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JEE Main Mathematics Mock - 1

25 Questions MCQ Test Mock Test Series for JEE Main & Advanced 2022 | JEE Main Mathematics Mock - 1

Description

Attempt JEE Main Mathematics Mock - 1 | 25 questions in 60 minutes | Mock test for JEE preparation | Free important questions MCQ to study Mock Test Series for JEE Main & Advanced 2022 for JEE Exam | Download free PDF with solutions

QUESTION: 1

In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer-

Assertion(A):If C_r is the coefficient of x^r in the expansion of (1 + x)²⁰

Reason(R) : Cr = C_{n − r} for any positive integer n

A.
Assertion is true but Reason is false.
B.
Both Assertion and Reason are true and Reason is not the correct explanation of Assertion.
C.
Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
D.
Assertion is false but Reason is true.

Solution:

QUESTION: 2

The area (in square units) bounded by the curves y² = 4x and x² = 4y in the plane is

A.
8/3
B.
16/3
C.
32/3
D.
64/3

Solution:

QUESTION: 3

If sin θ is real, then θ =

A.
π
B.
π/2
C.
π/3
D.
π/6

Solution:

QUESTION: 4

In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer-
Assertion(A) :The inverse of does not exist.
Reason(R) :The matrix is non singular.

A.
Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
B.
Both Assertion and Reason are true and Reason is not the correct explanation of Assertion.
C.
Assertion is true but Reason is false.
D.
Assertion is false but Reason is true.

Solution:

QUESTION: 5

If the line 3x-4y=λ touches the circle x²+y²-4x-8y-5=0, λ can have the values

A.
-35,15
B.
-35,-15
C.
35,-15
D.
35,15

Solution:

QUESTION: 6

The length of the tangent from (0,0) to the circle 2x² + 2y² + x - y + 5 = 0 is

A.
√5
B.
√5 ∕ 2
C.
√2
D.

Solution:

QUESTION: 7

The differential equation which represents the family of plane curves y=exp. (cx) is

A.
y'=cy
B.
xy'-logy=0
C.
x log y=yy'
D.
y log y=xy'

Solution:

y = e^cx
dy/dx = c. e^cx
y' = cy

QUESTION: 8

If sin y = x sin (a + y), then (dy/dx) =

A.
[sin²(a + y)/sin a]
B.
sin (a + y)
C.
sin² (a + y)
D.
[sin (a + y)/sin a]

Solution:

QUESTION: 9

The fundamental period of the function f(x) = 2 cos 1/3(x - π) is

A.
6π
B.
4π
C.
2π
D.
π

Solution:

QUESTION: 10

If A, B, C are represented by 3 + 4i, 5 - 2i, -1 + 16i respectively, then A, B, C are

A.
collinear
B.
vertices of equilateral triangle
C.
vertices of isosceles triangle
D.
vertices of right angled triangle

Solution:

QUESTION: 11

Which of the following is not a statement ?

A.
Every set is a finite set .
B.
8 is less than 6.
C.
Where are you going ?
D.
The sum of interior angles of a triangle is 180 degree.

Solution:

QUESTION: 12

In the following question, a Statement-1 is given followed by a corresponding Statement-2 just below it. Read the statements carefully and mark the correct answer-
Consider the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5.
Statement-1:
The parametric equations of the line of intersection of the given planes are
x = 3 + 14t, y = 1 + 2t, z = 15t.
Statement-2:
The vector 14î+2ĵ+15k̂ is parallel to the line of intersection of given planes

A.
Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B.
Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1
C.
Statement-1 is True, Statement-2 is False
D.
Statement-1 is False, Statement-2 is True

Solution:

QUESTION: 13

In the following question, a Statement-1 is given followed by a corresponding Statement-2 just below it. Read the statements carefully and mark the correct answer-
Tangents are drawn from the point (17,7) to the circle x²+y²=169.
Statement-1:
The tangents are mutually perpendicular.
Statement-2:
The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x²+y²=338.

A.
Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1
B.
Statement -1 is True, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1
C.
Statement -1 is True, Statement -2 is False
D.
Statement -1 is False, Statement -2 is True

Solution:

Clearly, m₁m₂ = - 1.
Hence, the two tangents arc mutually perpendicular.
Statement 1 is true.
Now, the locus of the point of intersection of two mutually perpendicular tangents to the circle x² + y² = r² is the director circle i.e. the circle x² +y²= 2r².
For the given circle r = 13. ..
Its director circle is x² + y² = 338.
Hence, statement 2 is true and a cogect explanation of statement as the point (17, 7) lies on the director circle of the circle (i).

QUESTION: 14

The value of a for which the system of equations
a³x+(a+1)³y+(a+2)³z = 0
ax+(a+1)y+(a+2)z = 0
x+y+z = 0
has a non-zero solution, is

A.
1
B.
0
C.
-1
D.
none of these

Solution:

The system of equation has a non-zero solution

QUESTION: 15

The pole of the line 2x + 3y − 4 = 0 with respect to the parabola y² = 4 x is

A.
(4, 6)
B.
(− 2, − 3)
C.
(1, 1)
D.
(2, − 3)

Solution:

QUESTION: 16

If ⁿC₁₂=ⁿC₈, then n=

A.
20
B.
12
C.
6
D.
30

Solution:

QUESTION: 17

The chance of getting a doublet with 2 dice is

A.
2/3
B.
1/6
C.
5/6
D.
5/36

Solution:

Total outcomes = 36
Doublet are 6 (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)

Probability of getting doublet = 6/36
= 1/6

QUESTION: 18

If cov. (x, y) = 0, then ρ(x, y) equals

A.
0
B.
1
C.
-1
D.
+1/2

Solution:

Covariance is a quantitative measure of the extent to which the deviation of one variable from its mean matches the deviation of the other from its mean. It is a mathematical relationship that is defined as:

Cov(X,Y) = E[(X − E[X])(Y − E[Y])]

Correlation between two random variables, ρ(X,Y) is the covariance of the two variables normalized by the variance of each variable. This normalization cancels the units out and normalizes the measure so that it is always in the range [0, 1]:

QUESTION: 19

The two opposite vertices of a square on xy-plane are A(-1,1) and B(5,3), the equation of other diagonal (not passing through A and B) is

A.
x-3y+4=0
B.
2x-y+3=0
C.
y+3x-8=0
D.
x+2y-1=0

Solution:

Given: Here,AB is the diagonal of square.
The vertices of a square A
Let the mid−point of AB be EThen coordinates of E are
Therefore equation of other diagonal is

QUESTION: 20

If the normal to the curve y=f(x) at the point (3,4) makes an angle 3π/4 with the positive x-axis, then f'(3)

A.
-1
B.
-3/4
C.
4/3
D.
1

Solution:

Given y = f(x)

differentiating w.r.t x

y' = f'(x) which is the slope of the tangent

Hence the slope of the normal is - 1/f'(x) = 3pi/4 = -1

therefore f'(x) = 1

Hence f'(3) = 1

*Answer can only contain numeric values

QUESTION: 21

Assume e^-4/5 = 2/5. If x, y satisfy, y = e^x and the minimum value of (x² + y²) is expressed in the form of m/n then (2m - n)/5 equals (where m & n are coprime natural numbers)

Solution:

OP² = x² + y²
y = e^x, y' = e^x,

*Answer can only contain numeric values

QUESTION: 22

Let ƒ(x) be non-constant thrice differentiable function defined on (–∞, ∞) such that ƒ(x) = ƒ(6 – x) and ƒ'(0) = 0 = ƒ'(2) = ƒ'(5). If 'n' is the minimum number of roots of (ƒ"(x))² + ƒ'(x)ƒ"'(x) = 0 in the interval x ∈ [0, 6] then sum of digits of n equals

Solution:

ƒ(x) = ƒ(6 – x)
⇒ ƒ'(x) = –ƒ'(6 – x) .... (1)
put x = 0, 2, 5
ƒ'(0) = ƒ'(6) = ƒ'(2) = ƒ'(4) = ƒ'(5) = ƒ'(1) = 0
and from equation (1) we get ƒ'(3) = –ƒ'(3)
⇒ ƒ'(3) = 0
So ƒ'(x) = 0 has minimum 7 roots in
x ∈ [0, 6] ⇒ ƒ"(x) has min 6 roots in x ∈ [0,6]
h(x) = ƒ'(x).ƒ"(x)
h'(x) = (ƒ"(x))² + ƒ'(x) ƒ"'(x)
h(x) = 0 has 13 roots in x ∈ [0, 6]
h'(x) = 0 has 12 roots in x ∈ [0, 6]

*Answer can only contain numeric values

QUESTION: 23

If the co-ordinate of the vertex of the parabola whose parametric equation is x = t² – t + 1 and y = t² + t + 1, t ∈ R is (a, b) then (2a + 4b) equals

Solution:

x = t² – t + 1 .... (1)
y = t² + t + 1 .... (2)
y – x = 2t & x + y = 2(t² + 1)
________on elminating 't' we get
⇒ (x + y – 2) = 2(y - x)/2²
(x – y)² = 2(x + y – 2)
Axis : x – y = 0
Tangent at vertex : x + y – 2 = 0
Vertex : (1, 1) = (x, y)