Question:
If p,q,r,s are statements, where: $p:A^{2}-B^{2}=(A-B)(A+B)$ for matrices $AB\ne BA$; $q:5\le5$; $r:{^{8}C_{1}+{}^{8}C_{2}+...+^{8}C_{8}=256}$; $s$: Max value of ${}^{8}C_{r}$ is 70. The statement with truth value true is:
If p,q,r,s are statements, where: $p:A^{2}-B^{2}=(A-B)(A+B)$ for matrices $AB\ne BA$; $q:5\le5$; $r:{^{8}C_{1}+{}^{8}C_{2}+...+^{8}C_{8}=256}$; $s$: Max value of ${}^{8}C_{r}$ is 70. The statement with truth value true is:
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Carefully evaluate individual statement values before checking logical combinations.
Updated On: Jun 19, 2026
- $(p\wedge\sim r)\vee(\sim q\wedge\sim s)$
- $(p\vee\sim q)\leftrightarrow(\sim r\rightarrow s)$
- $(p\leftrightarrow q)\wedge(\sim p\vee\sim q)$
- $(s\vee\sim p)\leftrightarrow(\sim p\wedge\sim r)$
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The Correct Option is D
Solution and Explanation
Step 1: Truth Values
- $p$: False (In matrices, formula holds only if $AB=BA$). - $q$: True ($5=5$ is true). - $r$: False ($\sum_{k=1}^8 {}^8C_k = 2^8 - {}^8C_0 = 256 - 1 = 255$). - $s$: True (${}^8C_4 = \frac{8\cdot7\cdot6\cdot5}{4\cdot3\cdot2\cdot1} = 70$).
Step 2: Analysis
- In (D): $(s \vee \sim p) \leftrightarrow (\sim p \wedge \sim r)$ - $(T \vee T) \leftrightarrow (T \wedge T) \implies T \leftrightarrow T$, which is True.
Step 3: Conclusion
Hence, (D) has the truth value True. Final Answer: (D)
- $p$: False (In matrices, formula holds only if $AB=BA$). - $q$: True ($5=5$ is true). - $r$: False ($\sum_{k=1}^8 {}^8C_k = 2^8 - {}^8C_0 = 256 - 1 = 255$). - $s$: True (${}^8C_4 = \frac{8\cdot7\cdot6\cdot5}{4\cdot3\cdot2\cdot1} = 70$).
Step 2: Analysis
- In (D): $(s \vee \sim p) \leftrightarrow (\sim p \wedge \sim r)$ - $(T \vee T) \leftrightarrow (T \wedge T) \implies T \leftrightarrow T$, which is True.
Step 3: Conclusion
Hence, (D) has the truth value True. Final Answer: (D)
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