Download App

Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
MCQ
Assertion (A) : The projection of the vector $3 \hat{i}-\hat{j}-2 \hat{k}$ on the vector $\hat{i}+2 \hat{j}-3 \hat{k}$ is $\frac{7}{\sqrt{14}}$.
Reason (R) : The projection of a vector $\vec{a}$ on another vector $\vec{b}$ is $\frac{(\vec{a} \cdot \vec{b})}{|\vec{b}|}$.
  • ✓
    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.
✓

Answer

Correct option: A.
Both (A) and (R) are true and (R) is the correct explanation of (A).
(a) : Required projection $=\frac{(3 \hat{i}-\hat{j}-2 \hat{k}) \cdot(\hat{i}+2 \hat{j}-3 \hat{k})}{\sqrt{1^2+2^2+(-3)^2}}$
$
\frac{3-2+6}{\sqrt{1+4+9}}=\frac{7}{\sqrt{14}}
$
Also, projection of vector $\vec{a}$ on $\vec{b}=(\vec{a} \cdot \hat{b})=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\right)$
Hence, both assertion and reason are true and reason is the correct explanation of assertion.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Assertion (A) & Reason (B) MCQ" from Vector Algebra→Vector Algebra — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: $\tan^{-1}\big(\frac{3}{4}\big)+\tan^{-1}\big(\frac{1}{7}\big)=\frac{\pi}{4}.$
Reason: For x > 0, y > 0, xy < 1, $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big).$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→
Assertion (A) : Both $\sin x$ and $\cos x$ are decreasing functions in $\left(\frac{\pi}{2}, \pi\right)$.
Reason (R): If a differentiable function decreases in $(a, b)$, then its derivative also decreases in $(a, b)$.
→
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: $\text{I}=\int_{0}^{1}\frac{\text{dx}}{3\sqrt{1+\text{x}^3}}=\int_{0}^{{2}^\frac{-1}{3}}\frac{\text{dt}}{1-\text{t}^3}$
Reason: The integrand of the integral I becomes rational by the substitution $\text{t}=\frac{\text{x}}{3\sqrt{1+\text{x}^3}}$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If $\text{A}=\begin{pmatrix}0&-2&3\\ 2&0&6\\-3&-6&0 \end{pmatrix},$ then A-1 does not exist.
Reason: If A is a skew symmetric matrix of odd order, then A is singular.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: For a matrix $\begin{bmatrix}2&-1\\-3&4\end{bmatrix},$ A. adj $\text{A}=\begin{bmatrix}4&0\\0&4\end{bmatrix}.$
Reason: For a square matrix A, $\text{A}(\text{adj}\text{A})=(\text{adj}\text{A})\text{A}=\mid\text{A}\mid.$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: $\sin^{-1}\frac{8}{17}+\sin^{-1}\frac{3}{5}=\sin^{-1}\frac{77}{85}.$
Reason: $\sin^{-1}\text{x}+\sin^{-1}\text{y}=\sin^{-1}\big(\text{x}\sqrt{1-\text{y}^{2}}+\text{y}\sqrt{1-\text{x}^{2}}\big)$ for $-1\leq\text{x},\text{y}\leq1,\text{x}^{2}+\text{y}^{2}\leq1.$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→
Assertion (A) : Relation $R$ defined in the set $A$ as $R\{(x, y): y-x$ is an integer, $x, y \in R\}$ is an equivalence relation.
Reason (R) : Relation $R$ defined in the set $B$ as $R\{(x, y): x=\alpha y$ for some rational number $\alpha$, $x, y \in R\}$ is an equivalence relation.
→
For any square matrix $A$ with real number entries, consider the following statements.
Assertion (A) : $A+A^{\prime}$ is a symmetric matrix.
Reason (R): $A-A^{\prime}$ is a skew-symmetric matrix.
→
Assertion (A) : Integrating factor of $\left(1+x^2\right) \frac{d y}{d x}+x y=\frac{1}{2 x}$ is given by $\sqrt{1+x^2}$.
Reason (R) : Integrating factor of $\frac{d y}{d x}+P(x) \cdot y=Q(x)$ is $e^{\int P(x) d x}$.
→
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Let $\overline{\text{a}}=\hat{\text{i}}+\hat{\text{j}}=3\hat{\text{k}}$ and  $\overline{\text{b}}=\hat{2\text{i}}+\hat{\text{j}}=\hat{\text{k}}$
Assertion: Vectors $\overline{\text{a}}$ and $\overline{\text{b}}$ are perpendicular to each other.
Reason: $\overline{\text{a}}.\overline{\text{b}}=0$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
→