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JEE Main

Vector Algebra and 3D Geometry

Mathematics

Previous Years Questions

Numerical

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Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-coplanar vectors such that $$\ov...
JEE Main 2022 (Online) 27th July Evening Shift
Let the line $$\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$$ intersect the plane containing the lines $$\frac{x-4}{1}=\f...
JEE Main 2022 (Online) 27th July Morning Shift
The plane passing through the line $$L: l x-y+3(1-l) z=1, x+2 y-z=2$$ and perpendicular to the plane $$3 x+2 y+z=6$$ is ...
JEE Main 2022 (Online) 26th July Evening Shift
The largest value of $$a$$, for which the perpendicular distance of the plane containing the lines $$ \vec{r}=(\hat{i}+\...
JEE Main 2022 (Online) 26th July Evening Shift
Let $$\mathrm{Q}$$ and $$\mathrm{R}$$ be two points on the line $$\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$$ at a dista...
JEE Main 2022 (Online) 26th July Morning Shift
The line of shortest distance between the lines $$\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$$ and $$\frac{x-3}{2}=\frac{y-...
JEE Main 2022 (Online) 25th July Morning Shift
Consider a triangle ABC whose vertices are A(0, $$\alpha$$, $$\alpha$$), B($$\alpha$$, 0, $$\alpha$$) and C($$\alpha$$, ...
JEE Main 2022 (Online) 30th June Morning Shift
If the shortest distance between the lines $$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda...
JEE Main 2022 (Online) 24th June Morning Shift
Let a line having direction ratios, 1, $$-$$4, 2 intersect the lines $${{x - 7} \over 3} = {{y - 1} \over { - 1}} = {{z ...
JEE Main 2022 (Online) 24th June Morning Shift
Let the lines $${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \i...
JEE Main 2022 (Online) 25th June Morning Shift
Let $$\theta$$ be the angle between the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$, where $$|\overrightar...
JEE Main 2022 (Online) 25th June Morning Shift
Let l1 be the line in xy-plane with x and y intercepts $${1 \over 8}$$ and $${1 \over {4\sqrt 2 }}$$ respectively, and l...
JEE Main 2022 (Online) 25th June Evening Shift
Let $$\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$$, $$\lambda$$ $$\in$$ R. If $$\overrightarrow a...
JEE Main 2022 (Online) 25th June Evening Shift
Let the mirror image of the point (a, b, c) with respect to the plane 3x $$-$$ 4y + 12z + 19 = 0 be (a $$-$$ 6, $$\beta$...
JEE Main 2022 (Online) 27th June Morning Shift
If $$\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 3\widehat i + 3\widehat j + \...
JEE Main 2022 (Online) 28th June Morning Shift
Let the image of the point P(1, 2, 3) in the line $$L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}$$ be Q. ...
JEE Main 2022 (Online) 28th June Evening Shift
Let  $$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$,   $$\overrightarrow b = \widehat i...
JEE Main 2022 (Online) 29th June Evening Shift
Let $${P_1}:\overrightarrow r \,.\,\left( {2\widehat i + \widehat j - 3\widehat k} \right) = 4$$ be a plane. Let P2 be a...
JEE Main 2022 (Online) 29th June Morning Shift
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 ...
JEE Main 2022 (Online) 29th June Morning Shift
Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j ...
JEE Main 2021 (Online) 1st September Evening Shift
Suppose the line $${{x - 2} \over \alpha } = {{y - 2} \over { - 5}} = {{z + 2} \over 2}$$ lies on the plane $$x + 3y - 2...
JEE Main 2021 (Online) 31st August Evening Shift
The square of the distance of the point of intersection of the line $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z + 1} \...
JEE Main 2021 (Online) 31st August Morning Shift
Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x $$-$$ y + z + 3 = 0 and let R(3, 5, $$\ga...
JEE Main 2021 (Online) 27th August Evening Shift
Let $$\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$$, $$\overrightarrow b = \widehat i + 3\widehat...
JEE Main 2021 (Online) 27th August Morning Shift
Let Q be the foot of the perpendicular from the point P(7, $$-$$2, 13) on the plane containing the lines $${{x + 1} \ove...
JEE Main 2021 (Online) 26th August Evening Shift
If the projection of the vector $$\widehat i + 2\widehat j + \widehat k$$ on the sum of the two vectors $$2\widehat i + ...
JEE Main 2021 (Online) 26th August Evening Shift
Let the line L be the projection of the line $${{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 4} \over 2}$$ in the plane ...
JEE Main 2021 (Online) 26th August Morning Shift
Let a plane P pass through the point (3, 7, $$-$$7) and contain the line, $${{x - 2} \over { - 3}} = {{y - 3} \over 2} =...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b $$ and $$\overrightarrow c = \widehat...
JEE Main 2021 (Online) 27th July Morning Shift
If the lines $${{x - k} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}$$ and $${{x + 1} \over 3} = {{y + 2} \over 2} =...
JEE Main 2021 (Online) 25th July Evening Shift
If $$\left( {\overrightarrow a + 3\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 5\o...
JEE Main 2021 (Online) 25th July Evening Shift
The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points. Q(3, $$-$$4, $$-$$5)...
JEE Main 2021 (Online) 27th July Evening Shift
Let $$\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$$, $$\overrightarrow b = 3\widehat i + \be...
JEE Main 2021 (Online) 27th July Evening Shift
Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j ...
JEE Main 2021 (Online) 25th July Morning Shift
For p > 0, a vector $${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$$ is obtained by rotating the vector $...
JEE Main 2021 (Online) 20th July Evening Shift
If the shortest distance between the lines $$\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + ...
JEE Main 2021 (Online) 20th July Morning Shift
Let P be a plane passing through the points (1, 0, 1), (1, $$-$$2, 1) and (0, 1, $$-$$2). Let a vector $$\overrightarrow...
JEE Main 2021 (Online) 20th July Morning Shift
Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three mutually perpendicular vectors of th...
JEE Main 2021 (Online) 20th July Morning Shift
Let P be a plane containing the line $${{x - 1} \over 3} = {{y + 6} \over 4} = {{z + 5} \over 2}$$ and parallel to the l...
JEE Main 2021 (Online) 18th March Evening Shift
Let the mirror image of the point (1, 3, a) with respect to the plane $$\overrightarrow r .\left( {2\widehat i - \wideha...
JEE Main 2021 (Online) 18th March Evening Shift
The equation of the planes parallel to the plane x $$-$$ 2y + 2z $$-$$ 3 = 0 which are at unit distance from the point (...
JEE Main 2021 (Online) 18th March Morning Shift
Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, $$-$$3, 1) and (2, 3, $$-$$5) at the right ang...
JEE Main 2021 (Online) 18th March Morning Shift
Let P be an arbitrary point having sum of the squares of the distances from the planes x + y + z = 0, lx $$-$$ nz = 0 an...
JEE Main 2021 (Online) 17th March Evening Shift
Let $$\overrightarrow x $$ be a vector in the plane containing vectors $$\overrightarrow a = 2\widehat i - \widehat j +...
JEE Main 2021 (Online) 17th March Evening Shift
If $$\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k$$,$$\overrightarrow b = - \beta \widehat ...
JEE Main 2021 (Online) 17th March Morning Shift
If the equation of the plane passing through the line of intersection of the planes 2x $$-$$ 7y + 4z $$-$$ 3 = 0, 3x $$-...
JEE Main 2021 (Online) 17th March Morning Shift
Let $$\overrightarrow c $$ be a vector perpendicular to the vectors, $$\overrightarrow a $$ = $$\widehat i$$ + $$\wideha...
JEE Main 2021 (Online) 16th March Evening Shift
If the distance of the point (1, $$-$$2, 3) from the plane x + 2y $$-$$ 3z + 10 = 0 measured parallel to the line, $${{x...
JEE Main 2021 (Online) 16th March Evening Shift
Let ($$\lambda$$, 2, 1) be a point on the plane which passes through the point (4, $$-$$2, 2). If the plane is perpendic...
JEE Main 2021 (Online) 26th February Morning Shift
Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alph...
JEE Main 2021 (Online) 25th February Evening Shift
A line 'l' passing through origin is perpendicular to the lines$${l_1}:\overrightarrow r = (3 + t)\widehat i + ( - 1 + ...
JEE Main 2021 (Online) 25th February Evening Shift
Let $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j$$ and...
JEE Main 2021 (Online) 25th February Morning Slot
Let three vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be such that is coplanar with $$\...
JEE Main 2021 (Online) 24th February Morning Slot
If $$\overrightarrow x $$ and $$\overrightarrow y $$ be two non-zero vectors such that $$\left| {\overrightarrow x + \o...
JEE Main 2020 (Online) 6th September Evening Slot
If $$\overrightarrow a $$ and $$\overrightarrow b $$ are unit vectors, then the greatest value of $$\sqrt 3 \left| {\ove...
JEE Main 2020 (Online) 6th September Morning Slot
Let the vectors $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be such that $$\left| {\overrigh...
JEE Main 2020 (Online) 5th September Evening Slot
If $$\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$$, then the value of $${\left| {\widehat i \times \lef...
JEE Main 2020 (Online) 4th September Evening Slot
If the equation of a plane P, passing through the intersection of the planes, x + 4y - z + 7 = 0 and 3x + y + 5z = 8 is ...
JEE Main 2020 (Online) 4th September Morning Slot
Let a plane P contain two lines $$\overrightarrow r = \widehat i + \lambda \left( {\widehat i + \widehat j} \right)$$, ...
JEE Main 2020 (Online) 3rd September Evening Slot
Let the position vectors of points 'A' and 'B' be $$\widehat i + \widehat j + \widehat k$$ and $$2\widehat i + \widehat ...
JEE Main 2020 (Online) 2nd September Evening Slot
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $${\left| ...
JEE Main 2020 (Online) 2nd September Morning Slot
If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane containing the lines $${{x + 1} \over 2} = {{y ...
JEE Main 2020 (Online) 9th January Evening Slot
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\left| {\over...
JEE Main 2020 (Online) 9th January Evening Slot
The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, ...
JEE Main 2020 (Online) 9th January Morning Slot
If the vectors, $$\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k$$, $$\overrightarrow...
JEE Main 2020 (Online) 9th January Morning Slot
If the system of linear equations, x + y + z = 6 x + 2y + 3z = 10 3x + 2y + $$\lambda $$z = $$\mu $$ has more than two s...
JEE Main 2020 (Online) 7th January Evening Slot

MCQ (Single Correct Answer)

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If the line of intersection of the planes $$a x+b y=3$$ and $$a x+b y+c z=0$$, a $$>0$$ makes an angle $$30^{\circ}$$ wi...
JEE Main 2022 (Online) 27th July Evening Shift
If the length of the perpendicular drawn from the point $$P(a, 4,2)$$, a $$>0$$ on the line $$\frac{x+1}{2}=\frac{y-3}{3...
JEE Main 2022 (Online) 27th July Evening Shift
If the plane $$P$$ passes through the intersection of two mutually perpendicular planes $$2 x+k y-5 z=1$$ and $$3 k x-k ...
JEE Main 2022 (Online) 27th July Morning Shift
$$ \text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text ...
JEE Main 2022 (Online) 27th July Morning Shift
Let $$\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$$ and $$\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$$ be two vectors, such ...
JEE Main 2022 (Online) 27th July Morning Shift
A vector $$\vec{a}$$ is parallel to the line of intersection of the plane determined by the vectors $$\hat{i}, \hat{i}+\...
JEE Main 2022 (Online) 26th July Evening Shift
Let $$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\...
JEE Main 2022 (Online) 26th July Morning Shift
The length of the perpendicular from the point $$(1,-2,5)$$ on the line passing through $$(1,2,4)$$ and parallel to the ...
JEE Main 2022 (Online) 26th July Morning Shift
Let $$\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$$ and let $$\vec{b}$$ be a vector such that $$\vec{a} \times \vec{b}=2 \hat{i}-\...
JEE Main 2022 (Online) 25th July Evening Shift
The shortest distance between the lines $$\frac{x+7}{-6}=\frac{y-6}{7}=z$$ and $$\frac{7-x}{2}=y-2=z-6$$ is
JEE Main 2022 (Online) 25th July Evening Shift
A plane $$E$$ is perpendicular to the two planes $$2 x-2 y+z=0$$ and $$x-y+2 z=4$$, and passes through the point $$P(1,-...
JEE Main 2022 (Online) 25th July Evening Shift
Let $$\mathrm{ABC}$$ be a triangle such that $$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow...
JEE Main 2022 (Online) 25th July Morning Shift
Let $$\mathrm{P}$$ be the plane containing the straight line $$\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$$ and perpend...
JEE Main 2022 (Online) 25th July Morning Shift
Let a vector $$\overrightarrow c $$ be coplanar with the vectors $$\overrightarrow a = - \widehat i + \widehat j + \wi...
JEE Main 2022 (Online) 30th June Morning Shift
The distance of the point (3, 2, $$-$$1) from the plane $$3x - y + 4z + 1 = 0$$ along the line $${{2 - x} \over 2} = {{y...
JEE Main 2022 (Online) 30th June Morning Shift
Let $$\widehat a$$, $$\widehat b$$ be unit vectors. If $$\overrightarrow c $$ be a vector such that the angle between $$...
JEE Main 2022 (Online) 24th June Morning Shift
Let $$\widehat a$$ and $$\widehat b$$ be two unit vectors such that $$|(\widehat a + \widehat b) + 2(\widehat a \times \...
JEE Main 2022 (Online) 24th June Evening Shift
Let the points on the plane P be equidistant from the points ($$-$$4, 2, 1) and (2, $$-$$2, 3). Then the acute angle bet...
JEE Main 2022 (Online) 24th June Evening Shift
If the shortest distance between the lines $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }$$ and $${{...
JEE Main 2022 (Online) 24th June Evening Shift
Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing throu...
JEE Main 2022 (Online) 25th June Morning Shift
Let $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$$ $${a_i} > 0$$, $$i = 1,2,3$$ be a vecto...
JEE Main 2022 (Online) 25th June Morning Shift
Let p be the plane passing through the intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + 3\wideha...
JEE Main 2022 (Online) 25th June Evening Shift
Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$, $$\overrightarrow b = 2\widehat i - 3\widehat j + \...
JEE Main 2022 (Online) 26th June Evening Shift
If the lines $$\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j ...
JEE Main 2022 (Online) 26th June Evening Shift
If the plane $$2x + y - 5z = 0$$ is rotated about its line of intersection with the plane $$3x - y + 4z - 7 = 0$$ by an ...
JEE Main 2022 (Online) 26th June Evening Shift
If $$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$ and $$\overrightarr...
JEE Main 2022 (Online) 26th June Morning Shift
Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x $$-$...
JEE Main 2022 (Online) 26th June Morning Shift
If the two lines $${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$$ and $${l_2}:{{x - 1} \over 1} = {{2y + 3} \o...
JEE Main 2022 (Online) 26th June Morning Shift
Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j +...
JEE Main 2022 (Online) 27th June Morning Shift
Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be the vectors along the diagonals of a parallelogram having area ...
JEE Main 2022 (Online) 27th June Evening Shift
The shortest distance between the lines $${{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}$$ and $${{x + 3...
JEE Main 2022 (Online) 27th June Evening Shift
Let the foot of the perpendicular from the point (1, 2, 4) on the line $${{x + 2} \over 4} = {{y - 1} \over 2} = {{z + 1...
JEE Main 2022 (Online) 27th June Evening Shift
Let the plane $$P:\overrightarrow r \,.\,\overrightarrow a = d$$ contain the line of intersection of two planes $$\over...
JEE Main 2022 (Online) 28th June Morning Shift
The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the plan...
JEE Main 2022 (Online) 28th June Morning Shift
If two distinct point Q, R lie on the line of intersection of the planes $$ - x + 2y - z = 0$$ and $$3x - 5y + 2z = 0$$ ...
JEE Main 2022 (Online) 28th June Morning Shift
Let $$\overrightarrow a $$ be a vector which is perpendicular to the vector $$3\widehat i + {1 \over 2}\widehat j + 2\wi...
JEE Main 2022 (Online) 28th June Evening Shift
Let the plane ax + by + cz = d pass through (2, 3, $$-$$5) and is perpendicular to the planes 2x + y $$-$$ 5z = 10 and 3...
JEE Main 2022 (Online) 28th June Evening Shift
Let $$\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$$ and $$\overrightarrow b = - 2\widehat i + \a...
JEE Main 2022 (Online) 28th June Evening Shift
Let A, B, C be three points whose position vectors respectively are $$\overrightarrow a = \widehat i + 4\widehat j + 3\...
JEE Main 2022 (Online) 29th June Evening Shift
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 t...
JEE Main 2022 (Online) 29th June Evening Shift
Let $${{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}}$$ lie on the plane $$px - qy + z = 5$$, for so...
JEE Main 2022 (Online) 29th June Evening Shift
Let $$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$$, $$\overrightarrow b = 3\widehat i - \beta \w...
JEE Main 2022 (Online) 29th June Morning Shift
If the mirror image of the point (2, 4, 7) in the plane 3x $$-$$ y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to:...
JEE Main 2022 (Online) 29th June Morning Shift
The distance of line $$3y - 2z - 1 = 0 = 3x - z + 4$$ from the point (2, $$-$$1, 6) is :
JEE Main 2021 (Online) 1st September Evening Shift
Let the acute angle bisector of the two planes x $$-$$ 2y $$-$$ 2z + 1 = 0 and 2x $$-$$ 3y $$-$$ 6z + 1 = 0 be the plane...
JEE Main 2021 (Online) 1st September Evening Shift
The distance of the point ($$-$$1, 2, $$-$$2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x $$-$$ 2...
JEE Main 2021 (Online) 31st August Evening Shift
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ three vectors mutually perpendicular to each other and ...
JEE Main 2021 (Online) 31st August Evening Shift
Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two vectors such that $$\left| {2\overrightarrow a + 3\overrig...
JEE Main 2021 (Online) 31st August Morning Shift
Let the equation of the plane, that passes through the point (1, 4, $$-$$3) and contains the line of intersection of the...
JEE Main 2021 (Online) 31st August Morning Shift
The equation of the plane passing through the line of intersection of the planes $$\overrightarrow r .\left( {\widehat i...
JEE Main 2021 (Online) 27th August Evening Shift
Equation of a plane at a distance $${2 \over {\sqrt {21} }}$$ from the origin, which contains the line of intersection o...
JEE Main 2021 (Online) 27th August Morning Shift
Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $$\overrightarrow r \,...
JEE Main 2021 (Online) 26th August Evening Shift
A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH betwee...
JEE Main 2021 (Online) 26th August Evening Shift
A plane P contains the line $$x + 2y + 3z + 1 = 0 = x - y - z - 6$$, and is perpendicular to the plane $$ - 2x + y + z ...
JEE Main 2021 (Online) 26th August Morning Shift
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = \widehat j - \widehat k$$. ...
JEE Main 2021 (Online) 26th August Morning Shift
Let the plane passing through the point ($$-$$1, 0, $$-$$2) and perpendicular to each of the planes 2x + y $$-$$ z = 2 a...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat ...
JEE Main 2021 (Online) 27th July Morning Shift
If $$\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$$ and $$\left| {\overrightarrow a ...
JEE Main 2021 (Online) 25th July Evening Shift
Let a, b and c be distinct positive numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i+\wideha...
JEE Main 2021 (Online) 25th July Evening Shift
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\overrightarr...
JEE Main 2021 (Online) 27th July Evening Shift
For real numbers $$\alpha$$ and $$\beta$$ $$\ne$$ 0, if the point of intersection of the straight lines$${{x - \alpha } ...
JEE Main 2021 (Online) 27th July Evening Shift
Let the foot of perpendicular from a point P(1, 2, $$-$$1) to the straight line $$L:{x \over 1} = {y \over 0} = {z \over...
JEE Main 2021 (Online) 25th July Morning Shift
Let the vectors$$(2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b...
JEE Main 2021 (Online) 25th July Morning Shift
If the shortest distance between the straight lines $$3(x - 1) = 6(y - 2) = 2(z - 1)$$ and $$4(x - 2) = 2(y - \lambda ) ...
JEE Main 2021 (Online) 22th July Evening Shift
Let three vectors $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarr...
JEE Main 2021 (Online) 22th July Evening Shift
Let a vector $${\overrightarrow a }$$ be coplanar with vectors $$\overrightarrow b = 2\widehat i + \widehat j + \wideha...
JEE Main 2021 (Online) 22th July Evening Shift
Let L be the line of intersection of planes $$\overrightarrow r .(\widehat i - \widehat j + 2\widehat k) = 2$$ and $$\ov...
JEE Main 2021 (Online) 22th July Evening Shift
In a triangle ABC, if $$\left| {\overrightarrow {BC} } \right| = 3$$, $$\left| {\overrightarrow {CA} } \right| = 5$$ and...
JEE Main 2021 (Online) 20th July Evening Shift
Consider the line L given by the equation $${{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}$$. Let Q be the mi...
JEE Main 2021 (Online) 20th July Evening Shift
The lines x = ay $$-$$ 1 = z $$-$$ 2 and x = 3y $$-$$ 2 = bz $$-$$ 2, (ab $$\ne$$ 0) are coplanar, if :
JEE Main 2021 (Online) 20th July Evening Shift
Let $$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$...
JEE Main 2021 (Online) 20th July Morning Shift
In a triangle ABC, if $$|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10$$, then the...
JEE Main 2021 (Online) 18th March Evening Shift
Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two non-zero vectors perpendicular to each other and $$|\overri...
JEE Main 2021 (Online) 18th March Evening Shift
A vector $$\overrightarrow a $$ has components 3p and 1 with respect to a rectangular cartesian system. This system is r...
JEE Main 2021 (Online) 18th March Morning Shift
Let O be the origin. Let $$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$$ and $$\overrightarrow {OQ} ...
JEE Main 2021 (Online) 17th March Evening Shift
If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line $${{x + 1} \over 2} ...
JEE Main 2021 (Online) 17th March Evening Shift
The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :
JEE Main 2021 (Online) 17th March Morning Shift
Let $$\overrightarrow a $$ = 2$$\widehat i$$ $$-$$ 3$$\widehat j$$ + 4$$\widehat k$$ and $$\overrightarrow b $$ = 7$$\wi...
JEE Main 2021 (Online) 17th March Morning Shift
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, ...
JEE Main 2021 (Online) 16th March Evening Shift
If the foot of the perpendicular from point (4, 3, 8) on the line $${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - ...
JEE Main 2021 (Online) 16th March Evening Shift
Let $$\overrightarrow a $$ = $$\widehat i$$ + 2$$\widehat j$$ $$-$$ 3$$\widehat k$$ and $$\overrightarrow b = 2\widehat...
JEE Main 2021 (Online) 16th March Evening Shift
If for a > 0, the feet of perpendiculars from the points A(a, $$-$$2a, 3) and B(0, 4, 5) on the plane lx + my + nz = ...
JEE Main 2021 (Online) 16th March Morning Shift
Let P be a plane lx + my + nz = 0 containing the line, $${{1 - x} \over 1} = {{y - 4} \over 2} = {{z + 2} \over 3}$$. If...
JEE Main 2021 (Online) 16th March Morning Shift
Let the position vectors of two points P and Q be 3$$\widehat i$$ $$-$$ $$\widehat j$$ + 2$$\widehat k$$ and $$\widehat ...
JEE Main 2021 (Online) 16th March Morning Shift
Let a vector $$\alpha \widehat i + \beta \widehat j$$ be obtained by rotating the vector $$\sqrt 3 \widehat i + \widehat...
JEE Main 2021 (Online) 16th March Morning Shift
Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P($$\alpha$$, $$\be...
JEE Main 2021 (Online) 26th February Evening Shift
If vectors $$\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k$$ and $$\overrightarrow {{a_2}} = \wideha...
JEE Main 2021 (Online) 26th February Evening Shift
If the mirror image of the point (1, 3, 5) with respect to the plane 4x $$-$$ 5y + 2z = 8 is ($$\alpha$$, $$\beta$$, $$\...
JEE Main 2021 (Online) 26th February Evening Shift
If (1, 5, 35), (7, 5, 5), (1, $$\lambda$$, 7), (2$$\lambda$$, 1, 2) are coplanar, then the sum of all possible values of...
JEE Main 2021 (Online) 26th February Morning Shift
Consider the three planesP1 : 3x + 15y + 21z = 9,P2 : x $$-$$ 3y $$-$$ z = 5, and P3 : 2x + 10y + 14z = 5Then, which one...
JEE Main 2021 (Online) 26th February Morning Shift
If $$\overrightarrow a $$ and $$\overrightarrow b $$ are perpendicular, then $$\overrightarrow a \times \left( {\overri...
JEE Main 2021 (Online) 26th February Morning Shift
A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, $$-$$1, 1), the...
JEE Main 2021 (Online) 25th February Evening Shift
Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations l + m $$-$$ n = 0 and l2 + m...
JEE Main 2021 (Online) 25th February Morning Slot
The equation of the line through the point (0, 1, 2) and perpendicular to the line $${{x - 1} \over 2} = {{y + 1} \over ...
JEE Main 2021 (Online) 25th February Morning Slot
The vector equation of the plane passing through the intersection of the planes $$\overrightarrow r .\left( {\widehat i ...
JEE Main 2021 (Online) 24th February Evening Slot
Let a, b$$ \in $$R. If the mirror image of the point P(a, 6, 9) with respect to the line $${{x - 3} \over 7} = {{y - 2} ...
JEE Main 2021 (Online) 24th February Evening Slot
The distance of the point (1, 1, 9) from the point of intersection of the line $${{x - 3} \over 1} = {{y - 4} \over 2} =...
JEE Main 2021 (Online) 24th February Morning Slot
The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes 3x + y - 2z = 5 and 2x - ...
JEE Main 2021 (Online) 24th February Morning Slot
Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, ...
JEE Main 2020 (Online) 6th September Evening Slot
A plane P meets the coordinate axes at A, B and C respectively. The centroid of $$\Delta $$ABC is given to be (1, 1, 2)....
JEE Main 2020 (Online) 6th September Evening Slot
The shortest distance between the lines $${{x - 1} \over 0} = {{y + 1} \over { - 1}} = {z \over 1}$$ and x + y + z + 1 =...
JEE Main 2020 (Online) 6th September Morning Slot
If for some $$\alpha $$ $$ \in $$ R, the lines L1 : $${{x + 1} \over 2} = {{y - 2} \over { - 1}} = {{z - 1} \over 1}$$ a...
JEE Main 2020 (Online) 5th September Evening Slot
If (a, b, c) is the image of the point (1, 2, -3) in the line $${{x + 1} \over 2} = {{y - 3} \over { - 2}} = {z \over { ...
JEE Main 2020 (Online) 5th September Morning Slot
If the volume of a parallelopiped, whose coterminus edges are given by the vectors $$\overrightarrow a = \widehat i + \...
JEE Main 2020 (Online) 5th September Morning Slot
The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line $${x \over 2} = {y \over...
JEE Main 2020 (Online) 4th September Evening Slot
Let x0 be the point of Local maxima of $$f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c }...
JEE Main 2020 (Online) 4th September Morning Slot
The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the p...
JEE Main 2020 (Online) 3rd September Evening Slot
Let a, b c $$ \in $$ R be such that a2 + b2 + c2 = 1. If $$a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}}...
JEE Main 2020 (Online) 3rd September Evening Slot
The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, –2, 3) and (1, 1, 0) lie...
JEE Main 2020 (Online) 3rd September Morning Slot
The lines $$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$$...
JEE Main 2020 (Online) 3rd September Morning Slot
A plane passing through the point (3, 1, 1) contains two lines whose direction ratios are 1, –2, 2 and 2, 3, –1 respecti...
JEE Main 2020 (Online) 2nd September Evening Slot
The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through t...
JEE Main 2020 (Online) 2nd September Morning Slot
The mirror image of the point (1, 2, 3) in a plane is $$\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)$$....
JEE Main 2020 (Online) 8th January Evening Slot
Let $$\overrightarrow a = \widehat i - 2\widehat j + \widehat k$$ and $$\overrightarrow b = \widehat i - \widehat j + ...
JEE Main 2020 (Online) 8th January Evening Slot
The shortest distance between the lines $${{x - 3} \over 3} = {{y - 8} \over { - 1}} = {{z - 3} \over 1}$$ and $${{x + 3...
JEE Main 2020 (Online) 8th January Morning Slot
Let the volume of a parallelopiped whose coterminous edges are given by $$\overrightarrow u = \widehat i + \widehat j +...
JEE Main 2020 (Online) 8th January Morning Slot
Let $$\overrightarrow a $$ , $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $$\over...
JEE Main 2020 (Online) 7th January Evening Slot
Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the im...
JEE Main 2020 (Online) 7th January Morning Slot
A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$...
JEE Main 2020 (Online) 7th January Morning Slot
Let $$\alpha $$ $$ \in $$ R and the three vectors $$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$,...
JEE Main 2019 (Online) 12th April Evening Slot
The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines $$\overrightarrow r =...
JEE Main 2019 (Online) 12th April Evening Slot
A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through...
JEE Main 2019 (Online) 12th April Evening Slot
If the line $${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 1} \over { - 1}}$$ intersects the plane 2x + 3y – z + 13 = ...
JEE Main 2019 (Online) 12th April Morning Slot
If the volume of parallelopiped formed by the vectors $$\widehat i + \lambda \widehat j + \widehat k$$, $$\widehat j + \...
JEE Main 2019 (Online) 12th April Morning Slot
Let $$\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j...
JEE Main 2019 (Online) 12th April Morning Slot
The distance of the point having position vector $$ - \widehat i + 2\widehat j + 6\widehat k$$ from the straight line p...
JEE Main 2019 (Online) 10th April Evening Slot
A perpendicular is drawn from a point on the line $${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {z \over 1}$$ to the p...
JEE Main 2019 (Online) 10th April Evening Slot
If the plane 2x – y + 2z + 3 = 0 has the distances $${1 \over 3}$$ and $${2 \over 3}$$ units from the planes 4x – 2y +...
JEE Main 2019 (Online) 10th April Evening Slot
Let A (3,0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM i...
JEE Main 2019 (Online) 10th April Morning Slot
If Q(0, –1, –3) is the image of the point P in the plane 3x – y + 4z = 2 and R is the point (3, –1, –2), then the area (...
JEE Main 2019 (Online) 10th April Morning Slot
If the length of the perpendicular from the point ($$\beta $$, 0, $$\beta $$) ($$\beta $$ $$ \ne $$ 0) to the line, $${x...
JEE Main 2019 (Online) 10th April Morning Slot
If a unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$ , $$\pi $$/ 4 with $$\widehat j$$ an...
JEE Main 2019 (Online) 9th April Evening Slot
Let P be the plane, which contains the line of intersection of the planes, x + y + z – 6 = 0 and 2x + 3y + z + 5 = 0 and...
JEE Main 2019 (Online) 9th April Evening Slot
The vertices B and C of a $$\Delta $$ABC lie on the line, $${{x + 2} \over 3} = {{y - 1} \over 0} = {z \over 4}$$ such t...
JEE Main 2019 (Online) 9th April Evening Slot
If the line, $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 2} \over 4}$$ meets the plane, x + 2y + 3z = 15 at a point ...
JEE Main 2019 (Online) 9th April Morning Slot
The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing ...
JEE Main 2019 (Online) 8th April Morning Slot
A plane passing through the points (0, –1, 0) and (0, 0, 1) and making an angle $${\pi \over 4}$$ with the plane y – z ...
JEE Main 2019 (Online) 9th April Morning Slot
Let $$\overrightarrow \alpha = 3\widehat i + \widehat j$$ and $$\overrightarrow \beta = 2\widehat i - \widehat j + 3...
JEE Main 2019 (Online) 9th April Morning Slot
If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R fr...
JEE Main 2019 (Online) 8th April Evening Slot
Let $$\mathop a\limits^ \to = 3\mathop i\limits^ \wedge + 2\mathop j\limits^ \wedge + x\mathop k\limits^ \wedge $...
JEE Main 2019 (Online) 8th April Evening Slot
The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y+ 4z = 5 which ...
JEE Main 2019 (Online) 8th April Evening Slot
The magnitude of the projection of the vector $$\mathop {2i}\limits^ \wedge + \mathop {3j}\limits^ \wedge + \mathop ...
JEE Main 2019 (Online) 8th April Morning Slot
The length of the perpendicular from the point (2, –1, 4) on the straight line, $${{x + 3} \over {10}}$$= $${{y - 2} \ov...
JEE Main 2019 (Online) 8th April Morning Slot
If an angle between the line, $${{x + 1} \over 2} = {{y - 2} \over 1} = {{z - 3} \over { - 2}}$$ and the plane, $$x - 2y...
JEE Main 2019 (Online) 12th January Evening Slot
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors, out of which vecto...
JEE Main 2019 (Online) 12th January Evening Slot
Let S be the set of all real values of $$\lambda $$ such that a plane passing through the points (–$$\lambda $$2, 1, 1),...
JEE Main 2019 (Online) 12th January Evening Slot
A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(–1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR i...
JEE Main 2019 (Online) 12th January Morning Slot
The perpendicular distance from the origin to the plane containing the two lines, $${{x + 2} \over 3} = {{y - 2} \over 5...
JEE Main 2019 (Online) 12th January Morning Slot
The sum of the distinct real values of $$\mu $$, for which the vectors, $$\mu \widehat i + \widehat j + \widehat k,$$ &n...
JEE Main 2019 (Online) 12th January Morning Slot
Let $$\sqrt 3 \widehat i + \widehat j,$$    $$\widehat i + \sqrt 3 \widehat j$$  and &nbsp...
JEE Main 2019 (Online) 11th January Evening Slot
If the point (2, $$\alpha $$, $$\beta $$) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and ...
JEE Main 2019 (Online) 11th January Evening Slot
Two lines $${{x - 3} \over 1} = {{y + 1} \over 3} = {{z - 6} \over { - 1}}$$ and $${{x + 5} \over 7} = {{y - 2} \over { ...
JEE Main 2019 (Online) 11th January Evening Slot
The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle $${\pi \ove...
JEE Main 2019 (Online) 11th January Morning Slot
The plane containing the line $${{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z - 1} \over 3}$$ and also containing it...
JEE Main 2019 (Online) 11th January Morning Slot
Let  $$\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,$$ $$\overrightarrow b = \widehat i +...
JEE Main 2019 (Online) 11th January Morning Slot
The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through wh...
JEE Main 2019 (Online) 10th January Evening Slot
If $$\overrightarrow \alpha $$ = $$\left( {\lambda - 2} \right)\overrightarrow a + \overrightarrow b $$  an...
JEE Main 2019 (Online) 10th January Evening Slot
On which of the following lines lies the point of intersection of the line,   $${{x - 4} \over 2} = {{y - 5} \...
JEE Main 2019 (Online) 10th January Evening Slot
Let A be a point on the line $$\overrightarrow r = \left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\wid...
JEE Main 2019 (Online) 10th January Morning Slot
The plane passing through the point (4, –1, 2) and parallel to the lines  $${{x + 2} \over 3} = {{y - 2} \over...
JEE Main 2019 (Online) 10th January Morning Slot
Let $$\overrightarrow a = 2\widehat i + {\lambda _1}\widehat j + 3\widehat k,\,\,$$   $$\overrightarrow b = ...
JEE Main 2019 (Online) 10th January Morning Slot
Let  $$\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,$$   $$\overrightarrow...
JEE Main 2019 (Online) 9th January Evening Slot
If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then :
JEE Main 2019 (Online) 9th January Evening Slot
The equation of the plane containing the straight line $${x \over 2} = {y \over 3} = {z \over 4}$$ and perpendicular to ...
JEE Main 2019 (Online) 9th January Evening Slot
The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis also pass...
JEE Main 2019 (Online) 9th January Morning Slot
The equation of the line passing through (–4, 3, 1), parallel to the plane x + 2y – z – 5 = 0 and intersecting the line...
JEE Main 2019 (Online) 9th January Morning Slot
Let $$\overrightarrow a $$ = $$\widehat i - \widehat j$$, $$\overrightarrow b $$ = $$\widehat i + \widehat j + \widehat ...
JEE Main 2019 (Online) 9th January Morning Slot
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow c = \widehat j - \widehat k$$ and a vec...
JEE Main 2018 (Online) 16th April Morning Slot
The sum of the intercepts on the coordinate axes of the plane passing through the point ($$-$$2, $$-2,$$ 2) and containi...
JEE Main 2018 (Online) 16th April Morning Slot
If the angle between the lines, $${x \over 2} = {y \over 2} = {z \over 1}$$ and $${{5 - x} \over { - 2}} = {{7y - 14} \...
JEE Main 2018 (Online) 16th April Morning Slot
A plane bisects the line segment joining the points (1, 2, 3) and ($$-$$ 3, 4, 5) at rigt angles. Then this plane also p...
JEE Main 2018 (Online) 15th April Evening Slot
If the position vectors of the vertices A, B and C of a $$\Delta $$ ABC are respectively $$4\widehat i + 7\widehat j + 8...
JEE Main 2018 (Online) 15th April Evening Slot
An angle between the lines whose direction cosines are gien by the equations, $$l$$ + 3m + 5n = 0 and 5$$l$$m $$-$$ 2m...
JEE Main 2018 (Online) 15th April Evening Slot
If $$\overrightarrow a ,\,\,\overrightarrow b ,$$ and $$\overrightarrow C $$ are unit vectors such that $$\overrightarr...
JEE Main 2018 (Online) 15th April Morning Slot
An angle between the plane, x + y + z = 5 and the line of intersection of the planes, 3x + 4y + z $$-$$ 1 = 0 and 5x + 8...
JEE Main 2018 (Online) 15th April Morning Slot
A variable plane passes through a fixed point (3,2,1) and meets x, y and z axes at A, B and C respectively. A plane is d...
JEE Main 2018 (Online) 15th April Morning Slot
Let $$\overrightarrow u $$ be a vector coplanar with the vectors $$\overrightarrow a = 2\widehat i + 3\widehat j - \wid...
JEE Main 2018 (Offline)
The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = ...
JEE Main 2018 (Offline)
If L1 is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L2 is the line of intersecti...
JEE Main 2018 (Offline)
If the vector $$\overrightarrow b = 3\widehat j + 4\widehat k$$ is written as the sum of a vector $$\overrightarrow {{...
JEE Main 2017 (Online) 9th April Morning Slot
If the line, $${{x - 3} \over 1} = {{y + 2} \over { - 1}} = {{z + \lambda } \over { - 2}}$$ lies in the plane, 2x−4y+3z=...
JEE Main 2017 (Online) 9th April Morning Slot
If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the lo...
JEE Main 2017 (Online) 9th April Morning Slot
If x = a, y = b, z = c is a solution of the system of linear equations x + 8y + 7z = 0 9x + 2y + 3z = 0 x + y + z = 0 su...
JEE Main 2017 (Online) 9th April Morning Slot
The line of intersection of the planes $$\overrightarrow r .\left( {3\widehat i - \widehat j + \widehat k} \right) = 1\,...
JEE Main 2017 (Online) 8th April Morning Slot
The area (in sq. units) of the parallelogram whose diagonals are along the vectors $$8\widehat i - 6\widehat j$$ and $$3...
JEE Main 2017 (Online) 8th April Morning Slot
The coordinates of the foot of the perpendicular from the point (1, $$-$$2, 1) on the plane containing the lines, $${{x ...
JEE Main 2017 (Online) 8th April Morning Slot
Let $$\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$...
JEE Main 2017 (Offline)
The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal perpendicular...
JEE Main 2017 (Offline)
If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line, $${x \over 1} =...
JEE Main 2017 (Offline)
Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are $$\overrightarrow a ,\ove...
JEE Main 2016 (Online) 10th April Morning Slot
The number of distinct real values of $$\lambda $$ for which the lines $${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + ...
JEE Main 2016 (Online) 10th April Morning Slot
ABC is a triangle in a plane with vertices A(2, 3, 5), B(−1, 3, 2) and C($$\lambda $$, 5, $$\mu $$). If the median thr...
JEE Main 2016 (Online) 10th April Morning Slot
The distance of the point (1, − 2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes...
JEE Main 2016 (Online) 9th April Morning Slot
In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively 3$$\widehat i$$ ...
JEE Main 2016 (Online) 9th April Morning Slot
The shortest distance between the lines $${x \over 2} = {y \over 2} = {z \over 1}$$ and $${{x + 2} \over { - 1}} = {{y ...
JEE Main 2016 (Online) 9th April Morning Slot
The distance of the point $$(1,-5,9)$$ from the plane $$x-y+z=5$$ measured along the line $$x=y=z$$ is :
JEE Main 2016 (Offline)
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $$\overrightarr...
JEE Main 2016 (Offline)
If the line, $${{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z + 4} \over 3}\,$$ lies in the planes, $$lx+my-z=9,$$ the...
JEE Main 2016 (Offline)
The distance of the point $$(1, 0, 2)$$ from the point of intersection of the line $${{x - 2} \over 3} = {{y + 1} \over ...
JEE Main 2015 (Offline)
The equation of the plane containing the line $$2x-5y+z=3; x+y+4z=5,$$ and parallel to the plane, $$x+3y+6z=1,$$ is :
JEE Main 2015 (Offline)
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of t...
JEE Main 2015 (Offline)
If $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\ov...
JEE Main 2014 (Offline)
The angle between the lines whose direction cosines satisfy the equations $$l+m+n=0$$ and $${l^2} = {m^2} + {n^2}$$ is
JEE Main 2014 (Offline)
The image of the line $${{x - 1} \over 3} = {{y - 3} \over 1} = {{z - 4} \over { - 5}}\,$$ in the plane $$2x-y+z+3=0$$ i...
JEE Main 2014 (Offline)
If the vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k$$ and $$\overrightarrow {AC} = 5\widehat i - 2\wideh...
JEE Main 2013 (Offline)
If the lines $${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$$ and $${{x - 1} \over k} = {{y - 4} \over...
JEE Main 2013 (Offline)
Distance between two parallel planes $$2x+y+2z=8$$ and $$4x+2y+4z+5=0$$ is
JEE Main 2013 (Offline)
Let $$ABCD$$ be a parallelogram such that $$\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrigh...
AIEEE 2012
If the line $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$$ and $${{x - 3} \over 1} = {{y - k} \over 2} = ...
AIEEE 2012
A equation of a plane parallel to the plane $$x-2y+2z-5=0$$ and at a unit distance from the origin is :
AIEEE 2012
Let $$\overrightarrow a $$ and $$\overrightarrow b $$ two unit vectors. If the vectors $$\,\overrightarrow c = \widehat...
AIEEE 2012
If $$\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)$$ and $$\overrightarrow b = ...
AIEEE 2011
Statement - 1: The point $$A(1,0,7)$$ is the mirror image of the point $$B(1,6,3)$$ in the line : $${x \over 1} = {{y -...
AIEEE 2011
If the angle between the line $$x = {{y - 1} \over 2} = {{z - 3} \over \lambda }$$ and the plane $$x+2y+3z=4$$ is $${\c...
AIEEE 2011
The vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ are not perpendicular and $$\overrightarrow c $$ and $$\ov...
AIEEE 2011
If the vectors $$\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i +...
AIEEE 2010
Let $$\overrightarrow a = \widehat j - \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k.$$ ...
AIEEE 2010
A line $$AB$$ in three-dimensional space makes angles $${45^ \circ }$$ and $${120^ \circ }$$ with the positive $$x$$-axi...
AIEEE 2010
Statement-1: The point $$A(3, 1, 6)$$ is the mirror image of the point $$B(1, 3, 4)$$ in the plane $$x-y+z=5.$$ Stateme...
AIEEE 2010
If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ are non-coplanar vectors and $$p,q$$ are real numbers, t...
AIEEE 2009
The projections of a vector on the three coordinate axis are $$6,-3,2$$ respectively. The direction cosines of the vecto...
AIEEE 2009
Let the line $$\,\,\,\,\,$$ $${{x - 2} \over 3} = {{y - 1} \over { - 5}} = {{z + 2} \over 2}$$ lie in the plane $$\,\,\,...
AIEEE 2009
If the straight lines $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $${{x - 1} \over k} = {{y - 2} \over 2} = {{z - 3} \over 3}$$ $$\,\,...
AIEEE 2008
The line passing through the points $$(5,1,a)$$ and $$(3, b, 1)$$ crosses the $$yz$$-plane at the point $$\left( {0,{{1...
AIEEE 2008
The non-zero vectors are $${\overrightarrow a ,\overrightarrow b }$$ and $${\overrightarrow c }$$ are related by $${\ove...
AIEEE 2008
The vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$$ lies in the plane of the vectors...
AIEEE 2008
Let $$L$$ be the line of intersection of the planes $$2x+3y+z=1$$ and $$x+3y+2z=2.$$ If $$L$$ makes an angle $$\alpha $$...
AIEEE 2007
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat...
AIEEE 2007
If $$(2,3,5)$$ is one end of a diameter of the sphere $${x^2} + {y^2} + {z^2} - 6x - 12y - 2z + 20 = 0,$$ then the coord...
AIEEE 2007
If $$\widehat u$$ and $$\widehat v$$ are unit vectors and $$\theta $$ is the acute angle between them, then $$2\widehat ...
AIEEE 2007
If a line makes an angle of $$\pi /4$$ with the positive directions of each of $$x$$-axis and $$y$$-axis, then the angle...
AIEEE 2007
The two lines $$x=ay+b, z=cy+d;$$ and $$x=a'y+b' ,$$ $$z=c'y+d'$$ are perpendicular to each other if
AIEEE 2006
The image of the point $$(-1, 3,4)$$ in the plane $$x-2y=0$$ is
AIEEE 2006
The values of a, for which points $$A, B, C$$ with position vectors $$2\widehat i - \widehat j + \widehat k,\,\,\wideha...
AIEEE 2006
If $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times...
AIEEE 2006
The plane $$x+2y-z=4$$ cuts the sphere $${x^2} + {y^2} + {z^2} - x + z - 2 = 0$$ in a circle of radius
AIEEE 2005
If non zero numbers $$a, b, c$$ are in $$H.P.,$$ then the straight line $${x \over a} + {y \over b} + {1 \over c} = 0$$ ...
AIEEE 2005
Let $$\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \wid...
AIEEE 2005
For any vector $${\overrightarrow a }$$ , the value of $${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\...
AIEEE 2005
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar vectors and $$\lambda $$ is a real numb...
AIEEE 2005
The angle between the lines $$2x=3y=-z$$ and $$6x=-y=-4z$$ is
AIEEE 2005
The distance between the line $$\overrightarrow r = 2\widehat i - 2\widehat j + 3\widehat k + \lambda \left( {i - j + ...
AIEEE 2005
If the plane $$2ax-3ay+4az+6=0$$ passes through the midpoint of the line joining the centres of the spheres $${x^2} + {...
AIEEE 2005
Let $$a, b$$ and $$c$$ be distinct non- negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\,\,\...
AIEEE 2005
If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB,$$ then
AIEEE 2005
If the angel $$\theta $$ between the line $${{x + 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2}$$ and the plane $$...
AIEEE 2005
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of t...
AIEEE 2004
If the straight lines $$x=1+s,y=-3$$$$ - \lambda s,$$ $$z = 1 + \lambda s$$ and $$x = {t \over 2},y = 1 + t,z = 2 - t,$...
AIEEE 2004
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be non-zero vectors such that $$\left( {\overri...
AIEEE 2004
The intersection of the spheres $${x^2} + {y^2} + {z^2} + 7x - 2y - z = 13$$ and $${x^2} + {y^2} + {z^2} - 3x + 3y + 4z...
AIEEE 2004
If $${\overrightarrow a ,\overrightarrow b ,\overrightarrow c }$$ are non-coplanar vectors and $$\lambda $$ is a real nu...
AIEEE 2004
A line with direction cosines proportional to $$2,1,2$$ meets each of the lines $$x=y+a=z$$ and $$x+a=2y=2z$$ . The co-...
AIEEE 2004
A particles is acted upon by constant forces $$ 4\widehat i + \widehat j - 3\widehat k$$ and $$ 3\widehat i + \widehat...
AIEEE 2004
Let $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ be such that $$\left| {\overrightarrow u } \right| = 1,...
AIEEE 2004
Distance between two parallel planes $$\,2x + y + 2z = 8$$ and $$4x + 2y + 4z + 5 = 0$$ is
AIEEE 2004
A particle acted on by constant forces $$4\widehat i + \widehat j - 3\widehat k$$ and $$3\widehat i + \widehat j - \wide...
AIEEE 2004
Two systems of rectangular axes have the same origin. If a plane cuts then at distances $$a,b,c$$ and $$a', b', c'$$ fro...
AIEEE 2003
If $$\overrightarrow u \,,\overrightarrow v $$ and $$\overrightarrow w $$ are three non-coplanar vectors, then $$\,\left...
AIEEE 2003
Consider points $$A, B, C$$ and $$D$$ with position vectors $$7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\wi...
AIEEE 2003
If $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {...
AIEEE 2003
A tetrahedron has vertices at $$O(0,0,0), A(1,2,1) B(2,1,3)$$ and $$C(-1,1,2).$$ Then the angle between the faces $$OAB$...
AIEEE 2003
The radius of the circle in which the sphere $${x^2} + {y^2} + {z^2} + 2x - 2y - 4z - 19 = 0$$ is cut by the plane $$x...
AIEEE 2003
$$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$ are $$3$$ vectors, such that $$\overrightarrow a + \o...
AIEEE 2003
The lines $${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$$ and $${{x - 1} \over k} = {{y - 4} \over 2}...
AIEEE 2003
The two lines $$x=ay+b,z=cy+d$$ and $$x = a'y + b',z = c'y + d'$$ will be perpendicular, if and only if
AIEEE 2003
The shortest distance from the plane $$12x+4y+3z=327$$ to the sphere $${x^2} + {y^2} + {z^2} + 4x - 2y - 6z = 155$$ is
AIEEE 2003
The vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\wi...
AIEEE 2003
Let $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarro...
AIEEE 2003
If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \t...
AIEEE 2002
The $$d.r.$$ of normal to the plane through $$(1, 0, 0), (0, 1, 0)$$ which makes an angle $$\pi /4$$ with plane $$x+y=3$...
AIEEE 2002
If the vectors $$\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$ and $$\widehat b = \w...
AIEEE 2002
$$\overrightarrow a = 3\widehat i - 5\widehat j$$ and $$\overrightarrow b = 6\widehat i + 3\widehat j$$ are two vector...
AIEEE 2002
If $$\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right|...
AIEEE 2002
If $$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$ are vectors show that $$\overrightarrow a + \overri...
AIEEE 2002
If $$\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c $$ are vectors such that $$\left[ {\overrig...
AIEEE 2002
A plane which passes through the point $$(3,2,0)$$ and the line $${{x - 4} \over 1} = {{y - 7} \over 5} = {{z - 4} \ove...
AIEEE 2002
If $$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$ and the angle between $${\overrig...
AIEEE 2002

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