1
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The set of all values of $$\lambda$$ for which the equation $${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda $$ has a real solution $$x$$, is :

A
$$\left[ { - 2, - 1} \right]$$
B
$$\left[ { - {3 \over 2}, - 1} \right]$$
C
$$\left[ { - 2, - {3 \over 2}} \right]$$
D
$$\left[ { - 1, - {1 \over 2}} \right]$$
2
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

If the tangent at a point P on the parabola $$y^2=3x$$ is parallel to the line $$x+2y=1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4}+\frac{y^2}{1}=1$$ are perpendicular to the line $$x-y=2$$, then the area of the triangle PQR is :

A
$$\frac{9}{\sqrt5}$$
B
$$3\sqrt5$$
C
$$5\sqrt3$$
D
$$\frac{3}{2}\sqrt5$$
3
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The area of the region $$A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}$$ is

A
$$\sqrt 5 + 2\sqrt 2 - 4.5$$
B
$$1 - {3 \over {\sqrt 2 }} + {4 \over {\sqrt 5 }}$$
C
$$\sqrt 5 - 2\sqrt 2 + 1$$
D
$${3 \over {\sqrt 5 }} - {3 \over {\sqrt 2 }} + 1$$
4
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\overrightarrow a = 4\widehat i + 3\widehat j$$ and $$\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$$. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightarrow b } \right) + 25 = 0,\overrightarrow c \,.(\widehat i + \widehat j + \widehat k) = 4$$, and projection of $$\overrightarrow c $$ on $$\overrightarrow a $$ is 1, then the projection of $$\overrightarrow c $$ on $$\overrightarrow b $$ equals :

A
$$\frac{3}{\sqrt2}$$
B
$$\frac{1}{\sqrt2}$$
C
$$\frac{1}{5}$$
D
$$\frac{5}{\sqrt2}$$
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