Questions number 19 and 20 are Assertion and Reason based questions.
Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).
Select the correct answer from the codes (A), (B), (C), and (D) as given below.
Assertion (A): For any non-zero unit vector \( \vec{a} \), \( \vec{a} \cdot (-\vec{a}) = (-\vec{a}) \cdot \vec{a} = -1 \).
Reason (R): Angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \).
Assertion (A): For any non-zero unit vector \( \vec{a} \), \( \vec{a} \cdot (-\vec{a}) = (-\vec{a}) \cdot \vec{a} = -1 \).
Reason (R): Angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \).
Show Hint
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
The Correct Option is C
Solution and Explanation
Step 1: Verify Assertion (A)
The dot product of a vector \( \vec{a} \) and its negative \( -\vec{a} \) is given by: \[ \vec{a} \cdot (-\vec{a}) = |\vec{a}| \cdot |-\vec{a}| \cdot \cos \theta. \] Here, \( |\vec{a}| = 1 \) (since \( \vec{a} \) is a unit vector) and \( |-\vec{a}| = 1 \). The angle \( \theta \) between \( \vec{a} \) and \( -\vec{a} \) is \( \pi \) (180°), so: \[ \cos \pi = -1. \] Thus: \[ \vec{a} \cdot (-\vec{a}) = 1 \cdot 1 \cdot (-1) = -1. \]
Therefore, Assertion (A) is true.
Step 2: Verify Reason (R)
The Reason (R) states that the angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \) (90°). However, the actual angle between \( \vec{a} \) and \( -\vec{a} \) is \( \pi \) (180°), as they are in opposite directions.
Hence, Reason (R) is false.
Conclusion: Assertion (A) is true, but Reason (R) is false.
Assertion (A): Every scalar matrix is a diagonal matrix.
Reason (R): In a diagonal matrix, all the diagonal elements are 0.
Reason (R): In a diagonal matrix, all the diagonal elements are 0.
Show Hint
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
The Correct Option is C
Solution and Explanation
Step 1: Analyze Assertion (A)
A scalar matrix is a diagonal matrix where all the diagonal elements are equal, and the non-diagonal elements are zero.
Hence, Assertion (A) is true.
Step 2: Analyze Reason (R)
In a diagonal matrix, the diagonal elements can have any value (not necessarily 0).
Therefore, Reason (R) is false.
Step 3: Conclude the result
Assertion (A) is true, but Reason (R) is false.
Hence, the correct option is (C).
Top Questions on Vectors
যদি \( \vec{a} = 4\hat{i} - \hat{j} + \hat{k} \) এবং \( \vec{b} = 2\hat{i} - 2\hat{j} + \hat{k} \) হয়, তবে \( \vec{a} + \vec{b} \) ভেক্টরের সমান্তরাল একটি একক ভেক্টর নির্ণয় কর।
যদি ভেক্টর \( \vec{\alpha} = a\hat{i} + a\hat{j} + c\hat{k}, \quad \vec{\beta} = \hat{i} + \hat{k}, \quad \vec{\gamma} = c\hat{i} + c\hat{j} + b\hat{k} \) একই সমতলে অবস্থিত (coplanar) হয়, তবে প্রমাণ কর যে \( c^2 = ab \)।
- If the sum of two vectors is a unit vector, then the magnitude of their difference is:
- Let \( \vec{a} \) be a position vector whose tip is the point (2, -3). If \( \overrightarrow{AB} = \vec{a} \), where coordinates of A are (–4, 5), then the coordinates of B are:
The respective values of \( |\vec{a}| \) and} \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:
Questions Asked in CBSE CLASS XII exam
- Find the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]
- CBSE CLASS XII - 2026
- Differential Equations
- A square loop of side 0.50 m is placed in a uniform magnetic field of 0.4 T perpendicular to the plane of the loop. The loop is rotated through an angle of 60° in 0.2 s. The value of emf induced in the loop will be:
- CBSE CLASS XII - 2026
- Electromagnetic induction
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following:
(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).
- CBSE CLASS XII - 2026
- Integration
- Consider a cylindrical conductor of length \( l \) and area of cross-section \( A \). Current \( I \) is maintained in the conductor and electrons drift with velocity \( \vec{v}_d \, (|\vec{v}_d| = \frac{eE}{m} \tau) \), where symbols have their usual meanings. Show that the conductivity of the material of the conductor is given by \[ \sigma = \frac{n e^2 \tau}{m}. \]
- CBSE CLASS XII - 2026
- Current Density
- The painting Hazrat Nizamuddin Auliya and Amir Khusro belongs to which sub-school of Deccan Miniature?
- CBSE CLASS XII - 2026
- Indian Miniature Paintings and Museums