and D is the common difference of B, then which one of the following is always correct ?

1. A.D > d
2. B.D < d
3. C.7D > 12d
4. D.None of the above

What is the minimum value of the function $f(x)={\log }_{10}(x^2+2x+11)$ ?

1. A.0
2. B.1
3. C.2
4. D.10

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

What is $\int e^x\{1+\ln x+x\ln x\}\mathrm{dx}$ equal to ?

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

What is $\int \frac{(\cos x{)}^{1.5}-(\sin x{)}^{1.5}}{\sqrt{\sin x\cdot \cos x}}\mathrm{dx}$ equal to ?

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

If $y=\frac{x\sqrt{x^2-16}}{2}-8\ln \left|x+\sqrt{x^2-16}\right|$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to ?

1. A.$x\sqrt{x^2-16}$
2. B.$x-\sqrt{x^2-16}$
3. C.$\sqrt{x^2-16}$
4. D.$4\sqrt{x^2-16}$

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

1. A.$\frac{\mathrm{dy}}{\mathrm{dx}}+xy(1+2\ln x)=0$
2. B.$\frac{\mathrm{dy}}{\mathrm{dx}}-xy(1+2\ln x)=0$
3. C.$\frac{\mathrm{dy}}{\mathrm{dx}}-2xy(1+\ln x)=0$
4. D.$\frac{\mathrm{dy}}{\mathrm{dx}}+2xy(1+\ln x)=0$

What is the maximum value of $3(\sin x-\cos x)+4({\cos }^{3}x-{\sin }^{3}x)$ ?

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

What is ${\lim }_{x\to 0}\frac{x}{\sqrt{1-\cos 4x}}$ equal to ?

1. A.$\frac{1}{2\sqrt{2}}$
2. B.$-\frac{1}{2\sqrt{2}}$
3. C.$\sqrt{2}$
4. D.Limit does not exist

What is ${\lim }_{x\to \frac{\pi }{2}}\frac{4x-2\pi }{\cos x}$ equal to ?

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

If $f(x)=\frac{x^2+x+|x|}{x}$, then what is ${\lim }_{x\to 0}f(x)$ equal to ?

1. A.0
2. B.1
3. C.2
4. D.${\lim }_{x\to 0}f(x)$ does not exist

Under which one of the following conditions does the function $f(x)=(p\sec x{)}^2+(q\mathrm{cosec}x{)}^2$ attain minimum value ?

1. A.${\tan }^{2}x=\frac{q}{p}$
2. B.${\cot }^{2}x=\frac{q}{p}$
3. C.${\tan }^{2}x=pq$
4. D.${\cot }^{2}x=pq$

Where does the function $f(x)={\sum }_{j=1}^7(x-j{)}^2$ attain its minimum value ?

1. A.$x=3.5$
2. B.$x=4$
3. C.$x=4.5$
4. D.$x=5$

Consider the following statements in respect of the function $f(x)=\{\begin{array}{ll}|x|+1,& 0<|x|\le 3\\ 1,& x=0\end{array}$ : 1. The function attains maximum value only at $x=3$ 2. The function attains local minimum only at $x=0$ Which of the statements given above 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 ${\int }_0^1\ln \left(\frac{1}{x}-1\right)\mathrm{dx}$ equal to ?

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

1. A.$k$
2. B.$10k$
3. C.$20k$
4. D.$40k$

What is the maximum value of n such that $5^n$ divides $(30!+35!)$, where n is a natural number?

1. A.4
2. B.6
3. C.7
4. D.8

What is the value of 2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 2 × 1) + .................. + 9(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) + 2 ?

1. A.11!
2. B.10!
3. C.10 + 10!
4. D.11 + 10!

Consider the following statements in respect of the function y = [x], x ∈ (-1, 1) where [.] is the greatest integer function: 1. Its derivative is 0 at x = 0.5 2. It is continuous 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