JEE Advance - Mathematics (2019 - Paper 1 Offline)
1
Let S be the set of all complex numbers z satisfying $$\left| {z - 2 + i} \right| \ge \sqrt 5 $$. If the complex number z0 is such that $${1 \over {\left| {{z_0} - 1} \right|}}$$ is the maximum of the set $$\left\{ {{1 \over {\left| {{z_0} - 1} \right|}}:z \in S} \right\}$$, then the principal argument of $${{4 - {z_0} - {{\overline z }_0}} \over {{z_0} - {{\overline z }_0} + 2i}}$$ is
where $$\alpha $$ = $$\alpha $$($$\theta $$) and $$\beta $$ = $$\beta $$($$\theta $$) are real numbers, and I is the 2 $$ \times $$ 2 identity matrix. If $$\alpha $$* is the minimum of the set {$$\alpha $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)} and {$$\beta $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)}, then the value of $$\alpha $$* + $$\beta $$* is
پاسخ دهید
(D)
$$ - {{29} \over {16}}$$
3
A line y = mx + 1 intersects the circle $${(x - 3)^2} + {(y + 2)^2}$$ = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate $$ - {3 \over 5}$$, then which one of the following options is correct?
پاسخ دهید
(D)
2 $$ \le $$ m < 4
4
The area of the region
{(x, y) : xy $$ \le $$ 8, 1 $$ \le $$ y $$ \le $$ x2} is
پاسخ دهید
(C)
$$16{\log _e}2 - {{14} \over 3}$$
5
Let $$\Gamma $$ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at YP. If PYP has length 1 for each point P on I`, then which of the following options is/are correct?
پاسخ دهید
A
C
6
Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :
R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1;
En : ellipse $${{{x^2}} \over {a_n^2}} + {{{y^2}} \over {b_n^2}} = 1$$ of the largest area inscribed in $${R_{n - 1}},n > 1$$;
Rn : rectangle of largest area, with sides parallel to the axes, inscribed in En, n > 1.
Then which of the following options is/are correct?
پاسخ دهید
C
D
7
In a non-right-angled triangle $$\Delta $$PQR, let p, q, r denote the lengths of the sides opposite to the angles At P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = $${\sqrt 3 }$$, q = 1, and the radius of the circumcircle of the $$\Delta $$PQR equals 1, then which of the following options is/are correct?
پاسخ دهید
A
B
D
8
Let $$\alpha $$ and $$\beta $$ be the roots of$${x^2} - x - 1 = 0$$, with $$\alpha $$ > $$\beta $$. For all positive integers n, define
Then which of the following options is/are correct?
پاسخ دهید
B
C
D
9
Let L1 and L2 denote the lines
$$r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)$$, $$\lambda $$$$ \in $$ R
and $$r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R$$
respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of the following options describe(s) L3?
پاسخ دهید
A
B
C
10
There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, and B3 contains 5 red and 3 green balls. Bags B1, B2 and B3 have probabilities $${3 \over {10}}$$, $${3 \over {10}}$$ and $${4 \over {10}}$$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
پاسخ دهید
A
C
11
Let $$M = \left[ {\matrix{
0 & 1 & a \cr
1 & 2 & 3 \cr
3 & b & 1 \cr
} } \right]$$ and
Then which of the following options is/are correct?
پاسخ دهید
B
C
D
13
Let S be the sample space of all 3 $$ \times $$ 3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given by
E1 = {A$$ \in $$S : det A = 0} and
E2 = {A$$ \in $$S : sum of entries of A is 7}.
If a matrix is chosen at random from S, then the conditional probability P(E1 | E2) equals ...............
پاسخ دهید
0.5
14
Let the point B be the reflection of the point A(2, 3) with respect to the line $$8x - 6y - 23 = 0$$. Let $$\Gamma_{A} $$ and $$\Gamma_{B} $$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles $$\Gamma_{A} $$ and $$\Gamma_{B} $$ such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is .................
پاسخ دهید
10
15
Let $$\omega \ne 1$$ be a cube root of unity. Then the minimum of the set $$\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$$ distinct non-zero integers} equals ..................
Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is $$\Delta $$ then the value of (6$$\Delta $$)2 equals ..............
پاسخ دهید
0.75
18
Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If $$AP(1;3) \cap AP(2;5) \cap AP(3;7)$$ = AP(a ; d), then a + d equals ..............
