1. Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A ∪ B.
a) 3
b) 6
c) 9
d) 18

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2. If the set A contains 5 elements, then the number of elements in the power set P(A) is equal to
a) 32
b) 25
c) 16
d) 8
e) 10

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1. If z = r(cosθ + isinθ), then the value of $\frac{z}{\bar{z}} + \frac{\bar{z}}{z}$ is
a) cos2θ
b) 2 cos2θ
c) 2 cosθ
d) 2 sinθ
e) 2 sin 2θ

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2. Find the value of (1 + 2ω + ω²)³ⁿ - (1 - ω + 2ω²)³ⁿ =
a) 0
b) 1
c) ω
d) ω²

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1. The number of times the digit 3 will be written when listing the integers from 1 to 1000 is
a) 269
b) 300
c) 271
d) 302

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2. There is a rectangular sheet of dimension (2m - 1) x (2n - 1), (where , m > 0 , n > 0). It has been divided into square of unit area by drawing lines perpendicular to the sides. Find number of rectangles having sides of odd unit length

a) (m + n + 1)²
b) mn(m + 1)(n + 1)
c) 4^m+n-2
d) m²n²

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1. In the expansion of $\left (x + \frac{2}{x^{2}} \right )^{15}$, the term independent of x is
a) ¹⁵C₆2⁶
b) ¹⁵C₅2⁵
c) ¹⁵C₄2⁵
d) ¹⁵C₈2⁸

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2. Let P(n) be a statement and let P(n) = P(n + 1) for all natural numbers n, then P(n) is true
a) For all n
b) For all n > 1
c) For all n > m, m being a fixed positive integer
d) Nothing can be said

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1. The equations y = ±$sqrt{3}$x, y = 1 are the sides of
a) An equilateral triangle
b) A right angled triangle
c) An isosceles triangle
d) An obtuse angled triangle

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2. The distance of the lines 2x - 3y = 4 from the point (1,1) measured parallel to the line x + y = 1 is
a) $\sqrt{2}$
b) $\frac{5}{\sqrt{2}}$
c) $\frac{1}{\sqrt{2}}$
d) 6

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1. Let P(6,3) be a point ob the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1.If the normal at the point P intersects the x-axis at (9,0) , then the eccentricity of the hyperbola is
a) $\sqrt{\frac{5}{2}}$
b) $\sqrt{\frac{3}{2}}$
c) $\sqrt{2}$
d) $\sqrt{3}$

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2. The line y = x intersects the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{25}$ = 1 at the points P and Q. The eccentricity of ellipse with PQ is major axis and minor axis of length $\frac{5}{\sqrt{2}}$, is
a) $\frac{\sqrt{5}}{3}$
b) $\frac{5}{\sqrt{3}}$
c) $\frac{5}{9}$
d) $\frac{2\sqrt{2}}{3}$

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1. The negition of the statement If I become a teacher , then I will open a school is
a) I will become a teacher and I will not open school
b) Either I will not become a teacher or I will not open a school
c) Neither I will become a teacher not I will open a school
d) I will not become a teacher or I will open a school

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2. Which of the following statements is a tautology
a) (∼q ^ p) ^ q
b) (∼ q ^ p) ^ (p ^ ∼ p)
c) (∼q ^ p) ∨ (p∨ ∼ p)
d) (p ^ q) ^ (∼(p ^ q)

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1. A particle is dropped under gravity from rest from a height h(g = 9.8m/sec²) and then it travels a distance $\frac{9h}{25}$ in the last second. The height h is
a) 100 metre
b) 122.5 metre
c) 145 metre
d) 167.5 metre

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2. A particle moves towards east from a point A to a point B at this rate of 4km/h . Of AB = 12km and BC to 5km, then its average speed for its hourney from A to C and resultant average velocity direction from A to C are respectively
a) $\frac{13}{9}km/h$ and $\frac{17}{9}km/h$
b) $\frac{13}{4}km/h$ and $\frac{17}{4}km/h$
c) $\frac{17}{9}km/h$ and $\frac{17}{9}km/h$
d) V = $\frac{u}{1 - kxu}$

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1. If α , β, γ are roots of equation x³ + ax² + bx + c = 0, then α^-1 + β^-1 + γ^-1 =
a) a/c
b) -b/c
c) b/a
d) c/a

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2. The value of k for which the quadratic equation, kx² + 1 = kx + 3x - 11x² has real and equal roots are
a) -11, -3
b) 5,7
c) 5,-7
d) None of these

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1. Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. . Then the common ratio of the G.P is
a) $2 - \sqrt{3}$
b) $2 + \sqrt{3}$
c) $\sqrt{2} + \sqrt{3}$
d) $3 + \sqrt{2}$

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2. The value of n for which $\frac{x^{n+1} + y^{n+1}}{x^{n} + y^{n}}$ is the geometric mean of x and y is
a) n = $-\frac{1}{2}$
b) n = $\frac{1}{2}$
c) n = 1
d) n = -1

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1. The coefficient of xⁿ in the expansion of log_e(1 + 3x + 2x²) is
a) $\left (-1 \right )^{n}\left [\frac{2^{n} + 1}{n} \right ]$
b) $\frac{\left (-1 \right )^{n+1}}{n}\left [2^{n} + 1 \right ]$
c) $\frac{2^{n} + 1}{n}$
d) None of these

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2. The sum to infinity to the given series $\frac{1}{n} - \frac{1}{2n^{2}} + \frac{1}{3n^{3}} - \frac{1}{4n^{4}} + ....$ is
a) $log_{e}\left (\frac{n + 1}{n} \right )$
b) $log_{e}\left (\frac{n}{n + 1} \right )$
c) $log_{e}\left (\frac{n - 1}{n} \right )$
d) $log_{e}\left (\frac{n}{n - 1} \right )$

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1. The value of $\frac{tanA + secA - 1}{tanA - secA + 1}$
a) $\frac{1 + cosA}{sinA}$
b) $\frac{1 + sinA}{cosA}$
c) $\frac{1 - cosA}{1 + cosA}$
d) $\frac{1 + sinsA}{1 - sinA}$

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2. If sinx + cosecx = 2, then sinⁿx + cosecⁿx is equal to
a) 2
b) 2ⁿ
c) 2^n-1
d) 2^n-2

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1. If cosθ + cos7θ + cos3θ + cos5θ = 0, then is
a) $\frac{n\pi}{4}$
b) $\frac{n\pi}{2}$
c) $\frac{n\pi}{8}$
d) None of these

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2. ABCD is a trapezium such that AB and CD are parallel and BC ⊥ CD. If ∠ADB = θ , BC = p and CD = q, then AB is equal to
a) $\frac{\left (p^{2} + q^{2} \right )sin\theta}{pcos\theta + q sin\theta}$
b) $\frac{p^{2} + q^{2}cos\theta}{pcos\theta + qsin\theta}$
c) $\frac{p^{2} + q^{2}}{p^{2}cos\theta + q^{2}sin\theta}$
d) $\frac{\left (p^{2} + q^{2} \right )sin\theta}{\left (pcos\theta + q sin\theta \right )^{2}}$

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1. tanh^-1x =
a) $\frac{1}{2}log\left (\frac{x + 1}{x - 1} \right )$
b) $\frac{1}{2}log\left (\frac{x - 1}{x + 1} \right )$
c) $\frac{1}{2}log\left (\frac{1 - x}{1 + x} \right )$
d) $\frac{1}{2}log\left (\frac{1 + x}{1 - x} \right )$

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2. The period of cosh(4x) is
a) 2πi
b) πi
c) $\frac{\pi i}{2}$
d) 2π

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1. If (0,β) lies on a inside the triangle with sides y + 3x + 2 = 0, 3y - 2x - 5 = 0 and 4y + x - 14 = 0, then
a) $0 \le \beta \le \frac{7}{2}$
b) $0 \le \beta \le \frac{5}{2}$
c) $\frac{5}{3} \le \beta \le \frac{7}{2}$
d) None of these

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2. Let A(2,-3) and B(-2,1) be vertices of a triangle ABC . If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line
a) 3x - 2y = 3
b) 2x - 3y =7
c) 3x + 2y =5
d) 2x + 3y =9

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1. The centre of the circle passing through (0,0) and (1,0) and touching the circle x² + y² = 9 is
a) $\left (\frac{1}{2}, \frac{1}{2} \right )$
b) $\left (\frac{1}{2}, -\sqrt{2} \right )$
c) $\left (\frac{3}{2}, \frac{1}{2} \right )$
d) $\left (\frac{1}{2}, \frac{3}{2} \right )$

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2. If the tangent to the circle x² + y² = r² at the point (a,b) meets the coordinate axes at the point A and B and O is the origin, then the area of the triangle OAB is
a) $\frac{r^{4}}{2ab}$
b) $\frac{r^{4}}{ab}$
c) $\frac{r^{2}}{2ab}$
d) $\frac{r^{2}}{ab}$

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1. The point of intersection of the lines of the lines represented by the equation 2(x + 2)² + 3(x + 2)(y - 2) - 2(y - 2)² = 0 is
a) (2,2)
b) (-2,-2)
c) (-2,2)
d) (-2,1)

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2. If the bisectors of the lines x² - 2pxy - y² = 0 be x² - 2qxy - y² = 0, then
a) pq + 1 = 0
b) pq - 1 = 0
c) p + q = 0
d) p - q = 0

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