Limits, Continuity & Derivatives · NDA previous year questions

What is the derivative of $\frac{x}{|x|}$ with respect to $x$, where $x<0$ ?

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

If $f(x)$ is differentiable at $x=a$, then consider the following statements : I. $f(x)$ is continuous at $x=a$ II. ${\lim }_{x\to a}f(x)=f(a)$ Which of the statements given above is/are correct ?

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

What is the length of the longest interval in which the function $f(x)=2{\cos }^{2}x-1$ is decreasing ?

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

Passage: Let $\int \frac{\sin \theta d\theta }{(2+\cos \theta )(3+4\cos \theta )}=A\ln |2+\cos \theta |+B\ln |3+4\cos \theta |$. Question: What is the value of $B$ ?

1. A.$-\frac{2}{5}$
2. B.$-\frac{1}{5}$
3. C.$\frac{1}{5}$
4. D.$\frac{2}{5}$

Passage: Let $\int \frac{\sin \theta d\theta }{(2+\cos \theta )(3+4\cos \theta )}=A\ln |2+\cos \theta |+B\ln |3+4\cos \theta |$. Question: What is the value of $A$ ?

1. A.$-\frac{2}{5}$
2. B.$-\frac{1}{5}$
3. C.$\frac{1}{5}$
4. D.$\frac{2}{5}$

Passage: Let $f(x)=\sin x$ and $g(x)-f(x)=f(4-x)$. Question: What is ${\int }_0^4\frac{f(4-x)}{g(4-x)}dx$ equal to ?

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

Passage: Let $f(x)=\sin x$ and $g(x)-f(x)=f(4-x)$. Question: What is ${\int }_0^4\frac{f(x)}{g(x)}dx$ equal to ?

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

1. A.$k$
2. B.$2k$
3. C.$k(a+b)$
4. D.$\frac{k}{a+b}$

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

For the following three (03) items: Consider the function f(x) = x|x| What is ${\lim }_{x\to -1}f(x)$ equal to?

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

For the following three (03) items: Consider the function f(x) = x|x| Consider the following statements: I. The function is increasing in the interval $(-\infty ,\infty )$, II. The function is differentiable at x = 0 Which of the statements given above is/are correct?

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

For the following three (03) items: Let $y=f(x)=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+\ln \sqrt{1-x^2}$. What is the slope of the tangent to the curve y = f(x) at x = 0.5

1. A.$4\pi \sqrt{3}/27$
2. B.$8\pi \sqrt{3}/27$
3. C.$4\pi$
4. D.$8\pi$

For the following three (03) items: Let $y=f(x)=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+\ln \sqrt{1-x^2}$. What is $\frac{d^{2}y}{d{x}^2}$ at x = 0 equal to?

1. A.0
2. B.0.5
3. C.1
4. D.1.5

For the following two (02) items: Consider the function $f(x)=1-\sqrt[3]{(x-1{)}^2}$ The function has

1. A.a minimum at x = 1
2. B.a maximum at x = 1
3. C.neither maximum nor minimum at x = 1
4. D.no extremum

For the following two (02) items: Consider the function $f(x)=\{\begin{array}{ll}4(5^x),& x<0\\ 8k+x,& x\ge 0\end{array}$ If the function is continuous, then what is the value of k?

1. A.0.5
2. B.1
3. C.1.5
4. D.2

For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5]. Consider the following statements: I. The function is differentiable at x = 3 II. The function is differentiable at x = 4 Which of the statements given above is/are correct?

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

Consider the following statements regarding the function f(x) = 1/(x - 5) Statement-I: f(x) is decreasing on the intervals x < 5 and x > 5 Statement-II: f''(x) > 0 for all x ≠ 5. Which one of the following is correct in respect of the above statements?

1. A.Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
2. B.Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
3. C.Statement-I is correct but Statement-II is not correct
4. D.Statement-I is not correct but Statement-II is correct

Consider the following statements: Statement-I: The function $f(x)=\frac{x^3+128}{x}$ has a minimum value 48 at x = 4 Statement-II: As x increases through 4, f'(x) changes sign from positive to negative. Which one of the following is correct in respect of the above statements?

1. A.Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
2. B.Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
3. C.Statement-I is correct but Statement-II is not correct
4. D.Statement-I is not correct but Statement-II is correct

Consider the following for the two (02) items that follow: Let p=|sina-sin(\alpha-90°)|. What is the minimum value of p?

1. A.0
2. B.1/2
3. C.$1/\sqrt{2}$
4. D.1

Consider the following for the two (02) items that follow: Let $p=|\sin \alpha -\sin (\alpha -90^{\circ })|$. What is the maximum value of p?

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

What is ∠A equal to if the area of the triangle is maximum?

1. A.π/6
2. B.π/4
3. C.π/3
4. D.5π/12

What is the maximum area of the triangle?

1. A.$\sqrt{3}/2$
2. B.$\sqrt{3}$
3. C.$\sqrt{6}/2$
4. D.$\sqrt{6}$

Consider the following for the two (02) items that follow: Let $(x+y{)}^{p+q}=x^p{y}^q$, where p,q are positive integers. The derivative of y with respect to x

1. A.depends on p only
2. B.depends on q only
3. C.depends on both p and q
4. D.is independent of both p and q

If p+q=10, then what is $\frac{dy}{dx}$ equal to?

1. A.$\frac{y}{x}$
2. B.$xy$
3. C.$x^{10}{y}^{10}$
4. D.${\left(\frac{y}{x}\right)}^{10}$

What is the nature of the curve?

1. A.A straight line passing through (1,4)
2. B.A straight line passing through (-1, 4)
3. C.A parabola with vertex at origin and focus at (2,0)
4. D.A parabola with vertex at origin and focus at (1, 0)