**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

*x*^{2 }− 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)

= a^{2}c – ac^{2} + a^{2}d – ad^{2} +b^{2}c – bc^{2}+ b^{2}d – bd^{2}

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

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

= {(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)

b)

c)

d)

Answer: 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. }m_{2 }= -1 ⇒ b^{2 }= 2a^{2}

for eccentricity of ellipse,

**Question 5.** Let C_{1} and C_{2} be two biased coins such that the probabilities of getting head in a single toss are

_{1}is tossed twice, independently, and suppose α is the number of heads that appear when C

_{2}is tossed twice, independently. Then the probability that the roots of the quadratic polynomial ??

^{2 }− ???? + ?? are real and equal, is

a) 40/81

b) 20/81

c) 1/2

d) 1/4

Answer: b

Now roots of equation x^{2}– αx + β = 0 are real and equal

∴ D= 0

α^{2} – 4β = 0

α^{2} = 4β

⇒ (α = 0, β = 0) or (α = 2, β = 1)

⇒

**Question 6.** Consider all rectangles lying in the region

a)

b) ??

c)

d)

Answer: c

Let sides of rectangle are a & b

Then, perimeter = 2a + 2b

p = 2(a + b)

Now, b = 2sin2x and b = 2sin(2x + 2a) ⇒ 2x + 2x + 2a = ??

For perimeter maximum

P = 2a + 2b

P = ?? – 4x + 4sin2x

**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) = x^{3}– x^{2}+ (x – 1) sinx ; g : R ? R

h(x) = f(x). g(x) = {x^{3} – x^{2}+ (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)|I_{3}

(adjM) = |M|^{2} .M …………(1)

|adj M| = ||M|^{2}.M|

|M|^{2} = |M^{6}|.|M|

|M|=1, |M| ≠ 0

From equation (1)

adj.M =M …………(2)

Multiply by matrix M

M.adj M = M^{2}

|M|I_{3} = M^{2}

M^{2} = I

From (2) adj M =M

(adj M)^{2} = M^{2} = 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)

d) ??

^{2 }+ ??

^{2 }− ??

^{2 }= ????

Answer: b, c

?^{2} = (S – x)^{2}(S – z)^{2}

S(S–y) = (S–x) (S–z)

(x + y + z) (x + z – y) = (y + z – x) (x + y –z)

(x + z)^{2} – y^{2} = y^{2} – (z – x)^{2}

(x + z)^{2} + (x – z)^{2 }= 2y^{2}

x ^{2} + z^{2 }= y^{2}

**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)

b)

c)

d)

Answer: a, b, d

**Question 13.** Let m be the minimum possible value of

_{1}, ??

_{2}, ??

_{3}are real numbers for which ??

_{1 }+ ??

_{2 }+ ??

_{3 }= 9. Let M be the maximum possible value of (log

_{3}??

_{1 }+ log

_{3}??

_{2 }+ log

_{3}??

_{3}), where x

_{1}, ??

_{2}, ??

_{3}are positive real numbers for which ??

_{1 }+ ??

_{2 }+ ??

_{3 }= 9. Then the value of log

_{2}(??

^{3}) + log

_{3}(??

^{2}) is _____

Answer: 8.00

Using AM = GM

_{1}+ y

_{2}+ y

_{3}=9}

m = log_{3}81 = log_{3}3^{4} = 4log_{3}3 = 4

Again, using A.M ≥ G.M

=

_{1}+x

_{2}+x

_{3}= 9}

⇒27 ≥ x_{1}x_{2}x_{3 }

M = log_{3}??_{1}+ log_{3}??_{2}+ log_{3}??_{3}

M = log_{3}(x_{1}x_{2}x_{3}) = log_{3}(27) =3

∴ log_{2}(m)^{3} + log_{3}(M)^{2} ⇒ log_{2}(2^{6}) + log_{3}(3^{2}) = 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}, b_{2}, ??_{3},… be a sequence of positive integers in geometric progression with common ratio 2. If a_{1 }= ??_{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(a_{1}+a_{2} +……+ a_{n}) = b_{1} + b_{2} + … + b_{n}

⇒ 2n[a_{1} + (n – 1)] = b_{1}(2^{n} – 1)

⇒ 2na_{1}+ 2n^{2} – 2n = a_{1}(2^{n} – 1) {? a_{1} = b_{1}}

⇒ 2n^{2} – 2n = a_{1}(2 – 1=2n)

_{1}= c}

? c ≥ 1

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

= 2n^{2}+ 1 ≥ 2^{n}

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 = {(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}∈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) = (x^{2} – 1)^{2}h(x); h(x) = a_{0} +a_{1}x +a_{2}x^{2}+ a_{3}x^{3}

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) =(x^{2} – 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

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