Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Math Assignment Class XII Ch-11: Three Dimensional Geometry

Three Dimensional Geometry
Math. Assignment Class XII Chapter 11  
Important questions on Three dimensional geometry chapter 11 class XII based on the board examination point of view with answers. Revision questions based on three dimensional geometry class XII 



Extra Questions of Chapter 11 Three Dimensional Geometry

Question 1: A line makes angle 𝜶, 𝛃, 𝛾 with x-axis, y-axis, z-axis. Then find cos2𝜶, cos2𝛃, cos2𝛾 ?

Ans. 1             -1

Question 2: Write vector equation of the line: \[\frac{2x-1}{\sqrt{3}} =\frac{y+2}{2} =\frac{z-3}{3}\]

Question 3: Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line \[\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}\] Hint: Two parallel lines have same direction ratio.  

Ans. 3  \[\frac{x+2}{3}=\frac{y-4}{-5}= \frac{z+5}{6}\]

Question 4: Find angle between pair of lines \[\frac{2-x}{-2}=\frac{y-1}{7}=\frac{z+3}{-3}\; and\]\[\frac{x+2}{-1}=\frac{2y-8}{4}=\frac{z-5}{4}\]  

Ans 4:               90o 

Question 5: Find the point on the given line at a distance 3√2 from the point (1, 2, 3). \[\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}\] 

Ans 5:     \[\lambda =0, \lambda =\frac{30}{17}\;\; and\; Point\; is\; A(-2,-1,3)\; or\; A\left (\frac{56}{17},\frac{43}{17},\frac{111}{17} \right )\]

Question 6: Find the length and foot of perpendicular drawn from the point (2, -1, 5) on the line \[\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}\]

Ans 6:  \[\lambda =-1, Q \left ( 1,2,3 \right ), \; \; \left | PQ \right |=\sqrt{14}\]

Question 7: Find the shortest distance between the lines :-  \[\vec{r}=(1+\lambda)\hat{i}+(2-\lambda)\hat{j}+(\lambda +1)\hat{k}\]\[\vec{r}=(2\hat{i}-\hat{j}-\hat{k})+\mu (2\hat{i}+\hat{j}+2\hat{k})\]

Ans 7:    \[\vec{b_{1}}\times \vec{b_{2}}=-3\hat{i}+3\hat{k},\; Distance\; =\frac{3\sqrt{2}}{2}\]

Question 8: Find the equation of a line passing through the point P(2,-1,3) and perpendicular to the lines:\[\vec{r}=(\hat{i}+\hat{j}+\hat{k})+\lambda (2\hat{i}-2\hat{j}+\hat{k})\]\[\vec{r}=(2\hat{i}-\hat{j}-3\hat{k})+\mu(\hat{i}+2\hat{j}+2\hat{k})\]

Ans 8: \[\vec{r}=\left ( 2\hat{i}-\hat{j}+3\hat{k} \right )+\lambda \left ( 2\hat{i}+\hat{j}-2\hat{k} \right )\]

Question 9: Show that the following lines are intersecting, also find their point of intersection. \[\vec{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda(\hat{i}+2\hat{j}+2\hat{k})\]\[\vec{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})\]Ans 9               (-1, -6, -12)

Question 10: Show that the following lines are coplaner. \[\frac{5-x}{-4}=\frac{y-7}{4}=\frac{z+3}{-5} \; and\]\[\frac{x-8}{7}=\frac{2y-8}{2}=\frac{z-5}{3}\]

Question 11: Find the distance of the given plane from the origin: \[\vec{r}.(2\hat{i}+3\hat{j}-6\hat{k})+2=0\] Ans 11 :            2/7

Question 12: Find the angle between the planes \[\vec{r}.(\hat{i}-2\hat{j}-2\hat{k})=1 \;\; and\]\[\vec{r}.(3\hat{i}-6\hat{j}+2\hat{k})=0\]Ans 12  \[cos^{-1}\left ( \frac{11}{21} \right )\]

Question 13: Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of planes x + 2y + 3z - 4 = 0 and 2x + y - z + 5 = 0

Ans 13              51x + 15y - 50z + 173 = 0 , λ = -29/7

Question 14: Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the plane x + 2y + 3z = 5 and 3x + 3y + z = 0.

Ans 14              7x - 8y + 3z + 25 = 0

Question 15: Find the equation of the plane through (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane    x -2y + 4z = 10.

Ans 15              18x + 17y + 4z -49 = 0

Question 16: \[ Find \; image\; p'\; of\; point\; p\; having\; position\; vector\; \; \hat{i}+3\hat{j}+4\hat{k}\;\; in\; the\; plane\]\[\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0\; and\; also\; find\; \; \left | pp' \right |\]

Ans. 16             Image (-3,5,2), Length = 2√6

Question 17: Find the distance of the point (1, -2, 3) from the plane x - y + z = 5 measured parallel to the line  \[\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}\]

Ans 17              λ = 1/7,    AB = 1

Question 18: Find the distance of the point (-2, 3, -4) from the given line measured to the plane 4x + 12y - 3z + 1 = 0 \[\frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}\]

Ans 18              λ = 2/3 , Point is (4, 5/2, 2),  Distance = 17/2

Question 19: Find the equation of the plane passing through the point P(4, 6, 2) and the point of intersection of the plane x + y - z = 8 and the line \[\frac{x-1}{3}=\frac{y}{2}= \;\frac{z+1}{7}\]Ans 19 \[\frac{x-4}{1}=\frac{y-6}{1}=\frac{z-2}{2}\]

Question 20: Find the vector equation of the plane through the line of intersection of planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0. Hence find whether plane thus obtained contains the line \[\frac{x+2}{5}=\frac{y-3}{4}= \;\frac{z}{5}\]Ans 20 \[\vec{r}\left ( \hat{i}-\hat{k} \right )+2=0,\; \; \;Yes\]

Question 21: Find the angle between the following line and plane : \[\vec{r}=\left (\hat{i}-\hat{j}+\hat{k} \right )+\lambda (2\hat{i}-\hat{j}+3\hat{k})\]\[and\; plane\; \; \vec{r}.\left (2\hat{i}+\hat{j}-\hat{k} \right )=4\] Ans 21              0o



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