jee-advanced

Q3 Forces, equilibrium and resultants View

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance '$a$' from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$, is
(A) $\frac{F}{2m}\frac{a}{\sqrt{a^2-x^2}}$
(B) $\frac{F}{2m}\frac{x}{\sqrt{a^2-x^2}}$
(C) $\frac{F}{2m}\frac{x}{a}$
(D) $\frac{F}{2m}\frac{\sqrt{a^2-x^2}}{x}$

Q47 Proof True/False Justification View

Let $f(x) = 2 + \cos x$ for all real $x$. STATEMENT-1: For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f'(c) = 0$. because STATEMENT-2: $f(t) = f(t+2\pi)$ for each real $t$.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

Q48 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $f(x) = x^x$ for $x > 0$. Then $f$ is
(A) increasing on $(0, \infty)$
(B) decreasing on $(0, \infty)$
(C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$
(D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$

Q49 Matrices Linear System and Inverse Existence View

The number of distinct real values of $\lambda$ for which the system of linear equations $$x + y + z = 0$$ $$x + \lambda y + z = 0$$ $$x + y + \lambda z = 0$$ has a non-trivial solution is
(A) 0
(B) 1
(C) 2
(D) 3

Q50 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View

Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ is
(A) $\frac{2}{9}(p-q)(2q-p)$
(B) $\frac{2}{9}(q-p)(2p-q)$
(C) $\frac{2}{9}(q-2p)(2q-p)$
(D) $\frac{2}{9}(2p-q)(2q-p)$

Q51 Quadratic trigonometric equations View

The number of solutions of the pair of equations $$2\sin^2\theta - \cos 2\theta = 0$$ $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $[0, 2\pi]$ is
(A) 0
(B) 1
(C) 2
(D) 4

Q52 Vector Product and Surfaces View

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Which one of the following is correct?
(A) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} = \vec{0}$
(B) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq \vec{0}$
(C) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{a} \times \vec{c} \neq \vec{0}$
(D) $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}$ are mutually perpendicular

Q53 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View

Let $O(0,0)$, $P(3,4)$, $Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR$, $PQR$, $OQR$ are of equal area. The coordinates of $R$ are
(A) $\left(\frac{4}{3}, 3\right)$
(B) $(3, \frac{2}{3})$
(C) $(3, \frac{4}{3})$
(D) $\left(\frac{4}{3}, \frac{2}{3}\right)$

Q54 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View

The tangent to the curve $y = e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$
(A) on the left of $x = c$
(B) on the right of $x = c$
(C) at no point
(D) at all points

Q55 Vector Product and Surfaces View

The number of distinct real values of $\lambda$ for which the vectors $-\lambda^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\lambda^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2\hat{k}$ are coplanar is
(A) 0
(B) 1
(C) 2
(D) 3

Q56 Complex Numbers Argand & Loci Algebraic Conditions for Geometric Properties (Real, Imaginary, Collinear) View

If $|z| = 1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on
(A) a line not passing through the origin
(B) $|z| = \sqrt{2}$
(C) the $x$-axis
(D) the $y$-axis

Q57 Conic sections Confocal or Related Conic Construction View

A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2 + 4y^2 = 12$. Then its equation is
(A) $x^2\csc^2\theta - y^2\sec^2\theta = 1$
(B) $x^2\sec^2\theta - y^2\csc^2\theta = 1$
(C) $x^2\sin^2\theta - y^2\cos^2\theta = 1$
(D) $x^2\cos^2\theta - y^2\sin^2\theta = 1$

Q58 Curve Sketching Function Properties from Symmetry or Parity View

STATEMENT-1: The curve $y = \frac{-x^2}{2} + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2: A parabola is symmetric about its axis.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

Q59 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

Let $f: \{1,2,3,4\} \to \{1,2,3,4\}$ and $g: \{1,2,3,4\} \to \{1,2,3,4\}$ be invertible functions such that $f \circ g = $ identity. Then
(A) $f = g^{-1}$
(B) $g = f^{-1}$
(C) $f \circ g \neq g \circ f$
(D) $f \circ g = g \circ f$

Q60 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then
(A) $f$ is an increasing function of $x$
(B) $f$ is a decreasing function of $x$
(C) $f$ is neither increasing nor decreasing function of $x$
(D) $f'$ is not a monotonic function of $x$

Q61 Conditional Probability Combinatorial Conditional Probability (Counting-Based) View

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{2}{5}$
(D) $\frac{1}{5}$

Q62 Conditional Probability Proof of a General Conditional Expectation or Independence Property View

Let $E^c$ denote the complement of an event $E$. Let $E$, $F$, $G$ be pairwise independent events with $P(G) > 0$ and $P(E \cap F \cap G) = 0$. Then $P(E^c \cap F^c | G)$ equals
(A) $P(E^c) + P(F^c)$
(B) $P(E^c) - P(F^c)$
(C) $P(E^c) - P(F)$
(D) $P(E) - P(F^c)$

Q63 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

The number of solutions of the equation $\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) = \sin^{-1}\left(\frac{x-1}{\sqrt{1+(x-1)^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+(x-1)^2}}\right)$ is
(A) 0
(B) 1
(C) 2
(D) infinite

Q64 Modulus function Differentiability of functions involving modulus View

Let $f(x) = a_0 + a_1|x| + a_2|x|^2 + a_3|x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then $f'(x)$ exists at $x = 0$ if and only if
(A) $a_1 = 0$
(B) $a_1 = 0$ and $a_2 = 0$
(C) $a_1 = 0$ and $a_3 = 0$
(D) $a_2 = 0$ and $a_3 = 0$

Q65 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then
(A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$
(B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$
(C) $f$ has local minima at $x = -1$ and at $x = 2$
(D) $f$ has local maxima at $x = -1$ and at $x = 2$

Q66 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $I = \int_0^1 \frac{\sin x}{\sqrt{x}}\,dx$ and $J = \int_0^1 \frac{\cos x}{\sqrt{x}}\,dx$. Then which one of the following is true?
(A) $I > \frac{2}{3}$ and $J > 2$
(B) $I < \frac{2}{3}$ and $J < 2$
(C) $I < \frac{2}{3}$ and $J > 2$
(D) $I > \frac{2}{3}$ and $J < 2$

Q67 Areas Between Curves Select Correct Integral Expression View

The area of the region between the curves $y = \sqrt{\frac{1+\sin x}{\cos x}}$ and $y = \sqrt{\frac{1-\sin x}{\cos x}}$ bounded by the lines $x = 0$ and $x = \frac{\pi}{4}$ is
(A) $\int_0^{\sqrt{2}-1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(B) $\int_0^{\sqrt{2}-1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(C) $\int_0^{\sqrt{2}+1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(D) $\int_0^{\sqrt{2}+1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$

Q68 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Match the statements in Column I with the values in Column II.
Column I
(A) $\int_{-\pi}^{\pi} \cos^2 x\,\frac{1}{1+a^x}\,dx$, $a > 0$
(B) $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$
(C) $\int_{-2}^{2} \frac{x^2}{1+5^x}\,dx$
(D) $\int_1^2 \frac{\sqrt{\ln(3-x)}}{\sqrt{\ln(3-x)}+\sqrt{\ln(x+1)}}\,dx$
Column II
(p) $\frac{1}{2}$
(q) $0$
(r) $\frac{\pi}{4}$
(s) $\frac{\pi}{2}$

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