## JEE Main 2020 Maths Paper With Solutions Sep 2 Shift 2

**Question 1.**Let f:R -> R be a function which satisfies f(x+y) = f(x)+f(y) for all x, y ∈ R. If f(1) = 2 and g(n) =

a) 9

b) 5

c) 4

d) 20

f(1) = 2

f(x+y) = f(x)+f(y)

x = y = 1

f(2) = 2+2 = 4

x = 2, y = 1

f(3) = 4+2 = 6

g(n) = f(1)+f(2)+………+f(n-1)

= 2+4+6+ ….+ 2(n-1)

= 2Ʃ(n-1)

= 2(n-1)n/2

= n(n-1)

= n^{2}-n

n^{2}-n = 20

n^{2}-n-20 = 0

(n-5)(n+4) = 0

n = 5

**Answer:**(b)

**Question 2.**If the sum of first 11 terms of an A.P, a

_{1},a

_{2},a

_{3}… is 0 (a

_{1}≠ 0) then the sum of the A.P, a

_{1}, a

_{3 }, a

_{5}…a

_{23}is ka

_{1}, where k is equal to

a) -121/10

b) -72/5

c) 72/5

d) 121/10

Given

11a+55d = 0

a+5d = 0

d = -a/5

Now a_{1}+a_{3}+….+ a_{23} = ka_{1}

12a+d (2+4+6+…..+22) = ka

12a+2d×66 = ka

12(a+11d) = ka

12(a+11(-a/5)) = ka

12(1-11/5) = k

k = -72/5

**Answer:** (b)

**Question 3.**Let E

^{C}denote the complement of an event E. Let E

_{1}, E

_{2}and E

_{3}be any pair wise independent events with P(E

_{1})>0 and P(E

_{1}∩ E

_{2}∩ E

_{3}) = 0. Then P(E

_{2}

^{C}∩ E

_{3}

^{C}/E

_{1}) is equal to:

a) P(E

_{3}

^{C})- P(E

_{2}

^{C})

b) P(E

_{3})- P(E

_{2}

^{C})

c) P(E

_{3}

^{C})- P(E

_{2})

d) P(E

_{2}

^{C})+ P(E

_{3})

**Answer:** (c)

**Question 4.**If the equation cos

^{4}θ+sin

^{4}θ+λ = 0 has real solutions for θ, then λ lies in the interval:

a) (-1/2, -1/4]

b) [-1,-1/2]

c) [-3/2, -5/4]

d) (-5/4, -1)

cos^{4}θ+sin^{4}θ+λ = 0

λ = -{1-(1/2)sin^{2}2θ}

2(λ+1) = sin^{2}2θ

0 ≤2(λ+1) ≤1

0 ≤λ+1 ≤1/2

-1≤ λ ≤ -1/2

**Answer:** (b)

**Question 5.**The area (in sq. units) of an equilateral triangle inscribed in the parabola y

^{2}= 8x, with

one of its vertices on the vertex of this parabola, is:

a) 128√3

b) 192√3

c) 64√3

d) 256√3

A: (a cos 30^{0}, a sin 30^{0}) lies on parabola

(a^{2}/4) = 8×a√3/2

a = 16√3

Area of equilateral triangle, Δ = (√3/4)a^{2}

Δ= (√3/4)×16×16×3

Δ= 192√3

**Answer:** (b)

**Question 6.**The imaginary part of (3+2√-54)

^{1/2}-(3-2√-54)

^{1/2}can be:

a) √6

b) -2√6

c) 6

d) -√6

(3+2i√54)^{1/2}-(3-i2√54)^{1/2}

= (9+6i^{2}+2×3i√6)^{1/2}-(9+6i^{2}-2×3i√6)^{1/2}

= ((3+√6i)^{2})^{1/2} -((3-√6i)^{2})^{1/2}

= (3+√6i)( 3-√6i)

= -2√6i

**Answer:** (b)

**Question 7.**A plane passing through the point (3,1,1) contains two lines whose direction ratios are (1, -2, 2) and (2, 3, -1) respectively. If this plane also passes through the point (α,-3,5), then α is equal to:

a) -5

b) 10

c) 5

d) -10

Required plane is

-4(x-3)+5(y-1)+7(z-1) = 0

4x-5y-7z = 0

(α, -3,5) lies on 4x-5y-7z = 0

α = 5

**Answer: **(c)

**Question 8.**Let A = {X= (x,y,z)

^{T}: PX = 0 and x

^{2}+y

^{2}+z

^{2}= 1}, where P =

a) contains more than two elements.

b) is a singleton.

c) contains exactly two elements.

d) is an empty set.

Clearly |P|= 0

PX = 0 has infinite solutions

The line concurrence passes through (0,0,0) which is centre of sphere x^{2}+y^{2}+z^{2} = 1.

Diameter will intersect at two points.

two solutions (exactly) exist.

**Answer: **(c)

**Question 9.**The equation of the normal to the curve y = (1+x)

^{2y}+cos

^{2}(sin

^{-1}x) at x = 0 is:

a) y+4x = 2

b) 2y+x = 4

c) x+4y = 8

d) y = 4x+2

At x = 0, y = 1+cos^{2}(0) = 2

P;(0,2)

y = (1+x)^{2y}+cos^{2}(sin^{-1}x)

y = (1+x)^{2y}+cos(cos^{-1}√(1-x^{2}))^{2}

= (1+x)^{2y}+ (√(1-x^{2}))^{2}

y = (1+x)^{2y}+1-x^{2}

Differentiating w.r.t x

Now y’ = (1+x)^{2y }{(2y/1+x)+ln(1+x)2y’}-2x

y’ at (0,2) = 4-0 = 4

N_{0}: y-2 = (-1/4)(x-0)

N_{0}: 4y-8 = -x

x+4y = 8

**Answer: **(c)

**Question 10.**Consider a region R = {(x,y)∈R

^{2}: x

^{2}≤y ≤ 2x}. If a line y = α divides the area of region R into two equal parts, then which of the following is true.?

a) α

^{3}-6α

^{2}+16 = 0

b) 3α

^{2}-8α

^{3/2}+8 = 0

c) α

^{3}-6α

^{3/2}-16 = 0

d) 3α

^{2}-8α+8 = 0

Area =

= x^{2}-(x^{3}/3)

= 4/3 [Applying limits]

So

8.α^{3/2}-3α^{2} = 8

3α^{2}-8α^{3/2}+8 = 0

**Answer:** (b)

**Question 11.**Let f: (-1, ∞)→R be defined by f(0) = 1 and f(x) = (1/x)log

_{e}(1+x), x ≠ 0. Then the function f:

a) increases in (-1, ∞)

b) decreases in (-1,0) and increases in (0, ∞)

c) increases in (-1,0) and decreases in (0, ∞)

d) decreases in (-1, ∞)

f(x) = (1/x)ln(1+x)

f’(x) = [x/(1+x) – ln(1 + x)]/x^{2} = [x – (1+x)ln (1+x)]/x^{2}(1+x)

f’<0 ∀ x ∈(-1,∞)

**Answer:** (d)

**Question 12.**Which of the following is a tautology?

a) (p→q) ˄( q→p)

b) (~p) ˄(p ˅q)→q

c) (q→p) ˅~(p→q)

d) (~q)˅( p˄q)→q

p | q | ~p | (p ˅q) | ~p ˄(p˅q) | ~p ˄(p˅q)->q |

T | T | F | T | F | T |

T | F | F | T | F | T |

F | T | T | T | T | T |

F | F | T | F | F | T |

**Answer: **(b)

**Question 13.**Let f(x) be a quadratic polynomial such that f(-1)+f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other roots lies in:

a) (0,1)

b) (1,3)

c) (-1,0)

d) (-3,-1)

Let f(x) = a (x-3) (x-α)

f(-1)+f(2) = 0

a[(-1-3) (-1-α)+(2-3)(2-α)] = 0

a[4+4α-2+α]=0

5α+2 = 0

α = -2/5

**Answer:** (c)

**Question 14.**Let S be the sum of the first 9 terms of the series:

{x+ka}+{x^{2}+(k+2)a}+{x^{3}+(k+4)a}+{x^{4}+(k+6)a}+… where a ≠ 0 and a ≠ 1.

If S = (x^{10}-x+45a(x-1))/(x-1), then k is equal to:

a) 3

b) -3

c) 1

d) 5

S = {x+ka} + {x^{2}+(k+2)a} +{x^{3}+(k+4)a} up to 9 terms

S = (x+x^{2}+….+x^{9})+a{k+(k+2)+(k+4)+……up to 9 term)}

S = x(1-x^{9})/(1-x) + a(9k+2×36)

= (x^{10}-x)/(x-1) +9ak+72a

S = (x^{10}-x+45a(x-1))/(x-1)

= (x^{10}-x+(9k+72)a(x-1))/(x-1)

45 = 9k+72

9k = -27

k = -3

**Answer: **(b)

**Question 15.**The set of all possible values of θ in the interval (0, π) for which the points (1, 2) and (sin θ, cos θ ) lie on the same side of the line x+y = 1 is:

a) (0, π/4)

b) (0, π/2)

c) (0, 3π/4)

d) (π/4, 3π/4)

L_{11}×L_{22} >0

sinθ +cosθ > 1

θ ∈(0, π/2)

**Answer: **(b)

**Question 16.**Let n>2 be an integer. Suppose that there are n metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is:

a) 201

b) 199

c) 101

d) 200

Red line = 99 blue line

^{n}C_{2}-n = 99n

n(n-1)/2 = 100n

n-1 = 200

n = 201

**Answer:** (a)

**Question 17.**If a curve y = f(x), passing through the point (1,2) is the solution of the differential equation, 2x

^{2}dy = (2xy+y

^{2})dx, then f(1/2) is equal to:

a) -1/(1+log

_{e}2)

b) 1+log

_{e}2

c) 1/(1+log

_{e}2)

d) 1/(1-log

_{e}2)

2(dy/dx) = 2(y/x)+(y/x)^{2} HDE

Since y = vx

2(v+x(dv/dx)) = 2v+v^{2}

2(dv/v^{2}) = dx/x

-(2/v) = ln x+c

-2x/y = ln x+c

c = -1

c: ln x+2x/y = 1

for f(1/2) ln(1/2) +(2/2y) = 1

y = 1/(1+ln 2)

**Answer** (c)

**Question 18.**For some θ∈(0, π/2) , if the eccentricity of the hyperbola, x

^{2}-y

^{2}sec

^{2}θ = 10 is √5 times the eccentricity of the ellipse, x

^{2}sec

^{2}θ+y

^{2}= 5, then the length of the latus rectum of the ellipse, is:

a) 4√5/3

b) 2√5/3

c) 2√6

d) √30

H: x^{2}-y^{2}sec^{2}θ = 10

E: x^{2}sec^{2}θ+y^{2} = 5

√(1+10cos^{2}θ/10)= √5√(1-5cos^{2}θ/5)

1+cos^{2}θ = 5-5cos^{2}θ

6 cos^{2}θ = 4

cosθ = √(2/3)

LR of ellipse = 2×5×cos^{2}θ/√5

= 2√5×2/3

= 4√5/3

**Answer** (c)

**Question 19.**

a) e

b) e

^{2}

c) 2

d) 1

**Answer:** (b)

**Question 20.**Let a, b, c ∈ R be all non-zero and satisfy a

^{3}+b

^{3}+c

^{3}= 2. If the matrix A =

^{T}A = I, then a value of abc can be:

a) 2/3

b) 3

c) -1/3

d) 1/3

a^{3}+b^{3}+c^{3} = 2

A^{T}A = I

a^{2}+b^{2}+c^{2} = 1

ab+bc+ca = 0

Now (a+b+c)^{2} = Ʃa^{2}+2Ʃab

(Ʃa)^{2} = 1+0 = 1

Ʃa = 1

Now Ʃa^{3}-3abc = (Ʃa)(Ʃa^{2}– Ʃab)

2-3abc = ± 1(1-0)

2-3abc = ± 1

abc = 1/3 or abc = 1

**Answer:** (d)

**Question 21.**Let the position vectors of points ‘A’ and ‘B’ be

Now (14λ+6)/(λ+1) -3(6λ^{2})/(λ+1)^{2 }= 6

(14λ+6) (λ+1)—18λ^{2} = 6(λ+1)^{2}

-4λ^{2} + 20λ + 6 = 6λ^{2} + 12λ + 6

10λ^{2}-8λ = 0

λ(10λ-8) = 0

Since λ >0

λ = 0.8

**Answer:** (0.8)

**Question 22.**Let [t] denote the greatest integer less than or equal to t. Then the value of

**Solution:**

To find

Put 3x = t

=

=

**Answer: **(1)

**Question 23.**If y =

y =

where tanθ = 4/3

cos^{-1}[cos(kx+a)] = kx+a as kx+a∈[0, π]

y = cos^{-1}(cos (x+θ)+2 cos^{-1}(2x+θ))….+6cos^{-1}(cos (6x+θ))

y = x+θ+2(2x+θ)+…6(6x+θ)

dy/dx = 1×1+2×2+3×3+….+6×6

dy/dx at x = 0

= 1×1+2×2+3×3+….+6×6

= Ʃ6^{2}

= 6×7×13/6

= 91

**Answer:** (91)

**Question 24.**If the variance of the terms in an increasing A.P., b

_{1}, b

_{2}, b

_{3}, …., b

_{11}is 90, then the common difference of this A.P. is

Var(x) = (Ʃbi^{2}/11)- (Ʃbi/11)^{2}

90 = [(11a^{2}+385d^{2}+110ad)/11]-[(11a+55d)/11]^{2}

90 = a^{2}+35d^{2}+10ad- (a+5d)^{2}

90 = a^{2}+35d^{2}+10ad- a^{2}-25d^{2}-10ad

90 = 10d^{2}

9 = d^{2}

d = 3

**Answer:** (3)

**Question 25.**For a positive integer n, (1+1/x)

^{n}is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2:5:12, then n is equal to

Let 3 consecutive coefficients are ^{n}C_{r-1} :^{ n}C_{r} : ^{n}C_{r+1}::2:5:12

^{n}C_{r-1}/^{ n}C_{r} = 2/5

and ^{n}C_{r}/^{ n}C_{r+1} = 5/12

r/(n-r+1) = 2/5

and (r+1)/(n-r) = 5/12

7r = 2n+2 and 17r = 5n-12

(2n+2)/7 = (5n-12)/17

34n+34 = 35n-84

n = 118

**Answer **(118)

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