JEE Advance - Mathematics (2018 - Paper 2 Offline)
- 1For any positive integer n, define
$${f_n}:(0,\infty ) \to R$$ as
$${f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)$$
for all x$$ \in $$(0, $$\infty $$). (Here, the inverse trigonometric function tan$$-$$1 x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$). Then, which of the following statement(s) is (are) TRUE?پاسخ دهید(D)For any fixed positive integer n, $$\mathop {\lim }\limits_{x \to \infty } {\sec ^2}({f_n}(x)) = 1$$ - 2Let T be the line passing through the points P($$-$$2, 7) and Q(2, $$-$$5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?پاسخ دهیدAD
- 3Let S be the set of all column matrices $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$ such that $${b_1},{b_2},{b_3} \in R$$ and the system of equations (in real variables)
$$\eqalign{ & - x + 2y + 5z = {b_1} \cr & 2x - 4y + 3z = {b_2} \cr & x - 2y + 2z = {b_3} \cr} $$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$$$ \in $$S?پاسخ دهیدAD - 4Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $$\sqrt 2 $$, then which of the following statement(s) is (are) TRUE?پاسخ دهیدAC
- 6Let f : (0, $$\pi $$) $$ \to $$ R be a twice differentiable function such that $$\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t - x}} = {\sin ^2}x$$ for all x$$ \in $$ (0, $$\pi $$).
If $$f\left( {{\pi \over 6}} \right) = - {\pi \over {12}}$$, then which of the following statement(s) is (are) TRUE?پاسخ دهیدBCD - 12Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P in the XY-plane, then the length of PR is ...............پاسخ دهید8
- 13Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let $$S\left( {{1 \over 2},{1 \over 2},{1 \over 2}} \right)$$ be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p = SP, q = SQ, r = SR and t = ST, then the value of |(p $$ \times $$ q) $$ \times $$ (r $$ \times $$ t)| is ............پاسخ دهید0.5
- 15Let $${E_1} = \left\{ {x \in R:x \ne 1\,and\,{x \over {x - 1}} > 0} \right\}$$ and
$${E_2} = \left\{ \matrix{ x \in {E_1}:{\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right) \hfill \cr is\,a\,real\,number \hfill \cr} \right\}$$
(Here, the inverse trigonometric function $${\sin ^{ - 1}}$$ x assumes values in $$\left[ { - {\pi \over 2},{\pi \over 2}} \right]$$.).
Let f : E1 $$ \to $$ R be the function defined by f(x) = $${{{\log }_e}\left( {{x \over {x - 1}}} \right)}$$ and g : E2 $$ \to $$ R be the function defined by g(x) = $${\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right)$$.
The correct option is :LIST-I LIST-II P. The range of $f$ is 1. $\left( -\infty, \frac{1}{1-e} \right] \cup \left[ \frac{e}{e-1}, \infty \right)$ Q. The range of $g$ contains 2. $(0, 1)$ R. The domain of $f$ contains 3. $\left[ -\frac{1}{2}, \frac{1}{2} \right]$ S. The domain of $g$ is 4. $(-\infty, 0) \cup (0, \infty)$ 5. $\left( -\infty, \frac{e}{e-1} \right)$ 6. $(-\infty, 0) \cup \left( \frac{1}{2}, \frac{e}{e-1} \right]$ پاسخ دهید(A)P $$ \to $$ 4; Q $$ \to $$ 2; R $$ \to $$ 1 ; S $$ \to $$ 1 - 16In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.
(i) Let $$\alpha $$1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $$\alpha $$2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
i) Let $$\alpha $$3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $$\alpha $$4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together.
The correct option isLIST-I LIST-II P. The value of $\alpha_1$ is 1. 136 Q. The value of $\alpha_2$ is 2. 189 R. The value of $\alpha_3$ is 3. 192 S. The value of $\alpha_4$ is 4. 200 5. 381 6. 461 پاسخ دهید(C)P $$ \to $$ 4; Q $$ \to $$ 6; R $$ \to $$ 5; S $$ \to $$ 2 - 17Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60$$^\circ $$ at one of its vertices N. Let the area of the $$\Delta $$LMN be $$4\sqrt 3 $$.
List - I List - II P. The length of the conjugate axis of H is 1. 8 Q. The eccentricity of H is 2. $${4 \over {\sqrt 3 }}$$ R. The distance between the foci of H is 3. $${2 \over {\sqrt 3 }}$$ S. The length of the latus rectum of H is 4. 4 پاسخ دهید(B)P $$ \to $$ 4 ; Q $$ \to $$ 3 ; R $$ \to $$ 1 ; S $$ \to $$ 2 - 18Let $${f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to R$$ and $${f_4}:R \to R$$ be functions defined by
(i) $${f_1}(x) = \sin (\sqrt {1 - {e^{ - {x^2}}}} )$$,
(ii) $${f_2}(x) = \left\{ \matrix{ {{|\sin x|} \over {\tan { - ^1}x}}if\,x \ne 0,\,where \hfill \cr 1\,if\,x = 0 \hfill \cr} \right.$$
the inverse trigonometric function tan$$-$$1x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$,
(iii) $${f_3}(x) = [\sin ({\log _e}(x + 2))]$$, where for $$t \in R,\,[t]$$ denotes the greatest integer less than or equal to t,
(iv) $${f_4}(x) = \left\{ \matrix{ {x^2}\sin \left( {{1 \over x}} \right)\,if\,x \ne 0 \hfill \cr 0\,if\,x = 0 \hfill \cr} \right.$$
LIST-I LIST-II P. The function $$ f_1 $$ is 1. NOT continuous at $$ x = 0 $$ Q. The function $$ f_2 $$ is 2. continuous at $$ x = 0 $$ and NOT differentiable at $$ x = 0 $$ R. The function $$ f_3 $$ is 3. differentiable at $$ x = 0 $$ and its derivative is NOT continuous at $$ x = 0 $$ S. The function $$ f_4 $$ is 4. differentiable at $$ x = 0 $$ and its derivative is continuous at $$ x = 0 $$ پاسخ دهید(D)P $$ \to $$ 2 ; Q $$ \to $$ 1 ; R $$ \to $$ 4 ; S $$ \to $$ 3