Limits, Continuity & Derivatives · NDA previous year questions

2023 29 questions

1. Q132023 · 3 Sept 2023

   What is the maximum number of points of intersection of 10 circles?

   1. A.45
   2. B.60
   3. C.90
   4. D.120
   Show answer

   Answer: C

2. Q442023 · 3 Sept 2023

   Consider the following for the next items that follow: Let a function f be defined on $\mathbb{R}-\{0\}$ and $2f(x)+f\left(\frac{1}{x}\right)=x+3$. If f is differentiable, then what is f'(0.5) equal to?

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

   Answer: C

3. Q452023 · 3 Sept 2023

   Consider the following for the next items that follow: A function is defined by $f(x)=\left|\begin{array}{ccc}x+1& 2& 3\\ 2& x+4& 6\\ 3& 6& x+9\end{array}\right|$. The function is decreasing on :

   1. A.$\left[-\frac{28}{3},0\right]$
   2. B.$\left[0,\frac{28}{3}\right]$
   3. C.$\left[0,\frac{50}{3}\right]$
   4. D.$\left[0,\frac{56}{3}\right]$
   Show answer

   Answer: A

4. Q462023 · 3 Sept 2023

   Consider the following for the next items that follow: A function is defined by $f(x)=\left|\begin{array}{ccc}x+1& 2& 3\\ 2& x+4& 6\\ 3& 6& x+9\end{array}\right|$. The function attains local minimum value at :

   1. A.$x=-\frac{28}{3}$
   2. B.x = -1
   3. C.x = 0
   4. D.$x=\frac{28}{3}$
   Show answer

   Answer: C

5. Q472023 · 3 Sept 2023

   Consider the following for the next items that follow: Given that $4x^2+y^2=9$. What is the maximum value of y?

   1. A.$\frac{3}{2}$
   2. B.3
   3. C.4
   4. D.6
   Show answer

   Answer: B

6. Q482023 · 3 Sept 2023

   Consider the following for the next items that follow: Given that $4x^2+y^2=9$. What is the maximum value of xy ?

   1. A.$\frac{9}{4}$
   2. B.$\frac{3}{2}$
   3. C.$\frac{4}{9}$
   4. D.$\frac{2}{3}$
   Show answer

   Answer: A

7. Q592023 · 3 Sept 2023

   Consider the following for the next items that follow: Let $f(x)=\{\begin{array}{ll}\frac{x-3}{|x-3|}+a;& x<3\\ a-b;& x=3\\ \frac{x-3}{|x-3|}+b;& x>3\end{array}$ and f(x) be continuous at x = 3. What is the value of a ?

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

   Answer: D

8. Q602023 · 3 Sept 2023

   Consider the following for the next items that follow: Let $f(x)=\{\begin{array}{ll}\frac{x-3}{|x-3|}+a;& x<3\\ a-b;& x=3\\ \frac{x-3}{|x-3|}+b;& x>3\end{array}$ and f(x) be continuous at x = 3. What is the value of b?

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

   Answer: B

9. Q612023 · 3 Sept 2023

   Consider the following for the next items that follow: Let $I={\int }_{-2\pi }^{2\pi }\frac{{\sin }^{4}x+{\cos }^{4}x}{1+3^x}dx$ What is ${\int }_0^{\pi }({\sin }^{4}x+{\cos }^{4}x)dx$ equal to?

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

   Answer: B

10. Q622023 · 3 Sept 2023

    Consider the following for the next items that follow: Let $I={\int }_{-2\pi }^{2\pi }\frac{{\sin }^{4}x+{\cos }^{4}x}{1+3^x}dx$ What is I equal to?

    1. A.0
    2. B.$\frac{3\pi }{4}$
    3. C.$\frac{3\pi }{2}$
    4. D.$3\pi$
    Show answer

    Answer: C

11. Q632023 · 3 Sept 2023

    Consider the following for the next items that follow: Let $f(x)=\{\begin{array}{ll}ax(x+1)+b,& x<1\\ x-1,& 1\le x\le 2\end{array}$ If the function f(x) is differentiable at x = 1, then what is the value of (a + b)?

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

    Answer: A

12. Q642023 · 3 Sept 2023

    Consider the following for the next items that follow: Let $f(x)=\{\begin{array}{ll}ax(x+1)+b,& x<1\\ x-1,& 1\le x\le 2\end{array}$ What is ${\lim }_{x\to 0}f(x)$ equal to ?

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

    Answer: B

13. Q652023 · 3 Sept 2023

    If f(x) = |ln|x|| where 0 < x < 1, then what is f'(0.5) equal to ?

    1. A.-2
    2. B.-1
    3. C.0
    4. D.2
    Show answer

    Answer: A

14. Q372023 · 16 Apr 2023

    Consider the following for the next two (02) items that follow : Let $x=\frac{{\sin }^{2}A+\sin A+1}{\sin A}$ where $0<A\le \frac{\pi }{2}$ What is the minimum value of x ?

    1. A.1
    2. B.2
    3. C.3
    4. D.4
    Show answer

    Answer: C

15. Q382023 · 16 Apr 2023
    
    1. A.$\frac{\pi }{6}$
    2. B.$\frac{\pi }{4}$
    3. C.$\frac{\pi }{3}$
    4. D.$\frac{\pi }{2}$
    Show answer

    Answer: D

16. Q412023 · 16 Apr 2023
    Passage

    Consider the following for the next two (02) items that follow : Consider the function f(x) = |x - 2| + |3 - x| + |4 - x|, where x ∈ R.

    At what value of x does the function attain minimum value ?

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

    Answer: B

17. Q422023 · 16 Apr 2023
    Passage

    Consider the following for the next two (02) items that follow : Consider the function f(x) = |x - 2| + |3 - x| + |4 - x|, where x ∈ R.

    What is the minimum value of the function ?

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

    Answer: A

18. Q432023 · 16 Apr 2023
    Passage

    Consider the following for the next two (02) items that follow : Consider the sum S = 0! + 1! + 2! + 3! + 4! +... .+ 100!

    If the sum S is divided by 8, what is the remainder ?

    1. A.0
    2. B.1
    3. C.2
    4. D.Cannot be determined
    Show answer

    Answer: C

19. Q442023 · 16 Apr 2023
    Passage

    Consider the following for the next two (02) items that follow : Consider the sum S = 0! + 1! + 2! + 3! + 4! +... .+ 100!

    If the sum S is divided by 60, what is the remainder ?

    1. A.1
    2. B.3
    3. C.17
    4. D.34
    Show answer

    Answer: D

20. Q862023 · 16 Apr 2023
    Passage

    Consider the following for the next three (03) items that follow : Let f(x) = |ln x|, x ≠ 1

    What is the derivative of f(x) at x = 0.5 ?

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

    Answer: A

21. Q872023 · 16 Apr 2023
    Passage

    Consider the following for the next three (03) items that follow : Let f(x) = |ln x|, x ≠ 1

    What is the derivative of f(x) at x = 2 ?

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

22. Q882023 · 16 Apr 2023
    Passage

    Consider the following for the next three (03) items that follow : Let f(x) = |ln x|, x ≠ 1

    What is the derivative of fof(x), where 1 < x < 2 ?

    1. A.$\frac{1}{\ln x}$
    2. B.$\frac{1}{x\ln x}$
    3. C.$-\frac{1}{\ln x}$
    4. D.$-\frac{1}{x\ln x}$
    Show answer

    Answer: D

23. Q892023 · 16 Apr 2023

    Consider the following for the next two (02) items that follow Let $f(x)=\{\begin{array}{ll}x+6,& x\le 1\\ px+q,& 1<x<2\\ 5x,& x\ge 2\end{array}$ and f(x) is continuous What is the value of p ?

    1. A.2
    2. B.3
    3. C.4
    4. D.5
    Show answer

    Answer: B

24. Q902023 · 16 Apr 2023
    
    1. A.2
    2. B.3
    3. C.4
    4. D.5
    Show answer

    Answer: C

25. Q912023 · 16 Apr 2023

    Consider the following statements : 1. $f(x)=\ln x$ is increasing in $(0,\infty )$ 2. $g(x)=e^x+e^{\frac{1}{x}}$ is decreasing in $(0,\infty )$ 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
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    Answer: A