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NCERT Exemplar · Class 12 Mathematics Vector Algebra

45 questions · 45 still being checked

EXERCISE 10.3 31–40 (part 4 of 5)

  1. Choose the correct answer from the given four options in each of the Exercises from $\displaystyle 19$ to $\displaystyle 33$ (M.C.Q)

    Exercise 31

    If a⃗,b⃗,c⃗\displaystyle \vec{a}, \vec{b}, \vec{c} are three vectors such that a⃗+b⃗+c⃗=0→\displaystyle \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} and ∣a⃗∣=2,∣b⃗∣=3,∣c⃗∣=5\displaystyle |\vec{a}|=2,|\vec{b}|=3,|\vec{c}|=5, then value of a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗\displaystyle \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a} is
    (A)
    0\displaystyle 0 (B) 1\displaystyle 1 (C) -19\displaystyle 19
    (D)
    38\displaystyle 38

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    NCERT’s answer
    C
    (C) \(\displaystyle -19\)\[|\vec a+\vec b+\vec c|^2=0 \]\[|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)=0 \]\[4+9+25+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)=0 \]\[\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a=-19 \]

  2. Exercise 32

    If ∣a⃗∣=4\displaystyle |\vec{a}|=4 and −3≤λ≤2\displaystyle -3 \leq \lambda \leq 2, then the range of ∣λa⃗∣\displaystyle |\lambda \vec{a}| is
    (A)
    [0\displaystyle 0, 8\displaystyle 8]
    (B)
    [-12,8\displaystyle 12{,}8]
    (C)
    [0\displaystyle 0, 12\displaystyle 12]
    (D)
    [8\displaystyle 8, 12\displaystyle 12]

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    NCERT’s answer
    C
    (C) \(\displaystyle [0,12]\)\[|\lambda\vec{a}|=|\lambda|\,|\vec{a}|=4|\lambda| \]\[-3\le\lambda\le 2 \quad\Rightarrow\quad 0\le|\lambda|\le 3 \]\[0\le 4|\lambda|\le 12 \]

  3. Exercise 33

    The number of vectors of unit length perpendicular to the vectors a⃗=2i^+j^+2k^\displaystyle \vec{a}=2 \hat{i}+\hat{j}+2 \hat{k} and b⃗=j^+k^\displaystyle \vec{b}=\hat{j}+\hat{k} is
    (A)
    one
    (B)
    two
    (C)
    three
    (D)
    infinite

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    NCERT’s answer
    B
    (B) two\[\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\2&1&2\\0&1&1\end{vmatrix}=-\hat i-2\hat j+2\hat k \]\[|\vec a\times\vec b|=\sqrt{1+4+4}=3 \]\[\hat n=\pm\frac{\vec a\times\vec b}{|\vec a\times\vec b|}=\pm\frac13\left(-\hat i-2\hat j+2\hat k\right) \]Only \(\displaystyle \hat n\) and \(\displaystyle -\hat n\) are unit vectors perpendicular to both.
  4. Fill in the blanks in each of the Exercises from $\displaystyle 34$ to 40.

    Exercise 34

    The vector a⃗+b⃗\displaystyle \vec{a}+\vec{b} bisects the angle between the non-collinear vectors a⃗\displaystyle \vec{a} and b⃗\displaystyle \vec{b} if ____

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    \(\displaystyle |\vec a|=|\vec b|\)NCERT_Solution_Class12_Maths_Exemplar_Ch10_Ex10-3_Q34\[\vec{OA}=\vec a,\quad \vec{OB}=\vec b,\quad \vec{OC}=\vec a+\vec b \]\[\cos\theta_1=\frac{\vec a\cdot(\vec a+\vec b)}{|\vec a||\vec a+\vec b|}=\frac{|\vec a|^2+\vec a\cdot\vec b}{|\vec a||\vec a+\vec b|} \]\[\cos\theta_2=\frac{\vec b\cdot(\vec a+\vec b)}{|\vec b||\vec a+\vec b|}=\frac{|\vec b|^2+\vec a\cdot\vec b}{|\vec b||\vec a+\vec b|} \]Bisector means \(\displaystyle \theta_1=\theta_2\):\[|\vec b|\left(|\vec a|^2+\vec a\cdot\vec b\right)=|\vec a|\left(|\vec b|^2+\vec a\cdot\vec b\right) \]\[\left(|\vec a|-|\vec b|\right)\left(|\vec a||\vec b|-\vec a\cdot\vec b\right)=0 \]\[\vec a\cdot\vec b\ne|\vec a||\vec b| \quad \text{(non-collinear)} \;\Rightarrow\; |\vec a|=|\vec b| \]

  5. Exercise 35

    If r⃗.a⃗=0,r⃗.b⃗=0\displaystyle \vec{r} . \vec{a}=0, \vec{r} . \vec{b}=0, and r⃗.c⃗=0\displaystyle \vec{r} . \vec{c}=0 for some non-zero vector r⃗\displaystyle \vec{r}, then the value of a⃗⋅(b⃗×c⃗)\displaystyle \vec{a} \cdot(\vec{b} \times \vec{c}) is ____

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    NCERT’s answer
    $\displaystyle 0$
    \(\displaystyle 0\)If \(\displaystyle \vec a\cdot(\vec b\times\vec c)\ne0\), then \(\displaystyle \vec a,\vec b,\vec c\) are not coplanar and span space:\[\vec r=x\vec a+y\vec b+z\vec c \]\[\vec r\cdot\vec r=x(\vec r\cdot\vec a)+y(\vec r\cdot\vec b)+z(\vec r\cdot\vec c)=0 \]\[\Rightarrow\ \vec r=\vec 0 \quad \text{(contradicts } \vec r\ne\vec 0\text{)} \]So \(\displaystyle \vec a,\vec b,\vec c\) are coplanar:\[\vec a\cdot(\vec b\times\vec c)=0 \]

  6. Exercise 36

    The vectors a⃗=3i−2j+2k^\displaystyle \vec{a}=3 i-2 j+2 \hat{k} and b⃗=−i^−2k^\displaystyle \vec{b}=-\hat{i}-\widehat{2 k} are the adjacent sides of a parallelogram. The acute angle between its diagonals is ____.

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    NCERT’s answer
    \(\displaystyle \frac{\pi}{4}\)
    \(\displaystyle \dfrac{\pi}{4}\)NCERT_Solution_Class12_Maths_Exemplar_Ch10_Ex10-3_Q36\[\vec a=3\hat i-2\hat j+2\hat k,\qquad \vec b=-\hat i-2\hat k \]\[\vec{OC}=\vec a+\vec b=2\hat i-2\hat j \]\[\vec{BA}=\vec a-\vec b=4\hat i-2\hat j+4\hat k \]\[\cos\theta=\frac{(\vec a+\vec b)\cdot(\vec a-\vec b)}{|\vec a+\vec b|\,|\vec a-\vec b|}=\frac{8+4+0}{2\sqrt2\cdot 6} \]\[\cos\theta=\frac1{\sqrt2}\;\Rightarrow\;\theta=\frac{\pi}{4} \]

  7. Exercise 37

    The values of k\displaystyle k for which ∣ka⃗∣<∣a⃗∣\displaystyle |k \vec{a}|<|\vec{a}| and ka⃗+12a⃗\displaystyle k \vec{a}+\frac{1}{2} \vec{a} is parallel to a⃗\displaystyle \vec{a} holds true are ____.

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    NCERT’s answer
    \(\displaystyle k \in ]-1,1[ k \neq -\frac{1}{2}\)
    \(\displaystyle -1<k<1,\ k\ne-\tfrac12\)\[|k\vec a|<|\vec a|\;\Rightarrow\;|k|\,|\vec a|<|\vec a|\;\Rightarrow\;|k|<1 \]\[k\vec a+\tfrac12\vec a=\left(k+\tfrac12\right)\vec a \]This is parallel to \(\displaystyle \vec a\) except when \(\displaystyle k=-\tfrac12\), where it is the zero vector.

  8. Exercise 38

    The value of the expression ∣a⃗×b⃗∣2+(a⃗⋅b⃗)2\displaystyle |\vec{a} \times \vec{b}|^2+(\vec{a} \cdot \vec{b})^2 is ____.

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    NCERT’s answer
    \(\displaystyle |\vec{a}|^2|\vec{b}|^2\)
    \(\displaystyle |\vec a|^2|\vec b|^2\)\[|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta,\qquad \vec a\cdot\vec b=|\vec a||\vec b|\cos\theta \]\[|\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2=|\vec a|^2|\vec b|^2\left(\sin^2\theta+\cos^2\theta\right) \]\[=|\vec a|^2|\vec b|^2 \]

  9. Exercise 39

    If ∣a⃗×b⃗∣2+∣a⃗⋅b⃗∣2=144\displaystyle |\vec{a} \times \vec{b}|^2+|\vec{a} \cdot \vec{b}|^2=144 and ∣a⃗∣=4\displaystyle |\vec{a}|=4, then ∣b⃗∣\displaystyle |\vec{b}| is equal to ____.

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    NCERT’s answer
    $\displaystyle 3$
    \(\displaystyle 3\)\[|\vec a\times\vec b|^2+|\vec a\cdot\vec b|^2=|\vec a|^2|\vec b|^2\left(\sin^2\theta+\cos^2\theta\right)=|\vec a|^2|\vec b|^2 \]\[144=4^2\,|\vec b|^2 \]\[|\vec b|^2=9\;\Rightarrow\;|\vec b|=3 \]

  10. Exercise 40

    If a⃗\displaystyle \vec{a} is any non-zero vector, then (a⃗.i^)i^+(a⃗.j^)j^+(a⃗.k^)k^\displaystyle (\vec{a} . \hat{i}) \hat{i}+(\vec{a} . \hat{j}) \hat{j}+(\vec{a} . \hat{k}) \hat{k} equals ____.

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    NCERT’s answer
    \(\displaystyle \vec{a}\)
    \(\displaystyle \vec a\)\[\vec a=a_1\hat i+a_2\hat j+a_3\hat k \]\[\vec a\cdot\hat i=a_1,\qquad \vec a\cdot\hat j=a_2,\qquad \vec a\cdot\hat k=a_3 \]\[(\vec a\cdot\hat i)\hat i+(\vec a\cdot\hat j)\hat j+(\vec a\cdot\hat k)\hat k=a_1\hat i+a_2\hat j+a_3\hat k=\vec a \]