Download JEE Advanced 2020 Mathematics Question Paper - 1

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SECTION 1 (Maximum Marks:18)

• This section contains SIX (06) questions.
• Each question has FOUR options. ONLY ONE of these four options is the correct answer.
• For each question, choose the option corresponding to the correct answer.
• Answer to each question will be evaluated according to the following marking scheme:
  
  
  • Full Marks: +3 ONLY if the correct option is chosen;
  • Zero Marks: 0 If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -1 In all other cases.

1. Suppose $𝑎$, $𝑏$ denote the distinct real roots of the quadratic polynomial $ 𝑥^2 + 20𝑥 − 2020$ and suppose $𝑐$, $𝑑$ denote the distinct complex roots of the quadratic polynomial $𝑥^2 − 20𝑥 + 2020$. Then the value of $𝑎𝑐(𝑎 − 𝑐)$ + $𝑎𝑑(𝑎 − 𝑑)$ + $𝑏𝑐(𝑏 − 𝑐)$ + $𝑏𝑑(𝑏 − 𝑑)$
   is
   1. 0
   2. 8000
   3. 8080
   4. 16000
2. If the function $𝑓: R $⟶ $R$ is defined by $𝑓(𝑥)$ = $|𝑥|$$(𝑥 − sin 𝑥)$, then which of the following statement is TRUE
   1. $f$ is one - one but NOT onto.
   2. $f$ is onto but NOT one - one.
   3. $f$ is BOTH one - one and onto.
   4. $f$ is NEITHER one - one NOR onto.
3. Let the functions $𝑓: R$⟶ $R$ and $𝑔: R$⟶ $R$ be defined by $𝑓(𝑥)$ =$ 𝑒^{𝑥−1} − 𝑒^{−|𝑥−1|} $ and $𝑔(𝑥)$ =$\frac{1} {2} (𝑒^{𝑥−1} + 𝑒^{1−𝑥} )$.
   Then the area of the region in the first quadrant bounded by the curves $𝑦$ = $𝑓(𝑥)$, $𝑦$ = $𝑔(𝑥)$ and $𝑥$ = 0 is
   1. $(2 - \sqrt{3})+ \frac{1}{2}(e - e^{-1})$
   2. $(2 + \sqrt{3})+ \frac{1}{2}(e - e^{-1})$
   3. $(2 - \sqrt{3})+ \frac{1}{2}(e + e^{-1})$
   4. $(2 + \sqrt{3})+ \frac{1}{2}$$(e + e^{-1})$
4. Let $𝑎$, $𝑏$ and $\lambda$ be positive real numbers. Suppose $𝑃$ is an end point of the latus rectum of the parabola $𝑦^2 = 4 \lambda 𝑥$, and suppose the ellipse $ \frac{𝑥^2} {𝑎^2} + \frac{𝑦^2} {𝑏^2} = 1$ passes through the point $𝑃$. If the tangents to the parabola and the ellipse at the point $𝑃$ are perpendicular to each other, then the eccentricity of the ellipse is
   1. $ \frac{1}{\sqrt{2}}$
   2. $ \frac{1}{2}$
   3. $ \frac{1}{3}$
   4. $ \frac{2}{5}$
5. Let $𝐶_1$ and $𝐶_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2} {3}$ and $\frac{1} {3}$ , respectively. Suppose $\alpha$ is the number of heads that appear when $𝐶_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $𝐶_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $ 𝑥^2 − \alpha 𝑥 + \beta$ are real and equal, is
   1. $ \frac{40}{81}$
   2. $ \frac{20}{81}$
   3. $ \frac{1}{2}$
   4. $ \frac{1}{4}$
6. Consider all rectangles lying in the region {$(𝑥, 𝑦) \in R × R ∶ 0 \leq 𝑥 \leq\frac{\pi} {2}$ and $ 0 \leq 𝑦 \leq 2 sin(2𝑥)$} and having one side on the $𝑥-$axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
   1. $ \frac{3 \pi}{2}$
   2. $ \pi $
   3. $ \frac{\pi}{2 \sqrt{3}}$
   4. $ \frac{\pi \sqrt{3}}{2}$

SECTION 2 (Maximum Marks:24)

• This section contains FOUR (06) questions.
• Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).
• For each question, choose the option(s) corresponding to (all) the correct answer(s).
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If only (all) the correct option(s) is(are) chosen;
  • Partial Marks: +3 If all the four options are correct but ONLY three options are chosen;
  • Partial Marks: +2 If three or more options are correct but ONLY two options are chosen and both of which are correct;
  • Partial Marks: +1 If two or more options are correct but ONLY one option is chosen and it is a correct option;
  • Zero Marks: 0 If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -2 In all other cases.

7. Let the function $𝑓: R$ → $R$ be defined by $ 𝑓(𝑥) $=$ 𝑥^3 − 𝑥^2 $+$ (𝑥 − 1) sin 𝑥 $ and let $𝑔: R$ → $R$ be an arbitrary function. Let $𝑓𝑔: R $→ $R$ be the product function defined by $(𝑓𝑔)(𝑥)$ = $𝑓(𝑥)𝑔(𝑥)$. Then which of the following statements is/are TRUE?
   1. If $𝑔$ is continuous at $𝑥$ = 1, then $𝑓𝑔$ is differentiable at $𝑥$ = 1
   2. If $𝑓𝑔$ is differentiable at $𝑥$ = 1, then $𝑔$ is continuous at $𝑥$ = 1
   3. If $𝑔$ is differentiable at $𝑥$ = 1, then $𝑓𝑔$ is differentiable at $𝑥$ = 1
   4. If $𝑓𝑔$ is differentiable at $𝑥$ = 1, then $𝑔$ is differentiable at $𝑥$ = 1
8. Let $𝑀$ be a 3 × 3 invertible matrix with real entries and let $𝐼$ denote the 3 × 3 identity matrix. If$𝑀^{−1} $= adj (adj $𝑀$), then which of the following statements is/are ALWAYS TRUE?
   1. $𝑀$ = $𝐼$
   2. det $𝑀$ = 1
   3. $𝑀^2 = 𝐼$
   4. (adj $𝑀$)$^ 2 = 𝐼$
9. Let $𝑆$ be the set of all complex numbers $𝑧$ satisfying $|𝑧^2 + 𝑧 + 1|$ = 1. Then which of the following statements is/are TRUE?
   1. $ |z + \frac{1}{2}| \leq \frac{1}{2} $ for all $z \in S$
   2. $ |z| \leq 2 $ for all $z \in S$
      
      $\hspace{0.5cm}$
   3. $ |z + \frac{1}{2}| \geq \frac{1}{2} $ for all $z \in S$
   4. The set has exactly four elements
10. Let $𝑥$, $𝑦$ and $𝑧$ be positive real numbers. Suppose $𝑥$, $𝑦$ and $𝑧$ are the lengths of the sides of a triangle opposite to its angles $𝑋$, $𝑌$ and $𝑍$, respectively. If $tan\frac{𝑋} {2}$ + $tan \frac{𝑍} {2} $=$\frac{2𝑦} {𝑥 + 𝑦 + 𝑧} $,then which of the following statements is/are TRUE?
    1. $2𝑌$ = $𝑋$ + $𝑍$
    2. $𝑌$ = $𝑋$ + $𝑍$
    3. $tan \frac{X} {2}$ = $\frac{𝑥} {𝑦+𝑧} $
    4. $𝑥^2$ + $𝑧^2 − 𝑦^2$ = $𝑥𝑧$
11. Let $𝐿_1$ and $𝐿_2$ be the following straight lines.$𝐿_1 :\frac{𝑥 − 1} {1} = \frac{𝑦} {−1} = \frac{𝑧 − 1} {3} $and $𝐿_2 : \frac{𝑥 − 1} {−3} = \frac{𝑦} {−1} = \frac{𝑧 − 1} {1} $.Suppose the straight line $𝐿:\frac{𝑥 − \alpha} {𝑙} = \frac{𝑦 − 1} {𝑚} = \frac{𝑧 − \gamma} {−2} $lies in the plane containing $ 𝐿_1$ and $ 𝐿_2$, and passes through the point of intersection of $𝐿_1$ and $𝐿_2$. Ifthe line 𝐿 bisects the acute angle between the lines $𝐿_1$ and $𝐿_2$, then which of the followingstatements is/are TRUE?
    1. $\alpha - \gamma = 3$
    2. $l$ + $m$ = 2
    3. $tan \alpha - \gamma = 1$
    4. $l$ + $m$ = 0
12. Which of the following inequalities is/are TRUE?
    1. $\int \limits_0^1 x cos x dx \geq \frac{3}{8}$
    2. $\int \limits_0^1 x sin x dx \geq \frac{3}{10}$
    3. $\int \limits_0^1 x^2 cos x dx \geq \frac{1}{2}$
    4. $\int \limits_0^1 x^2 sin x dx \geq \frac{2}{9}$

SECTION 3 (Maximum Marks:24)

• This section contains SIX (06) questions.
• The answer to each question is a NUMERICAL VALUE.
• For each question, enter the correct numerical value of the answer using the mouse and the on screenvirtual numeric keypad in the place designated to enter the answer. If the numerical value has morethan two decimal places, truncate/round off the value to TWO decimal places.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 ONLY if the correct numerical value is entered;
  • Zero Marks: 0 In all other cases.

13. Let $𝑚$ be the minimum possible value of $log_3(3^{𝑦_1}$ + $3^{𝑦_2}$ + $3^{𝑦_3})$, where $𝑦_1$, $𝑦_2$, $𝑦_3$ are real numbers for which $𝑦_1$ + $𝑦_2$ + $𝑦_3 = 9$. Let $𝑀$ be the maximum possible value of ($log_3 𝑥_1$ + $log_3 𝑥_2$ + $log_3 𝑥_3$),where $𝑥_1$, $𝑥_2$, $𝑥_3$ are positive real numbers for which $𝑥_1$ +$ 𝑥_2 $+ $𝑥_3 = 9$. Then the value of $log_2(𝑚^3 )$ + $log_3(𝑀^2) $ is _____
14. Let $𝑎_1$, $𝑎_2$, $𝑎_3$, … be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $𝑏_1$, $𝑏_2$, $𝑏_3$, … be a sequence of positive integers in geometric progression with common ratio 2. If $𝑎_1$ = $𝑏_1$ = $𝑐$, then the number of all possible values of $𝑐$, for which the equality $2(𝑎_1 + 𝑎_2 + ⋯ + 𝑎_𝑛)$ = $𝑏_1 + 𝑏_2 + ⋯ + 𝑏_𝑛$ holds for some positive integer $𝑛$, is _____
15. Let $𝑓:[0, 2]$ ⟶ $R$ be the function defined by$𝑓(𝑥)$ = $( 3 − sin(2 \pi 𝑥))$ $ sin ( \pi 𝑥 −\frac{\pi} {4})$ − $sin (3 \pi 𝑥 +\frac{ \pi} {4} ) $.If $\alpha, \beta \in [0, 2]$ are such that $ {𝑥 \in [0, 2] ∶ 𝑓(𝑥) ≥ 0} = [\alpha, \beta]$, then the value of $ \beta − \alpha $ is ____
16. In a triangle $𝑃𝑄𝑅$, let $\vec{ 𝑎} = \vec{𝑄𝑅} , \vec{𝑏}$ = $\vec{𝑅P}$ and $\vec{𝑐} = \vec{𝑃𝑄} $. If$| \vec{𝑎} | = 3$, $| \vec{𝑏} | = 4$ and $\frac{\vec{𝑎} ⋅ ( \vec{c} − \vec{ 𝑏} )}{\vec{𝑐} ⋅ ( \vec{𝑎} − \vec{𝑏})}$=$\frac{| \vec{𝑎} |}{| \vec{𝑎} | + | \vec{𝑏} |}$,then the value of $| \vec{𝑎} × \vec{ 𝑏} |^2$ is _____
17. For a polynomial $𝑔(𝑥)$ with real coefficients, let $𝑚_𝑔$ denote the number of distinct real roots of $𝑔(𝑥)$. Suppose $𝑆$ is the set of polynomials with real coefficients defined by $𝑆$ = {$(𝑥^2 − 1)^2$ ($𝑎_0 + 𝑎_1 𝑥 + 𝑎_2 𝑥^2$ + $𝑎_3 𝑥^3)$ ∶ $𝑎_0, 𝑎_1, 𝑎_2, 𝑎_3$ $ \in R$}. For a polynomial $𝑓$, let $𝑓′$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $(𝑚_{𝑓′} + 𝑚_{𝑓′′} )$, where $𝑓 \in 𝑆$, is _____
18. Let $𝑒$ denote the base of the natural logarithm. The value of the real number $𝑎$ for which the right hand limit $\lim \limits_{x \to 0^{+}} \frac{(1 − 𝑥)^{\frac{1} {𝑥}} − 𝑒^{−1}} {𝑥^𝑎}$ is equal to a nonzero real number, is _____

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