Limits, Continuity & Derivatives · NDA previous year questions

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). How much money (in crores) is allocated to Education?

1. A.3000
2. B.6000
3. C.9000
4. D.10800

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). How much money (in crores) is allocated to both Agriculture and Employment?

1. A.20000
2. B.21000
3. C.24000
4. D.27000

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). How much excess money (in crores) is allocated to Miscellaneous over Education?

1. A.3600
2. B.4200
3. C.4500
4. D.4800

Consider the following statements in respect of the function $f(x)=x^3-1,x\in [-1,1]$ 1. $f(x)$ is increasing in $[-1,1]$ 2. $f(x)$ has no root in $(-1,1)$ 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

The largest value of $2x^3-3x^2-12x+5$ for $-2\le x\le 2$ occurs when

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

What is ${\int }_0^{\pi }\frac{dx}{1+2{\sin }^{2}x}$ equal to?

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

Which one of the following is correct in respect of the function $f(x)=|x|+x^2$?

1. A.$f(x)$ is not continuous at $x=0$
2. B.$f(x)$ is differentiable at $x=0$
3. C.$f(x)$ is continuous but not differentiable at $x=0$
4. D.None of the above

The function $f(x)=k\sin x+\frac{1}{3}\sin 3x$ has maximum value at $x=\frac{\pi }{3}$. What is the value of $k$?

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

What is the value of $k$ for which the following function $f(x)$ is continuous for all $x$? $$f(x)=\{\begin{array}{ll}\frac{x^3-3x+2}{(x-1{)}^2}& \text{for }x\ne 1\\ k& \text{for }x=1\end{array}$$

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

What is the maximum number of straight lines that can be drawn with any four points in a plane such that each line contains at least two of these points?

1. A.2
2. B.4
3. C.6
4. D.12

If $x+iy=\left|\begin{array}{ccc}6i& -3i& 1\\ 4& 3i& -1\\ 20& 3& i\end{array}\right|$, then what is $x-iy$ equal to?

1. A.$3+i$
2. B.$1+3i$
3. C.$3i$
4. D.0

A square is drawn by joining mid-points of the sides of a square. Another square is drawn inside the second square in the same way and the process is continued indefinitely. If the side of the first square is 16 cm, then what is the sum of the areas of all the squares?

1. A.256 sq. cm
2. B.512 sq. cm
3. C.1024 sq. cm
4. D.512/3 sq. cm

If the function $f(x)=\frac{x(x-2)}{x^2-4}$, $x\ne \pm 2$, is continuous at $x=2$, then what is $f(2)$ equal to?

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

If $x=\cos (2t)$ and $y={\sin }^{2}t$, then what is $\frac{d^{2}y}{d{x}^2}$ equal to?

1. A.0
2. B.$\sin (2t)$
3. C.$-\cos (2t)$
4. D.$-1/2$

At how many points is the function $f(x)=[x]$ discontinuous?

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

What is the interval over which the function $f(x)=6x-x^2,x>0$ is increasing?

1. A.$(0,3)$
2. B.$(3,6)$
3. C.$(6,9)$
4. D.None of the above

If $f$ and $g$ are two increasing functions such that $f\circ g$ is defined, then which one of the following is correct?

1. A.$f\circ g$ is always an increasing function
2. B.$f\circ g$ is always a decreasing function
3. C.$f\circ g$ is neither an increasing nor a decreasing function
4. D.None of the above

What is the value of ${\int }_1^2{e}^x\left(\frac{1}{x}-\frac{1}{x^2}\right)dx$ ?

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

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

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

Consider the following statements : 1. $f(x)=|x-3|$ is continuous at $x=0$. 2. $f(x)=|x-3|$ is differentiable 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 $(1111{)}_2+(1001{)}_2-(1010{)}_2$ equal to?

1. A.$(111{)}_2$
2. B.$(1100{)}_2$
3. C.$(1110{)}_2$
4. D.$(1010{)}_2$

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

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

If $x=k(\theta +\sin \theta )$ and $y=k(1+\cos \theta )$, then what is the derivative of $y$ with respect to $x$ at $\theta =\pi /2$ ?

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

What is the value of $1+i^2+i^4+i^6+\dots +i^{100}$ where $i=\sqrt{-1}$ ?

1. A.$0$
2. B.$1$
3. C.$-1$
4. D.None of the above

Directions : For the next 3 (three) questions to follow : Consider the following lists : Each item under List I is associated with one or more items under List II. $$\begin{array}{l}\text{List I (Function)}\\ \text{A. }\sin x\\ \text{B. }\cos x\\ \text{C. }\tan x\end{array}$$ $$\begin{array}{l}\text{List II (Property)}\\ \text{1. Periodic function}\\ \text{2. Non-periodic function}\\ \text{3. Continuous at every point on }(-\infty ,\infty )\\ \text{4. Discontinuous function}\\ \text{5. Differentiable at every point on }(-\infty ,\infty )\\ \text{6. Not differentiable at every point on }(-\infty ,\infty )\\ \text{7. has period }\pi \\ \text{8. has period }2\pi \\ \text{9. increases on }(0,\pi /2)\\ \text{10. decreases on }(0,\pi /2)\\ \text{11. increases on }(\pi /2,\pi )\\ \text{12. decreases on }(\pi /2,\pi )\end{array}$$ A is associated with

1. A.1, 3, 5, 8, 9, 12
2. B.2, 4, 6, 8, 10, 11
3. C.1, 3, 5, 7, 10, 11
4. D.None of the above