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This mock test of JEE(MAIN) Mathematics Mock Test - 2 for JEE helps you for every JEE entrance exam.
This contains 30 Multiple Choice Questions for JEE JEE(MAIN) Mathematics Mock Test - 2 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

The orthocentre of the traingle whose vertices are (5, -2), (-1, 2) and (1,4) is

Solution:

QUESTION: 2

If the equation [(k(x+1)^{2}/3)]+[(y+2)^{2}/4]=1 represents a circle, then k=

Solution:
The given equation can be write as

=> 4k(x+1) Â²+3(y+2) Â²=12

on expanding wee get xÂ² coefficient as 4k

and yÂ² coefficient as 3

but in equation of circle xÂ² coefficient is equal to yÂ² coefficient

therefore 4k=3

=> k=3/4

=> 4k(x+1) Â²+3(y+2) Â²=12

on expanding wee get xÂ² coefficient as 4k

and yÂ² coefficient as 3

but in equation of circle xÂ² coefficient is equal to yÂ² coefficient

therefore 4k=3

=> k=3/4

QUESTION: 3

The eccentric angles of the extremities of the latus-rectum intersecting positive x-axis of the ellipse ((x^{2}/a^{2}) + (y^{2}/b^{2}) = 1) are given by

Solution:

QUESTION: 4

If arg (z) = θ, then arg(z̅) =

Solution:

QUESTION: 5

Solution:

QUESTION: 6

If N^{ N+} denotes the set of all positive integers and if f : N^{N+} → N is defined by f(n) = the sum of positive divisors of (n) then f (2^{k} . 3), where *k* is a positive integer is

Solution:

f(2^{k}. 3) = The sum of positive divisors of 2^{k} . 3

QUESTION: 7

If *a, b, c * are different and

Solution:

**Correct Answer : b**

**Explanation : **A = {(a, a^{2}, a^{3}-1) (b, b^{2}, b^{3}-1) (c, c^{2}, c^{3}-1)}

=> {(a, a^{2}, a^{3}) (b, b^{2}, b^{3}) (c, c^{2}, c^{3})} - {(a, a^{2}, 1) (b, b^{2}, 1) (c, c^{2}, 1)} = 0

=> abc{(1, a, a^{2}) (1, b, b^{2}) (1, c, c^{2})} - {(a, a^{2}, 1) (b, b^{2}, 1) (c, c^{2}, 1)} = 0

=> abc{(a, a^{2}, 1) (b, b^{2}, 1) (c, c^{2}, 1)} - {(a, a^{2}, 1) (b, b^{2} 1) (c, c^{2}, 1)} = 0

=> (abc-1){(a, a^{2}, 1) (b, b^{2}, 1) (c, c^{2}, 1)} = 0

abc - 1 = 0

=> abc = 1

QUESTION: 8

Solution:

QUESTION: 9

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)__: Angle between is acute angle

__Reason (R)__: If is acute then is obtuse then

Solution:

QUESTION: 10

The point on the curve *y = x*^{2} which is nearest to (3, 0) is

Solution:

QUESTION: 11

The degree of the differential equation

Solution:

QUESTION: 12

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)__: are non zero vectors then is a vector perpendicular to all the vectors a → , b → , c →

__Reason (R)__: are perpendicular to both

Solution:

QUESTION: 13

If i^{2} = -1, then the sum i + i^{2} + i^{3} + ..... upto 1000 terms is equal to

Solution:
There will equal n opposite signed terms that is 500 +ve one and 500-ve one therefore it's value comes to zero.

QUESTION: 14

A parallelogram is cut by two sets of m lines parallel to the sides, the number of parallelogram thus formed is

Solution:

Parallelogram is cut by two sets of m parallel lines to its sides.

then we have 2 sets of (m+2) parallel lines ( 2 lines of the parallelogram)

so parallelogram is formed by taking 2 lines from each set

= m+2C2 * m+2C2

= [(m+2)(m+1)/2 ]2

this also include 1 original parallelogram

so total number of new parallelogram formed is = (m + 2)2(m + 1)2/4

QUESTION: 15

If *A* and *B* are two events such that

Solution:

QUESTION: 16

The probability that a leap year will have exactly 52 Tuesdays is

Solution:

The probability of a year being a leap year is 1/4 and being non-leap is 3/4.A leap year has 366 days or 52 weeks and 2 odd days. The two odd days can be {Sunday,Monday},{Monday,Tuesday},{Tuesday,Wednesday}, Wednesday,Thursday},{Thursday,Friday},{Friday,Saturday},{Saturday,Sunday}.So there are 7 possibiliyies out of which 2 have a Sunday. So the probability of 53 Sundays in a leap year is 2/7.

So, the probability of 52 sundays is 1-2/7 = 5/7.

QUESTION: 17

Product of the real roots of the equation t^{2}x^{2} + ∣x∣ + 9 = 0

Solution:

QUESTION: 18

If A.M. between two numbers is 5 and their G.M. is 4, then their H.M. is

Solution:

If x, y and z respectively represent AM, GM and HM between two numbers a and b, then

y^{2} = xz

Here x = 5, y = 4

then 16 = 5 x z

z = 16/5

QUESTION: 19

If the coefficient of correlation between x and y is 0.28, covariance between x and y is 7.6, and the variance of x is 9, then the standard deviation of the y series is

Solution:
N the given problem it is SD of x is 3: (or Variance of x is 9). As Variance = (Sx)^2. We know the relation : correlation coefficient (r) = Cov (x,y) / (Sx * Sy) so, 0.28 = 7.6 / (3 * Sy) From here we get the value of SD of Y : Sy = 9.05.

QUESTION: 20

The equation line passing through the point P(1,2) whose portion cut by axes is bisected at P, is

Solution:

QUESTION: 21

The equation of the plane through intersection of planes x + 2y + 3z = 4 and 2x + y - z = -5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

Solution:

QUESTION: 22

The projections of a line segment on the co-ordinate axes are 12,4,3 respectively. The length of the line segment is

Solution:

QUESTION: 23

sin 120° cos 150° − cos 240° sin 330° =

Solution:

Sin120= sin(90+30)=cos30= sqrt(3)/2

Cos150= cos(180–30)= - cos30= -sqrt(3)/2

Cos240= cos(180+60)= - cos60= -1/2

Sin330= sin(360–30)= - sin30= - 1/2

((Sqrt(3)/2) * (-sqrt(3)/2))-((-1/2) * (- 1/2))

= (- 3/4)-(1/4)

{ since ‘- ‘multiplied with ‘+’ equals ‘-’ ,

And ‘-’ multiplied with ‘-’ gives ‘+’}

=-1

Therefore the answer is - 1

QUESTION: 24

Let be three non-coplanar vectors such and be vectors defined by the relations :

then the value of the expression

equal to

Solution:

QUESTION: 25

The average of the number n sin n° for n = 2, 4, 6, ...180 is

Solution:

180/2 = 90.

The total no of terms = 90.

We can write the terms as:

(2 * sin(2) + 4 * sin(4) + 6 sin(6) + ... + 180 * sin(180)) / 90

as we know the relation sin Î¸ = sin (180 - Î¸)

So, we can write: sin(2) = sin(178), sin(4) = sin(176) , all the way to sin(88) = sin(92)

And we know some of the values as well, like sin 180 is zero, sin90 is one and so on.

(2 * sin(2) + 178 * sin(2) + 4 * sin(4) + 176 * sin(4) + ... + 88 * sin(88) + 92 * sin(92) + 90 * sin(90)) / 90

or, (180 * (sin(2) + sin(4) + sin(6) + ... + sin(88)) + 90) / 90

Now most importantly, we have to know which deduction to be used where and we know sin/cos = tan. and cos/sin = cot.

So, the series can be written as sin(a) + sin(2a) + sin(3a) + ... + sin(na) = (1/2) * cot(a/2) - cos((n + 1/2) * a) / (2 * sin(a/2))

Therefore, sin(2) + sin(4) + ... + sin(88) =>

(1/2) * cot(2/2) - cos((44 + 1/2) * 2) / (2 * sin(2/2)) [ applying the above formula ]

or, (1/2) * (cos(1) - cos(89)) / sin(1)

(180 * (sin(2) + sin(4) + sin(6) + ... + sin(88)) + 90) / 90

or, (180 * (1/2) * (cos(1) - cos(89)) / sin(1) + 90) / 90

or, 90 * ((cos(1) - cos(89)) / sin(1) + 1) / 90

or, (cos(1) - cos(89) + sin(1)) / sin(1)

or, (cos(1) - sin(1) + sin(1)) / sin(1)

or, cos(1)/sin(1)

or, cot(1) = [RHS] Proved.

QUESTION: 26

The number of permutation of the letters PPPPQQQR in which the P's appear together in a block of four letters or the Q's appear in a block of 3 letters, is

Solution:

QUESTION: 27

Let f x be defined in *R* such that f (1) = 2, f (2) = 8 and f (u + v) = f (u) + *kuv* - 2*v*^{2} for all u , v ∈ R and *k* is a fixed constant. Then

Solution:

**Correct Answer : a**

**Explanation :** f(1) = 2, f(2) = 8

f(u+v) = f(u) + kuv - 2v^{2}

u = 1, v = 1

f(1+1) = f(1) +k(1)(1) - 2(1)^{2}

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

8 = 2 + k - 2

=> k = 8

f'(x) = lim(h-->0) [f(x+h) - f(x)]/h

= lim(h-->0) [f(x) + k*x*h - 2h^{2} - f(x)]/h

= lim(h-->0) h[kx - 2h]/h

f'(x) = kx - 2(0)

f'(x) = kx

f'(x) = 8x

QUESTION: 28

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)__:

__Reason(R)__:

Solution:

QUESTION: 29

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 orthocentre of a given triangle is coincident with the incentre of the pedal triangle of the given triangle.

__Reason(R)__: Pedal triangle is the ex-central triangle of the given triangle.

Solution:

QUESTION: 30

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 determinants are not identically equal.

__Reason(R)__: If two rows (or columns) of a determinant are identical, then value of the determinant is zero.

Solution:

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