**Question 1:**Let M =

a) -37/16

b) -29/16

c) -31/16

d) -17/16

Let M =

M^{2} = αM + βI

a_{11} = sin θ – 1 – sin^{2} θ – cos^{2} θ – cos^{2} θ sin^{2} θ = β + α sin^{4} θ sin^{8} θ – 2 – cos^{2} θsin^{2} θ

= β + α sin^{4} θ

a_{21} = sin^{4} θ + cos^{2} θ sin^{4} θ + cos^{4} θ + cos^{6} θ = α (1 + cos^{2} θ)

(1 + cos^{2} θ) α = sin^{4} θ (1 + cos^{2} θ) + cos^{4} θ (1 + cos^{2} θ)

α = sin^{4} θ + cos^{4} θ = 1 – 2 sin^{2} θ cos^{2} θ = 1 – ((sin^{2} θ)/2)

α_{min} = ½

β = sin^{8} θ – 2 – cos^{2} θsin^{2} θ – sin^{8} θ – cos^{4} θsin^{4} θ

= -2 – (sin^{2} 2θ)/4 – (sin^{4} 2θ)/16

β_{min} = -2 – 1/16 [4t^{2} + t^{4}]

= -2 – 1/16 [t^{2} + 2]^{2} + ¼

= -7/4 – 1/16(9) = -37/16

Therefore, α + β = -37/16 + 1/2 = -29/16

**Question 2:**A line y = mx + 1 intersects the circle (x – 3)

^{2}+ (y + 2)

^{2}= 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate -3/5, then which one of the following options is correct?

a) 6 ≤ m < 8

b) 2 ≤ m < 4

c) 4 ≤ m < 6

d) -3 ≤ m < -1

y = mx + 1

(*x* – 3)^{2} + ((*m**x* +1) + 2)^{2} = 25

=> *x*^{2} (1+ *m*^{2} ) + 6(*m *-1) *x *– 7 = 0

1 + m^{2} = 5m – 5

Or m = 2, 3

**Question 3:**Let S be the set of all complex numbers z satisfying |z – 2 + i| ≥ √5. If the complex number z

_{0}is such that 1/|z

_{0}– 1| is the maximum of the set

a) π/4

b) -π/2

c) 3π/4

d) π/2

|z – 2 + i| ≥ √5

P is along

**Question 4:**The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x

^{2}} is

{(x, y) : xy ≤ 8, 1 ≤ y ≤ x^{2}}

8/x = x^{2}

= 16 log_{e} 2 – 14/3

**Question 5:**There are three bags B

_{1}, B

_{2}and B

_{3}. The bag B

_{1}contains 5 red and 5 green balls, B

_{2}contains 3 red and 5 green balls, and B

_{3}contains 5 red and 3 green balls, Bags B

_{1}, B

_{2}and B

_{3}have probabilities 3/10, 3/10 and 4/10 respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

a) Probability that the selected bag is B

_{3}and the chosen ball is green equals 3/10

b) Probability that the chosen ball is green equals 39/80

c) Probability that the chosen ball is green, given that the selected bag is B

_{3}, equals 3/8

d) Probability that the selected bag is B

_{3}, given that the chosen balls is green, equals 5/13

**Question 6:**Define the collections E

_{1}, E

_{2}, E

_{3},….. of ellipses and {R

_{1}, R

_{2}, R

_{3}, …..} of rectangles as follows:

R_{1}: Rectangle of largest area, with sides parallel to the axes, inscribed in E_{1};

E_{n}: Ellipse

_{n-1}, n > 1; R

_{n}: Rectangle of largest area, with sides parallel to the axes, inscribed in E

_{n}, n > 1.

Then which of the following options is/are correct?

a) The eccentricities of E_{18} and E_{19} are NOT equal.

b) The distance of a focus from the centre in E_{9} is √5/32

c) The length of latus rectum of E_{9} is 1/6

d)

_{n}) < 24, for each positive integer N

A = 6 cosθ . 4 sin θ = 12 sin 2θ -> max

θ = π/4

E_{2} = a_{2} = 3 cos(π/4) = 3/√2

b_{2} = 2 sin θ = √2

So, e_{18} = 1 – b_{2}/a_{2} = √5/3

e of all ellipses -> same

Difference of *f* from conic in equation a_{q}e

Length of latus rectum = [2×4]/4(√2) = 1/6

**Question 7:**Let

a) a + b = 3

b) det(adj M

^{2}) = 81

c) (adj M)

^{-1}+ adj M

^{-1}= -M

d) If

(a)

adj M =

b – 6 = -5

=>b = 1

and ab – 1 = 1

or a = 2

Now,

|M| = -2

And a + b = 2

(b) |adj M^{2}|= |M^{2}|^{2} = |M|^{4} = 16

(c) (adj M)^{-1} + adj M^{-1} = 2 (adj M)^{-1}

= 2 M^{-1} M

= 2 x (-1/2) x M

= -M

(d)

This implies,

β + 2γ = 1

α + 2β + 3γ = 2

3α + β + γ = 1

α – β + γ = 3

Solving all the equations, we have α = 1, β = -1 and γ = 1

**Question 8:**

Then which of the following options is/are correct?

a) f’ has a local maximum at x = 1

b) f is onto

c) f is increasing on (-∞, 0)

d) f’ is NOT differentiable at x = 1

Range: -∞, +1

Therefore, Not monotonic.

f(x) = x^{2} – x + 1

f’(x) = 2x – 1

Max at x = 0 and 1

x ≥ 3

f(3) = 1 log 1 – 3 + 10/3 = 1/3

f(∞) -> ∞

Loc. Max at x = 1

**Question 9:**Let α and β be the roots of x

^{2}– x – 1 = 0, with α > β. For all positive integers n, define

b_{1} = 1 and b_{n} = a_{n-1} + a_{n+1}, n ≥ 2.

Then which of the following options is/are correct?

a)

b)

c)

d)

x^{2} – x – 1 = 0

α = [1+ √5]/2 and β = [1- √5]/2

As, b_{1} = 1 and b_{n} = a_{n-1} + a_{n+1}, n ≥ 2

a_{1} + a_{2} + …+ a_{n} =

=> a_{n} + a_{n+1} = a_{n+2}

=>

= a_{n+2} – [α^{2} – β^{2}]/[α -β]

= a_{n+2} – [α + β]

= a_{n+2} – 1

**Question 10:**Let denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to at a point P intersect the y-axis at Y

_{P}. If PY

_{P}has length 1 for each point P on , then which of the following is options is/are correct?

a)

b) xy’ – √[1-x

^{2}] = 0

c)

d) xy’ + √[1-x

^{2}] = 0

y – y_{1} = m(x – x_{1})

y_{p} – y_{1} = -mx_{1}

=> 1 = x_{1}^{2} + m^{2} x_{1}^{2}

1 = x^{2}[1 + (dy/dx)^{2}]

Or (dy/dx)^{2} = 1/x^{2} – 1

dy/dx = ± √[1-x^{2}]/x

Here x = sinθ, dx = cos θ dθ

=>

And the correct differential equation will be xy’ + √[1-x^{2}] = 0

**Question 11**In a non-right-angle triangle ΔPQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at 0. If p = √3, q = 1, and the radius of the circumcircle of the ΔPQR equals 1, then which of the following options is/are correct?

a) Area of ΔSOE = √3/12

b) Radius of incircle of ΔPQR = √3/2 [2 – √3]

c) Length of RS = √7/2

d) Length of OE = 1/6

(sin P)/√3 = (sin Q)/1 = 1/2R = ½ ; [Given R = 1]

∠P = π/3 or 2π/3

∠Q = π/6 or 5π/6

p>q=> ∠P = ∠Q

If ∠P = π/3 and ∠Q = π/6 => ∠R = π/2

Therefore, ∠P = 2π/3 and ∠Q = ∠R = π/6

**Question 12:**Let L

_{1}and L

_{2}denotes the lines

respectively. If L_{3} is a line which is perpendicular to both L_{1} and L_{2} and cuts both of them, then which of the following options describe(s) L_{3}?

L_{1} : [x-1]/-1 = [y-0]/2 = [z-0]/2

L_{2} : x/2 = y/-1 = z/2

L_{3} : x/a = y/b = z/c

L_{3} = L_{1} x L_{2}

Similarly,

A on L_{1} : (-λ + 1, 2λ, 2λ)

B on L_{2} : (2μ, -μ, 2μ)

AB Δrs : (2μ, + λ – 1 -μ – 2λ, 2μ – 2λ) = (6, 6, -3) or (2, 2, -1)

=>

λ = [3k+1]/3 and μ = -4k – 2/3

Now, 2 μ – 2 λ + k = 0

Putting μ and λ values, we have

k = -2/9=> μ = 2/9 and λ = 1/9

So, A: (8/9, 2/9, 2/9) and B : (4/9, -2/9, 4/9)

Mid point is (2/3, 0, 1/3)

For equation of L_{3};

vector A is the position vector of any point on L_{3}.

**Answer: a, b, d**

**Question 13:**If

^{2}equals _________

27I^{2} = 4

**Question 14:**Let the point B be the reflection of the point A(2, 3) with respect to the line 8x – 6y – 23 = 0. Let

_{A}and

_{B}be circle of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circle

_{A}and

_{B}such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is ________.

ΔAPC and ΔBQC are similar.

BC/AC = 1/2 => 2(AC – AB) = AC

AC = 2AB = 10

**Question 15: **Let AP (a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If AP(1; 3) ∩ AP (2; 5) ∩ AP (3; 7) = AP (a; d) then a + d equals_________

I: 1, 4, 7, 10, 13, 16 ….. 32

II: 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52

III: 3, 10, 17, 24, 31, 38, 45, 52

52 <– a + d –> LCM of 3, 5, 7 = 105

So, a + d = 157

**Question 16:**Let S be the sample space of all 3 × 3 matrices with entries from the set {0, 1}. Let the events E

_{1}and E

_{2}be given by

E_{1} = {A ϵ S : det A = 0} and

E_{2} = {A ϵ S : sum of entries of A is 7}.

If a matrix is chosen at random from S, then the conditional probability P(E_{1}|E_{2}) equals …….

S: 2^{9}

E_{2} : Sum of entries 7, we need 7 – 1s and 2 0’s

Total E_{2} = 9!/7!2! = 36

Per |A| to be zero, both zeroes should be in the same row/column.

Therefore, 3 x 3 x 2 = 18 cases

**Question 17:**Three lines are given by

Let the lines cut the plane x + y + z 1 at the points A, B and C respectively. If the area of the triangle ABC is Δ then the value of (6Δ)^{2} equals __________

1^{st} line: x = λ, y = 0, z = 0

x + y + z = 1 => λ = 1

A(1, 0, 0)

2^{nd} line: x = μ, y = μ, z = 0

2μ = 1 or μ = ½

B(1/2, ½, 0)

Parallels, C(1/3, 1/3, 1/3)

**Question 18:**Let ω ≠ 1 be a cube root of unity. Then the minimum of the set {|a + bω + cω

^{2}|

^{2}; Where, a, b, c distinct non-zero integers }equals ________

| a + bω + cω^{2}|^{2}

= (a + bω + cω^{2}) (a + cω + bω^{2})

= (a^{2} + b^{2} + c^{2} – ab – bc – ca)

^{2}) = (a + cω + bω

^{2})]

= > (1/2)[(a-b)^{2} + (b-c)^{2} + (c-a)^{2}] = ½(1+1+4) = 3

## Video Lessons – Paper 1 Maths

## JEE Advanced 2019 Maths Paper 1 Solutions

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