JEE Mains Previous Year Questions (PYQ)

JEE Main 2023 30th Jan Evening Shift

For $ \alpha, \beta \in \mathbb{R} $, suppose the system of linear equations
$ x - y + z = 5 $
$ 2x + 2y + \alpha z = 8 $
$ 3x - y + 4z = \beta $
has infinitely many solutions. Then $ \alpha $ and $ \beta $ are the roots of:

6.

Let the image of the point $ P(0,-5,0) $ in the line $ \dfrac{x-1}{2}=\dfrac{y}{1}=\dfrac{z+1}{-2} $ be the point $ R $ and the image of the point $ Q\!\left(0,\dfrac{-1}{2},0\right) $ in the line $ \dfrac{x-1}{-1}=\dfrac{y+9}{4}=\dfrac{z+1}{1} $ be the point $ S $. Then the square of the area of the parallelogram $ PQRS $ is ________.

7.

Let $ P(\alpha, \beta, \gamma) $ be the point on the line $ \frac{x-1}{2} = \frac{y+1}{-3} = z $ at a distance $ 4\sqrt{14} $ from the point $ (1, -1, 0) $ and nearer to the origin. Then the shortest distance, between the lines $ \frac{x - \alpha}{1} = \frac{y - \beta}{2} = \frac{z - \gamma}{3} $ and $ \frac{x + 5}{2} = \frac{y - 10}{1} = \frac{z - 3}{1} $, is equal to:

8.

Let the shortest distance between the lines $ L:\; \dfrac{x-5}{-2}=\dfrac{y-\lambda}{0}=\dfrac{z+\lambda}{1},\; \lambda \ge 0 $ and $ L_{1}:\; x+1=y-1=4-z $ be $ 2\sqrt{6} $. If $ (\alpha,\beta,\gamma) $ lies on $ L $, then which of the following is NOT possible?

9.

A line with direction ratios $ 2,1,2 $ meets the lines $ x=y+2=z $ and $ x+2=2y=2z $ respectively at the points $ P $ and $ Q $. If the length of the perpendicular from the point $ (1,2,12) $ to the line $ PQ $ is $ l $, then $ l^{2} $ is __________.

10.

If the foot of the perpendicular from the point $ A(-1,4,3) $ on the plane $ P: 2x+my+nz=4 $ is $ \left(-2,\dfrac{7}{2},\dfrac{3}{2}\right) $, then the distance of the point $ A $ from the plane $ P $, measured parallel to a line with direction ratios $ 3,-1,-4 $, is equal to:

11.

For $ a, b \in \mathbb{Z} $ and $ |a-b| \leq 10 $, let the angle between the plane $ P: ax + y - z = b $ and the line $ l: x - 1 = a - y = z + 1 $ be $ \cos^{-1}\!\left(\dfrac{1}{3}\right) $. If the distance of the point $ (6,-6,4) $ from the plane $ P $ is $ 3\sqrt{6} $, then $ a^{4} + b^{2} $ is equal to:

12.

Let the plane $ P $ pass through the intersection of the planes $ 2x+3y-z=2 $ and $ x+2y+3z=6 $, and be perpendicular to the plane $ 2x+y-z+1=0 $. If $ d $ is the distance of $ P $ from the point $ (-7,1,1) $, then $ d^{2} $ is equal to:

13.

Let the lines $\displaystyle \frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2} $ and $\displaystyle \frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda} $ be coplanar and $ P $ be the plane containing these two lines. Then which of the following points does NOT lie on $ P $?

14.

The shortest distance between the lines x+1=2y=-12z and x=y+2=6z-6 is:

15.

The shortest distance between the lines $ \dfrac{x-4}{4}=\dfrac{y+2}{5}=\dfrac{z+3}{3} $ and $ \dfrac{x-1}{3}=\dfrac{y-3}{4}=\dfrac{z-4}{2} $ is:

16.

If two straight lines whose direction cosines are given by the relations $ l + m - n = 0 $, $ 3l^2 + m^2 + cnl = 0 $ are parallel, then the positive value of $ c $ is:

17.

Let P be the point of intersection of the line $ \dfrac{x+3}{3}=\dfrac{y+2}{1}=\dfrac{1-z}{2} $ and the plane $ x+y+z=2 $. If the distance of the point P from the plane $ 3x-4y+12z=32 $ is $ q $, then $ q $ and $ 2q $ are the roots of the equation.

18.

The distance of the point $ P(3,4,4) $ from the point of intersection of the line joining the points $ Q(3,-4,-5) $ and $ R(2,-3,1) $ and the plane $ 2x+y+z=7 $, is equal to ____________.

19.

If the shortest distance between the straight lines $ 3(x-1)=6(y-2)=2(z-1) $ and $ 4(x-2)=2(y-\lambda)=(z-3), \lambda \in \mathbb{R} $ is $ \dfrac{1}{\sqrt{38}} $, then the integral value of $ \lambda $ is equal to:

20.

If a plane passes through the points $( -1, k, 0)$, $( 2, k, -1)$, $( 1, 1, 2)$ and is parallel to the line $ \dfrac{x-1}{1}=\dfrac{2y+1}{2}=\dfrac{z+1}{-1} $, then the value of $ \dfrac{k^{2}+1}{(k-1)(k-2)} $ is :

21.

If $ A(1,-1,2), \ B(5,7,-6), \ C(3,4,-10) $ and $ D(-1, -4, -2) $ are the vertices of a quadrilateral $ ABCD $, then its area is:

22.

The equation of the planes parallel to the plane $ x - 2y + 2z - 3 = 0 $ which are at unit distance from the point $ (1, 2, 3) $ is $ ax + by + cz + d = 0 $. If $ (b - d) = K(c - a) $, then the positive value of $ K $ is ______________ .

23.

If the shortest distance between the lines $ \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda} $ and $ \frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5} $ is $ \frac{1}{\sqrt{3}} $, then the sum of all possible values of $ \lambda $ is:

24.

If the image of the point $ P(1, 0, 3) $ in the line joining the points $ A(4, 7, 1) $ and $ B(3, 5, 3) $ is Q$(\alpha, \beta, \gamma)$, then $ \alpha + \beta + \gamma $ is equal to:

25.

Let the plane $ 2x + 3y + z + 20 = 0 $ be rotated through a right angle about its line of intersection with the plane $ x - 3y + 5z = 8 $. If the mirror image of the point $ \left( 2, -\dfrac{1}{2}, 2 \right) $ in the rotated plane is $ B(a,b,c) $, then:

26.

The angle between the straight lines, whose direction cosines are given by the equations $ 2l + 2m - n = 0 $ and $ mn + nl + lm = 0 $, is:

27.

Let the acute angle bisector of the two planes $ x - 2y - 2z + 1 = 0 $ and $ 2x - 3y - 6z + 1 = 0 $ be the plane P. Then which of the following points lies on P?

28.

Let the image of the point $ \left(\dfrac{5}{3},\dfrac{5}{3},\dfrac{8}{3}\right) $ in the plane $ x-2y+z-2=0 $ be $ P $. If the distance of the point $ Q(6,-2,\alpha),\ \alpha>0 $, from $ P $ is $ 13 $, then $ \alpha $ is equal to _____.

29.

If the equation of the plane containing the line $ x+2y+3z-4=0=2x+y-z+5 $ and perpendicular to the plane $ \vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k}) $ is $ ax+by+cz=4 $, then $ (a-b+c) $ is equal to:

30.

If the foot of the perpendicular drawn from $ (1, 9, 7) $ to the line passing through the point $ (3, 2, 1) $ and parallel to the planes $ x + 2y + z = 0 $ and $ 3y - z = 3 $ is $ (\alpha, \beta, \gamma) $, then $ \alpha + \beta + \gamma $ is equal to:

31.

If the equation of the line passing through the point $ \left(0,-\frac{1}{2},0\right) $ and perpendicular to the lines $ \vec{r}=\lambda(\hat{i}+a\hat{j}+b\hat{k}) $ and $ \vec{r}=(\hat{i}-\hat{j}+6\hat{k})+\mu(-b\hat{i}+a\hat{j}+5\hat{k}) $ is $ \frac{x-1}{-2}=\frac{y-4}{d}=\frac{z-c}{-4} $, then $ a+b+c+d $ is equal to:

32.

The line of shortest distance between the lines $\displaystyle \frac{x-2}{0} = \frac{y-1}{1} = \frac{z}{1} $ and $\displaystyle \frac{x-3}{2} = \frac{y-5}{2} = \frac{z-1}{1} $ makes an angle of $\displaystyle \cos^{-1}\!\left(\sqrt{\frac{2}{27}}\right) $ with the plane $ P: ax - y - z = 0, \ (a>0) $. If the image of the point $ (1,1,-5) $ in the plane $ P $ is $ (\alpha,\beta,\gamma) $, then $ \alpha + \beta - \gamma $ is equal to ______.

33.

Let $ \vec a=\hat i+2\hat j+\hat k $ and $ \vec b=2\hat i+7\hat j+3\hat k $. Let $ L_1:\ \vec r=(-\hat i+2\hat j+\hat k)+\lambda \vec a,\, \lambda\in \mathbb{R} $ and $ L_2:\ \vec r=(\hat j+\hat k)+\mu \vec b,\, \mu\in \mathbb{R} $ be two lines. If the line $ L_3 $ passes through the point of intersection of $ L_1 $ and $ L_2 $, and is parallel to $ \vec a+\vec b $, then $ L_3 $ passes through the point:

34.

Let $ Q $ and $ R $ be two points on the line $\displaystyle \frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2} $ at a distance $ \sqrt{26} $ from the point $ P(4,2,7) $. Then the square of the area of the triangle $ PQR $ is ________.

35.

Let the plane passing through the point $ (-1,0,-2) $ and perpendicular to each of the planes $ 2x+y-z=2 $ and $ x-y-z=3 $ be $ ax+by+cz+8=0 $. Then the value of $ a+b+c $ is equal to:

36.

A line passes through $ A(4,-6,-2) $ and $ B(16,-2,4) $. The point $ P(a,b,c) $, where $ a,b,c $ are non‑negative integers, on the line $ AB $ lies at a distance of $ 21 $ units from the point $ A $. The distance between the points $ P(a,b,c) $ and $ Q(4,-12,3) $ is equal to ________.

37.

If the shortest distance between the lines $ \frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1} $ and $ \frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1} $ is $ 1 $, then the sum of all possible values of $ \lambda $ is:

38.

Let $d$ be the distance of the point of intersection of the lines $ \dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1} $ and $ \dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2} $ from the point $ (7,8,9) $. Then $ d^{2}+6 $ is equal to:

39.

Consider the lines $ L_{1} $ and $ L_{2} $ given by:
$ L_{1}:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2} $
$ L_{2}:\ \dfrac{x-2}{2}=\dfrac{y-2}{2}=\dfrac{z-3}{3} $
A line $ L_{3} $ having direction ratios $ 1,-1,-2 $, intersects $ L_{1} $ and $ L_{2} $ at the points $ P $ and $ Q $ respectively. Then the length of line segment $ PQ $ is:

40.

If the lines $ \dfrac{x-k}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3} $ and $ \dfrac{x+1}{3} = \dfrac{y+2}{2} = \dfrac{z+3}{1} $ are co-planar, then the value of $ k $ is ________.

41.

Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1:\; \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} $ at the point $ A(\alpha,\beta,\gamma) $ and the line $ L_2:\; x-6=y=-z+4 $ at the point $ B(a,b,c) $.

42.

If the shortest distance between the lines;

$ L_{1} : \ \vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}, \ \lambda \in \mathbb{R} $

$ L_{2} : \ \vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}, \ \mu \in \mathbb{R} $

is $ \dfrac{m}{\sqrt{n}} $, where $ \gcd(m, n) = 1 $, then the value of $ m + n $ equals:

43.

Let the values of $ p $, for which the shortest distance between the lines $ \frac{x+1}{3}=\frac{y}{4}=\frac{z}{5} $ and $ \vec{r}= (p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k}) $ is $ \frac{1}{\sqrt{6}} $, be $ a, b $, $ (a<b)$.Then the length of the latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is:

44.

A plane $ E $ is perpendicular to the two planes $ 2x-2y+z=0 $ and $ x-y+2z=4 $, and passes through the point $ P(1,-1,1) $. If the distance of the plane $ E $ from the point $ Q(a,a,2) $ is $ 3\sqrt{2} $, then $ (PQ)^{2} $ is equal to:

45.

Let a triangle $ \triangle PQR $ be such that $ P $ and $ Q $ lie on the line $ \frac{x+3}{8} = \frac{y-4}{2} = \frac{x+1}{2} $ and are at a distance of $ 6 $ units from $ R(1, 2, 3) $. If $ (\alpha, \beta, \gamma) $ is the centroid of $ \triangle PQR $, then $ \alpha + \beta + \gamma $ is equal to:

46.

Let $a,b\in \mathbb{R}$. If the mirror image of the point $P(a,6,9)$ with respect to the line $\dfrac{x-3}{7}=\dfrac{y-2}{5}=\dfrac{z-1}{-9}$ is $Q(20,b,-a-9)$, then $|a+b|$ is equal to:

47.

Let $ A(x, y, z) $ be a point in $ xy $-plane, which is equidistant from three points $ (0, 3, 2) $, $ (2, 0, 3) $ and $ (0, 0, 1) $. Let $ B = (1, 4, -1) $ and $ C = (2, 0, -2) $. Then among the statements: $ (S1) $: $ \triangle ABC $ is an isosceles right angled triangle, and $ (S2) $: the area of $ \triangle ABC $ is $ \dfrac{9\sqrt{2}}{2} $.

48.

Let the foot of perpendicular from the point $ (\lambda,2,3) $ on the line $ \dfrac{x-4}{1}=\dfrac{y-9}{2}=\dfrac{z-5}{1} $ be the point $ (1,\mu,2) $. Then the distance between the lines $ \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z+4}{6} $ and $ \dfrac{x-\lambda}{2}=\dfrac{y-\mu}{3}=\dfrac{z+5}{6} $ is equal to:

49.

Let the plane $ a x + b y + c z + d = 0 $ bisect the line joining the points $ (4, -3, 1) $ and $ (2, 3, -5) $ at the right angles. If $ a, b, c, d $ are integers, then the minimum value of $ (a^2 + b^2 + c^2 + d^2) $ is ____________.

50.

Let the line passing through the points $ (-1, 2, 1) $ and parallel to the line $ \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4} $ intersect the line $ \frac{x+2}{3} = \frac{y-3}{2} = \frac{z-4}{1} $ at the point $ P $. Then the distance of $ P $ from the point $ Q (4, -5, 1) $ is:

51.

If the shortest distance between the lines $ \frac{x-\lambda}{2}=\frac{y-4}{4}=\frac{z-3}{4} $ and $ \frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8} $ is $ \frac{13}{\sqrt{29}} $, then a value of $ \lambda $ is:

52.

Let $ P $ be the foot of the perpendicular from the point $ Q(10,-3,-1) $ on the line $ \dfrac{x-3}{7}=\dfrac{y-2}{-1}=\dfrac{z+1}{-2} $. Then the area of the right-angled triangle $ PQR $, where $ R $ is the point $ (3,-2,1) $ is:

53.

The shortest distance between the lines $ \frac{x-3}{2} = \frac{y-2}{3} = \frac{z-1}{-1} $ and $ \frac{x+3}{2} = \frac{y-6}{1} = \frac{z-5}{3} $, is

54.

If the equation of the plane passing through the point $ (1, 1, 2) $ and perpendicular to the line $ x - 3y + 2z - 1 = 0 = 4x - y + z $ is $ Ax + By + Cz = 1 $, then $ 140(C - B + A) $ is equal to _______.

55.

Let in a $ \triangle ABC $, the length of the side $ AC $ be $ 6 $, the vertex $ B $ be $ (1, 2, 3) $ and the vertices $ A, C $ lie on the line $ \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} $. Then the area (in sq. units) of $ \triangle ABC $ is:

56.

Let $\theta$ be the angle between the planes $P_1 : \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2 : \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $(\tan^2 \theta)(\cot^2 \alpha)$ is equal to _____.

57.

Let $ Q $ be the mirror image of the point $ P(1,2,1) $ with respect to the plane $ x+2y+2z=16 $. Let $ T $ be a plane passing through the point $ Q $ and contains the line $ \vec{r}=-\hat{k}+ \lambda(\hat{i}+\hat{j}+2\hat{k}), \ \lambda \in \mathbb{R} $. Then which of the following points lies on $ T $?

58.

Let $ \lambda $ be an integer. If the shortest distance between the lines $ x-\lambda = 2y - 1 = -2z $ and $ x = y + 2\lambda = z - \lambda $ is $ \dfrac{\sqrt{7}}{2\sqrt{2}} $, then the value of $ |\lambda| $ is ________.

59.

Let the direction cosines of two lines satisfy the equations: $ 4l + m - n = 0 $ and $ 2mn + 10nl + 3lm = 0 $. Then the cosine of the acute angle between these lines is:

60.

If the shortest distance between the lines $ \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} $ and $ \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2} $ is $ \frac{38}{3\sqrt{5}} \, k $, and $ \int_{0}^{k} [ x^{2} ]dx = \alpha - \sqrt{\alpha} $, where $ [x] $ denotes the greatest integer function, then $ 6\alpha^{3} $ is equal to ________.

61.

Line $ L_{1} $ passes through the point $ (1,2,3) $ and is parallel to $ z $-axis. Line $ L_{2} $ passes through the point $ (\lambda,5,6) $ and is parallel to $ y $-axis. Let for $ \lambda = \lambda_{1}, \lambda_{2}, \lambda_{2} < \lambda_{1} $, the shortest distance between the two lines be $ 3 $. Then the square of the distance of the point $ (\lambda_{1}, \lambda_{2}, 7) $ from the line $ L_{1} $ is:

62.

Let $ \dfrac{x-2}{3}=\dfrac{y+1}{-2}=\dfrac{z+3}{-1} $ lie on the plane $ px-qy+z=5 $, for some $ p,q\in\mathbb{R} $. The shortest distance of the plane from the origin is:

63.

Let a line $ l $ pass through the origin and be perpendicular to the lines $ l_{1}:\ \vec{r}=(\hat{i}-11\hat{j}-7\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k}),\ \lambda\in\mathbb{R} $ and $ l_{2}:\ \vec{r}=(-\hat{i}+\hat{k})+\mu(2\hat{i}+2\hat{j}+\hat{k}),\ \mu\in\mathbb{R} $. If $ P $ is the point of intersection of $ l $ and $ l_{1} $, and $ Q(\alpha,\beta,\gamma) $ is the foot of perpendicular from $ P $ on $ l_{2} $, then $ 9(\alpha+\beta+\gamma) $ is equal to __________ .

64.

If the point of intersection of the lines $ \frac{x+1}{3} = \frac{y+a}{5} = \frac{z+b+1}{7} $ and $ \frac{x-2}{1} = \frac{y-b}{4} = \frac{z-2a}{7} $ lies on $ xy $-plane, then the value of $ a+b $ is:

65.

If the mirror image of the point $ P(3, 4, 9) $ in the line $ \dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-2}{1} $ is $ (\alpha,\beta,\gamma) $, then $ 14(\alpha + \beta + \gamma) $ is:

66.

If the distance of the point $ (a, 2, 5) $ from the image of the point $ (1, 2, 7) $ in the line $ \frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2} $ is $ 4 $, then the sum of all possible value of $ a $ is equal to:

67.

If the distance of the point $ Q(1,-2,3) $ from the plane $ x+2y-3z+10=0 $ measured parallel to the line $ \dfrac{x-1}{3}=\dfrac{2-y}{m}=\dfrac{z+3}{1} $ is $ \sqrt{\dfrac{{7}}{{2}}} $, then the value of $ |m| $ is equal to ____________.

68.

Let the equation of the plane passing through the line $ x-2y-z-5=0 = x+y+3z-5 $ and parallel to the line $ x+y+2z-7=0 = 2x+3y+z-2 $ be $ ax+by+cz=65 $. Then the distance of the point $ (a,b,c) $ from the plane $ 2x+2y-z+16=0 $ is __________.

69.

The line $ l_{1} $ passes through the point $ (2, 6, 2) $ and is perpendicular to the plane $ 2x + y - 2z = 10 $. Then the shortest distance between the line $ l_{1} $ and the line $ \frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2} $ is:

70.

Let $ P $ be the plane containing the straight line $ \displaystyle\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5} $ and perpendicular to the plane containing the straight lines $\displaystyle \frac{x}{2}=\frac{y}{3}=\frac{z}{5} $ & $\displaystyle \frac{x}{3}=\frac{y}{7}=\frac{z}{8} $. If $ d $ is the distance of $ P $ from the point $ (2,-5,11) $, then $ d^2 $ is equal to:

71.

The shortest distance between the lines $ \dfrac{x+2}{1}=\dfrac{y}{-2}=\dfrac{z-5}{2} $ and $ \dfrac{x-4}{1}=\dfrac{y-1}{2}=\dfrac{z-3}{0} $ is:

72.

Let the line $ L_1 $ be parallel to the vector $ -3\hat{\text{i}} + 2\hat{\text{j}} + 4\hat{\text{k}} $ and pass through the point $ (2,\, 6,\, 7) $, and the line $ L_2 $ be parallel to the vector $ 2\hat{\text{i}} + \hat{\text{j}} + 3\hat{\text{k}} $ and pass through the point $ (4,\, 3,\, 5) $. If the line $ L_3 $ is parallel to the vector $ -3\hat{\text{i}} + 5\hat{\text{j}} + 16\hat{\text{k}} $ and intersects the lines $ L_1 $ and $ L_2 $ at the points $ C $ and $ D $, respectively, then $ |\overline{CD}|^2 $ is equal to:

73.

If the image of the point $ (4, 4, 3) $ in the line $ \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} $ is $ (\alpha, \beta, \gamma) $, then $ \alpha + \beta + \gamma $ is equal to:

74.

Let the image of the point $ P(1, 2, 3) $ in the plane $ 2x - y + z = 9 $ be $ Q $. If the coordinates of the point $ R $ are $ (6, 10, 7) $, then the square of the area of the triangle $ PQR $ is __________.

75.

Let $ P $ be the plane passing through the line $ \dfrac{x-1}{1}=\dfrac{y-2}{-3}=\dfrac{z+5}{7} $ and the point $ (2,4,-3) $. If the image of the point $ (-1,3,4) $ in the plane $ P $ is $ (\alpha,\beta,\gamma) $ then $ \alpha+\beta+\gamma $ is equal to:

76.

Let $ Q $ be the mirror image of the point $ P(1,0,1) $ with respect to the plane $ S:x+y+z=5 $. If a line $ L $ passing through $ (1,-1,-1) $, parallel to the line $ PQ $ meets the plane $ S $ at $ R $, then $ QR^{2} $ is equal to:

77.

Let $ (\alpha, \beta, \gamma) $ be the co-ordinates of the foot of the perpendicular drawn from the point $ (5, 4, 2) $ on the line $ \vec{r} = (\hat{-i} + 3\hat{j} + \hat{k}) + \lambda (2\hat{i} + 3\hat{j} - \hat{k}) $. Then the length of the projection of the vector $ \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} $ on the vector $ 6\hat{i} + 2\hat{j} + 3\hat{k} $ is:

78.

If the line $ \frac{2 - x}{3} = \frac{3y - 2}{4\lambda + 1} = 4 - z $ makes a right angle with the line $ \frac{x + 3}{3\mu} = \frac{1 - 2y}{6} = \frac{5 - z}{7} $, then $ 4\lambda + 9\mu $ is equal to:

79.

The square of the distance of the point $ (-2,-8,6) $ from the line $ \dfrac{x-1}{1}=\dfrac{y-1}{2}=\dfrac{z}{-1} $ along the line $ \dfrac{x+5}{1}=\dfrac{y+5}{-1}=\dfrac{z}{2} $ is equal to:

80.

The foot of perpendicular of the point $ (2,0,5) $ on the line $ \frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1} $ is $ (\alpha,\beta,\gamma) $. Then, which of the following is NOT correct?

81.

Let $Q$ be the foot of the perpendicular from the point $P(7, -2, 13)$ on the plane containing the lines
$ \frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8} \quad \text{and} \quad \frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}. $
Then $(PQ)^2$, is equal to ________.

82.

The plane, passing through the points $ (0,-1,2) $ and $ (-1,2,1) $ and parallel to the line passing through $ (5,1,-7) $ and $ (1,-1,-1) $, also passes through the point:

83.

Let a line having direction ratios $ 1, -4, 2 $ intersect the lines
$ \dfrac{x-7}{3} = \dfrac{y-1}{-1} = \dfrac{z+2}{1} $ and $ \dfrac{x}{2} = \dfrac{y-7}{3} = \dfrac{z}{1} $ at the point A and B. Then $ (AB)^2 $ is equal to ______________.

84.

Let $ L $ be the line $ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} $ and let $ S $ be the set of all points $ (a, b, c) $ on $ L $, whose distance from the line $ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} $ along the line $ L $ is $ 7 $. Then $ \sum_{(a,b,c)\in S} (a + b + c) $ is equal to:

85.

Let the line $ L $ pass through $ (1, 1, 1) $ and intersect the lines $ \dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4} $ and $ \dfrac{x-3}{1} = \dfrac{y-4}{2} = \dfrac{z}{1} $. Then, which of the following points lies on the line $ L $?

86.

Let $ ABCD $ be a tetrahedron such that the edges $ AB $, $ AC $ and $ AD $ are mutually perpendicular. Let the areas of the triangles $ ABC $, $ ACD $ and $ ADB $ be $ 5 $, $ 6 $ and $ 7 $ square units respectively. Then the area (in square units) of the $ \triangle BCD $ is equal to:

87.

If $ \left(2\alpha+1,\alpha^2-3\alpha,\frac{\alpha-1}{2}\right) $ is the image of $ (\alpha,2\alpha,1) $ in the line $ \frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1} $, then the possible value(s) of $ \alpha $ is (are):

88.

Let the point, on the line passing through the points $ P(1, -2, 3) $ and $ Q(5, -4, 7) $, farther from the origin and at a distance of $ 9 $ units from the point $ P $, be $ (\alpha, \beta, \gamma) $. Then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

89.

Let the values of $ \lambda $ for which the shortest distance between the lines $ \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} $ and $ \dfrac{x-\lambda}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} $ is $ \dfrac{1}{\sqrt{6}} $ be $ \lambda_{1} $ and $ \lambda_{2} $. Then the radius of the circle passing through the points $ (0,0) $, $ (\lambda_{1},\lambda_{2}) $ and $ (\lambda_{2},\lambda_{1}) $ is:

90.

The shortest distance between the lines $ \frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} $ and $ \frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1} $ is:

91.

The distance of the point $ (-1,\,2,\,-2) $ from the line of intersection of the planes $ 2x+3y+2z=0 $ and $ x-2y+z=0 $:

92.

Let the image of the point $ P(1, 2, 3) $ in the line $ L:\ \dfrac{x - 6}{3} = \dfrac{y - 1}{2} = \dfrac{z - 2}{3} $ be $ Q $. Let $ R(\alpha, \beta, \gamma) $ be a point that divides internally the line segment $ PQ $ in the ratio $ 1:3 $. Then the value of $ 22(\alpha + \beta + \gamma) $ is equal to $ \_ \_ \_ \_ \_ \_ $.

93.

Let the line $ L : \dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{1} $ intersect the plane $ 2x + y + 3z = 16 $ at the point $ P $. Let the point $ Q $ be the foot of perpendicular from the point $ R(1,-1,-3) $ on the line $ L $. If $ \alpha $ is the area of triangle $ PQR $, then $ \alpha^{2} $ is equal to _____.

94.

Let $ \alpha x + \beta y + yz = 1 $ be the equation of the plane passing through the point $ (3, -2, 5) $ and perpendicular to the line joining the points $ (1, 2, 3) $ and $ (-2, 3, 5) $. Then the value of $ \alpha \beta \gamma $ is equal to ______.

95.

Let the position vectors of three vertices of a triangle be $ 4\vec{p} + \vec{q} - 3\vec{r} $, $ -5\vec{p} + \vec{q} + 2\vec{r} $ and $ 2\vec{p} - \vec{q} + 2\vec{r} $. If the position vectors of the orthocenter and the circumcenter of the triangle are $ \frac{\vec{p} + \vec{q} + \vec{r}}{4} $ and $ \alpha \vec{p} + \beta \vec{q} + \gamma \vec{r} $ respectively, then $ \alpha + 2\beta + 5\gamma $ is equal to:

96.

Consider the lines $ L_{1}: {x-1} = {y-2} = {z} $ and $ L_{2}: {x-2} = {y} = {z-1} $. Let the feet of the perpendiculars from the point $ P(5,1,-3) $ on the lines $ L_{1} $ and $ L_{2} $ be $ Q $ and $ R $ respectively. If the area of the triangle $ PQR $ is $ A $, then $ 4A^{2} $ is equal to:

97.

For the system of linear equations:
$ x - 2y = 1 $
$ x - y + kz = -2 $
$ ky + 4z = 6, \; k \in \mathbb{R} $,
Consider the following statements:
(A) The system has unique solution if $ k \neq 2, \, k \neq -2 $.
(B) The system has unique solution if $ k = -2 $.
(C) The system has unique solution if $ k = 2 $.
(D) The system has no-solution if $ k = 2 $.
(E) The system has infinite number of solutions if $ k \neq -2 $.
Which of the following statements are correct?

98.

The distance of the line $\displaystyle \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4} $ from the point $ (1, 4, 0) $ along the line $\displaystyle \frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} $ is:

99.

Let a plane $ P $ pass through the point $ (3,7,-7) $ and contain the line, $ \dfrac{x-2}{-3}=\dfrac{y-3}{2}=\dfrac{z+2}{1} $. If distance of the plane $ P $ from the origin is $ d $, then $ d^2 $ is equal to $ \underline{\quad} $.

100.

Let the mirror image of the point $ (a, b, c) $ with respect to the plane $ 3x - 4y + 12z + 19 = 0 $ be $ (a - 6, \beta, \gamma) $. If $ a + b + c = 5 $, then $ 7\beta - 9\gamma $ is equal to ____________.

101.

Suppose the line $ \dfrac{x-2}{\alpha}=\dfrac{y-2}{-5}=\dfrac{z+2}{2} $ lies on the plane $ x+3y-2z+\beta=0 $. Then $ (\alpha+\beta) $ is equal to ______.

102.

If the equation of the plane passing through the line of intersection of the planes $ 2x - 7y + 4z - 3 = 0 $, $ 3x - 5y + 4z + 11 = 0 $ and the point $ (-2, 1, 3) $ is $ ax + by + cz - 7 = 0 $, then the value of $ 2a + b + c - 7 $ is _______.

103.

Let the lines $ l_1 : \dfrac{x+5}{3} = \dfrac{y+4}{1} = \dfrac{z-\alpha}{-2} $ and $ l_2 : 3x + 2y + z - 2 = 0 = x - 3y + 2z - 13 $ be coplanar. If the point $ P(a,b,c) $ on $ l_1 $ is nearest to point $ Q(-4,-3,2) $, then $ |a| + |b| + |c| $ is equal to:

104.

If the shortest between the lines: $ \dfrac{x+\sqrt{6}}{2}=\dfrac{y-\sqrt{6}}{3}=\dfrac{z-\sqrt{6}}{4} $ and $ \dfrac{x-\lambda}{3}=\dfrac{y-2\sqrt{6}}{4}=\dfrac{z+2\sqrt{6}}{5} $ is $ 6 $, then the square of sum of all possible values of $ \lambda $ is:

105.

Let the line $ L $ pass through the point $ (0, 1, 2) $, intersect the line $ \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4} $ and be parallel to the plane $ 2x + y - 3z = 4 $. Then the distance of the point $ P(1, -9, 2) $ from the line $ L $ is:

106.

Let a line $ L_1 $ pass through the origin and be perpendicular to the lines $ L_2:\ \mathbf{\vec r}=(3+t)\mathbf{\hat i}+(2t-1)\mathbf{\hat j}+(2t+4)\mathbf{\hat k} $ and $ L_3:\ \mathbf{\vec r}=(3+2s)\mathbf{\hat i}+(3+2s)\mathbf{\hat j}+(2+s)\mathbf{\hat k},\ t,s\in \mathbb{R} $. If $ (a,b,c),\ a\in \mathbb{Z} $, is the point on $ L_3 $ at a distance of $ \sqrt{17} $ from the point of intersection of $ L_1 $ and $ L_2 $, then $ (a+b+c)^2 $ is equal to $ \_\_\_\_\_ $.

107.

If the square of the shortest distance between the lines $\displaystyle \frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{-3} $ and $\displaystyle \frac{x+1}{2} = \frac{y+3}{4} = \frac{z+5}{-5} $ is $\displaystyle \frac{m}{n} $, where $ m, n $ are coprime numbers, then $ m + n $ is equal to:

108.

The shortest distance between the lines $ \dfrac{x+7}{-6}=\dfrac{y-6}{7}=z $ and $ \dfrac{7-x}{2}=y-2=z-6 $ is

109.

One vertex of a rectangular parallelopiped is at the origin $ O $ and the lengths of its edges along $ x, y $ and $ z $ axes are $ 3, 4 $ and $ 5 $ units respectively. Let $ P $ be the vertex $ (3,4,5) $. Then the shortest distance between the diagonal $ OP $ and an edge parallel to $ z $-axis, not passing through $ O $ or $ P $ is:

110.

Consider the line $ L $ given by the equation $\displaystyle \frac{x-3}{2} = \frac{y-1}{1} = \frac{z-2}{1} $. Let $ Q $ be the mirror image of the point $ (2, 3, -1) $ with respect to $ L $. Let a plane $ P $ be such that it passes through $ Q $, and the line $ L $ is perpendicular to $ P $. Then which of the following points is on the plane $ P $?

111.

The sum of all values of $ \alpha $, for which the shortest distance between the lines $ \frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha} $ and $ \frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha} $ is $ \sqrt{2} $, is:

112.

Let $ L_{1}:\ \vec{r} = (\hat{\imath}-\hat{\jmath}+2\hat{k}) + \lambda(\hat{\imath}-\hat{\jmath}+2\hat{k}),\ \lambda \in \mathbb{R}; \ L_{2}:\ \vec{r} = (j-\hat{k}\,) + \mu(3\hat{\imath}+\hat{\jmath}+p\hat{k}),\ \mu \in \mathbb{R}; $ and $ L_{3}:\ \vec{r} = \delta(\hat{\imath}+m\hat{\jmath}+n\hat{k}),\ \delta \in \mathbb{R} $ be three lines such that $ L_{1} $ is perpendicular to $ L_{2} $ and $ L_{3} $ is perpendicular to both $ L_{1} $ and $ L_{2} $. Then, the point which lies on $ L_{3} $ is:

113.

If the length of the perpendicular drawn from the point $ P(a,4,2), a>0 $ on the line $ \displaystyle\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1} $ is $ 2\sqrt{6} $ units and $ Q(\alpha_1,\alpha_2,\alpha_3) $ is the image of the point $ P $ in this line, then $ a + \displaystyle\sum_{i=1}^{3}\alpha_i $ is equal to:

114.

If the line of intersection of the planes $ ax+by=3 $ and $ ax+by+cz=0,\, a>0 $ makes an angle $ 30^\circ $ with the plane $ y-z+2=0 $, then the direction cosines of the line are:

115.

If the shortest distance between the line joining the points $ (1,2,3) $ and $ (2,3,4) $, and the line $ \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0} $ is $ \alpha $, then $ 28\alpha^2 $ is equal to ______.

116.

The distance of the point $Q(0, 2, -2)$ from the line passing through the point $P(5, -4, 3)$ and perpendicular to the lines $\vec{r} = (-3\hat{i} + 2\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 5\hat{k}), \lambda \in R$ and $\vec{r} = (\hat{i} - 2\hat{j} + \hat{k}) + \mu(-\hat{i} + 3\hat{j} + 2\hat{k}), \mu \in R$ is:

117.

If $ \lambda_{1}<\lambda_{2} $ are two values of $ \lambda $ such that the angle between the planes $ P_{1}:\ \vec{r}\cdot(3\hat{i}-5\hat{j}+\hat{k})=7 $ and $ P_{2}:\ \vec{r}\cdot(\lambda \hat{i}+\hat{j}-3\hat{k})=9 $ is $ \sin^{-1}\!\left(\dfrac{2\sqrt{6}}{5}\right) $, then the square of the length of perpendicular from the point $ (38\lambda_{1},\,10\lambda_{2},\,2) $ to the plane $ P_{1} $ is $ \_\_\_\_\_ $.

118.

Let the vertices $ Q $ and $ R $ of the triangle $ PQR $ lie on the line $ \dfrac{x+3}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} $, $ QR = 5 $ and the coordinates of the point $ P $ be $ (0, 2, 3) $. If the area of the triangle $ PQR $ is $ \dfrac{m}{n} $ then:

119.

For $ \alpha, \beta \in \mathbb{R} $, suppose the system of linear equations
$ x - y + z = 5 $
$ 2x + 2y + \alpha z = 8 $
$ 3x - y + 4z = \beta $
has infinitely many solutions. Then $ \alpha $ and $ \beta $ are the roots of:

120.

Let the mirror image of the point $ (1, 3, a) $ with respect to the plane $ \vec{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) - b = 0 $ be $ (-3, 5, 2) $. Then, the value of $ |\, a + b \,| $ is equal to:

121.

Let $ P(x, y, z) $ be a point in the first octant, whose projection in the $ xy $-plane is the point $ Q $. Let $ OP = \gamma $; the angle between $ OQ $ and the positive $ x $-axis be $ \theta $; and the angle between $ OP $ and the positive $ z $-axis be $ \phi $, where $ O $ is the origin. Then the distance of $ P $ from the $ x $-axis is:

122.

Let the equation of plane passing through the line of intersection of the planes $ x+2y+az=2 $ and $ x-y+z=3 $ be $ 5x-11y+bz=6a-1 $. For $ c\in \mathbb{Z} $, if the distance of this plane from the point $ (a,-c,c) $ is $ \dfrac{2}{\sqrt{a}} $ then $ \dfrac{a+b}{c} $ is equal to:

123.

The shortest distance between the lines $ \frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} $ and $ \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3} $ is:

124.

The lines $ \dfrac{x-2}{2}=\dfrac{y}{-2}=\dfrac{z-7}{16} $ and $ \dfrac{x+3}{4}=\dfrac{y+2}{3}=\dfrac{z+2}{1} $ intersect at the point $ P $. If the distance of $ P $ from the line $ \dfrac{x+1}{2}=\dfrac{y-1}{3}=\dfrac{z-1}{1} $ is $ l $, then $ 14\, l^{2} $ is equal to __________ .

125.

If the mirror image of the point $ (2,4,7) $ in the plane $ 3x-y+4z=2 $ is $ (a,b,c) $, then $ 2a+b+2c $ is equal to:

126.

If the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x}{1} = \frac{y}{\alpha} = \frac{z-5}{1}$ is $\frac{5}{\sqrt{6}}$, then the sum of all possible values of $\alpha$ is:

127.

Let $ N $ be the foot of perpendicular from the point $ P(1,-2,3) $ on the line passing through the points $ (4,5,8) $ and $ (1,-7,5) $. Then the distance of $ N $ from the plane $ 2x-2y+z+5=0 $ is:

128.

Let the lines
$ L_1:\ \vec{r}=\lambda(\hat{i}+2\hat{j}+3\hat{k}),\ \lambda\in\mathbb{R} $ and
$ L_2:\ \vec{r}=(\hat{i}+3\hat{j}+\hat{k})+\mu(\hat{i}+\hat{j}+5\hat{k}),\ \mu\in\mathbb{R} $
Intersect at the point $ S $. If a plane $ ax+by-z+d=0 $ passes through $ S $ and is parallel to both the lines $ L_1 $ and $ L_2 $, then the value of $ a+b+d $ is equal to ________.

129.

If the system of linear equations.
$ 8x + y + 4z = -2 $
$ x + y + z = 0 $
$ \lambda x - 3y = \mu $
has infinitely many solutions, then the distance of the point $ \left(\lambda, \mu, -\dfrac{1}{2}\right) $ from the plane $ 8x + y + 4z = 0 $ is:

130.

Let $ P $ be the plane passing through the intersection of the planes $ \vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 5 $ and $ \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 3 $, and the point $ (2, 1, -2) $. Let the position vectors of the points $ X $ and $ Y $ be $ \hat{i} - 2\hat{j} + 4\hat{k} $ and $ 5\hat{i} - \hat{j} + 2\hat{k} $ respectively. Then the points

131.

Let $ P $ be the point $ (10, -2, -1) $ and $ Q $ be the foot of the perpendicular drawn from the point $ R(1, 7, 6) $ on the line passing through the points $ (2, -5, 11) $ and $ (-6, 7, -5) $. Then the length of the line segment $ PQ $ is equal to:

132.

The point of intersection $ C $ of the plane $ 8x + y + 2z = 0 $ and the line joining the points $ A(-3, -6, 1) $ and $ B(2, 4, -3) $ divides the line segment $ AB $ internally in the ratio $ k : 1 $. If $ a, b, c $ ( $ |a|, |b|, |c| $ are coprime) are the direction ratios of the perpendicular from the point $ C $ on the line $ \frac{1 - x}{1} = \frac{y + 4}{2} = \frac{z + 2}{3} $, then $ |a + b + c| $ is equal to ______.

133.

Let the points on the plane $ P $ be equidistant from the points $ (-4,2,1) $ and $ (2,-2,3) $. Then the acute angle between the plane $ P $ and the plane $ 2x+y+3z=1 $ is:

134.

Let the line $ L $ intersect the lines $ x - 2 = -y = z - 1 $, $ 2(x + 1) = 2(y - 1) = z + 1 $ and be parallel to the line $ \dfrac{x - 2}{3} = \dfrac{y - 1}{1} = \dfrac{z - 2}{2} $. Then which of the following points lies on $ L $?

135.

Let two vertices of a triangle $ ABC $ be $ (2, 4, 6) $ and $ (0, -2, -5) $, and its centroid be $ (2, 1, -1) $. If the image of the third vertex in the plane $ x + 2y + 4z = 11 $ is $ (\alpha, \beta, \gamma) $, then $ \alpha\beta + \beta\gamma + \gamma\alpha $ is equal to:

136.

The distance of the point $ (-1, 2, 3) $ from the plane $ \vec r \cdot (\hat i - 2\hat j + 3\hat k) = 10 $ parallel to the line of the shortest distance between the lines $ \vec r = (\hat i - \hat j) + \lambda (2\hat i + \hat k) $ and $ \vec r = (2\hat i - \hat j) + \mu (\hat i - \hat j + \hat k) $ is:

137.

Let $ P(3, 2, 3) $, $ Q(4, 6, 2) $ and $ R(7, 3, 2) $ be the vertices of $ \triangle PQR $. Then, the angle $ \angle QPR $ is:

138.

Let $ P $ be a plane passing through the points $ (1,0,1) $, $ (1,-2,1) $ and $ (0,1,-2) $. Let a vector $ \vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k} $ be such that $ \vec{a} $ is parallel to the plane $ P $, perpendicular to $ (\hat{i}+2\hat{j}+3\hat{k}) $ and $ \vec{a}\cdot(\hat{i}+\hat{j}+2\hat{k})=2 $, then $ (\alpha-\beta+\gamma)^{2} $ equals _______.

139.

The length of the perpendicular from the point $ (1,-2,5) $ on the line passing through $ (1,2,4) $ and parallel to the line $ x+y-z=0=x-2y+3z-5 $ is:

140.

The line $ L_{1} $ is parallel to the vector $ \vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} $ and passes through the point $ (7, 6, 2) $ and the line $ L_{2} $ is parallel to the vector $ \vec{b} = 2\hat{i} + \hat{j} + 3\hat{k} $ and passes through the point $ (5, 3, 4) $. The shortest distance between the lines $ L_{1} $ and $ L_{2} $ is:

141.

The distance of line $ 3y - 2z - 1 = 0 = 3x - z + 4 $ from the point $ (2, -1, 6) $ is:

142.

Let a line $ L $ passing through the point $ P(1, 1, 1) $ be perpendicular to the lines $ \frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1} $ and $ \frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0} $. Let the lines $ L $ intersect the $ yz $-plane at the point $ Q $. Another line parallel to $ L $ and passing through the point $ S(1, 0, -1) $ intersects the $ yz $-plane at the point $ R $. Then the square of the area of the parallelogram $ PQRS $ is equal to _______.

143.

Let the plane $P : \vec{r} \cdot \vec{a} = d$ contain the line of intersection of two planes $\vec{r} \cdot \left(\hat{i} + 3\hat{j} - \hat{k}\right) = 6$ and $\vec{r} \cdot \left(-6\hat{i} + 5\hat{j} - \hat{k}\right) = 7$. If the plane $P$ passes through the point $\left(2, 3, \frac{1}{2}\right)$, then the value of $\frac{|13\vec{a}|^2}{d^2}$ is equal to:

144.

The square of the distance of the image of the point $ (6,1,5) $ in the line $ \dfrac{x-1}{3}=\dfrac{y}{2}=\dfrac{z-2}{4} $, from the origin is_________.

145.

Let $ d $ be the distance between the foot of perpendiculars of the points $ P(1,2,-1) $ and $ Q(2,-1,3) $ on the plane $ -x + y + z = 1 $. Then $ d^2 $ is equal to __________.

146.

Each of the angles $ \beta $ and $ \gamma $ that a given line makes with the positive $ y $- and $ z $-axes, respectively, is half of the angle that this line makes with the positive $ x $-axes. Then the sum of all possible values of the angle $ \beta $ is:

147.

Let $ (\alpha,\beta,\gamma) $ be the mirror image of the point $ (2,3,5) $ in the line $ \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} $. Then $ 2\alpha+3\beta+4\gamma $ is equal to:

148.

Let $ P_{1} $ be the plane $ 3x-y-7z=11 $ and $ P_{2} $ be the plane passing through the points $ (2,-1,0) $, $ (2,0,-1) $, and $ (5,1,1) $. If the foot of the perpendicular drawn from the point $ (7,4,-1) $ on the line of intersection of the planes $ P_{1} $ and $ P_{2} $ is $ (\alpha,\beta,\gamma) $, then $ \alpha+\beta+\gamma $ is equal to _______.

149.

Let $ A $ be the point of intersection of the lines $ L_1 : \, \frac{x-7}{1} = \frac{y-5}{0} = \frac{z-3}{-1} $ and $ L_2 : \, \frac{x-1}{3} = \frac{y+3}{4} = \frac{z+7}{5} $.Let $ B $ and $ C $ be the points on the lines $ L_1 $ and $ L_2 $ respectively such that $ AB = AC = \sqrt{15} $. Then the square of the area of the triangle $ ABC $ is:

150.

Let the equation of the plane $P$ containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point $(1, 27, 7)$ be $c$. Then $a^2+b^2+c^2$ is equal to _______.

151.

The vertices $ B $ and $ C $ of a triangle $ ABC $ lie on the line $ \dfrac{x}{1} = \dfrac{1 - y}{-2} = \dfrac{z - 2}{3} $. The coordinates of $ A $ and $ B $ are $ (1, 6, 3) $ and $ (4, 9, \alpha) $ respectively and $ C $ is at a distance of $ 10 $ units from $ B $. The area (in sq. units) of $ \triangle ABC $ is:

152.

If $ d_1 $ is the shortest distance between the lines $ x + 1 = 2y = -12z $ and $ x = y + 2 = 6z - 6 $ and $ d_2 $ is the shortest distance between the lines $ \dfrac{x - 1}{2} = \dfrac{y + 8}{-7} = \dfrac{z - 4}{5} $ and $ \dfrac{x - 1}{2} = \dfrac{y - 2}{1} = \dfrac{z - 6}{-3} $, then the value of $ \dfrac{32\sqrt{3}\, d_1}{d_2} $ is:

153.

The distance of the point $ (-1,9,-16) $ from the plane $ 2x+3y-z=5 $ measured parallel to the line $ \frac{x+4}{3}=\frac{2-y}{4}=\frac{z-3}{12} $ is:

154.

The distance of the point $ P(4,6,-2) $ from the line passing through the point $ (-3,2,3) $ and parallel to a line with direction ratios $ 3, 3, -1 $ is equal to:

155.

The shortest distance between the lines $ \dfrac{x-1}{2}=\dfrac{y+8}{-7}=\dfrac{z-4}{5} $ and $ \dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-6}{-3} $ is:

156.

Let the image of the point $ (1,0,7) $ in the line $\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} $ be the point $ (\alpha,\beta,\gamma) $. Then which one of the following points lies on the line passing through $ (\alpha,\beta,\gamma) $ and making angles $\displaystyle \frac{2\pi}{3} $ and $\displaystyle \frac{3\pi}{4} $ with $ y $-axis and $ z $-axis respectively and an acute angle with $ x $-axis?

157.

Let the line passing through the points $ P(2,-1,2) $ and $ Q(5,3,4) $ meet the plane $ x-y+z=4 $ at the point $ R $. Then the distance of the point $ R $ from the plane $ x+2y+3z+2=0 $ measured parallel to the line $\displaystyle \frac{x-7}{2}=\frac{y+3}{2}=\frac{z-2}{1} $ is equal to:

158.

If the foot of the perpendicular from point $ (4, 3, 8) $ on the line $ L_1 : \dfrac{x-a}{1}=\dfrac{y-2}{3}=\dfrac{z-b}{4} $,$\displaystyle 1 \neq 0$ is $ (3,5,7) $, then the shortest distance between the line $ L_1 $ and line $ L_2 : \dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} $ is equal to:

159.

If the distances of the point $ (1,2,a) $ from the line $ \dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1} $ along the lines $ L_{1}: \ \dfrac{x-1}{3} = \dfrac{y-2}{4} = \dfrac{z-a}{b} $ and $ L_{2}: \ \dfrac{x-1}{1} = \dfrac{y-2}{4} = \dfrac{z-a}{c} $ are equal, then $ a+b+c $ is equal to:

160.

The shortest distance between the lines $ \frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2} $ and $ \frac{x-6}{3} = \frac{1-y}{2} = \frac{z+8}{0} $ is equal to ________.

161.

The foot of perpendicular from the origin $ O $ to a plane $ P $ which meets the co-ordinate axes at the points $ A, B, C $ is $ (2,a,4) $, $ a\in \mathbb{N} $. If the volume of the tetrahedron $ OABC $ is $ 144 \,\text{unit}^{3} $, then which of the following points is NOT on $ P $?

162.

Let the plane $ ax + by + cz = d $ pass through $ (2,3,-5) $ and is perpendicular to the planes $ 2x + y - 5z = 10 $ and $ 3x + 5y - 7z = 12 $. If $ a,b,c,d $ are integers $ d>0 $ and $ \gcd\!\left(|a|,|b|,|c|,d\right)=1 $, then the value of $ a + 7b + c + 20d $ is equal to:

163.

Let the image of the point $ P(1, 2, 6) $ in the plane passing through the points $ A(1, 2, 0) $, $ B(1, 4, 1) $ and $ C(0, 5, 1) $ be $ Q(\alpha, \beta, \gamma) $. Then $ \left( \alpha^{2} + \beta^{2} + \gamma^{2} \right) $ is equal to:

164.

Let the plane containing the line of intersection of the planes $ P1: x + (\lambda + 4)y + z = 1 $ and $ P2: 2x + y + z = 2 $ pass through the points $ (0, 1, 0) $ and $ (1, 0, 1) $. Then the distance of the point $ (2\lambda, \lambda, -\lambda) $ from the plane $ P2 $ is:

165.

The lines $ x = ay - 1 = z - 2 $ and $ x = 3y - 2 = bz - 2 $, $ (ab \neq 0) $ are coplanar, if:

166.

The acute angle between the planes $ P_1 $ and $ P_2 $, when $ P_1 $ and $ P_2 $ are the planes passing through the intersection of the planes $ 5x + 8y + 13z - 29 = 0 $ and $ 8x - 7y + z - 20 = 0 $ and the points $ (2,1,3) $ and $ (0,1,2) $, respectively, is:

167.

Let the point $ A $ be the foot of perpendicular drawn from the point $ P(a,b,0) $ on the line $ \dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-\alpha}{3} $. If the midpoint of the line segment $ PA $ is $ \left(0,\dfrac{3}{4},-\dfrac{1}{4}\right) $, then the value of $ a^2+b^2+\alpha^2 $ is equal to:

168.

If the plane $ 2x+y-5z=0 $ is rotated about its line of intersection with the plane $ 3x-y+4z-7=0 $ by an angle of $ \frac{\pi}{2} $, then the plane after the rotation passes through the point:

169.

Let the point $ (+1, \alpha, \beta) $ lie on the line of the shortest distance between the lines $ \dfrac{x + 2}{-3} = \dfrac{y - 2}{4} = \dfrac{z - 5}{2} $ and $ \dfrac{x + 2}{1} = \dfrac{y + 6}{2} = \dfrac{z - 1}{0} $.Then $ (\alpha - \beta)^{2} $ is equal to $ \_\_\_\_\_\_ $.

170.

Let the area of the triangle formed by the lines $ x+2=y-1=z $, $ \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1} $, and $ \frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1} $ be $ A $. Then $ A^{2} $ is equal to ______.

171.

Let $ P_1 : \vec{r} \cdot (2\vec{i} + \vec{j} - 3\vec{k}) = 4 $ be a plane. Let $ P_2 $ be another plane which passes through the points $ (2, -3, 2) $, $ (2, -2, -3) $ and $ (1, -4, 2) $. If the direction ratios of the line of intersection of $ P_1 $ and $ P_2 $ be $ 16, \alpha, \beta $, then the value of $ \alpha + \beta $ is equal to _________.

172.

Let the lines $L_1:\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda(2\hat{i}+3\hat{j}+4\hat{k}),\lambda\in R$ and $L_2:\vec{r}=(4\hat{i}+\hat{j})+\mu(5\hat{i}+2\hat{j}+\hat{k}),\mu\in R$, intersect at the point $R$. Let $P$ and $Q$ be the point lying on lines $L_1$ and $L_2$, respectively, such that $\left|\overrightarrow{PR}\right|=\sqrt{29}$ and $\left|\overrightarrow{PQ}\right|=\sqrt{\frac{47}{3}}$. If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to:

173.

Let a straight line $ L $ pass through the point $ P(2,-1,3) $ and be perpendicular to the lines $\displaystyle \frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2} $ and $\displaystyle \frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4} $. If the line $ L $ intersects the $ yz $-plane at the point $ Q $, then the distance between the point $ P $ and $ Q $ is:

174.

Let $ (\alpha,\beta,\gamma) $ be the image of the point $ (8,5,7) $ in the line: $ \dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{5} $. Then $ \alpha+\beta+\gamma $ is equal to:

175.

Let $ \lambda_1, \lambda_2 $ be the values of $ \lambda $ for which the points $ \left(\dfrac{5}{2},1,\lambda\right) $ and $ (-2,0,1) $ are at equal distance from the plane $ 2x+3y-6z+7=0 $. If $ \lambda_1>\lambda_2 $, then the distance of the point $ (\lambda_1-\lambda_2,\lambda_2,\lambda_1) $ from the line $ \dfrac{x-5}{1}=\dfrac{y-1}{2}=\dfrac{z+7}{2} $ is ________.

176.

Let $ P $ be the plane passing through the point $ (1, 2, 3) $ and the line of intersection of the planes $ \vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16 $ and $ \vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6 $. Then which of the following points does NOT lie on $ P $?

177.

Let the foot of the perpendicular from the point $ (1, 2, 4) $ on the line $ \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} $ be $ P $. Then the distance of $ P $ from the plane $ 3x + 4y + 12z + 23 = 0 $ is

178.

Let the image of the point $ P(2,-1,3) $ in the plane $ x+2y-z=0 $ be $ Q $. Then the distance of the plane $ 3x+2y+z+29=0 $ from the point $ Q $ is :

179.

Let the line $ l : x = \dfrac{1-y}{-2} = \dfrac{z-3}{\lambda}, \ \lambda \in \mathbb{R} $ meet the plane $ P : x + 2y + 3z = 4 $ at the point $ (\alpha,\beta,\gamma) $. If the angle between the line $ l $ and the plane $ P $ is $ \cos^{-1}\!\left(\sqrt{\dfrac{5}{14}}\right) $, then $ \alpha + 2\beta + 6\gamma $ is equal to __________.

180.

Let the co-ordinates of one vertex of $ \triangle ABC $ be $ A(0, 2, \alpha) $ and the other two vertices lie on the line $ \dfrac{x+\alpha}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} $. For $ \alpha \in $ $\mathbb{Z}$ , if the area of $ \triangle ABC $ is $ 21 $ sq. units and the line segment $ BC $ has length $ 2\sqrt{21} $ units, then $ \alpha^{2} $ is equal to ________.

181.

Let $ P(-2,-1,1) $ and $ Q\!\left( \dfrac{56}{17}, \dfrac{43}{17}, \dfrac{111}{17} \right) $ be the vertices of the rhombus $ PRQS $. If the direction ratios of the diagonal $ RS $ are $ \alpha, -1, \beta $, where both $ \alpha $ and $ \beta $ are integers of minimum absolute values, then $ \alpha^{2} + \beta^{2} $ is equal to __________.

182.

Let $ (\alpha, \beta, \gamma) $ be the image of the point $ P (2, 3, 5) $ in the plane $ 2x + y - 3z = 6 $. Then $ \alpha + \beta + \gamma $ is equal to:

183.

If the lines $ \dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z+3}{1} $ and $ \dfrac{x-a}{2}=\dfrac{y+2}{3}=\dfrac{z-3}{1} $ intersect at the point $ P $, then the distance of the point $ P $ from the plane $ z=a $ is:

184.

Let a line pass through two distinct points $ P(-2,-1,3) $ and $ Q $, and be parallel to the vector $ 3\hat{i} + 2\hat{j} + 2\hat{k} $. If the distance of the point $ Q $ from the point $ R(1,3,3) $ is $ 5 $, then the square of the area of $ \triangle PQR $ is equal to:

185.

Let $ P $ be the plane, passing through the point $ (1,-1,-5) $ and perpendicular to the line joining the points $ (4,1,-3) $ and $ (2,4,3) $. Then the distance of $ P $ from the point $ (3,-2,2) $ is:

186.

Let $ Q $ and $ R $ be the feet of perpendiculars from the point $ P(a,a,a) $ on the lines $ x=y, \ z=1 $ and $ x=-y, \ z=-1 $ respectively. If $ \angle QPR $ is a right angle, then $ 12a^2 $ is equal to:

187.

If the image of the point $ P(1, 2, a) $ in the line $ \frac{x-6}{3} = \frac{y-7}{2} = \frac{7 - z}{2} $ is $ Q(5, b, c) $, then $ a^{2} + b^{2} + c^{2} $ is equal to:

188.

Let $ L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2} $ and $ L_2 : \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1} $ be two lines. Let $ L_3 $ be a line passing through the point $ (\alpha, \beta, \gamma) $ and be perpendicular to both $ L_1 $ and $ L_2 $. If $ L_3 $ intersects $ L_1 $, then $ |\,5\alpha - 11\beta - 8\gamma\,| $ equals:

189.

The equation of the plane which contains the y-axis and passes through the point $ (1, 2, 3) $ is;

190.

The distance of the point $ (1, -2, 3) $ from the plane $ x - y + z = 5 $ measured parallel to a line, whose direction ratios are $ 2, 3, -6 $, is:

191.

Let $ P $ be the point of intersection of the lines $ \dfrac{x-2}{1} = \dfrac{y-4}{5} = \dfrac{z-2}{1} $ and $ \dfrac{x-3}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{2} $. Then, the shortest distance of $ P $ from the line $ 4x = 2y = z $ is:

192.

A plane $ P $ is parallel to two lines whose direction ratios are $ -2, 1, -3 $ and $ -1, 2, -2 $ and it contains the point $ (2, 2, -2) $. Let $ P $ intersect the co-ordinate axes at the points $ A, B, C $ making the intercepts $ \alpha, \beta, \gamma $. If $ V $ is the volume of the tetrahedron $ OABC $, where $ O $ is the origin, and $ p = \alpha + \beta + \gamma $, then the ordered pair $ (V, \ p) $ is equal to:

193.

If the two lines $ l_1 : \dfrac{x-2}{3} = \dfrac{y+1}{-2}, \; z = 2 $ and $ l_2 : \dfrac{x-1}{1} = \dfrac{2y+3}{\alpha} = \dfrac{z+5}{2} $ are perpendicular, then an angle between the lines $ l_2 $ and $ l_3 : \dfrac{1-x}{3} = \dfrac{2y-1}{-4} = \dfrac{z}{4} $ is:

194.

Let the equation of the plane, that passes through the point $ (1,4,-3) $ and contains the line of intersection of the planes $ 3x-2y+4z-7=0 $ and $ x+5y-2z+9=0 $, be $ \alpha x + \beta y + \gamma z + 3 = 0 $, then $ \alpha + \beta + \gamma $ is equal to:

195.

Consider a line $ L $ passing through the points $ P(1,2,1) $ and $ Q(2,1,-1) $. If the mirror image of the point $ A(2,2,2) $ in the line $ L $ is $ (\alpha,\beta,\gamma) $, then $ \alpha+\beta+6\gamma $ is equal to ________________.

196.

If $ (x,y,z) $ be an arbitrary point lying on a plane $ P $ which passes through the points $ (42,0,0) $, $ (0,42,0) $ and $ (0,0,42) $, then the value of the expression $\displaystyle 3 + \frac{x-11}{(y-19)^{2}(z-12)^{2}} + \frac{y-19}{(x-11)^{2}(z-12)^{2}} + \frac{z-12}{(x-11)^{2}(y-19)^{2}} + \frac{x+y+z}{14(x-11)(y-19)(z-12)} $ is equal to:

197.

The plane $ 2x - y + z = 4 $ intersects the line segment joining the points $ A(a, -2, 4) $ and $ B(2, b, -3) $ at the point $ C $ in the ratio $ 2 : 1 $ and the distance of the point $ C $ from the origin is $ \sqrt{5} $. If $ ab < 0 $ and $ P $ is the point $ (a - b, 2b - a) $ then $ CP^2 $ is equal to:

198.

Let a line $ L $ be perpendicular to both the lines $ L_{1} : \, \dfrac{x+1}{3} = \dfrac{y+3}{5} = \dfrac{z+5}{7} $ and $ L_{2} : \, \dfrac{x-2}{1} = \dfrac{y-4}{4} = \dfrac{z-6}{7} $. If $ \theta $ is the acute angle between the lines $ L $ and $ L_{3} : \, \dfrac{x-8/7}{2} = \dfrac{y-4/7}{1} = \dfrac{z}{2} $, then $ \theta $ is equal to:

199.

Let a line $ L $ pass through the point $ P(2,3,1) $ and be parallel to the line $ x + 3y - 2z - 2 = 0 = x - y + 2z $. If the distance of $ L $ from the point $ (5,3,8) $ is $ \alpha $, then $ 3\alpha^{2} $ is equal to __________.

200.

If two distinct points $ Q, R $ lie on the line of intersection of the planes $ -x + 2y - z = 0 $ and $ 3x - 5y + 2z = 0 $ and $ PQ = PR = \sqrt{18} $ where the point $ P $ is $ (1,-2,3) $, then the area of the triangle $ PQR $ is equal to:

201.

Let the foot of perpendicular from the point $ A(4, 3, 1) $ on the plane $ P : x - y + 2z + 3 = 0 $ be $ N $. If $ B(5, \alpha, \beta), \ \alpha, \beta \in \mathbb{Z} $ is a point on plane $ P $ such that the area of the triangle $ ABN $ is $ 3\sqrt{2} $, then $ \alpha^{2} + \beta^{2} + \alpha\beta $ is equal to ________.

202.

Let $ P(\alpha,\beta,\gamma) $ be the image of the point $ Q(1,6,4) $ in the line $\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} $. Then $ 2\alpha+\beta+\gamma $ equal to ________.

203.

Let a line with direction ratios $ a,-4a,-7 $ be perpendicular to the lines with direction ratios $ 3,-1,2b $ and $ b,a,-2 $. If the point of intersection of the line $\displaystyle \frac{x+1}{a^2+b^2}=\frac{y-2}{a^2-b^2}=\frac{z}{1} $ and the plane $ x-y+z=0 $ is $ (\alpha,\beta,\gamma) $, then $ \alpha+\beta+\gamma $ is equal to __________.

204.

Let a unit vector $\widehat{OP}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $OX, OY, OZ$ respectively, where $\beta \in \left(0, \frac{\pi}{2}\right)$. If $\widehat{OP}$ is perpendicular to the plane through points $(1, 2, 3)$, $(2, 3, 4)$ and $(1, 5, 7)$, then which one of the following is true?

205.

Let $ P $ and $ Q $ be the points on the line $ \dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2} $ which are at a distance of $ 6 $ units from the point $ R(1, 2, 3) $. If the centroid of the triangle $ PQR $ is $ (\alpha,\beta,\gamma) $, then $ \alpha^{2} + \beta^{2} + \gamma^{2} $ is:

206.

Let $ O $ be the origin, and $ M $ and $ N $ be the points on the lines $ \dfrac{x-5}{4}=\dfrac{y-4}{1}=\dfrac{z-5}{3} $ and $ \dfrac{x+8}{12}=\dfrac{y+2}{5}=\dfrac{z-11}{9} $ respectively such that $ MN $ is the shortest distance between the given lines. Then $ \overrightarrow{OM}\cdot\overrightarrow{ON} $ is equal to ________.

207.

Let the plane $ P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0 $ be parallel to the line $ L : \frac{x+2}{2} = \frac{y-3}{3} = \frac{z+4}{5} $. If the intercept of $ P $ on the $ y $-axis is $ 1 $, then the distance between $ P $ and $ L $ is:

208.

Let $ P $ be the foot of the perpendicular from the point $ (1, 2, 2) $ on the line $ L : \dfrac{x-1}{1}=\dfrac{y+1}{-1}=\dfrac{z-2}{2} $.
Let the line $ \vec{r}=(\hat{-i}+\hat{j}-2\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}),\, \lambda\in\mathbb{R} $, intersect the line $ L $ at $ Q $. Then $ 2(PQ)^2 $ is equal to:

209.

Let $ P $ be the plane passing through the points $ (5,3,0) $, $ (13,3,-2) $ and $ (1,6,2) $. For $ \alpha \in \mathbb{N} $; if the distances of the points $ A(3,4,\alpha) $ and $ B(2,\alpha,a) $ from the plane $ P $ are $ 2 $ and $ 3 $ respectively, then the positive value of $ a $ is:

210.

Consider the line $ L $ passing through the points $ (1, 2, 3) $ and $ (2, 3, 5) $. The distance of the point $\displaystyle \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) $ from the line $ L $ along the line $\displaystyle \frac{3x-11}{2} = \frac{3y-11}{1} = \frac{3z-19}{2} $ is equal to

211.

The foot of the perpendicular from a point on the circle $ x^2 + y^2 = 1, \, z = 0 $ to the plane $ 2x + 3y + z = 6 $ lies on which one of the following curves?

212.

If the plane $ P $ passes through the intersection of two mutually perpendicular planes $ 2x + ky - 5z = 1 $ and $ 3kx - ky + z = 5, \; k < 3 $ and intercepts a unit length on positive $ x $-axis, then the intercept made by the plane $ P $ on the $ y $-axis is:

213.

The distance of the point $ (7,-3,-4) $ from the plane passing through the points $ (2,-3,1) $, $ (-1,1,-2) $ and $ (3,-4,2) $ is:

214.

The square of the distance of the point $\displaystyle \left(\frac{15}{7},\frac{32}{7},7\right) $ from the line $\displaystyle \frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} $ in direction of the vector $ \hat{i}+4\hat{j}+7\hat{k} $ is:

215.

The largest value of $ a $, for which the perpendicular distance of the plane containing the lines $ \vec{r} = (\hat{i}+\hat{j}) + \lambda(\hat{i}+a\hat{j}-\hat{k}) $ and $ \vec{r} = (\hat{i}+\hat{j}) + \mu(-\hat{i}+\hat{j}-a\hat{k}) $ from the point $ (2,1,4) $ is $ \sqrt{3} $, is ______.

216.

Let a line passing through the point $ (-1, 2, 3) $ intersects the lines $ L_1 : \dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2} $ at $ M(\alpha,\beta,\gamma) $ and $ L_2 : \dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4} $ at $ N(a,b,c) $. Then, the value of $ \dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2} $ equals _______.

217.

A line with direction ratios $ 1,-1,2 $ intersects the lines $ \dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z+1}{3} $ and $ \dfrac{x+1}{-1}=\dfrac{y-2}{1}=\dfrac{z}{4} $ at the points $ P $ and $ Q $, respectively. If the length of the line segment $ PQ $ is $ a $, then $ 225a^{2} $ is equal to:

218.

Let $ S $ be the set of all values of $ \lambda $, for which the shortest distance between the lines $ \dfrac{x-\lambda}{0}=\dfrac{y-3}{4}=\dfrac{z+6}{1} $ and $ \dfrac{x+\lambda}{3}=\dfrac{y}{-4}=\dfrac{z-6}{0} $ is $ 13 $. Then $ 8\left|\sum\limits_{\lambda\in S}\lambda\right| $ is equal to:

219.

If the shortest distance between the lines $ \vec{r}_{1}=\alpha\hat{i}+2\hat{j}+2\hat{k}+\lambda(\hat{i}-2\hat{j}+2\hat{k}),\, \lambda\in\mathbb{R},\, \alpha>0 $ and $ \vec{r}_{2}=-4\hat{i}-\hat{k}+\mu(3\hat{i}-2\hat{j}-2\hat{k}),\, \mu\in\mathbb{R} $ is $ 9 $, then $ \alpha $ is equal to ______.

220.

If the equation of plane passing through the mirror image of a point $ (2, 3, 1) $ with respect to line $ \dfrac{x + 1}{2} = \dfrac{y - 3}{1} = \dfrac{z - 2}{-1} $ and containing the line $ \dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z + 1}{1} $ is $ \alpha x + \beta y + \gamma z = 24 $, then $ \alpha + \beta + \gamma $ is equal to:

221.

Let the plane $ x + 3y - 2z + 6 = 0 $ meet the co-ordinate axes at the points $ A, B, C $. If the orthocenter of the triangle $ ABC $ is $ \left(\alpha, \beta, \dfrac{6}{7}\right) $, then $ 98(\alpha+\beta)^2 $ is equal to____.

222.

If equation of the plane that contains the point $ (-2, 3, 5) $ and is perpendicular to each of the planes $ 2x + 4y + 5z = 8 $ and $ 3x - 2y + 3z = 5 $ is $ \alpha x + \beta y + \gamma z + 97 = 0 $ then $ \alpha + \beta + \gamma = $

223.

Let the plane $ P : 4x - y + z = 10 $ be rotated by an angle $ \dfrac{\pi}{2} $ about its line of intersection with plane $ x + y - z = 4 $. If $ \alpha $ is the distance of the points $ (2, 3, -4) $ from the new position of the plane $ P $, then $ 35\alpha $ is equal to:

224.

A plane $ P $ contains the line $ x+2y+3z+1=0 = x-y-z-6 $, and is perpendicular to the plane $ -2x+y+z+8=0 $. Then which of the following points lies on $ P $?

225.

Let the line of the shortest distance between the lines $ L_1 : \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) $ and $ L_2 : \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k}) $ intersect $ L_1 $ and $ L_2 $ at $ P $ and $ Q $ respectively. If $ (\alpha, \beta, \gamma) $ is the mid point of the line segment $ PQ $, then $ 2(\alpha + \beta + \gamma) $ is equal to ______.

226.

The perpendicular distance, of the line $\displaystyle \frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2} $ from the point $ P(2,-10,1) $, is:

227.

Let $ L_{1} : \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4} $ and $ L_{2} : \dfrac{x-2}{3} = \dfrac{y-4}{4} = \dfrac{z-5}{5} $ be two lines. Then which of the following points lies on the line of the shortest distance between $ L_{1} $ and $ L_{2} $?

228.

If the equation of the plane passing through the line of intersection of the planes $ 2x - y + z = 3 $, $ 4x - 3y + 5z + 9 = 0 $ and parallel to the line $ \dfrac{x+1}{-2} = \dfrac{y+3}{4} = \dfrac{z-2}{5} $ is $ ax + by + cz + 6 = 0 $, then $ a + b + c $ is equal to:

229.

A vector $ \vec{v} $ in the first octant is inclined to the x-axis at $ 60^\circ $, to the y-axis at $ 45^\circ $ and to the z-axis at an acute angle. If a plane passing through the points $ (\sqrt{2},-1,1) $ and $ (a,b,c) $, is normal to $ \vec{v} $, then:

230.

Let a line $ L $ passing through the point $ (1,1,1) $ be perpendicular to both the vectors $ 2\hat{i}+2\hat{j}+\hat{k} $ and $ \hat{i}+2\hat{j}+2\hat{k} $. If $ P(a,b,c) $ is the foot of perpendicular from the origin on the line $ L $, then the value of $ 34(a+b+c) $ is:

231.

Let the line $ L $ be the projection of the line $ \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{2} $ in the plane $ x - 2y - z = 3 $. If $ d $ is the distance of the point $ (0, 0, 6) $ from $ L $, then $ d^2 $ is equal to ______.

232.

Equations of a plane at a distance $ \sqrt{\dfrac{2}{21}} $ from the origin, which contains the line of intersection of the planes $ x - y - z - 1 = 0 $ and $ 2x + y - 3z + 4 = 0 $, is:

233.

The distance of the point $ (7, 10, 11) $ from the line $ \frac{x-4}{1} = \frac{y-4}{0} = \frac{z-2}{3} $ along the line $ \frac{x-9}{2} = \frac{y-13}{3} = \frac{z-17}{6} $ is:

234.

Let $ l_1 $ be the line in $ xy $-plane with $ x $ and $ y $ intercepts $ \frac{1}{8} $ and $ \frac{1}{4\sqrt{2}} $ respectively, and $ l_2 $ be the line in $ zx $-plane with $ x $ and $ z $ intercepts $ -\frac{1}{8} $ and $ -\frac{1}{6\sqrt{3}} $ respectively. If $ d $ is the shortest distance between the line $ l_1 $ and $ l_2 $, then $ d^{-2} $ is equal to __________.

235.

The equation of the plane passing through the line of intersection of the planes $ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 $ and $ \vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 $ and parallel to the $ x $-axis is:

236.

The shortest distance, between lines $ L_1 $ and $ L_2 $, where $ L_1:\; \dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+4}{2} $ and $ L_2 $ s the line, passing through the line, passing through the points $ A=(-4,4,3), B=(-1,6,3) $ and perpendicular to the line $ \dfrac{x-3}{-2}=\dfrac{y}{3}=\dfrac{z-1}{1} $ is:

237.

The shortest distance between the lines $ \frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3} $ and $ \frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5} $ is:

238.

Let $ PQR $ be a triangle with $ R(-1, 4, 2) $. Suppose $ M(2, 1, 2) $ is the mid point of $ PQ $. The distance of the centroid of $ \Delta PQR $ from the point of intersection of the lines $ \frac{x-2}{0} = \frac{y}{2} = \frac{z+3}{-1} $ and $ \frac{x-1}{1} = \frac{y+3}{-3} = \frac{z+1}{1} $ is:

239.

The distance, of the point $ (7, -2, 11) $ from the line $ \dfrac{x-6}{1} = \dfrac{y-4}{0} = \dfrac{z-8}{3} $ along the line $ \dfrac{x-5}{2} = \dfrac{y-1}{-3} = \dfrac{z-5}{6} $, is:

240.

Let $ (\alpha, \beta, \gamma) $ be the foot of perpendicular from the point $ (1, 2, 3) $ on the line $ \frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} $. Then $ 19(\alpha + \beta + \gamma) $ is equal to:

241.

Let $ S $ be the mirror image of the point $ Q(1,\,3,\,4) $ with respect to the plane $ 2x-y+z+3=0 $ and let $ R(3,\,5,\gamma) $ be a point of this plane. Then the square of the length of the line segment $ SR $ is ______.

242.

Let the image of the point $ P(1,6,a) $ in the line $ L : \, \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}, \, b>0 $, be $ \left(\dfrac{a}{3},0,a+c\right) $. If $ S(\alpha,\beta,\gamma), \, \alpha>0 $, is the point on $ L $ such that the distance of $ S $ from the foot of perpendicular from the point $ P $ on $ L $ is $ 2\sqrt{14} $, then $ \alpha + \beta + \gamma $ is equal to:

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JEE Mains | PYQ