Practice Test-I
1. Let f(x) = x2 + bx + c, where b, c ∈ R. If f(x) is a factor of both x4 + 6x2 + 25 and 3x4 + 4x2 + 28x + 5, then the least value of f(x) is
(A) 2
(B) 3
(C) 5/2
(D) 4
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2. Let a, b, c be the sides of a triangle. No two of them are equal and λ ∈ R. If the roots of the equation x2 + 2(a + b+ c) x + 3λ (ab + bc + ca) = 0 are real, then
(A) λ < 4/3
(B) λ > 5/3
(C) λ ∈(1/3 , 5/3)
(D) λ ∈(4/3 , 5/3)
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3. Let f(x) = x2 + ax + b be a quadratic polynomial in which a and b are integers. If for a given integer n, f(n) f(n + 1) = f(m) for some integer m, then the value of m is
(A) n(a + b) + ab
(B) n2 + an + b
(C) n(n + 1) + an + b
(D) n2 + n + a + b
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4. If the equations x2 + ax + b=0 and x2 + bx + a = 0 have exactly one common root, then the numerical value of a + b is
(A) 1
(B) –1
(C) 0
(D) none of these
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5. The number of ordered pairs of positive integers x, y such that x2 + 3y and y2 + 3x are both perfect squares is
(A) 2
(B) 3
(C) 4
(D) 5
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6. For the equations x2 + bx + c = 0 and 2x2 + (b + 1)x + c + 1 = 0 select the correct alternative
(A) both the equations can have integral roots
(B) both the equations can’t have integral roots simultaneously
(C) none of the equations can have integral roots
(D) nothing can be said
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7. If x2 +ax +b is an integer for every integer x then
(A) ‘ a ‘ is always an integer but ‘ b ‘ need not be an integer.
(B) ‘ b ‘ is always an integer but ‘ a ‘ need not be an integer.
(C) a + b is always an integer.
(D) none of these.
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8. If a , b , c be the sides of ΔABC and equations ax2 + bx + c=0 and 5x2 + 12x + 13=0 have a common root, then ∠C is
(A) 60°
(B) 90°
(C) 120°
(D) 45°
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9. The equation x2 + nx + m = 0, n, m ∈ I, can not have
(A) integral roots
(B) non-integral rational roots
(B) irrational roots
(D) complex roots
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10. If $\large log_{(3x+5)}(ax^2 + 8x + 2) > 2 $ then x lies in the interval
(A) (-4/3 , -20/11)
(B) (-4/3 , -23/22)
(C) (-5/3 , -23/22)
(D) None of these
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11. The number of ordered pairs (a, b) such that the equations ax + by = 1 and x2 + y2 = 50 have all solutions integral is
(A) 72
(B) 66
(C) 84
(D) 36
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12. If the roots of the equation (a2 + b2) x2 + 2x (ac + bd) + c2 + d2 = 0, are real, then these are equal. This statement is (a, b, c, d ∈ R)
(A) true
(B) false
(C) can’t say
(D) none of these
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13. If equation x2 – (2 + m)x + (m2 – 4m + 4) = 0 has coincident roots then
(A) m = 0, 1
(B) m = 2/3 , 1
(C) m = 0, 2
(D) m = 2/3 , 6
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14. A root of the equation, sinx + x – 1 = 0, lies in the interval
(A) (0, π/2)
(B) (- π/2, 0)
(C) (π/2, π)
(D) ( -π, -π/2)
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15. If p, q, r ∈ R and the quadratic equation px2 + qx + r = 0 has no real root, then
(A) p(p + q + r) < 0
(B) p(p – 2q + 4r)
(C) p(p + 4q + 2r) < 0
(D) None of these
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16. If x2 – 4x + log1/2 a = 0 does not have two distinct real roots, then maximum value of a is
(A) 1/4
(B) 1/ 16
(C) –1/4
(D) none of these
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17. The least value of |a| for which tanθ and cotθ are the roots of the equation x2 + ax + b = 0 is
(A) 2
(B) 1
(C) 1/2
(D) 0
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18. If the equation x3 – 3ax2 + 3bx – c = 0 has positive and distinct roots, then
(A) a2 > b
(B) ab > c
(C) a3 > c
(D) a3 > b2 > c
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19. The value of a for which exactly one root of the equation eax2 – e2ax + ea – 1 = 0 lies between 1 and 2 are given by
(A) $\large ln(\frac{5-\sqrt{13}}{4}) < a < ln(\frac{5+\sqrt{13}}{4})$
(B) 0 < a < 100
(C) $ln\frac{5}{4} < a < \frac{10}{3}$
(D) None of these
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20. If α, β be the roots of x2 – a(x – 1) – b = 0, then the value of $\large \frac{1}{\alpha^2 – a \alpha} + \frac{1}{\beta^2 – a \beta} + \frac{2}{a+b}$ is
(A) 4/(a+b)
(B) 1/(a+b)
(C) 0
(D) 1
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21. Consider the equation x3 – nx + 1 = 0 , n ∈ N , n ≥ 3 . Then
(A) Equation has atleast one rational root .
(B) Equation has exactly one rational root.
(C) Equation has all real roots belonging to (0, 1).
(D) Equation has no rational root.
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22. If sina, sin b and cosa are in GP, then roots of x2 + 2xcotβ + 1 = 0 are always
(A) equal
(B) real
(C) imaginary
(D) greater than 1
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23. If a, b ∈ (0, 2) and the equation $\frac{x^2 + 5}{2} = x-2 cos(ax+b)$ has at least one solution then a + b is
(A) 1
(B) 2
(C) e
(D) π
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24. Let P(x) and Q(x) be two polynomials. If f(x) = P(x4) + xQ(x4) is divisible by x2 +1, then (A) P(x) is divisible by (x-1)
(B) Q(x) is divisible by (x-1)
(C) f(x) is divisible by (x-1)
(D) all of them
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25. If α, β be the roots of 4x2 – 16x + λ = 0, λ ∈ R such that 1 < α < 2 and 2 < β < 3, then the number of integral solutions of λ is
(A) 5
(B) 6
(C) 3
(D) 2
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26. The solution of the equation |x + 1|2 – 2|x + 2| – 26 = 0 is
(A) ±7
(B) –7, √29
(C) ±√29
(D) –7, 29
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27. The roots of the equation (c2 –ab)x2 – 2(a2 –bc)x + (b2 – ac) =0 are equal then
(A) a2 + b3 + c3 = 3abc or a = 0
(B) a + b + c = 0
(C) a2 + b3 + c3 = 3abc or a = 1
(D) none of these
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28. If (λ2 + λ – 2)x2 + (λ + 2)x < 1, x ∈ R, then λ belongs to the interval
(A) (-2, 1)
(B) (-2 , 2/5)
(C) (2/5 , 1)
(D) None of these
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29. If expression x2 – 4cx + b2 > 0 ∀ x ∈ R and a2 + c2 < ab, then range of the function $\frac{x+a}{x^2 + bx+ c^2}$ is
(A) (- ∞, 0)
(B) (0, ∞)
(C) (- ∞, ∞)
(D) None of these
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30. If for all real values of x , $ \frac{4x^2 + 1}{64x^2 – 32x sin\alpha + 29} > \frac{1}{32}$
then α lies in the interval
(A) (0, π/3)
(B) (π/3 , 2π/3)
(C) (4π/3 , 5π/3)
(D) none of these
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Click to See All Answers :
11. B 12. A 13. D 14. A 15. B 16. B 17. A 18. A 19. D 20. C
21. A 22. B 23. D 24. D 25. C 26. B 27. A 28. D 29. C 30. A