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2014 paper2

20 maths questions

Q41 First order differential equations (integrating factor) View
The function $y = f(x)$ is the solution of the differential equation
$$\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^4 + 2x}{\sqrt{1 - x^2}}$$
in $(-1,1)$ satisfying $f(0) = 0$. Then
$$\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x)\, dx$$
is
(A) $\frac{\pi}{3} - \frac{\sqrt{3}}{2}$
(B) $\frac{\pi}{3} - \frac{\sqrt{3}}{4}$
(C) $\frac{\pi}{6} - \frac{\sqrt{3}}{4}$
(D) $\frac{\pi}{6} - \frac{\sqrt{3}}{2}$
Q42 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The following integral
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (2\operatorname{cosec} x)^{17}\, dx$$
is equal to
(A) $\int_{0}^{\log(1+\sqrt{2})} 2\left(e^{u} + e^{-u}\right)^{16} du$
(B) $\int_{0}^{\log(1+\sqrt{2})} \left(e^{u} + e^{-u}\right)^{17} du$
(C) $\int_{0}^{\log(1+\sqrt{2})} \left(e^{u} - e^{-u}\right)^{17} du$
(D) $\int_{0}^{\log(1+\sqrt{2})} 2\left(e^{u} - e^{-u}\right)^{16} du$
Q43 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
Coefficient of $x^{11}$ in the expansion of $\left(1 + x^2\right)^4 \left(1 + x^3\right)^7 \left(1 + x^4\right)^{12}$ is
(A) 1051
(B) 1106
(C) 1113
(D) 1120
Q44 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
Let $f : [0,2] \rightarrow \mathbb{R}$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0) = 1$. Let
$$F(x) = \int_{0}^{x^2} f(\sqrt{t})\, dt$$
for $x \in [0,2]$. If $F'(x) = f'(x)$ for all $x \in (0,2)$, then $F(2)$ equals
(A) $e^2 - 1$
(B) $e^4 - 1$
(C) $e - 1$
(D) $e^4$
Q45 Circles Area and Geometric Measurement Involving Circles View
The common tangents to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$. Then the area of the quadrilateral $PQRS$ is
(A) 3
(B) 6
(C) 9
(D) 15
Q46 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
For $x \in (0, \pi)$, the equation $\sin x + 2\sin 2x - \sin 3x = 3$ has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution
Q47 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
In a triangle the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2 = y$, where $c$ is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is
(A) $\frac{3y}{2x(x+c)}$
(B) $\frac{3y}{2c(x+c)}$
(C) $\frac{3y}{4x(x+c)}$
(D) $\frac{3y}{4c(x+c)}$
Q48 Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
Six cards and six envelopes are numbered $1,2,3,4,5,6$ and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
(A) 264
(B) 265
(C) 53
(D) 67
Q49 Probability Definitions Finite Equally-Likely Probability Computation View
Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$
Q50 Complex Numbers Arithmetic True/False or Property Verification Statements View
The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x)) = 0$$
has
(A) only purely imaginary roots
(B) all real roots
(C) two real and two purely imaginary roots
(D) neither real nor purely imaginary roots
Q51 Circles Circle-Related Locus Problems View
Let $a, r, s, t$ be nonzero real numbers. Let $P(at^2, 2at)$, $Q$, $R(ar^2, 2ar)$ and $S(as^2, 2as)$ be distinct points on the parabola $y^2 = 4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a, 0)$.
The value of $r$ is
(A) $-\frac{1}{t}$
(B) $\frac{t^2+1}{t}$
(C) $\frac{1}{t}$
(D) $\frac{t^2-1}{t}$
Q52 Tangents, normals and gradients Normal or perpendicular line problems View
If $st = 1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is
(A) $\frac{(t^2+1)^2}{2t^3}$
(B) $\frac{a(t^2+1)^2}{2t^3}$
(C) $\frac{a(t^2+1)^2}{t^3}$
(D) $\frac{a(t^2+2)^2}{t^3}$
Q53 Exponential Distribution View
Given that for each $a \in (0,1)$,
$$\lim_{h \rightarrow 0^+} \int_{h}^{1-h} t^{-a}(1-t)^{a-1}\, dt$$
exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
The value of $g\left(\frac{1}{2}\right)$ is
(A) $\pi$
(B) $2\pi$
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{4}$
Q54 Exponential Distribution View
The value of $g'\left(\frac{1}{2}\right)$ is
(A) $\frac{\pi}{2}$
(B) $\pi$
(C) $-\frac{\pi}{2}$
(D) 0
Q55 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Box 1 contains three cards bearing numbers $1,2,3$; box 2 contains five cards bearing numbers $1,2,3,4,5$; and box 3 contains seven cards bearing numbers $1,2,3,4,5,6,7$. A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{\text{th}}$ box, $i = 1,2,3$.
The probability that $x_1 + x_2 + x_3$ is odd, is
(A) $\frac{29}{105}$
(B) $\frac{53}{105}$
(C) $\frac{57}{105}$
(D) $\frac{1}{2}$
Q56 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
The probability that $x_1, x_2, x_3$ are in an arithmetic progression, is
(A) $\frac{9}{105}$
(B) $\frac{10}{105}$
(C) $\frac{11}{105}$
(D) $\frac{7}{105}$
Q57 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
Let $z_k = \cos\left(\frac{2k\pi}{10}\right) + i\sin\left(\frac{2k\pi}{10}\right)$; $k = 1,2,\ldots,9$.
List I P. For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j = 1$ Q. There exists a $k \in \{1,2,\ldots,9\}$ such that $z_1 \cdot z = z_k$ has no solution $z$ in the set of complex numbers. R. $\frac{|1-z_1||1-z_2|\cdots|1-z_9|}{10}$ equals S. $1 - \sum_{k=1}^{9} \cos\left(\frac{2k\pi}{10}\right)$ equals
List II
1. True
2. False
3. 1
4. 2
P Q R S
(A) 1243
(B) 2134
(C) 1234
(D) 2143
Q58 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
List IList II
P. The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\leq 2$, satisfying $f(0) = 0$ and $\int_{0}^{1} f(x)\,dx = 1$, is1. 8
Q. The number of points in the interval $[-\sqrt{13}, \sqrt{13}]$ at which $f(x) = \sin(x^2) + \cos(x^2)$ attains its maximum value, is2. 2
R. $\int_{-2}^{2} \frac{3x^2}{1+e^x}\,dx$ equals3. 4
S. $\dfrac{\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} \cos 2x \log\left(\frac{1+x}{1-x}\right)dx}{\displaystyle\int_{0}^{\frac{1}{2}} \cos 2x \log\left(\frac{1+x}{1-x}\right)dx}$ equals4. 0

P Q R S
(A) 3241
(B) 2341
(C) 3214
(D) 2314
Q59 Differentiating Transcendental Functions Higher-order or nth derivative computation View
List I P. Let $y(x) = \cos\left(3\cos^{-1}x\right)$, $x \in [-1,1]$, $x \neq \pm\frac{\sqrt{3}}{2}$. Then $\frac{1}{y(x)}\left\{\left(x^2-1\right)\frac{d^2y(x)}{dx^2} + x\frac{dy(x)}{dx}\right\}$ equals Q. Let $A_1, A_2, \ldots, A_n$ $(n > 2)$ be the vertices of a regular polygon of $n$ sides with its centre at the origin. Let $\overrightarrow{a_k}$ be the position vector of the point $A_k$, $k = 1,2,\ldots,n$. If $\left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \times \overrightarrow{a_{k+1}}\right)\right| = \left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \cdot \overrightarrow{a_{k+1}}\right)\right|$, then the minimum value of $n$ is R. If the normal from the point $P(h,1)$ on the ellipse $\frac{x^2}{6} + \frac{y^2}{3} = 1$ is perpendicular to the line $x + y = 8$, then the value of $h$ is S. Number of positive solutions satisfying the equation $\tan^{-1}\left(\frac{1}{2x+1}\right) + \tan^{-1}\left(\frac{1}{4x+1}\right) = \tan^{-1}\left(\frac{2}{x^2}\right)$ is
List II
1. 1
2. 2
3. 3
4. 4
P Q R S
(A) 4321
(B) 2431
(C) 4312
(D) 2413
Q60 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $f_1 : \mathbb{R} \rightarrow \mathbb{R}$, $f_2 : [0,\infty) \rightarrow \mathbb{R}$, $f_3 : \mathbb{R} \rightarrow \mathbb{R}$ and $f_4 : \mathbb{R} \rightarrow [0,\infty)$ be defined by
$$f_1(x) = \begin{cases} |x| & \text{if } x < 0 \\ e^x & \text{if } x \geq 0 \end{cases}$$
$$f_2(x) = x^2;$$
$$f_3(x) = \begin{cases} \sin x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$
and
$$f_4(x) = \begin{cases} f_2(f_1(x)) & \text{if } x < 0 \\ f_2(f_1(x)) - 1 & \text{if } x \geq 0 \end{cases}$$
List I (functions) P. $f_4$ is Q. $f_3$ is R. $f_2 \circ f_1$ is S. $f_2$ is
List II (properties)
1. onto but not one-one
2. neither continuous nor one-one
3. differentiable but not one-one
4. continuous and one-one
P Q R S
(A) 3142
(B) 1342
(C) 3124
(D) 1324