Calculus

Questions

1. GATE CS 2026 Set 1NAT1 markEasy (estimate)

   GATE CS 2026 Set 1, Q.32

   Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined as follows…

   Calculus

2. GATE CS 2026 Set 1MSQ2 marksMedium (estimate)

   GATE CS 2026 Set 1, Q.46

   Let $f:\mathbb{R}\to \mathbb{R}$ be defined as follows: $f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right)$ Which of the following statements is/ar...

   Calculus

3. GATE CS 2026 Set 2MSQ1 markEasy (estimate)

   GATE CS 2026 Set 2, Q.27

   For a real number $a$, let $I(a)={\int }_{-1}^1(3x^2-ax+1)dx$. Which of the following statements is/are true?

   Calculus

4. GATE CS 2026 Set 2NAT2 marksMedium (estimate)

   GATE CS 2026 Set 2, Q.64

   Consider a function $f:(0,1)\to \{0,1\}$ defined as follows. For a real number $r\in (0,1)$, $f(r)=1$ if the second digit after the decimal point in $r$ is one of...

   Calculus

5. GATE CS 2025 Set 1NAT1 markEasy (estimate)

   GATE CS 2025 Set 1, Q.31

   Consider the given function $f(x)$. $f(x)=\{\begin{array}{ll}ax+b& \text{for }x<1\\ x^3+x^2+1& \text{for }x\ge 1\end{array}$ If the function is differentiable ever...

   Calculus

6. GATE CS 2025 Set 2MCQ1 markEasy (estimate)

   GATE CS 2025 Set 2, Q.12

   The value of $x$ such that $x>1$, satisfying the equation ${\int }_1^{x}t\ln tdt=\frac{1}{4}$ is

   Calculus

7. GATE CS 2025 Set 2NAT2 marksMedium (estimate)

   GATE CS 2025 Set 2, Q.65

   The unit interval $(0,1)$ is divided at a point chosen uniformly distributed over $(0,1)$ in $\mathbb{R}$ into two disjoint subintervals. The expected length of the subinterva...

   Probability and Statistics · Calculus

8. GATE CS 2024 Set 1MCQ1 markEasy (estimate)

   GATE CS 2024 Set 1, Q.11

   Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x)=\max \{x,x^3\},x\in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of al...

   Calculus

9. GATE CS 2024 Set 1NAT2 marksMedium (estimate)

   GATE CS 2024 Set 1, Q.55

   The baseline execution time of a program on a 2 GHz single core machine is 100 nanoseconds (ns). The code corresponding to 90% of the execution time can be fully parallelized. T...

   Instruction Pipelining and Hazards · Calculus

10. GATE CS 2024 Set 2MCQ1 markEasy (estimate)

    GATE CS 2024 Set 2, Q.16

    Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x)=1-f(2-x)$ Which one of the following options is the CORRECT value of…

    Calculus

11. GATE CS 2023MSQ1 markEasy (estimate)

    GATE CS 2023, Q.28

    Let $f(x)=x^3+15x^2-33x-36$ be a real-valued function. Which of the following statements is/are TRUE?

    Calculus

12. GATE CS 2023NAT1 markEasy (estimate)

    GATE CS 2023, Q.31

    The value of the definite integral ${\int }_{-3}^3{\int }_{-2}^2{\int }_{-1}^1(4x^{2}y-z^3)dzdydx$ is _______. (Rounded off to the nearest integer)

    Calculus

13. GATE CS 2022NAT1 markEasy (estimate)

    GATE CS 2022, Q.34

    The value of the following limit is _____________. ${\lim }_{x\to 0^{+}}\frac{x}{1-e^{2x}}$

    Calculus

14. GATE CS 2021 Set 1NAT1 markEasy (estimate)

    GATE CS 2021 Set 1, Q.30 (CS Q.20)

    Consider the following expression. ${\lim }_{x\to -3}\frac{\sqrt{2x+22}-4}{x+3}$ The value of the above expression (rounded to 2 decimal places) is __________

    Calculus

15. GATE CS 2021 Set 2NAT1 markEasy (estimate)

    GATE CS 2021 Set 2, Q.35 (CS Q.25)

    Suppose that $f:\mathbb{R}\to \mathbb{R}$ is a continuous function on the interval $[-3,3]$ and a differentiable function in the interval $(-3,3)$ such that for every $x$ in...

    Calculus

16. GATE CS 2020MCQ1 markEasy (estimate)

    GATE CS 2020, Q.11 (CS Q.1)

    Consider the functions I. $e^{-x}$ II. $x^2-\sin x$ III. $\sqrt{x^3+1}$ Which of the above functions is/are increasing everywhere in $[0,1]$?

    Calculus

17. GATE CS 2019MCQ1 markEasy (estimate)

    GATE CS 2019, Q.23 (CS Q.13)

    Compute ${\lim }_{x\to 3}\frac{x^4-81}{2x^2-5x-3}$

    Calculus

18. GATE CS 2018NAT1 markEasy (estimate)

    GATE CS 2018, Q.26 (CS Q.16)

    The value of ${\int }_0^{\pi /4}x\cos (x^2)dx$ correct to three decimal places (assuming that $\pi =3.14$) is _____.

    Calculus

19. GATE CS 2017 Set 1MCQ2 marksEasy (estimate)

    GATE CS 2017 Set 1, Q.28 (CS Q.28)

    The value of ${\lim }_{x\to 1}\frac{x^7-2x^5+1}{x^3-3x^2+2}$

    Calculus

20. GATE CS 2017 Set 2MCQ1 markMedium (estimate)

    GATE CS 2017 Set 2, Q.10 (CS Q.10)

    If $f(x)=R\sin \left(\frac{\pi x}{2}\right)+S$, $f^{\prime }\left(\frac{1}{2}\right)=\sqrt{2}$ and ${\int }_0^{1}f(x)dx=\frac{2R}{\pi }$, then the constants $R$ and $S$ are, respe...

    Calculus

21. GATE CS 2017 Set 2MCQ2 marksMedium (estimate)

    GATE CS 2017 Set 2, Q.64 (GA Q.9)

    The number of roots of $e^x+0.5x^2-2=0$ in the range $[-5,5]$ is

    Quantitative Aptitude · Calculus

22. GATE CS 2016 Set 1NAT1 markEasy (estimate)

    GATE CS 2016 Set 1, Q.13 (CS Q.3)

    The value of ${\lim }_{x\to 4}\frac{\sin (x-4)}{x-4}$ is _____.

    Calculus

23. GATE CS 2016 Set 2NAT1 markMedium (estimate)

    GATE CS 2016 Set 2, Q.12 (CS Q.2)

    Let $f(x)$ be a polynomial and $g(x)=f^{\prime }(x)$ be its derivative. If the degree of $(f(x)+f(-x))$ is 10, then the degree of $(g(x)-g(-x))$ is ________.

    Calculus

24. GATE CS 2015 Set 1MCQ1 markEasy (estimate)

    GATE CS 2015 Set 1, Q.14 (CS Q.4)

    ${\lim }_{x\to \infty }{x}^{1/x}$ is:

    Calculus

25. GATE CS 2015 Set 1NAT2 marksEasy (estimate)

    GATE CS 2015 Set 1, Q.36 (CS Q.26)

    ${\sum }_{x=1}^{99}\frac{1}{x(x+1)}$ The value of the summation is _____.

    Calculus

26. GATE CS 2015 Set 1NAT2 marksEasy (estimate)

    GATE CS 2015 Set 1, Q.54 (CS Q.44)

    ${\int }_{1/\pi }^{2/\pi }\frac{\cos (1/x)}{x^2}dx$ The value of the integral is _____.

    Calculus

27. GATE CS 2015 Set 2MCQ2 marksMedium (estimate)

    GATE CS 2015 Set 2, Q.36 (CS Q.26)

    Let $f(x)=x^{-(1/3)}$ and $A$ denote the area of the region bounded by $f(x)$ and the X-axis, when $x$ varies from $-1$ to 1. Which of the following statements is/are TRUE? I)...

    Calculus

28. GATE CS 2015 Set 2MCQ2 marksEasy (estimate)

    GATE CS 2015 Set 2, Q.49 (CS Q.39)

    The secant method is used to find the root of an equation $f(x)=0$. It is started from two distinct estimates $x_a$ and $x_b$ for the root. It is an iterative procedure involv...

    Calculus

29. GATE CS 2015 Set 3MCQ1 markEasy (estimate)

    GATE CS 2015 Set 3, Q.19 (CS Q.9)

    The value of ${\lim }_{x\to \infty }(1+x^2{)}^{e^{-x}}$ is

    Calculus

30. GATE CS 2015 Set 3MCQ2 marksHard (estimate)

    GATE CS 2015 Set 3, Q.55 (CS Q.45)

    If for non-zero $x$, $af(x)+bf\left(\frac{1}{x}\right)=\frac{1}{x}-25$ where $a\ne b$ then ${\int }_1^{2}f(x)dx$ is

    Calculus

31. GATE CS 2015 Set 3NAT2 marksMedium (estimate)

    GATE CS 2015 Set 3, Q.60 (CS Q.50)

    The velocity $v$ (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time $t$ (in minutes) as follows: t 2 4 6 8 10 12 14 16 18 20 v 10 1...

    Calculus

32. GATE CS 2014 Set 1MCQ1 markMedium (estimate)

    GATE CS 2014 Set 1, Q.16 (CS Q.6)

    Let the function…

    Calculus

33. GATE CS 2014 Set 1NAT2 marksEasy (estimate)

    GATE CS 2014 Set 1, Q.56 (CS Q.46)

    The function $f(x)=x\sin x$ satisfies the following equation: $f^{\prime \prime }(x)+f(x)+t\cos x=0$. The value of $t$ is ______.

    Calculus

34. GATE CS 2014 Set 1MCQ2 marksMedium (estimate)

    GATE CS 2014 Set 1, Q.57 (CS Q.47)

    A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0)=f(2)=-1$ and $f(1)=1$. Which one of the following statements must be true?

    Calculus

35. GATE CS 2014 Set 2MCQ1 markEasy (estimate)

    GATE CS 2014 Set 2, Q.15 (CS Q.5)

    A non-zero polynomial $f(x)$ of degree 3 has roots at $x=1,x=2$ and $x=3$. Which one of the following must be TRUE?

    Calculus

36. GATE CS 2014 Set 2MCQ2 marksMedium (estimate)

    GATE CS 2014 Set 2, Q.56 (CS Q.46)

    In the Newton-Raphson method, an initial guess of $x_0=2$ is made and the sequence $x_0,x_1,x_2\dots$ is obtained for the function $0.75x^3-2x^2-2x+4=0$ Consider...

    Calculus

37. GATE CS 2014 Set 3NAT1 markMedium (estimate)

    GATE CS 2014 Set 3, Q.16 (CS Q.6)

    If ${\int }_0^{2\pi }|x\sin x|dx=k\pi$, then the value of $k$ is equal to _____.

    Calculus

38. GATE CS 2014 Set 3MCQ2 marksMedium (estimate)

    GATE CS 2014 Set 3, Q.56 (CS Q.46)

    With respect to the numerical evaluation of the definite integral, $K={\int }_a^b{x}^{2}dx$, where $a$ and $b$ are given, which of the following statements is/are TRUE? I) The v...

    Calculus

39. GATE CS 2014 Set 3MCQ2 marksMedium (estimate)

    GATE CS 2014 Set 3, Q.57 (CS Q.47)

    The value of the integral given below is ${\int }_0^{\pi }{x}^2\cos xdx$

    Calculus

40. GATE CS 2013MCQ1 markEasy (estimate)

    GATE CS 2013, Q.22 (CS Q.22)

    Which one of the following functions is continuous at $x=3$?

    Calculus

41. GATE CS 2013MCQ1 markEasy (estimate)

    GATE CS 2013, Q.23 (CS Q.23)

    Function $f$ is known at the following points…

    Calculus

42. GATE CS 2012MCQ1 markMedium (estimate)

    GATE CS 2012, Q.9 (CS Q.9)

    Consider the function $f(x)=\sin (x)$ in the interval $x\in [\pi /4,7\pi /4]$. The number and location(s) of the local minima of this function are

    Calculus

43. GATE CS 2012MCQ2 marksEasy (estimate)

    GATE CS 2012, Q.28 (CS Q.28)

    The bisection method is applied to compute a zero of the function $f(x)=x^4-x^3-x^2-4$ in the interval $[1,9]$. The method converges to a solution after ----- iterations.

    Calculus

44. GATE CS 2011MCQ2 marksEasy (estimate)

    GATE CS 2011, Q.31 (CS Q.31)

    Given $i=\sqrt{-1}$, what will be the evaluation of the definite integral ${\int }_0^{\pi /2}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx\text{ ?}$

    Calculus

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