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2014 paper1

20 maths questions

Q41 Matrices Matrix Algebra and Product Properties View
Let $M$ and $N$ be two $3 \times 3$ matrices such that $MN = NM$. Further, if $M \neq N^2$ and $M^2 = N^4$, then
(A) determinant of $\left(M^2 + MN^2\right)$ is 0
(B) there is a $3 \times 3$ non-zero matrix $U$ such that $\left(M^2 + MN^2\right)U$ is the zero matrix
(C) determinant of $\left(M^2 + MN^2\right) \geq 1$
(D) for a $3 \times 3$ matrix $U$, if $\left(M^2 + MN^2\right)U$ equals the zero matrix then $U$ is the zero matrix
Q42 Curve Sketching Number of Solutions / Roots via Curve Analysis View
For every pair of continuous functions $f, g : [0,1] \rightarrow \mathbb{R}$ such that $$\max\{f(x) : x \in [0,1]\} = \max\{g(x) : x \in [0,1]\}$$ the correct statement(s) is(are):
(A) $(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(B) $(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(C) $(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $c \in [0,1]$
(D) $(f(c))^2 = (g(c))^2$ for some $c \in [0,1]$
Q43 Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions View
Let $f : (0, \infty) \rightarrow \mathbb{R}$ be given by $$f(x) = \int_{\frac{1}{x}}^{x} e^{-\left(t + \frac{1}{t}\right)} \frac{dt}{t}$$ Then
(A) $f(x)$ is monotonically increasing on $[1, \infty)$
(B) $f(x)$ is monotonically decreasing on $(0,1)$
(C) $f(x) + f\left(\frac{1}{x}\right) = 0$, for all $x \in (0, \infty)$
(D) $f\left(2^x\right)$ is an odd function of $x$ on $\mathbb{R}$
Q44 Stationary points and optimisation Count or characterize roots using extremum values View
Let $a \in \mathbb{R}$ and let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $$f(x) = x^5 - 5x + a$$ Then
(A) $f(x)$ has three real roots if $a > 4$
(B) $f(x)$ has only one real root if $a > 4$
(C) $f(x)$ has three real roots if $a < -4$
(D) $f(x)$ has three real roots if $-4 < a < 4$
Q45 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
Let $f : [a,b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_{a}^{x} f(t)\, dt & \text{if } a \leq x \leq b \\ \int_{a}^{b} f(t)\, dt & \text{if } x > b \end{cases}$$ Then
(A) $g(x)$ is continuous but not differentiable at $a$
(B) $g(x)$ is differentiable on $\mathbb{R}$
(C) $g(x)$ is continuous but not differentiable at $b$
(D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both
Q46 Function Transformations View
Let $f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathbb{R}$ be given by $$f(x) = (\log(\sec x + \tan x))^3$$ Then
(A) $f(x)$ is an odd function
(B) $f(x)$ is a one-one function
(C) $f(x)$ is an onto function
(D) $f(x)$ is an even function
Q47 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
From a point $P(\lambda, \lambda, \lambda)$, perpendiculars $PQ$ and $PR$ are drawn respectively on the lines $y = x, z = 1$ and $y = -x, z = -1$. If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda$ is(are)
(A) $\sqrt{2}$
(B) $1$
(C) $-1$
(D) $-\sqrt{2}$
Q48 Vectors Introduction & 2D True/False or Multiple-Statement Verification View
Let $\vec{x}, \vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi}{3}$. If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x}$ and $\vec{y} \times \vec{z}$ and $\vec{b}$ is a nonzero vector perpendicular to $\vec{y}$ and $\vec{z} \times \vec{x}$, then
(A) $\vec{b} = (\vec{b} \cdot \vec{z})(\vec{z} - \vec{x})$
(B) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{y} - \vec{z})$
(C) $\vec{a} \cdot \vec{b} = -(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z})$
(D) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{z} - \vec{y})$
Q49 Circles Circle Equation Derivation View
A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2 + y^2 = 16$ and $x^2 + y^2 = 1$. Then
(A) radius of $S$ is 8
(B) radius of $S$ is 7
(C) centre of $S$ is $(-7, 1)$
(D) centre of $S$ is $(-8, 1)$
Q50 Matrices Structured Matrix Characterization View
Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if
(A) the first column of $M$ is the transpose of the second row of $M$
(B) the second row of $M$ is the transpose of the first column of $M$
(C) $M$ is a diagonal matrix with nonzero entries in the main diagonal
(D) the product of entries in the main diagonal of $M$ is not the square of an integer
Q51 Solving quadratics and applications Evaluating an algebraic expression given a constraint View
Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b + 2$, then the value of $$\frac{a^2 + a - 14}{a + 1}$$ is
Q52 Combinations & Selection Geometric Combinatorics View
Let $n \geq 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$ is
Q53 Combinations & Selection Counting Integer Solutions to Equations View
Let $n_1 < n_2 < n_3 < n_4 < n_5$ be positive integers such that $n_1 + n_2 + n_3 + n_4 + n_5 = 20$. Then the number of such distinct arrangements $(n_1, n_2, n_3, n_4, n_5)$ is
Q54 Stationary points and optimisation Composite or piecewise function extremum analysis View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be respectively given by $f(x) = |x| + 1$ and $g(x) = x^2 + 1$. Define $h : \mathbb{R} \rightarrow \mathbb{R}$ by $$h(x) = \begin{cases} \max\{f(x), g(x)\} & \text{if } x \leq 0 \\ \min\{f(x), g(x)\} & \text{if } x > 0 \end{cases}$$ The number of points at which $h(x)$ is not differentiable is
Q55 Integration by Parts Definite Integral Evaluation by Parts View
The value of $$\int_{0}^{1} 4x^3 \left\{\frac{d^2}{dx^2}\left(1 - x^2\right)^5\right\} dx$$ is
Q56 Tangents, normals and gradients Find tangent line equation at a given point View
The slope of the tangent to the curve $\left(y - x^5\right)^2 = x\left(1 + x^2\right)^2$ at the point $(1, 3)$ is
Q57 Sign Change & Interval Methods View
The largest value of the nonnegative integer $a$ for which $$\lim_{x \rightarrow 1} \left\{\frac{-ax + \sin(x-1) + a}{x + \sin(x-1) - 1}\right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}$$ is
Q58 Curve Sketching Number of Solutions / Roots via Curve Analysis View
Let $f : [0, 4\pi] \rightarrow [0, \pi]$ be defined by $f(x) = \cos^{-1}(\cos x)$. The number of points $x \in [0, 4\pi]$ satisfying the equation $$f(x) = \frac{10 - x}{10}$$ is
Q59 Areas by integration View
For a point $P$ in the plane, let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x - y = 0$ and $x + y = 0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_1(P) + d_2(P) \leq 4$, is
Q60 Vectors Introduction & 2D Dot Product Computation View
Let $\vec{a}, \vec{b}$, and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p\vec{a} + q\vec{b} + r\vec{c}$, where $p$, $q$ and $r$ are scalars, then the value of $\frac{p^2 + 2q^2 + r^2}{q^2}$ is