Download JEE Advanced 2014 Mathematics Question Paper - 1

Please wait! Loading...

 Download as PDF

2013:Paper-1

2015:Paper-1

All the Papers

Marking Scheme

• For each question in SECTION 1, you will be awarded 3 marks if you darken all the bubbles(s) corresponding to the correct answer(s) and zero mark if no bubbles are darkened. No negative marks will be awarded for incorrect answers in this section.
• For each question in Section 2, you will be awarded 3 marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. No negative marks will be awarded for incorrect answer in this section.

SECTION - 1 (One or More than One Options Correct Type)

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.

1. Let $𝑀$ and $𝑁$ be two 3 × 3 matrices such that $𝑀N$= $𝑁M$. Further, if $𝑀N \neq 𝑁^2$ and $𝑀^2 = 𝑁^4$, then
   1. determinant of ($𝑀^2 + 𝑀𝑁^2$) is 0
   2. there is a 3 × 3 non-zero matrix $𝑈$ such that $(𝑀^2 + 𝑀𝑁^2)𝑈$ is the zero matrix
   3. determinant of $(𝑀^2 + 𝑀𝑁^2)$ ≥ 1
   4. for a 3 × 3 matrix $𝑈$, if $(𝑀^2 + 𝑀𝑁^2)𝑈$ equals the zero matrix then $𝑈$ is the zero matrix
2. For every pair of continuous functions $𝑓$, $𝑔$:[0, 1] → $ℝ$ such that max {$𝑓(𝑥)$: $𝑥$ ∈ [0,1]} = max {$𝑔(𝑥)$: $𝑥$ ∈ [0,1]}, the correct statement(s) is(are) :
   1. $(𝑓(𝑐))^2$ + $3𝑓(𝑐)$ = $(𝑔(𝑐))^ 2$ + $3𝑔(𝑐)$ for some $𝑐$ ∈ [0, 1]
   2. $(𝑓(𝑐))^2$ + $𝑓(𝑐)$ = $(𝑔(𝑐))^ 2$ + $3𝑔(𝑐)$ for some $𝑐$ ∈ [0, 1]
   3. $(𝑓(𝑐))^2$ + $3𝑓(𝑐)$ = $(𝑔(𝑐))^ 2$ + $𝑔(𝑐)$ for some $𝑐$ ∈ [0, 1]
   4. $(𝑓(𝑐))^ 2$ =$ (𝑔(𝑐))^ 2$ for some $𝑐$ ∈ [0, 1]
3. Let $𝑓:$ (0, ∞) → $ℝ$ be given by 𝑓(𝑥) = $\int \limits_{\frac{1}{x}}^xe^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t}$. Then
   1. $𝑓(𝑥)$ is monotonically increasing on [1, ∞)
   2. $𝑓(𝑥)$ is monotonically decreasing on (0, 1)
   3. $𝑓(𝑥)$ + $𝑓(\frac{1}{x})$ = 0, for all $𝑥$ ∈ (0, ∞)
   4. $𝑓(2^𝑥 )$ is an odd function of $𝑥$ on $ℝ$
4. Let $𝑎$ ∈ $ℝ$ and let $𝑓$: $ℝ$ → $ℝ$ be given by
   $f(x)$=$ x^5$-$5x$+$a$. Then
   1. $𝑓(𝑥)$ has three real roots if $𝑎$ > 4
   2. $𝑓(𝑥)$ has only one real root if $𝑎$ > 4
   3. $𝑓(𝑥)$ has three real roots if $𝑎$ < −4
   4. $𝑓(𝑥)$ has three real roots if −4 < $𝑎$ < 4
5. Let $𝑓:[𝑎, 𝑏]$ → [1, ∞) be a continuous function and let $𝑔: ℝ → ℝ$ be defined as
   $g(x)$=$\left\{\begin{array}{lr}0 & \text { if } x < a \\ \int \limits_a^x f(t) d t & \text { if } a \leq x \leq b \\ \int \limits_a^b f(t) d t & \text { if } x > b\end{array}\right.$ Then
   1. $𝑔(𝑥)$ is continuous but not differentiable at $𝑎$
   2. $𝑔(𝑥)$ is differentiable on $ℝ$
   3. $𝑔(𝑥)$ is continuous but not differentiable at $𝑏$
   4. $𝑔(𝑥)$ is continuous and differentiable at either $𝑎$ or $𝑏$ but not both
6. Let $𝑓$: $(− \frac{\pi}{2} , \frac{\pi}{2})$ → $ℝ$ be given by
   $f(x)$=$(log (secx+tanx))^3$. Then
   1. $𝑓(𝑥)$ is an odd function
   2. $𝑓(𝑥)$ is a one-one function
   3. $𝑓(𝑥)$ is an onto function
   4. $𝑓(𝑥)$ is an even function
7. From a point $𝑃(𝜆, 𝜆, 𝜆)$, perpendiculars $𝑃Q$ and $𝑃R$ are drawn respectively on the lines $𝑦$ = $𝑥$, $𝑧$ = 1 and $𝑦$ = $−𝑥$, $𝑧$ = −1. If $𝑃$ is such that $∠𝑄PR$ is a right angle, then the possible value(s) of $𝜆$ is(are)
   1. $\sqrt{2}$
   2. 1
   3. -1
   4. $-\sqrt{2}$
8. 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 $\vec{y}$ × $\vec{z}$ and $\vec{b}$ is a nonzero vector perpendicular to $\vec{y}$ and $\vec{z}$ × $\vec{x}$, then
   1. $\vec{b}$ = ($\vec{b}$ ∙ $\vec{z}$)($\vec{z}$ − $\vec{x}$)
   2. $\vec{a}$ = ($\vec{a}$ ∙ $\vec{y}$)($\vec{y}$ − $\vec{z}$)
   3. $\vec{a}$ ∙ $\vec{b}$=-($\vec{a}$ ∙ $\vec{y}$)($\vec{b}$ ∙ $\vec{z}$)
   4. $\vec{a}$ = ($\vec{a}$ ∙ $\vec{y}$)($\vec{z}$ − $\vec{y}$)
9. 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
   1. radius of $𝑆$ is 8
   2. radius of $𝑆$ is 7
   3. centre of $𝑆$ is (−7, 1)
   4. centre of $𝑆$ is (−8, 1)
10. Let $𝑀$ be a 2 × 2 symmetric matrix with integer entries. Then $𝑀$ is invertible if
    1. the first column of $𝑀$ is the transpose of the second row of $𝑀$
    2. the second row of $𝑀$ is the transpose of the first column of $𝑀$
    3. $𝑀$ is a diagonal matrix with nonzero entries in the main diagonal
    4. the product of entries in the main diagonal of $𝑀$ is not the square of an integer

SECTION - 2 (One Integer Value Correct Type)

This section contains 10 questions. Each question, when worked out will result in one integer from 0 to 9 (both inclusive).

11. 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
12. Let 𝑛 ≥ 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
13. 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
14. Let $𝑓: ℝ → ℝ$ and $𝑔: ℝ → ℝ$ be respectively given by $𝑓(𝑥)$ = $|𝑥|$ + 1 and $𝑔(𝑥)$ = $𝑥^2 $+ 1. Define $ℎ: ℝ → ℝ$ by
    $h(x)$=$\left\{\begin{array}{ll}\max \{f(x), g(x)\} & \text { if } x \leq 0, \\ \min \{f(x), g(x)\} & \text { if } x>0 .\end{array}\right.$ The number of points at which $h(x)$ is not differentiable is
15. The value of
    $\int \limits_0^1 4x^3 \left \{ \frac{d^2}{dx^2}(1-x^2)^5 \right\}dx$ is
16. The slope of the tangent to the curve $(𝑦 − 𝑥^5 )^2$ = $𝑥(1 + 𝑥^2)^2$ at the point (1, 3) is
17. The largest value of the nonnegative integer $𝑎$ for which
    $\lim \limits_{x \to 1}\left \{ \frac{-ax+sin(x-1)+a}{x + sin(x-1)-1}\right \}^{\frac{1-x}{1-\sqrt{x}}}$=$\frac{1}{4}$
    is
18. Let $𝑓:[0, 4𝜋]$ → $[0, 𝜋]$ be defined by $𝑓(𝑥)$ = $cos^{−1}(cos 𝑥)$. The number of points $𝑥$ ∈ $[0, 4𝜋]$ satisfying the equation
    $f(x)$=$\frac{10-x}{10}$
    is
19. 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 ≤ $𝑑_1(𝑃)$ + $𝑑_2 (𝑃)$ ≤ 4 , is
20. Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{𝜋}{3}$. If $\vec{a}$ × $\vec{b}$+ $\vec{b}$× $\vec{c}$ = $𝑝\vec{a}$+ $𝑞\vec{b}$+ $𝑟\vec{c}$, where $𝑝$, $𝑞$ and $𝑟$ are scalars, then the value of$\frac{𝑝^2+ 2𝑞^2+ 𝑟^2}{ 𝑞^2}$ is

2014:Paper-2

2014:Ans-Keys

All the Papers Ans-Keys

Please wait! Loading...

 Download as PDF