Question 20 - CBSE Class 12 Sample Paper for 2026 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Question 20 Assertion(A): Given two non-zero vectors 𝑎 ⃗ and 𝑏 ⃗. If 𝑟 ⃗ is another non-zero vector such that 𝑟 ⃗ × (𝑎 ⃗+𝑏 ⃗)=0 ⃗. Then 𝑟 ⃗ is perpendicular to 𝑎 ⃗ × 𝑏 ⃗. Reason (R): The vector ( 𝑎 ⃗+𝑏 ⃗ ) is perpendicular to the plane of 𝑎 ⃗ and 𝑏 ⃗ Select the correct answer from the options (A), (B), (C) and (D) as given below.) (A) Both (A) and (R) are true and (R) is the correct explanation of (A). (B) Both (A) and (R) are true but (R) is not the correct explanation of (A). (C) (A) is true but (R) is false. (D) (A) is false but (R) is true.Checking Assertion Assertion(A): Given two non-zero vectors 𝑎 ⃗ and 𝑏 ⃗. If 𝑟 ⃗ is another non-zero vector such that 𝑟 ⃗ × (𝑎 ⃗+𝑏 ⃗)=0 ⃗. Then 𝑟 ⃗ is perpendicular to 𝑎 ⃗ × 𝑏 ⃗. Since 𝒓 ⃗ × (𝒂 ⃗+𝒃 ⃗)=𝟎 ⃗ This is only possible when 𝒓 ⃗ and (𝒂 ⃗+𝒃 ⃗) are parallel Since (𝑎 ⃗+𝑏 ⃗) lies on the plane of 𝑎 ⃗ and 𝑏 ⃗ ∴ 𝒓 ⃗ , 𝒂 ⃗ and 𝒃 ⃗ are parallel Now, 𝑎 ⃗ × 𝑏 ⃗ is perpendicular to both 𝑎 ⃗ and 𝑏 ⃗ Since 𝑟 ⃗ , 𝑎 ⃗ and 𝑏 ⃗ are parallel ∴ 𝒂 ⃗ × 𝒃 ⃗ is perpendicular to 𝒓 ⃗ Hence, Assertion is true Checking Reason Reason (R): The vector ( 𝑎 ⃗+𝑏 ⃗ ) is perpendicular to the plane of 𝑎 ⃗ and 𝑏 ⃗ This is false. As 𝑎 ⃗+ 𝑏 ⃗ is in same plane to both 𝑎 ⃗ and 𝑏 ⃗ Hence, Reason is false We don’t need to check explanation here So, Assertion is true Reasoning is false So, the correct answer is (c)