1. If X = {8ⁿ - 7n - 1 : n ε N} and Y = {49(n - 1) : n ε N}, then
a) X ⊆ Y
b) Y ⊆ X
c) X = Y
d) None of these

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2. If A = [(x,y) : x² + y² = 25]
and B = [(x,y) : x² + 9y² = 144], then A ∩ B contains
a) One point
b) Three points
c) Two points
d) Four points

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1. If z = 1 + i, then the multiplicative inverse of z² is (where I = $\sqrt{-1}$)
a) 2i
b) 1 - i
c) -i/2
d) i/2

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2. The number of solutions of the system of equations Re(z² = 0, |z| = 2 is
a) 4
b) 3
c) 2
d) 1

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1. 20 persons are invited for a party. In how many different ways can they and the host be seated at a circular table. If the two particular persons are to be seated on either side of the host
a) 20!
b) 2 . 18!
c) 18!0
d) None of these

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2. The total number of natural numbers of six digits that can be made with digits 1,2,3,4, if all digits are to appear in the same number at least once is
a) 1560
b) 840
c) 1080
d) 480

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1. $\binom{n}{0} + 2\binom{n}{1} + 2^{2}\binom{n}{2} + .... + 2^{n \binom{n}{n}}$ is equal to
a) 2ⁿ
b) 0
c) 3ⁿ
d) None of these

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2. For r = 0,1,…..,10, let A_r, B_r and C_r denote, respectively, the coefficient of xr in the expansion of (1+x)¹⁰ , (1 + x)²⁰ and (1 + x)³⁰. Then , $\sum_{r = 1}^{10} A_{r}\left (B_{10}B_{r} - C_{10}A_{r} \right )$ is equal to
a) B₁₀ - C₁₀
b) A₁₀($B_{10}^{2}$ - C₁₀A₁₀)
c) 0
d) C₁₀ - B₁₀

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1. The coordinates of the foot of the perpendicular from (0,0) upon the line x + y = 2 are
a) (2,-1)
b) (-2,1)
c) (1,1)
d) (1,2)

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2. Equation of angle bisectors between x and y axes are
a) y = ± x
b) y = ±3x
c) $x = \pm\frac{1}{\sqrt{2}}x$
d) y = ±3x

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1. A tangent to a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1 intercepts a length of unity from each if the co-ordinate axes, then the point (a,b) lies on the rectanglar hyperbola.
a) x² - y² = 2
b) x² -y² = 1
c) x² - y² = -1
d) None of these

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2. If the area of the auxillary circle of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}$ = 1(a > b) is twice the area of the ellipse , then the eccentricity of the ellipse is
a) $\frac{1}{\sqrt{2}}$
b) $\frac{\sqrt{3}}{2}$
c) $\frac{1}{\sqrt{3}}$
d) $\frac{1}{2}$

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1. Let p : roses and q : the sun is a star. Then the verbal translation is (∼) ∨ q is:
a) Roses are not real and the sun is not a star
b) It is not true that roses are red or the sun is not a star
c) It is not true that roses are red and the sun is not a star
d) Roses are not red or the sun is a star
e) It is not true that roses are red and the sun is a star

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2. Let S be a non-empty subset of R. consider the following statement.
p : There is a rational number x ε S such that x > 0 .
Which of the following statements is the negation of the statement p
a) There is a rational number x ε S such that x ≤ 0
b) There is no rational number x ε S such that x ≤ 0
c) Every rational number x ε S satisfies x ≤ 0
d) x ε S and x ≤ 0 ⇒ is not rationsl

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1. Forces M and N acting at a point P makes an angle 150⁰ . Their resultant acts at O magnitude 2 unit and is perpendicular to M. Then, in the same unit, the magnitudes of M and N are
a) 2$\sqrt{3}$,4
b) $\sqrt{\frac{3}{2}}$,2
c) 3,4
d) 4.5

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2. If the time taken in slipping down on smooth inclined plane is twice to the time taken in falling from the vertical height of that plane, the inclination ofp lane will be
a) 45⁰
b) 60⁰
c) 75⁰
d) 30⁰

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1. The value of a and b for which equation x⁴ - 4x³ + ax² + bx + 1 = 0 have four real roots
a) -6, -4
b) -6,5
c) -6,4
d) 6,-4

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2. For k > 0 , the set of all values of k for which ke^x - x = 0 has two distinct roots is
a) $\left (0, \frac{1}{e} \right )$
b) $\left (\frac{1}{e} , 1\right )$
c) $\left (\frac{1}{e} , \infty \right )$
d) (0,1)

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1. n^th term of the series $1 + \frac{4}{5}+ \frac{7}{5^{2}} + \frac{10}{5^{3}} + .....$ will be
a) $\frac{3n + 1}{5^{n-1}}$
b) $\frac{3n - 1}{5^{n}}$
c) $\frac{3n - 2}{5^{n-1}}$
d) $\frac{3n + 2}{5^{n-1}}$

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2. If a, ,b, c, d be in H.P. then
a) a² + c² > b² + d²
b) a² + d² > b² + c²
c) ac + bd > b² + c²
d) ac + bd > b² + d²

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1. In the expansion of $\frac{a + bx}{e^{x}}$ , the coefficient of x^r is
a) $\frac{a - b}{r!}$
b) $\frac{a - br}{r!}$
c) $\left (-1 \right )^{r}\frac{a - br}{r!}$
d) None of these

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2. The coefficient of xⁿ in the expansion of log_e(1+x) is
a) $\frac{\left (-1 \right )^{n - 1}}{n}$
b) $\frac{\left (-1 \right )^{n - 1}}{n}log_{a}e$
c) $\frac{\left (-1 \right )^{n - 1}}{n}log_{e}a$
d) $\frac{\left (-1 \right )^{n}}{n}log_{a}e$

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1. In any triangle ABC, $sin^{2}\frac{A}{2} + sin^{2}\frac{B}{2} + sin^{2}\frac{C}{2}$ is equal to
a) $1 - 2cos\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$
b) $1 - 2sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$
c) $1 - 2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}$
d) $1 - 2cos\frac{A}{2}cos\frac{B}{2}sin\frac{C}{2}$

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2. If sinθ + cosθ = x, then s⁶θ + cos⁶θ = $\frac{1}{4}\left [4 - 3\left (x^{2} - 1 \right )^{2} \right ]$ for
a) All real x
b) x² ≤ 2
c) x² ≥ 2
d) None of these

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1. The value of n ε Z for which the function f(x) = $\frac{sin nx}{sin\left (x/n \right )}$ has 4π as its period , is
a) 2
b) 3
c) 4
d) 5

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2. In a triangle with one angle of 120⁰ the lengths of the sides form an A.P. .If the length of the greatest side is 7cm then, the area triangle is
a) $\frac{3\sqrt{15}}{4}cm^{2}$
b) $\frac{15\sqrt{3}}{4}cm^{2}$
c) $\frac{15}{4}cm^{2}$
d) $\frac{3\sqrt{3}}{4}cm^{2}$

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1. If sinx coshy = cosθ and cosx sinhy = sinθ, then sinh²y equals
a) sin²x
b) cosh²x
c) cos²x
d) 1

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2. sin²(ix) + cosh²x is equal to
a) 1
b) -1
c) 2cosh²x
d) cosh2x

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1. The following points A(2a,4a), B(2a,6a) and C(2a, $\sqrt{3}a$,5a), (a>0) are the vertices of
a) An acute angled triangle
b) A right angles triangle
c) An isosceles triangle
d) None of these

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2. Point Q is symmetric to P(4,-1) with respect to the bisector of the first quadrant. The length of PQ is
a) 3$\sqrt{2}$
b) 5$\sqrt{2}$
c) 7$\sqrt{2}$
d) 9$\sqrt{2}$

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1. The equation of the circle of radius 5 and touching the coordinate axes in third quadrant is
a) (x - 5)² + (y + 5)² = 25
b) (x + 4)² + (y + 4)² = 25
c) (x + 6)² + (y + 6)² = 25
d) (x + 5)² + (y + 5)² = 25

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2. The equation of the circle with origin as centre passing through the vertices of an equilateral triangle whose median is of length 3a is
a) x² + y² = 9a²
b) x² + y² = 16a²
c) x² + y² = a²
d) None of these

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1. The joint equation of the straight lines x + y = 1 and x - y = 4 is
a) x² - y² = -4
b) x² - y² = 4
c) (x + y - 1)(x - y - 4) = 0
d) (x + y + 1)(x - y + 4) = 0

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2. Two of the lines represented by the equation ay⁴ + bxy³ + cx²y² + dx³y + ex⁴ = - will be perpendicular, then
a) (b + d)(ad + be) + (e - a)²(a + c + e) = 0
b) (b + d)(ad + be) + (e + a)²(a + c + e) = 0
c) (b - d)(ad - be) + (e - a)²(a + c + e) = 0
d) (b - d)(ad - be) + (e + a)²(a + c + e) = 0

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