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Mathematical Reasoning: JEE Main & Advanced PYQs

95 previous year questions on Mathematical Reasoning from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-95 of 95

  1. Negation of the Boolean expression $p\Leftrightarrow(q\Rightarrow p)$ is
  2. The statement $(p\Rightarrow q)\vee(p\Rightarrow r)$ is NOT equivalent to
  3. Which of the following is a tautology ?
  4. Which of the following statements is a tautology ?
  5. Let the operations $*,\odot\in\{\wedge,\vee\}$. If $(p*q)\odot(p\odot\sim q)$ is a tautology, then the ordered pair $(*,\odot)$ is :
  6. The statement $(\sim(p\Leftrightarrow\sim q))\wedge q$ is :
  7. The proposition $p\to\sim(p\wedge\sim q)$ is equivalent to :
  8. If the truth value of the statement $(P\wedge(\sim R))\to((\sim R)\wedge Q)$ is F, then the truth value of which of the following is F ?
  9. Given the following two statements : ($S_1$) : $(q\vee p)\to(p\leftrightarrow\sim q)$ is a tautology. ($S_2$) : $\sim q\wedge(\sim p\leftrightarrow q)$ is a fallacy. …
  10. The negation of the Boolean expression $x\leftrightarrow\sim y$ is equivalent to :
  11. Given $n^{4}<10^{n}$ for a fixed positive integer $n\ge2$, prove that $(n+1)^{4}<10^{n+1}$.
  12. Use mathematical induction to prove: if $n$ is any odd positive integer, then $n\left(n^{2}-1\right)$ is divisible by $24$.
  13. Use the method of mathematical induction to show that $2\cdot7^{n}+3\cdot5^{n}-5$ is divisible by $24$ for all $n>0$.
  14. If $p$ be a natural number then prove that $p^{n+1}+(p+1)^{2n-1}$ is divisible by $p^{2}+p+1$ for every positive integer $n$.
  15. Prove by mathematical induction that $\dfrac{(2n)!}{2^{2n}(n!)^{2}}\le\dfrac{1}{(3n+1)^{1/2}}$ for all positive integers $n$.
  16. Prove that $\dfrac{n^{7}}{7}+\dfrac{n^{5}}{5}+\dfrac{2n^{3}}{3}-\dfrac{n}{105}$ is an integer for every positive integer $n$.
  17. Let $p\ge3$ be an integer and $\alpha,\beta$ be the roots of $x^{2}-(p+1)x+1=0$. Using mathematical induction show that $\alpha^{n}+\beta^{n}$ (i) is an integer and (ii) …
  18. If $x$ is not an integral multiple of $2\pi$, use mathematical induction to prove that $$\cos x+\cos2x+\cdots+\cos …
  19. Using mathematical induction prove that for every integer $n\ge1$, $\left(3^{2^{n}}-1\right)$ is divisible by $2^{n+2}$ but not by $2^{n+3}$.
  20. Use mathematical induction to show that $(25)^{n+1}-24n+5735$ is divisible by $(24)^{2}$ for all $n=1,2,\ldots$
  21. Which of the following statements is a tautology ?
  22. If $p\to(p\wedge\sim q)$ is false, then the truth values of $p$ and $q$ are respectively :
  23. The compound statement $(P\vee Q)\wedge(\sim P)\Rightarrow Q$ is equivalent to :
  24. Which one of the following is a tautology ?
  25. Let $p$ : Ramesh listens to music. $q$ : Ramesh is out of his village. $r$ : It is Sunday. $s$ : It is Saturday. Then the statement ``Ramesh listens to music only if he …
  26. Contrapositive of the statement : `If a function $f$ is differentiable at $a$, then it is also continuous at $a$', is :
  27. Consider the statement : ``For an integer $n$, if $n^3-1$ is even, then $n$ is odd.'' The contrapositive statement of this statement is :
  28. The contrapositive of the statement ``If I reach the station in time, then I will catch the train'' is :
  29. The negation of the statement "If I become a teacher, then I will open a school", is :
  30. Negation of the statement : `$\sqrt5$ is an integer or $5$ is irrational' is :
  31. Which of the following is the negation of the statement ``for all $M>0$, there exists $x\in S$ such that $x\ge M$'' ?
  32. Consider the following statements : $p$ : Amit plays cricket $q$ : Amit is out of Delhi. $r$ : It is Sunday Then the statement "Amit plays cricket only if he is in Delhi …
  33. Which of the following Boolean expression is a tautology ?
  34. The statement $(p\rightarrow(q\rightarrow p))\rightarrow(p\rightarrow(p\vee q))$ is :
  35. The negation of the Boolean expression $p\vee(\sim p\wedge q)$ is equivalent to :
  36. Let $S(K)=1+3+5+\cdots+(2K-1)=3+K^{2}$. Then which of the following is true?
  37. Statement - 1 : $\sim(p\leftrightarrow\sim q)$ is equivalent to $p\leftrightarrow q$. Statement - 2 : $\sim(p\leftrightarrow\sim q)$ is a tautology.
  38. Let $p$ be the statement ``$x$ is an irrational number'', $q$ be the statement ``$y$ is a transcendental number'', and $r$ be the statement ``$x$ is a rational number …
  39. The statement $p\rightarrow(q\rightarrow p)$ is equivalent to
  40. Let $S$ be a non-empty subset of $\mathbb{R}$. Consider the following statement: $P$ : There is a rational number $x\in S$ such that $x>0$. Which of the following …
  41. The statement $\sim(p\leftrightarrow\sim q)$ is :
  42. The negation of $\sim s\vee(\sim r\wedge s)$ is equivalent to :
  43. Consider : Statement - I : $(p\wedge\sim q)\wedge(\sim p\wedge q)$ is a fallacy. Statement - II : $(p\rightarrow q)\leftrightarrow(\sim q\rightarrow\sim p)$ is a …
  44. Among the statements: $(S1)$: $((p\vee q)\Rightarrow r)\Leftrightarrow(p\Rightarrow r)$ $(S2)$ …
  45. Consider the following three statements : (A) If $3+3=7$ then $4+3=8$ (B) If $5+3=8$ then earth is flat. (C) If both (A) and (B) are true then $5+6=17$. Then, which of …

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