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QUESTION: 1

In the expansion of the number of terms free from radicals is

Solution:

General term,

are integers for

0 ≤ r ≤ 6561

∴ r = 0, 9, 18, 27, ...6561

0, 9, 18, ...6561 is in A.P.

First term, a = 0

Common - difference = 18 - 9 = 9

Last term, t = 6561

l = a + (n - 1)d, where n = number of terms

⇒ 6561 = 0 + (n - 1)9

⇒ 9(n - 1) = 6561

⇒ n - 1 = 729

⇒ n = 729 + 1

⇒ n = 730

QUESTION: 2

The coefficient of x^{-9} in the expansion of ((x^{2}/2) - (2/x))^{9} is

Solution:

QUESTION: 3

The coefficient of x^{3} in ((√x^{5}) + (3/√x^{3}))^{6} is

Solution:

QUESTION: 4

Equation *x*^{2}+2ax-b^{2}=0 has real roots $\alpha $,$\beta $ and equation *x*^{2}+2px-q^{2}=0 has real roots $\gamma $,$\delta $. If the circle *C* is drawn with the points ($\alpha $,$\gamma $), ($\beta $,$\delta $) as extremities of a diameter, then the equation of *C* is

Solution:

QUESTION: 5

If the cube roots of unity are 1,ω,ω^{2} then the roots of the equation (x - 2)^{3}+27 = 0 are

Solution:

QUESTION: 6

The equation of circle which passes through (4,5) and whose centre is (2,2) is

Solution:

QUESTION: 7

Let z_{1} and z_{2} be two complex numbers whose principle argument are α and β then, arg (z_{1}.z_{2}) =

Solution:
Let z_{1} and z_{2} have polar representations z_{1} = r_{1}( cos α + isinα)

and z_{2} = r_{2} (cosβ + isinβ ).

Then z_{1} z_{2} = r_{1} (cosα+i sinα) r_{2} (cosβ + isinβ )

= r_{1} r_{2} (cosαcosβ − sinαsinβ) + i(cosαsinβ + sinαcosβ )

= r_{1} r_{2} (cos (α+β ) + isin (α+β )),

which is the polar representation of z_{1} z_{2} , as r_{1} r_{2} = |z_{1} ||z_{2} |=|z_{1} z_{2} |.

Hence α+β is an argument of z_{1} z_{2}

and z

Then z

= r

= r

which is the polar representation of z

Hence α+β is an argument of z

QUESTION: 8

If *m* and *n* are integers, then what is the value of sin mx sin nxdx . If m ≠ n

Solution:

Since sin mx, sin nx is an odd function if m ≠ n, then sin mx . sin nx dx = 0

QUESTION: 9

Solution:

QUESTION: 10

Solution:

QUESTION: 11

Solution:

QUESTION: 12

Solution:

QUESTION: 13

The area bounded by the parabola y^{2}=4ax and the straight line y=2ax is

Solution:

QUESTION: 14

The solution of the differential equation 2xy(dy/dx)=x^{2}+3y^{2} is (where c is a constant)

Solution:

QUESTION: 15

What is the solution of x^{2}y^{2}dy = (1−xy^{3}) dx?

Solution:

QUESTION: 16

If

then f ′(1) =

Solution:

QUESTION: 17

The curve represented by x = 2(cos t + sin t), y = 5 (cos t - sin t) is

Solution:

QUESTION: 18

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: 19

If

Solution:

QUESTION: 20

The slopes of the common tangents to the hyperbola x^{2}/y - y^{2}/16 = 1 and y^{2}/9 - x^{2}/16 = 1 are

Solution:

QUESTION: 21

The foci of the hyperbola *9x*^{2} - 16y^{2} + 18x + 32y - 151 = 0 are

Solution:

QUESTION: 22

The function is increasing in

Solution:

QUESTION: 23

The value of cos⁻^{1}(cos 5π/3) + sin⁻^{1} (sin 5π/3) is

Solution:

QUESTION: 24

is equal to

Solution:

QUESTION: 25

For all real x, the minimum value of (1 - x + x^{2})/(1 + x + x^{2}) is

Solution:

QUESTION: 26

The system of linear equation x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = 4 has a unique solution, then

Solution:

Augmented matrix is

∴ Given system of equations has a unique solution ⇒ k ≠ 0

QUESTION: 27

If the vertices O,A,B of any equilateral triangle are situated at z=0, z=z₁ and z=z₂ respectively, then which of the following are true ?

Solution:

QUESTION: 28

Which of the following points lie on the parabola x^{2}=4ay ?

Solution:

QUESTION: 29

P,Q,R and S have to deliver lecture, then in how many ways can the lectures be arranged?

Solution:
Total no. of ways delivering lectures = ^{4}P_{4} = 4! = 24

QUESTION: 30

The length of the normal chord to the parabola y^{2} = 4x which subtends a right angle at the vertex is

Solution:

Again AB subtends a right angle at the vertex O (0, 0) of the parabola

QUESTION: 31

In how many different ways can the letters of the word 'AUCTION' be arranged so that the vowels always come together?

Solution:

QUESTION: 32

If ^{20}C_{n+2} = ^{n}C_{16}, then n =

Solution:

QUESTION: 33

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is

Solution:

QUESTION: 34

What is the chance that a leap year should have 53 sundays?

Solution:

QUESTION: 35

A problem in EAMCET examination is given to 3 students *A, B* and *C* whose chances of solving it are respectively. The probability that the problem will be solved is

Solution:

QUESTION: 36

f(x) = 1 + 2 sin x + 3 cos^{2}x, 0 ≤ x ≤ (2π/3) is

Solution:

QUESTION: 37

In ΔABC ,(a + b + c)

Solution:

QUESTION: 38

If R denotes circumradius then in a ΔABC , is equal to

Solution:

QUESTION: 39

If the roots of x^{2} + x + a = 0 exceed a, then

Solution:

QUESTION: 40

If f(x) is continuous and differentiable over [−2, 5] and −4 ≤ f′ (x) ≤ 3 for all x in (−2, 5) then the greatest possible value of f(5) − f(−2), is

Solution:

QUESTION: 41

If y = x - x^{2} + x^{3} - x^{4} + ... to ∞, then the value of x will be (-1 < x < 1)

Solution:

QUESTION: 42

The sum of an infinite number of G.P. is 20, and the sum of their squares is 100. The first term of the G.P. is

Solution:

QUESTION: 43

The two geometric means between 1 and 64 are

Solution:

QUESTION: 44

If a, b, c of a triangle are in A.P., then cot C/2 =

Solution:

QUESTION: 45

Let A = {a,b,c}, B = {b,c,d}, C = {a,b,d,e}, then A ∩ (B ∪ C) is

Solution:

QUESTION: 46

The equation of bisectors between the lines 3x+4y-7=0 and 12x+5y+17=0 are

Solution:

QUESTION: 47

If f(x) = (x - 3)/(x + 1), then f [f{f(x)}] =

Solution:

QUESTION: 48

The equations of two lines which pass through the point (3,2) and make angle of 45º with the line x-2y=3 are

Solution:

QUESTION: 49

The roots of the equation

Solution:

Given equations is

QUESTION: 50

Solution of 7 sin^{2}x + 3 cos^{2} x = 4 is

Solution:

QUESTION: 51

If α_{1} , α_{2} , α_{3} , … … , α n are the *n*^{th} roots of unity, then equals

Solution:

QUESTION: 52

If the hypotenuse of right angled triangle is four times the length of perpendicular drawn from opposite vertex to it, then the difference of two acute angle will be

Solution:

QUESTION: 53

*ABC* is an isosceles triangle with *AB = AC*. If *B* (1, 3), *C*(- 2, 7) then vertex *A* may be

Solution:

QUESTION: 54

The area bounded by the curve f(x) = x + sinx and its inverse function between the ordinates x = 0 to x = 2 π is

Solution:

QUESTION: 55

Area bounded by the curve y = sin^{-1} |(sinx)| and

Solution:

QUESTION: 56

Let f (x) = e^{x } sin *x*, be the equation of a given curve. If at *x = a*, 0 ≤ a ≤ 2 π , the slope of the tangent is the maximum, then the value of *a* is

Solution:

QUESTION: 57

For real *x*, the function will assume all real values provided :

Solution:

QUESTION: 58

The ellipse x^{2}+4y^{2}=4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point(4, 0). Then the equation of the ellipse is:

Solution:

QUESTION: 59

n non-zero real numbers (n ≥ 2) are written on a board. Ritu erases any two numbers, say a and b, and then writes the numbers and instead. Then which of the following is true?

Solution:

QUESTION: 60

If a ∗ b = a+b-2 and if x ∗ 3 = 7 then what is the value of x^{-1}?

Solution:

QUESTION: 61

If a polynomial g(x) satisfies x g(x + 1) = (x - 3)g(x) for all x and g(3) = 6, then the value of g(25) is

Solution:

QUESTION: 62

If ∫ tan^{7 } xdx = f (x) + log | cos x | then

Solution:

QUESTION: 63

Three roots of the equation, x^{4} − px^{3} + qx^{2} − rx + s = 0 are tan A, tan B and tan C where A, B, C are the angles of a triangle. The fourth root of the biquadratic is:

Solution:

QUESTION: 64

If *I*^{k} means logloglog....log*x*, the log being repeated *K* times, then is equal to

Solution:

QUESTION: 65

The value of , where [.] is greatest integer function is

Solution:

*Multiple options can be correct

QUESTION: 66

Which of the following function(s) from f : A → A are not invertible, where A=[-1,1]:

Solution:

*Multiple options can be correct

QUESTION: 67

A tangent is drawn at point P (x_{1} , y_{1}) on the hyperbola If pair of tangents are drawn from any point on this tangent to the circle x^{2} + y^{2} = 16 such that chords of contact are concurrent at the point ( x_{2} , y_{2} ) then

Solution:

*Multiple options can be correct

QUESTION: 68

A particle is moving in a straight line such that its distance at any time t is given by Then

Solution:

*Multiple options can be correct

QUESTION: 69

Let f (x) = sin x + cos x be defined in [0 , 2π] , then f (x)

Solution:

*Multiple options can be correct

QUESTION: 70

If the tangents drawn from the point (0, 2) to the parabola *y ^{2} = 4ax* are inclined at angle 3 π 4 , then the value of '

Solution:

*Multiple options can be correct

QUESTION: 71

Circles are drawn on chords of the rectangular hyperbola xy = c^{2} parallel to the line y = x as diameters. All such circles pass through two fixed points whose co-ordinates are

Solution:

*Multiple options can be correct

QUESTION: 72

In the triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Then which of the following statements is true?

Solution:

*Multiple options can be correct

QUESTION: 73

If the line ax + by + c = 0 is a normal to the hyperbola xy = 1, then

Solution:

*Multiple options can be correct

QUESTION: 74

The value of x satisfying are

Solution:

*Multiple options can be correct

QUESTION: 75

If M ans N are two events, the pobability that exactly one of them occurs is

Solution:

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