JEE Advanced 2020 Maths Paper 1

Question 1. Suppose ??, ?? denote the distinct real roots of the quadratic polynomial ??² + 20?? − 2020 and suppose ??, ?? denote the distinct complex roots of the quadratic polynomial

x² − 20?? + 2020. Then the value of ???? (?? − ??) + ???? (?? − ??) + ???? (?? − ??) + ???? (?? − ??) is

a) 0
b) 8000
c) 8080
d) 16000

Answer: d

Now

= ac (a – c) + ad (a – d) +bc (b – c) + bd (b – d)

= a²c – ac² + a²d – ad² +b²c – bc²+ b²d – bd² 

= a² (c +d) + b² (c + d) – c² (a + b) – d² (a + b)

= (a² + b²) (c + d) – (a + b) (c² + d²)

= {(a+b)² – 2ab} (c + d) – {(c + d)² –2cd} (a + b)

Put value a, b, c and d then,

= {(-20)² – 2(-2020)} (20) – {(20)² – 2(2020)} (-20)}

= ((400 + 4040) (20) – (–20) ((20)²– 4040)

= 20[4440 – 3640]

20[800] = 16000

Question 2. If the function ??: R ? R is defined by (??) = |??| (??– sin??), then which of the following statements is TRUE?

a) ?? is one-one, but NOT onto
b) ?? is onto, but NOT one-one
c) ?? is BOTH one-one and onto
d) ?? is NEITHER one-one NOR onto

Answer: c

Given, f(x) = |x|(x – sinx)

f(–x) = –(|x|(x – sinx))

f(–x) = –f(x) ⇒ f(x) is odd, non-periodic and continuous function.

Now,

Question 3. Let the functions: R ? R and g : R ? R be defined by

a)

Question 4. Let ??, ?? and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola ??² = 4λ??, and suppose the ellipse

a) 1/√2
b) 1/2
c) 1/3
d) 2/5

Answer: a

P(λ, 2λ)

Now E:

From eq. (1) and (2)

m_1. m₂ = -1 ⇒ b² = 2a²

for eccentricity of ellipse,

Question 5. Let C₁ and C₂ be two biased coins such that the probabilities of getting head in a single toss are

a) 40/81
b) 20/81
c) 1/2
d) 1/4

Answer: b

Now roots of equation x²– αx + β = 0 are real and equal

∴ D= 0

α² – 4β = 0

α² = 4β

⇒ (α = 0, β = 0) or (α = 2, β = 1)

⇒

Question 6. Consider all rectangles lying in the region

a)

Question 7. Let the function ??: R → R be defined by (??) = ??³ − ??² + (?? − 1) sin ?? and let ??: R → R be an arbitrary function. Let ??g : R → R be the product function defined by (????)(??) = ??(??)??(??). Then which of the following statements is/are TRUE?

a) If g is continuous at ?? = 1, then ???? is differentiable at ?? = 1
b) If ???? is differentiable at ?? = 1, then g is continuous at ?? = 1
c) If g is differentiable at ?? = 1, then ???? is differentiable at ?? = 1
d) If ???? is differentiable at ?? = 1, then g is differentiable at ?? = 1

Answer: a, c

f : R → R

a) f(x) = x³– x²+ (x – 1) sinx ; g : R ? R

h(x) = f(x). g(x) = {x³ – x²+ (x – 1)sinx}. g(x)

as g(x) is constant at x =1

∴ g(1+h) = g(1 – h) = g(1)

h'(1⁺) = h'(1^–) = (1 + sin1) g(1)

‘a’ is correct.

Question 8. Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If ??⁻¹ = adj (adj ??), then which of the following statements is/are ALWAYS TRUE?

a) ?? = ??
b) det ?? = 1
c) ??² = ??
d) (adj ??)² = ??

Answer: b, c, d

M^–1 = adj(adj(M))

(adj M)M^–1 = (adjM)(adj(adj(M)))

(adj M)M^–1 = N. adj(N) { Let adj(M) = N }

(adj M)M^–1 = |N|I

(adjM)M^–1 = |adj(M)|I₃

(adjM) = |M|² .M …………(1)

|adj M| = ||M|².M|

|M|² = |M⁶|.|M|

|M|=1, |M| ≠ 0

From equation (1)

adj.M =M …………(2)

Multiply by matrix M

M.adj M = M²

|M|I₃ = M²

M² = I

From (2) adj M =M

(adj M)² = M² = I

Question 9. Let S be the set of all complex numbers z satisfying |??² + ?? + 1| = 1. Then which of the following statements is/are TRUE?

Answer: b, c

Question 10. Let x, ?? and z be positive real numbers. Suppose ??, ?? and z are the lengths of the sides of a triangle opposite to its angles ??, ?? and Z, respectively. If

a) 2?? = ?? + ??
b) Y = ?? + ??
c)

Question 11. Let ??₁ and ??₂ be the following straight lines.

Suppose the straight line is

Lies in the plane containing ??₁ and ??₂, and passes through the point of intersection of ??₁ and

??₂. If the line L bisects the acute angle between the lines ??₁ and ??₂, then which of the following statements is/are TRUE?

a) ?? − ?? = 3
b) l + ?? = 2
c) ?? − ?? = 1
d) ?? + ?? = 0

Answer: a, b

Solution:

Question 12. Which of the following inequalities is/are TRUE?

a)

Question 13. Let m be the minimum possible value of

Answer: 8.00

Using AM = GM

m = log₃81 = log₃3⁴ = 4log₃3 = 4

Again, using A.M ≥ G.M

=

⇒27 ≥ x₁x₂x₃

M = log₃??₁+ log₃??₂+ log₃??₃

M = log₃(x₁x₂x₃) = log₃(27) =3

∴ log₂(m)³ + log₃(M)² ⇒ log₂(2⁶) + log₃(3²) = 6 + 2 = 8

Question 14. Let ??₁, ??₂, ??₃,… be a sequence of positive integers in arithmetic progression with common difference 2. Also, let ??₁, b₂, ??₃,… be a sequence of positive integers in geometric progression with common ratio 2. If a₁ = ??₁ = ??, then the number of all possible values of c, for which the equality 2(??₁ + ??₂ + ? + ??_??) = ??₁ + ??₂ + ? + ??_?? holds for some positive integer n, is _____

Answer: 1.00

2(a₁+a₂ +……+ a_n) = b₁ + b₂ + … + b_n

⇒ 2n[a₁ + (n – 1)] = b₁(2ⁿ – 1)

⇒ 2na₁+ 2n² – 2n = a₁(2ⁿ – 1) {? a₁ = b₁}

⇒ 2n² – 2n = a₁(2 – 1=2n)

? c ≥ 1

2(n²– n) ≥ 2ⁿ– 1 – 2n {? n²– n ≥ 0 for n ≥ 1}

= 2n²+ 1 ≥ 2ⁿ

Therefore, n =1, 2,3,4,5,6

n = 1 ⇒ c= 0 (rejected)

n = 2 ⇒ c < 0 (rejected)

n = 3 ⇒ c = 12 (correct)

n = 4 ⇒ c = not Integer

n = 5 ⇒ c = not Integer

n = 6 ⇒ c = not Integer

? c = 12 for n = 3

Hence, no. of such c = 1

Question 15. Let f: [0, 2] ? R be the function defined by

If ??, ?? ∈ [0,2] are such that {?? ∈ [0, 2] : f(??) ≥ 0} = [??, ??], then the value of ?? − ?? is _____

Answer: 1.00

Question 16. In a triangle PQR, let 

Then the value of is _____

Answer: 108.00

Question 17. For a polynomial g(??) with real coefficients, let ??_?? denote the number of distinct real roots of g(??). Suppose S is the set of polynomials with real coefficients defined by S = {(x² – 1)²(a₀ + a₁x + a₂x² + a₃x³) : a₀, a₁, a₂, a₃∈R}

For a polynomial ??, let ??’and ??” denote its first and second order derivatives, respectively.

Then the minimum possible value of (??_??’ + ??_??”), where ?? ∈ ??, is _____

Answer: 5.00

f(x) = (x² – 1)²h(x); h(x) = a₀ +a₁x +a₂x²+ a₃x³

Now, f(1) = f(–1) = 0

⇒ f'(α) = 0, α∈ (–1,1) [Rolle’s Theorem]

Also, f'(1) = f'(–1) = 0 ⇒ f'(x) = 0 has at least 3 root –1, α,1 with –1 <α< 1

⇒ f”(x) = 0 will have at least 2 roots, say β, γ such that

–1 < β < α < γ < 1 [Rolle’s Theorem]

So, min= 2

and we find (m_f’ + m_f”) = 5 for f(x) =(x² – 1)²

Thus, Ans = 5

Question 18. Let e denote the base of the natural logarithm. The value of the real number a for which the right-hand limit is equal to a nonzero real number, is _____

Answer: 1.00