2020 math

Section A

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1. A line parallel to the straight line 2x – y = O is tangent to the hyperbola X²/4-Y²/2 = 1 at the point (x₁, y₁). Then x₁² + 5y₁² is equal to :

(1) 6
(2) 10
(3) 8
(4) 5

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2.The domain of the function 🎄🎄🎄🎄Then a is equal to :

(1)√17-1/2
(2)√17/2
(3)√17+1/2
(4)√17/2+1

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3. If a function ƒ(x) defined by
🎄🎄🎄🎄 be continuous for some a, b, c ε R and ƒ'(0) + ƒ'(2) = e, then the value of a is :

(1) 1/e² – 3e + 13
(2) e/e² – 3e – 13
(3) e/e² + 3e + 13
(4) e/e² – 3e + 13

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4. The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

(1) (-∞,-9] – [3, ∞]
(2) [1–3,∞)
(3) (-∞, 9]
(4) (-∞,-3] → [9, ∞]

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5. If R = {(x, y) : x, y ε = Z, x² + 3y²≤8} is a relation on the set of integers Z, then the domain of R-1 is :

(1) {-1,0,1}
(2) {-2,-1,1,2}
(3) {0,1}
(4) {-2,-1,0,1, 2}

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6. The value of 🎄🎄🎄🎄 is

(1) -1/2(1-i√3)
(2) 1/2(1-i√3)
(3) -1/2(√3-i)
(4) 1/2(√3-i)

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7. Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at Pis:

(1) x + 3y – 62 = 0
(2) x – 3y – 11 = 0
(3) x – 3y + 22 = 0
(4) x + 3y + 26 = 0

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8. Let A be a 2 x 2 real matrix with entries from {0, 1} and | A | ≠ 0. Consider the following two statements:

(P) If A¹ I₂, then | A |=-1
(Q) If |A|=1, then tr(A) = 2,
where I₂ denotes 2 x 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then:

(1) Both (P) and (Q) are false
(2) (P) is true and (Q) is false
(3) Both (P) and (Q) are true
(4) (P) is false and (Q) is true

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9. Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a nonprime number. The probability that the card was drawn from Box I is :

(1) 4/17
(2) 8/17
(3) 2/5
(4) 2/3

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10. If p(x) be a polynomial of degree three that
has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :

(1) 12
(2) -12
(3) -24
(4) 6

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11. The contrapositive of the statement “If I reach the station in time, then I will catch the train” is :

(1) If I will catch the train, then I reach the station in time.
(2) If I do not reach the station in time, then I will catch the train.
(3) If I do not reach the station in time, then I will not catch the train.
(4) If I will not catch the train, then I do not reach the station in time.

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12. Let α and β be the roots of the equation,
5x² + 6x – 2 = 0. If Sn = αn + βn, n = 1, 2, 3, ….., then:

(1) 5S₆ + 6S₅ + 2S₄ = 0
(2) 6S₆ + 5S₅ = 2S₄
(3) 6S₆ + 5S₅ + 2S₄ = 0
(4) 5S₆ + 6S₅ = 2S₄

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13. If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining
🎄🎄🎄🎄then:

(1) b = π/2 + a
(2) |a + b| = 1
(3) |6-a| = 1
(4) b = a

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14. Area (in sq. units) of the region outside
|x|/2+ |y|/3 = 1 and inside the ellipse x²/4 +y²/9 = 1 is :

(1) 3(π – 2)
(2) 6(π – 2)
(3) 6(4 – π)
(4) 3(4 – π)

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15. If |x|< 1,|y| < 1 and x ≠ y, then the sum to
infinity of the following series :

(1)🎄🎄
(2)🎄🎄
(3)🎄🎄
(4)🎄🎄

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16. Let α > 0, β > O be such that α² + β²
= 4. If the maximum value of the term independent of x in the binomial expansion of 🎄🎄 is 10k, then k is equal to:

(1) 176
(2) 336
(3) 352
(4) 84

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17. Let S be the set of all λ ε R for which the
system of linear equations
2x – y + 2z = 2
x – 2y + λz=-4
x + λy + z = 4
has no solution. Then the set S

(1) is an empty set.
(2) is a singleton
(3) contains more than two elements.
(4) contains exactly two elements.

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18. Let X = {x ε N:1 ≤ x ≤ 17} and Y = {ax. b:x ε X and a, b ε R, a > 0}. If mean ana variance of elements of Y are 17 and respectively then a + b is equal to :

(1) -27
(2) 7
(3) -7
(4) 9

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19. Let y = y(x) be the solution of the differentia 2+sinx/y+1 dy/dx =-cosx. If y > 0, y(0)
dx
1. If y(π) = a, and dy/dx at x = π is b, then the ordered pair (a, b) is equal to :

(1) (2, 3)
(2) (1,1)
(3) (2, 1)
(4) (1,-1)

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20. The plane passing through the points (1,2,1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point :

(1) (0, -6, 2)
(2) (0,6,-2)
(3) (-2,0,1)
(4) (2,0,-1)

Section B

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21. The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² +
y² – 2x – 4y + 4 = 0 at two distinct points is :

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22. Let 🎄🎄 be three unit vectors such
that 🎄🎄 Then find the
value of 🎄🎄

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23. If the letters of the word ‘MOTHER’be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word
‘MOTHER’ is :

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24. If 🎄🎄

then the value of n is equal to :

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25. The integral 🎄🎄 is equal to :