Attempting question papers is best way to practice. In this blog you have JEE Mains 2019 question paper with solutions for 9^{th} January shift 1.

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## JEE Mains 2019 Question Paper with Solutions- 9 January Shift 1

### Mathematics

**Question 1 **

The area (in sq. units) bounded by the parabola y=x^{2}-1, the tangent at the point (2,3) to it and y-axis is:

- 14/3
- 56/3
- 8/3
- 32/3

Correct answer: 3

**Question 2 **

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:

- 3√3 π
- 6 π
- 2√3 π
- 4/3 π

Correct answer: 3

**Question 3**

For x^{2} ≠ nπ + 1, n∊N (the set of natural numbers), the integral

is equal to:

(where c is a constant of integration)

1. | |

2. | |

3. | |

4. |

Correct answer: 1

**Question 4 **

Let α and β be two roots of the equations x^{2}+2x+2=0. Then, α^{15}+β^{15} is equal to:

- -256
- 512
- -512
- 256

Correct answer: 1

**Question 5 **

If y=y(x) is the solution of the differential equation, xdy/dx + 2y = x^{2} satisfying y(1) =1, then y(1/2) is equal to

- 7/64
- 13/16
- 49/16
- 1/4

Correct answer: 3

**Question 6 **

Equation of a common tangent to the circle, x^{2}+y_{2}-6x=0 and the parabola y^{2}=4x is:

- 2√3 y = 12x + 1
- √3 y = x + 3
- 2√3 y = -x – 12
- √3 y = 3x + 1

Correct answer: 2

**Question 7 **

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be members of the same team, is:

- 200
- 300
- 500
- 350

Correct answer: 2

**Question 8 **

Three circles of radii a,b,c (a<b<c) touch each other externally. If they have x-axis as a common tangent, then:

- 1/√a = 1/√b + 1/√c
- 1/√b = 1/√a + 1/√c
- a,b,c are in AP
- √a, √b, √c are in AP

Correct answer: 1

**Question 9 **

If the fractional part of the number 2^{403}/3 is k/15. Then k is equal to:

- 14
- 6
- 4
- 8

Correct answer: 4

**Question 10 **

Axis of a parabola lies along the x-axis. It its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?

- (4,-4)
- (5,2√6)
- (8,6)
- (6,4√2)

Correct answer: 3

**Question 11 **

The plane passing through the intersection of the planes x+y+z=0 and 2x+3y-z+4=0 and parallel to y axis also passes through the point:

- (-3,0,-1)
- (-3,1,1)
- (3,3,-1)
- (3,2,1)

Correct answer: 4

**Question 12 **

If a, b and c be three distinct real numbers in G.P and a+b+c=xb. Then x cannot be:

- 4
- -3
- -2
- 2

Correct answer: 4

**Question 13 **

Consider the set of all lines px+qy+r=0 such that 3p+2q+4r=0. Which one of the following statements is true?

- The lines are all parallel
- Each line passes through the origin.
- The lines are not concurrent.
- The lines concurrent at the point (0.75,0.5)

Correct answer: 4

**Question 14 **

The system of linear equations

x + y + z = 2

2x + 3y + 2z = 5

2x + 3y + (a^{2}-1)z = a+1

- has infinitely many solutions for a=4
- is inconsistent when |a|=√3
- is inconsistent when a=4
- has a unique solution for |a|=√3

Correct answer: 2

**Question 15 **

Let a,b,c be vectors. Let a=i-j, b=i+j+k and c be the vector such that a✖c+b=0 and a.c=4. Then c^{2} is equal to

- 19/2
- 8
- 17/2
- 9

Correct answer: 1

**Question 16 **

Let a_{1}, a_{2},………….., a_{30} be an AP,

and

If a_{5}=27 and S-2T=75. Then a_{10} is equal to:

- 57
- 47
- 42
- 52

Correct answer: 4

**Question 17 **

5 students of a class have an average height of 150 cm and variance of 18cm^{2} . A new student whose height is 156 cm, joined them. The variance (in cm^{2}) of the height of these six students is:

- 22
- 20
- 16
- 18

Correct answer: 2

**Question 18 **

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of the number of aces obtained in the two drawn cards. Then P(X=1) + P(X=2) equals:

- 52/169
- 25/169
- 49/169
- 24/169

Correct answer: 2

**Question 19 **

For x∊R-{0,1}, let f_{1}(x)=1/x , f_{2}(x)=1-x anf f_{3}(x)=1/(1-x) be three given functions. If a function, J(x) satisfies (f_{2}∘J∘f_{1})(x)=f_{3}(x). Then J(x) is equal to

1. f_{3}(x)

2. f_{1}(x)

3. f_{2}(x)

4. x^{-1}f(x)

Correct answer: 1

**Question 20 **

Let

Then the sum of the elements in A is:

- 5π/6
- 2π/3
- 3π/4
- π

Correct answer: 2

**Question 21 **

If θ denotes the acute angle between the curves, y=10-x^{2} and y=2+x^{2} at a point of their intersection, then |tanθ| is equal to:

- 4/9
- 7/17
- 8/17
- 8/15

Correct answer: 4

**Question 22 **

If

, then the matrix A^{-50} when θ = π/12, is equal to:

1. | |

2. | |

3. | |

4. |

Correct answer: 1

**Question 23 **

Let 0<θ<π/2. If the eccentricity of the hyperbola

is greater than 2, then the length of its latus rectum lies in the interval

- (2,3]
- (3,∞)
- (3/2,2]
- (1,3/2]

Correct answer: 2

**Question 24 **

The equation of the line passing through (-4,3,1), parallel to the plane x+2y-z=0 and intersecting the line (x+1)/-3 = (y-3)/2 = (z-2)/-1 is:

- (x+4)/-1 = (y-3)/1 = (z-1)/1
- (x+4)/3 = (y-3)/-1 = (z-1)/1
- (x+4)/1 = (y-3)/1 = (z-1)/3
- (x-4)/2 = (y+3)/1 = (z+1)/4

Correct answer: 2

**Question 25 **

For any θ ∈ (π/4, π/2), the expression 3(sinθ – cosθ)^{4} + 6(sinθ + cosθ)^{2} + sin^{6}θ equals:

1. 13 – 4cos^{2}θ + 6sin^{2}θcos^{2}θ

2. 13 – 4cos^{6}θ

3. 13 – 4cos^{2}θ + 6cos^{4}θ

4. 13 – 4cos^{4}θ + 2sin^{2}θcos^{2}θ

Correct answer: 2

**Question 26 **

If cos^{-1}(2/(3x)) + cos^{-1}(3/(4x))=π/2 (x > 3/4). Then x is equal to:

- 1√145/12
- 2√145/10
- 3√146/12
- 4√145/11

Correct answer: 1

**Question 27 **

The value of

is :

- 2/3
- 0
- -4/3
- 4/3

Correct answer:

**Question 28 **

If the Boolean expression (p⨁q) ∧ (~p⨀q) is equivalent to p∧q, where ⨁, ⨀ ∊ {∧, ∨}. Then the ordered pair (⨁, ⨀) is:

- (∧, ∨)
- (∨, ∨)
- (∨, ∧)
- (∧, ∧)

Correct answer: 1

**Question 29 **

- exists and equals 1/4√2
- does not exist
- exists and equals 1/2√2
- exists and equals 1/(4+2√2)

Correct answer: 1

**Question 30 **

Let f:R→R be a function defined as

then f is:

1. continuous if a=5 and b=5

2. continuous if a=-5 and b=10

3. continuous if a=0 and b=5

4. not continuous for any values of a and b

Correct answer: 4

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