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Short Quiz - Chapter 11 Class 12 Three Dimensional Geometry

Chapter 11 Class 12 Three Dimensional Geometry | 5 questions

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Question 1 of 5
Question 1 of 5
CBSE Board Exam 2025
The equation of a line parallel to the vector \(3 \hat{i}+\hat{j}+2 \hat{k}\) and passing through the point \((4,-3,7)\) is :

Correct option: C

Answer: Line is parallel to:

$$ 3 \hat{i}+\hat{j}+2 \hat{k} $$


So direction ratios are:

$$ (3,1,2) $$


Line passes through:

$$ (4,-3,7) $$


Therefore parametric equation is:

$$ \begin{aligned} & x=4+3 t \\ & y=-3+t \\ & z=7+2 t \end{aligned} $$


So,

$$ x=3 t+4, \quad y=t-3, \quad z=2 t+7 $$


Answer: C

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Question 2 of 5
CBSE Board Exam 2024
If \(\alpha, \beta\) and \(\gamma\) are the angles which a line makes with positive directions of \(\mathrm{x}, \mathrm{y}\) and z axes respectively, then which of the following is not true ?

Correct option: D

Answer: For a line making angles \(\alpha, \beta, \gamma\) with positive \(x, y, z\)-axes:

$$ \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 $$


So:
Option A

$$ \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 $$


True.
Option B

$$ \begin{gathered} \sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma \\ =\left(1-\cos ^2 \alpha\right)+\left(1-\cos ^2 \beta\right)+\left(1-\cos ^2 \gamma\right) \\ =3-1=2 \end{gathered} $$


True.
Option C

$$ \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma $$


Using:

$$ \cos 2 \alpha=2 \cos ^2 \alpha-1 $$


So,

$$ \begin{gathered} =2\left(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma\right)-3 \\ =2(1)-3=-1 \end{gathered} $$


True.
Option D

$$ \cos \alpha+\cos \beta+\cos \gamma=1 $$


This is not always true.

Answer: D

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Question 3 of 5
CBSE Board Exam 2025
If a line makes angles of \(\frac{3 \pi}{4}, \frac{\pi}{3}\) and \(\theta\) with the positive directions of \(x, y\) and z -axis respectively, then \(\theta\) is

Correct option: D

Answer: Given angles:

$$ \alpha=\frac{3 \pi}{4}, \quad \beta=\frac{\pi}{3}, \quad \gamma=\theta $$


Using:

$$ \begin{gathered} \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \theta=1 \\ \cos ^2 \frac{3 \pi}{4}+\cos ^2 \frac{\pi}{3}+\cos ^2 \theta=1 \\ \left(-\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{2}\right)^2+\cos ^2 \theta=1 \\ \frac{1}{2}+\frac{1}{4}+\cos ^2 \theta=1 \\ \cos ^2 \theta=\frac{1}{4} \\ \cos \theta= \pm \frac{1}{2} \end{gathered} $$


So mathematically:

$$ \theta=\frac{\pi}{3} \quad \text { or } \quad \frac{2 \pi}{3} $$


But from the given options, the intended answer is:

$$ \text { Answer: D) } \pm \frac{\pi}{3} $$


Strictly speaking, an angle with an axis should lie between 0 and \(\pi\), so the cleaner answer is:

$$ \theta=\frac{\pi}{3} \text { or } \frac{2 \pi}{3} $$

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Question 4 of 5
CBSE Board Exam 2024
If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both \(x\)-axis and \(z\)-axis, then the angle which it makes with the positive direction of \(y\)-axis is :

Correct option: C

Answer: Given:

$$ \alpha=\frac{\pi}{4}, \quad \gamma=\frac{\pi}{4} $$


Using:

$$ \begin{gathered} \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ \cos ^2 \frac{\pi}{4}+\cos ^2 \beta+\cos ^2 \frac{\pi}{4}=1 \\ \frac{1}{2}+\cos ^2 \beta+\frac{1}{2}=1 \\ \cos ^2 \beta=0 \\ \cos \beta=0 \\ \beta=\frac{\pi}{2} \end{gathered} $$


Answer: C) \(\frac{\pi}{2}\)

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Question 5 of 5
CBSE Board Exam 2026
Direction ratios of lines \(l_{1}\) and \(l_{2}\) respectively are \(<1,-2,3>\) and \(<-2, \mathrm{p},-6>\). The value of p for which \(l_{1} \| l_{2}\), is:

Correct option: B

Answer: Direction ratios are:

$$ (1,-2,3) $$

and

$$ (-2, p,-6) $$


For parallel lines, direction ratios must be proportional:

$$ (-2, p,-6)=k(1,-2,3) $$


From first component:

$$ \begin{aligned} -2 & =k(1) \\ k & =-2 \end{aligned} $$


So,

$$ \begin{gathered} p=k(-2)=(-2)(-2)=4 \\ p=4 \end{gathered} $$


Answer: B

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