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QUESTION: 1

y-axis divides the segment joining points (-3,-4) and (1,-2) in the ratio

Solution:

QUESTION: 2

A pair of tangent lines are drawn from the origin to the circle x^{2}+y^{2}+20(x+y)+20 = 0. The equation of pair of tangents is

Solution:

QUESTION: 3

If the polar w.r.t.y^{2} = 4ax touches the ellipse (x^{2}/a^{2}) + (y^{2}/b^{2}) = 1, the locus of its pole is

Solution:

Let coordinates of the pole be (*h*,*k*) then equation of the polar of *y*2=4*a**x* is *k**y*=2*a*(*x*+*h*)

*y*=(2*a**/k)x*+(2*a**h**)/K*

Since line (1) is touching the ellipse

*x*^{2}/*a*^{2}+*y*^{2}/*b*^{2}=1

4*a*^{2}*b*^{2}/*k*^{2}=*a*^{2}.4*a*^{2}/*K*^{2}+*b*^{2}

Required locus is 4*a*^{2}*x*^{2}=4*a*^{4}+*b*^{2}*y*^{2}

=> 4*a*^{2}*x*^{2}−*b*^{2}*y*^{2}=4*a*^{4}

Hence A is the correct answer.

QUESTION: 4

If ω = (-1 + √3i)/2,then (3 + ω + 3ω^{2})^{4} is

Solution:

(3+w+3w²)^{4 }= [(1+w+w²)+2w²+2]^{4}= 2^{4}(w²+1)=16(-w)^{4} = 16w^{4}= 16w

QUESTION: 5

The area of the curve |x|+|y|=4 is

Solution:

If you draw a graph it will be clear let us consider 4 coordinates (4,0),(-4,0),(0,-4),(0,4) if you join this points you will get a square.this square can again be divided into 4 symmetric triangles.therefore the area = 4 times area of one triangle

area of triangle = 1/2bh=1/2*4*4=8

therefore total area = 4*8=32

QUESTION: 6

If f(x) = sinx-(x/2) is increasing function, then

Solution:

QUESTION: 7

Solution:

QUESTION: 8

The principal value of sin⁻^{1}(sin 2π/3) is

Solution:

Sine function is a periodic function. Its value repeat after an interval of 2π. Since 2π/3 is out of its inverse principle range (which is from -π/2 to π/2) so we will write sin 2π/3 = sin π/3. Now its in the principle range. Therefore sin⁻1(sin 2π/3) = sin⁻1(sin π/3) = π/3

QUESTION: 9

The equation of the curve passing through the origin and satisfying the differential equation dy/dx = (x − y)^{2} is

Solution:

We have, dy/dx=(x−y)^{2}

Let (x−y)=v. Then,

1−dy/dx=dv/dx⇒dy/dx=1−dv/dx

∴ dy/dx=(x−y)^{2}

⇒ 1−dv/dx=v^{2}

⇒ 1−v^{2}=dv/dx

⇒ dx=1/1−v^{2}dv

⇒ 2∫dx=2∫1/1−v^{2}dv

⇒ 2x=log(1+v/1−v)+logC

⇒ C(1+v/1−v)=e^{2x}

⇒C(x−y+1/y−x+1)=e^{2x}⇒C(x−y+1)=e^{2x}(y−x+1)

Taking C = 1, we find that option (a) is correct.

QUESTION: 10

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 system of equations

ax+by = 0, cx+dy = 0, where a, b, c, d ∈ {0,1}

Statement-1: The probability that the system of equations has a unique solution is 3/8 .

Statement-2: The probability that the system of equations has a solution is 1.

Solution:

For unique solution where a, b, c, d {0, 1}

total cases = 16

Favorable cases = 6 (either ad =1, bc = 0 or ad = 0, bc =1).

Probability that system of equation has unique solution is

and system of equations has either unique solution or infinite solutions so that probability for system to have a solution is 1.

QUESTION: 11

The maximum area of the rectangle that can be inscribed in a circle of radius r is

Solution:

QUESTION: 12

If z ≠ 0 is a complex number such that Arg (z) = π/4, then

Solution:

QUESTION: 13

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 the vectors

Reason (R): If θ is the angle between

Solution:

Solution :- A = i^+ j^ and B = j^+k^

A = (2)^1/2 , B = (2)^1/2

θ=(cos^−1)(A.B/AB)=(cos^−1)(1/2).

or, θ=(cos^−1)(cosπ/3) = π/3

Therefore, both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

QUESTION: 14

The number of distinct permutations of the letters of the word STATISTICS that begin and end with the letter S is

Solution:

QUESTION: 15

If *A* and *B* are two events and *B* is a subset of *A*, then *P(A/B)* is equal to

Solution:

Since, B is a subset of A so B ∩ A = B

now, P(A|B) = P(A ∩ B)/P(B) = P(B)/P(B) = 1

QUESTION: 16

If A and B are such events that P(A)>0 and P(B)≠1, then P(A̅/B̅) is equal to

Solution:

QUESTION: 17

If x^{2} + px + q is the quadratic equation whose roots are a - 2 and b - 2 where a and b are the roots of x^{2} - 3x + 1 = 0, then

Solution:

QUESTION: 18

The A.M., H.M. and G.M. between two numbers are 144/15, 15 and 12, but not necessarily in this order. Then H.M., G.M. and A.M. respectively are

Solution:

A.M. ,G.M., H.M. between two numbers are 144/15 , 15 , 12.

We know that A.M > G.M > H.M

So, the order is

Therefore, A.M = 15 , G.M = 12 , H.M = 144/15

Then,

H.M, G.M and A.M = 144/15 ,12, 15

QUESTION: 19

The measure of dispersion is

Solution:

Standard deviation (SD) is the most commonly used measure of dispersion. It is a measure of spread of data about the mean. SD is the square root of sum of squared deviation from the mean divided by the number of observations.

QUESTION: 20

If lines y = 3x+1 and 2y = x+3, are equally inclined with y = mx+4, m =

Solution:

Given,

y = 3x + 1 , slope of it {m1} = 3

2y = x + 3 , slope of it {m2} = 1/2

angle between y = 3x + 1 and y = mx + 4 is ∅

then , tan∅ = |3 - m|/| 1 + 3m| ____(1)

it is given that angle between the lines 2y = x + 3 and y = mx + 4 is also ∅ .

then, tan∅ = |1/2 - m|/| 1 +m/2|________(2)

from equations (1) and (2),

|3 - m|/|1 + 3m| = | 1/2 - m|/| 1 + m/2|

|3 - m|/|1 + 3m| = |1 - 2m|/|2 + m|

taking positive sign ,

(3 - m)(2 + m) = (1 - 2m)( 1 + 3m)

6 + 3m -2m -m^2 = 1 + 3m -2m - 6m^2

5m^2 = -5

m^2 = -1 it's not possible .

taking negative sign,

(3 - m)(2 + m) = -(1 - 2m)(1 + 3m)

6 + 3m - 2m - m^2 = -1 - 3m + 2m + 6m^2

7m^2 - 2m -7 = 0

m = { 2 ± √(4 + 49 × 4)}/14

= { 2 ± 2√50}/14

= { 1 ± √50}/7

= {1 ± 5√2}/7

hence, m = { 1 ± 5√2}/7

QUESTION: 21

The direction ratios of the diagonals of a cube which joins the origin to the opposite corner are (when the 3 concurrent edges of the cube are co-ordinate axes)

Solution:

Let the length of the sides of cube is a, then coordinates of corner (P) opposite to origin are (a,a,a)

∴ Direction ratios of diagonal OP are (a−0,a−0,a−0)

∴(a,a,a)i.e.(1,1,1)

QUESTION: 22

The points (1, 2, 3), (-1, -1, -1) and (3, 5, 7) are the vertices of

Solution:

**Correct Answer :- d**

**Explanation:- Using distance formula **

**AB=√29. BC=√116=2√29**

**AC=√29**

**BA+AC=BC**

**The points are collinear, so it doesn't make any triangle.**

QUESTION: 23

cosA+sin(270°+A)-sin(270°-A)+cos(180°+A) =

Solution:

Sin (180) = 0 cos (180) = - 1 sin (270) = - 1 cos (270) = 0 ∴ LHS = cosA + sin (270 + A) - sin (270 - A) + cos (180 + A) = cosA + sin(270)cos(A) + cos(270)sin (A) - [sin (270)cos (A) - cos (270)sin (A)] + cos (180)cos (A) - sin (180)sin (A) = cos(A) + 2cos (270)sin (A) - cosA = 0 = RHS

QUESTION: 24

The number of vectors of unit length perpendicular to vectors

Solution:

Since the first vector is in x-y plane and other is in y-z plane so if the resultant 2 vector seen as a plane then then perpendicular to them can be in both front and back

QUESTION: 25

Define a real valued function of a real variable given by where *a* and *b* are real numbers. f(log_{10}(log_{5} 10)) = 7, then the value of f(log_{10}(log_{10} 5)), is

Solution:

QUESTION: 26

The value of λ for which the area bounded between *y* = *x ^{2}* - 1 and

Solution:

QUESTION: 27

Let B and C lie on the circle with OA as a diameter, where O is the origin. If ∠ AOB = ∠ BOC = θ and z_{1}, z_{2}, z_{3} representing the points A, B, C respectively, then which one of the following is true?

Solution:

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): and if these vectors be coplanar, then λ is -8.

**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): Area common to the curve

**Reason**(R):

Solution:

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