1
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
For any 3 $$\times$$ 3 matrix M, let |M| denote the determinant of M. Let I be the 3 $$\times$$ 3 identity matrix. Let E and F be two 3 $$\times$$ 3 matrices such that (I $$-$$ EF) is invertible. If G = (I $$-$$ EF)$$-$$1, then which of the following statements is (are) TRUE?
A
| FE | = | I $$-$$ FE| | FGE |
B
(I $$-$$ FE)(I + FGE) = I
C
EFG = GEF
D
(I $$-$$ FE)(I $$-$$ FGE) = I
2
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
For any positive integer n, let Sn : (0, $$\infty$$) $$\to$$ R be defined by $${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)}$$, where for any x $$\in$$ R, $${\cot ^{ - 1}}(x) \in (0,\pi )$$ and $${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$. Then which of the following statements is (are) TRUE?
A
$${S_{10}}(x) = {\pi \over 2} - {\tan ^{ - 1}}\left( {{{1 + 11{x^2}} \over {10x}}} \right)$$, for all x > 0
B
$$\mathop {\lim }\limits_{n \to \infty } \cot ({S_n}(x)) = x$$, for all x > 0
C
The equation $${S_3}(x) = {\pi \over 4}$$ has a root in (0, $$\infty$$)
D
$$tan({S_n}(x)) \le {1 \over 2}$$, for all n $$\ge$$ 1 and x > 0
3
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
For any complex number w = c + id, let $$\arg (w) \in ( - \pi ,\pi ]$$, where $$i = \sqrt { - 1}$$. Let $$\alpha$$ and $$\beta$$ be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}$$, the ordered pair (x, y) lies on the circle $${x^2} + {y^2} + 5x - 3y + 4 = 0$$, Then which of the following statements is (are) TRUE?
A
$$\alpha$$ = $$-$$1
B
$$\alpha$$$$\beta$$ = 4
C
$$\alpha$$$$\beta$$ = $$-$$4
D
$$\beta$$ = 4
4
JEE Advanced 2021 Paper 1 Online
Numerical
+4
-0
For x $$\in$$ R, the number of real roots of the equation $$3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0$$ is ________.