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1

JEE Main 2021 (Online) 27th August Evening Shift

Numerical
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 ___________.
Your Input ________

Answer

Correct Answer is 72

Explanation



Since R(3, 5, $$\gamma$$) lies on the plane 2x $$-$$ y + z + 3 = 0.

Therefore, 6 $$-$$ 5 + $$\gamma$$ + 3 = 0

$$\Rightarrow$$ $$\gamma$$ = $$-$$4

Now,

dr's of line QS are 2, $$-$$1, 1

equation of line QS is

$${{x - 1} \over 2} = {{y - 3} \over { - 1}} = {{z - 4} \over 1} = \lambda $$ (say)

$$ \Rightarrow F(2\lambda + 1, - \lambda + 3,\lambda + 4)$$

F lies in the plane

$$ \Rightarrow 2(2\lambda + 1) - ( - \lambda + 3) + (\lambda + 4) + 3$$ = 0

$$ \Rightarrow 4\lambda + 2 + \lambda - 3 + \lambda + 7 = 0$$

$$ \Rightarrow 6\lambda + 6 = 0 \Rightarrow \lambda = - 1$$

$$\Rightarrow$$ F($$-$$1, 4, 3)

Since, F is mid-point of QS.

Therefore, coordinated of S are ($$-$$3, 5, 2).

So, SR = $$\sqrt {36 + 0 + 36} = \sqrt {72} $$

SR2 = 72.
2

JEE Main 2021 (Online) 27th August Morning Shift

Numerical
Let $$\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$$, $$\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$$ and $$\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k$$ be three vectors such that, $$\left| {\overrightarrow b \times \overrightarrow c } \right| = 5\sqrt 3 $$ and $${\overrightarrow a }$$ is perpendicular to $${\overrightarrow b }$$. Then the greatest amongst the values of $${\left| {\overrightarrow a } \right|^2}$$ is _____________.
Your Input ________

Answer

Correct Answer is 90

Explanation

Since, $$\overrightarrow a .\,\overrightarrow b = 0$$

$$1 + 15 + \alpha \beta = 0 \Rightarrow \alpha \beta = - 16$$ .... (1)

Also,

$${\left| {\overrightarrow b \, \times \overrightarrow c } \right|^2} = 75 \Rightarrow (10 + {\beta ^2})14 - {(5 - 3\beta )^2} = 75$$

$$\Rightarrow$$ 5$$\beta$$2 + 30$$\beta$$ + 40 = 0

$$\Rightarrow$$ $$\beta$$ = $$-$$4, $$-$$2

$$\Rightarrow$$ $$\alpha$$ = 4, 8

$$ \Rightarrow \left| {\overrightarrow a } \right|_{\max }^2 = {(26 + {\alpha ^2})_{\max }} = 90$$
3

JEE Main 2021 (Online) 26th August Evening Shift

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

Answer

Correct Answer is 96

Explanation

Containing the line $$\left| {\matrix{ {x + 1} & {y - 1} & {z - 3} \cr 6 & 7 & 8 \cr 3 & 5 & 7 \cr } } \right| = 0$$

$$9(x + 1) - 18(y - 1) + 9(z - 3) = 0$$

$$x - 2y + z = 0$$

$$PQ = \left| {{{7 + 4 + 13} \over {\sqrt 6 }}} \right| = 4\sqrt 6 $$

$$P{Q^2} = 96$$
4

JEE Main 2021 (Online) 26th August Evening Shift

Numerical
If the projection of the vector $$\widehat i + 2\widehat j + \widehat k$$ on the sum of the two vectors $$2\widehat i + 4\widehat j - 5\widehat k$$ and $$ - \lambda \widehat i + 2\widehat j + 3\widehat k$$ is 1, then $$\lambda$$ is equal to __________.
Your Input ________

Answer

Correct Answer is 5

Explanation

$$\overrightarrow a = \widehat i + 2\widehat j + \widehat k$$

$$\overrightarrow b = (2 - \lambda )\widehat i + 6\widehat j - 2\widehat k$$

$${{\overrightarrow a \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = 1,\overrightarrow a \,.\,\overrightarrow b = 12 - \lambda $$

$$\left( {\overrightarrow a \,.\,\overrightarrow b } \right) = |\overrightarrow b {|^2}$$

$$\lambda$$2 $$-$$ 24$$\lambda$$ + 144 = $$\lambda$$2 $$-$$ 4$$\lambda$$ + 4 + 40

20$$\lambda$$ = 100 $$\Rightarrow$$ $$\lambda$$ = 5

Questions Asked from Vector Algebra and 3D Geometry

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JEE Main 2021 (Online) 27th August Evening Shift (1)
JEE Main 2021 (Online) 27th August Morning Shift (1)
JEE Main 2021 (Online) 26th August Evening Shift (2)
JEE Main 2021 (Online) 26th August Morning Shift (1)
JEE Main 2021 (Online) 27th July Evening Shift (2)
JEE Main 2021 (Online) 27th July Morning Shift (2)
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JEE Main 2021 (Online) 26th February Morning Shift (1)
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