Limits, Continuity & Derivatives · NDA previous year questions

1. A.2, 3, 5, 8, 9, 12
2. B.1, 3, 5, 8, 10, 12
3. C.1, 3, 5, 8, 9, 12
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. $$\begin{array}{l}\text{List I (Function)}\\ \text{A. }\sin x\\ \text{B. }\cos x\\ \text{C. }\tan x\end{array}$$ $$\begin{array}{l}\text{List II (Property)}\\ \text{1. Periodic function}\\ \text{2. Non-periodic function}\\ \text{3. Continuous at every point on }(-\infty ,\infty )\\ \text{4. Discontinuous function}\\ \text{5. Differentiable at every point on }(-\infty ,\infty )\\ \text{6. Not differentiable at every point on }(-\infty ,\infty )\\ \text{7. has period }\pi \\ \text{8. has period }2\pi \\ \text{9. increases on }(0,\pi /2)\\ \text{10. decreases on }(0,\pi /2)\\ \text{11. increases on }(\pi /2,\pi )\\ \text{12. decreases on }(\pi /2,\pi )\end{array}$$ C is associated with

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

If p and q are positive integers, then which one of the following equations has $p-\sqrt{q}$ as one of its roots?

1. A.$x^2-2px-(p^2-q)=0$
2. B.$x^2-2px+(p^2-q)=0$
3. C.$x^2+2px-(p^2-q)=0$
4. D.$x^2+2px+(p^2-q)=0$

Given two squares of sides x and y such that $y=x+x^2$. What is the rate of change of area of the second square with respect to the area of the first square?

1. A.$1+3x+2x^2$
2. B.$1+2x+3x^2$
3. C.$1-2x+3x^2$
4. D.$1-2x-3x^2$

What is the maximum point on the curve $x=e^{x}y$?

1. A.$(1,e)$
2. B.$(1,e^{-1})$
3. C.$(e,1)$
4. D.$(e^{-1},1)$

What is the derivative of $x\sqrt{a^2-x^2}+a^2{\sin }^{-1}\left(\frac{x}{a}\right)$?

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

If $(-5,4)$ divides the line segment between the coordinate axes in the ratio $1:2$, then what is its equation?

1. A.$8x+5y+20=0$
2. B.$5x+8y-7=0$
3. C.$8x-5y+60=0$
4. D.$5x-8y+57=0$

What is the value of $\left|\begin{array}{cc}\cos 15^{\circ }& \sin 15^{\circ }\\ \cos 45^{\circ }& \sin 45^{\circ }\end{array}\right|\times \left|\begin{array}{cc}\cos 45^{\circ }& \cos 15^{\circ }\\ \sin 45^{\circ }& \sin 15^{\circ }\end{array}\right|$?

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

Out of 32 persons, 30 invest in National Savings Certificates and 17 invest in shares. What is the number of persons who invest in both?

1. A.13
2. B.15
3. C.17
4. D.19

The function $f(x)=x\mathrm{cosec}x$ is

1. A.continuous for all values of x
2. B.discontinuous everywhere
3. C.continuous for all x except at $x=n\pi$, where n is an integer
4. D.continuous for all x except at $x=n\pi /2$, where n is an integer

What is the value of ${\lim }_{x\to 1}\frac{(x-1{)}^2}{|x-1|}$?

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

If $f(x)=k{x}^3-9x^2+9x+3$ is monotonically increasing in every interval, then which one of the following is correct?

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

If ${\sin }^{-1}\frac{5}{x}+{\sin }^{-1}\frac{12}{x}=\frac{\pi }{2}$, then what is the value of x?

1. A.1
2. B.7
3. C.13
4. D.17

Statement I: $y=-{\tan }^{-1}(x^{-1})+1$ is an increasing function of x. Statement II: $\frac{dy}{dx}$ is positive for all values of x. Which one of the following is correct in respect of the above statements?

1. A.Both statements I and II are independently correct and statement II is the correct explanation of statement I
2. B.Both statements I and II are independently correct but statement II is not the correct explanation of statement I
3. C.Statement I is correct but statement II is false
4. D.Statement I is false but statement II is correct

What is the maximum value of $\sin \theta \cos \theta$ ?

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

If $\sin x+\mathrm{cosec}x=2$, then what is the value of ${\sin }^{4}x+{\mathrm{cosec}}^{4}x$ ?

1. A.2
2. B.4
3. C.8
4. D.16

What is the value of $\tan 15^{\circ }+\cot 15^{\circ }$ ?

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

If $A+B+C=\pi /2$, then what is the value of $\tan A\tan B+\tan B\tan C+\tan C\tan A$ ?

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

If angles of a triangle are in the ratio $1:2:3$, then what is the ratio of its corresponding sides ?

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

If $C(n,12)=C(n,8)$, then what is the value of $C(22,n)$ ?

1. A.131
2. B.231
3. C.256
4. D.292

The function $y={\tan }^{-1}x-x$

1. A.is always decreasing
2. B.is always increasing
3. C.first increases and then decreases
4. D.first decreases and then increases

If $n(A)=115$, $n(B)=326$, $n(A-B)=47$, then what is $n(A\cup B)$ equal to ?

1. A.373
2. B.165
3. C.370
4. D.394

Consider the following function $f:R\to R$ such that $f(x)=x$ if $x\ge 0$ and $f(x)=-x^2$ if $x<0$. Then which one of the following is correct ?

1. A.$f(x)$ is continuous at every $x\in R$.
2. B.$f(x)$ is continuous at $x=0$ only.
3. C.$f(x)$ is discontinuous at $x=0$ only.
4. D.$f(x)$ is discontinuous at every $x\in R$.

What is the derivative of ${\log }_{x}5$ with respect to ${\log }_{5}x$ ?

1. A.$-({\log }_{5}x{)}^{-2}$
2. B.$({\log }_{5}x{)}^{-2}$
3. C.$-({\log }_{x}5{)}^{-2}$
4. D.$({\log }_{x}5{)}^{-2}$

The velocity of telegraphic communication is given by $v=x^2\ln (1/x)$ where $x$ is the displacement. For maximum velocity, $x$ equals to ?

1. A.$e^{1/2}$
2. B.$e^{-1/2}$
3. C.$(2e{)}^{-1}$
4. D.$2e^{-1/2}$