General Mathematics I — salang Past Questions

University of Ilorin MTH101: 50 questions.

Topics: Sequence and Series, Trigonometry Function of Angle, Quadratic Equation, Set Theory and Venn Diagram, Complex Number, Binomial Theorem, Circular Measure

  1. If 8, X, Y, Z, 20 are in AP. Find the values of X, Y, and Z.
    1. X = 11, Y = 14, Z = 17
    2. X = 10, Y = 12, Z = 14
    3. X = 12, Y = 14, Z = 16
    4. X = 9, Y = 12, Z = 15
  2. If cot A = 12/5, evaluate (cot 2A)/(sin A + cos A). Given that A is an acute angle.
    1. 120/221
    2. 60/221
    3. 240/221
    4. 30/221
  3. Factorise 25x² - 15x + 2 = 0.
    1. (5x - 1)(5x - 2)
    2. (5x + 1)(5x + 2)
    3. (25x - 1)(x - 2)
    4. (25x + 1)(x + 2)
  4. Let P = {1, 2, 3, 4, 5, 6, 7} and Q = {2, 4, 6, 8, 10}. Find P - Q.
    1. {1, 3, 5, 7}
    2. {2, 4, 6}
    3. {8, 10}
    4. {1, 2, 3, 4, 5, 6, 7, 8, 10}
  5. Write the general nth term of the sequence 3, 4, 6, 10, 18, ...
    1. 2ⁿ⁻¹ + 2
    2. 2ⁿ + 1
    3. 3ⁿ⁻¹ + 1
    4. n² - n + 3
  6. In a class of 20 students, 10 like Mathematics, 14 like English. If 6 students like both, how many students like neither Math nor English?
    1. 2
    2. 4
    3. 6
    4. 8
  7. List the elements of the set A = {x | x > 50 and x is odd}.
    1. {51, 53, 55, ...}
    2. {49, 51, 53, ...}
    3. {50, 51, 52, ...}
    4. {52, 54, 56, ...}
  8. Find three numbers in G.P whose sum is 7 and product is 8.
    1. 1, 2, 4
    2. 2, 2, 3
    3. 1, 3, 3
    4. 2, 3, 2
  9. Express π/2 in degrees.
    1. 90°
    2. 180°
    3. 45°
    4. 360°
  10. Let U and V be the roots of the equation 3a² + 4a + 5 = 0. Find the value of U² + V².
    1. -14/9
    2. 14/9
    3. -26/9
    4. 26/9
  11. The first and second terms of a sequence are 9 and 44 respectively. The general nth term is A + B·3ⁿ. Find A and B.
    1. A = 5, B = 4
    2. A = 4, B = 5
    3. A = 3, B = 6
    4. A = 6, B = 3
  12. Find the Arithmetic mean of 4 and 10.
    1. 7
    2. 6
    3. 8
    4. 9
  13. Express sin 4x + sin 2x as a product of two trigonometric ratios.
    1. 2 sin 3x cos x
    2. 2 cos 3x sin x
    3. sin 6x
    4. cos 6x
  14. Express -2 - 2i in polar form.
    1. 2√2 (cos 225° + i sin 225°)
    2. 2 (cos 45° + i sin 45°)
    3. 2√2 (cos 45° + i sin 45°)
    4. 2 (cos 225° + i sin 225°)
  15. Find the value of (0.98)⁴ using the expansion of (1 - x)⁴.
    1. 0.9224
    2. 0.9412
    3. 0.9086
    4. 0.9604
  16. List 3 geometric means between 3 and 48.
    1. 6, 12, 24
    2. 4, 8, 16
    3. 5, 10, 20
    4. 7, 14, 28
  17. Evaluate cos 2A in terms of angle A.
    1. cos²A - sin²A
    2. 1 - 2sin²A
    3. 2cos²A - 1
    4. All of the above
  18. Write the general nth term for the sequence 1, -1, 1, -1, 1, -1, ...
    1. (-1)ⁿ⁻¹
    2. (-1)ⁿ
    3. cos(nπ)
    4. sin(nπ/2)
  19. If © and ® are the roots of x² + 5x - 6 = 0, find the value of 1/(© + 1) + 1/(® + 1).
    1. 5/7
    2. 7/5
    3. -5/7
    4. -7/5
  20. Evaluate tan 3A in terms of angle A.
    1. (3tanA - tan³A)/(1 - 3tan²A)
    2. (tanA + tan2A)/(1 - tanA tan2A)
    3. (3tanA + tan³A)/(1 + 3tan²A)
    4. (tanA - tan2A)/(1 + tanA tan2A)
  21. Find the values of x for which the expression 1/(x² + 8x + 15) is undefined.
    1. x = -3, -5
    2. x = 3, 5
    3. x = -8, -15
    4. x = 0
  22. Express cos 9x sin 3x as a sum of two trigonometric functions.
    1. (1/2)(sin 12x - sin 6x)
    2. (1/2)(cos 12x + cos 6x)
    3. (1/2)(sin 12x + sin 6x)
    4. (1/2)(cos 12x - cos 6x)
  23. The 2nd term of an exponential sequence is 35 and the 4th term is 875. Find the 6th term.
    1. 21875
    2. 4375
    3. 17500
    4. 6125
  24. If \( A = 2 - 3i \) and \( B = 3 + 5i \), find \( A - B \).
    1. \(-1 - 8i\)
    2. \(5 + 2i\)
    3. \(-1 + 2i\)
    4. \(1 - 8i\)
  25. Express \( \cos^4 A \) in multiple angles.
    1. \(\frac{3 + 4\cos 2A + \cos 4A}{8}\)
    2. \(\frac{1 + 2\cos 2A + \cos 4A}{4}\)
    3. \(\frac{1 + \cos 4A}{2}\)
    4. \(\frac{3 + \cos 4A}{4}\)
  26. Expand \( (x + 5y)^4 \).
    1. \(x^4 + 20x^3y + 150x^2y^2 + 500xy^3 + 625y^4\)
    2. \(x^4 + 10x^3y + 75x^2y^2 + 250xy^3 + 625y^4\)
    3. \(x^4 + 15x^3y + 100x^2y^2 + 300xy^3 + 625y^4\)
    4. \(x^4 + 25x^3y + 125x^2y^2 + 500xy^3 + 625y^4\)
  27. The third term of a GP is 63 and the fifth term is 576. Find the sum of the first 6 terms of the GP.
    1. 1701
    2. 1890
    3. 2016
    4. 2187
  28. An arc PQ of length 20 cm is marked on a circle of radius 6 cm. Find the area of the sector bounded by this arc PQ and the radius.
    1. 60 cm²
    2. 72 cm²
    3. 84 cm²
    4. 96 cm²
  29. Find the values of \( x \) given that \( x+1, 2x, 2x+3 \) are consecutive terms of an AP.
    1. 2
    2. 3
    3. 4
    4. 5
  30. Find \( P \) if the equation \( (5a + 1)x^2 - 8ax + 3a = 0 \) has equal roots.
    1. 1
    2. 2
    3. 3
    4. 4
  31. Obtain the expansion of \( (2 + x + x^2)^4 \) as far as the terms in \( x^3 \).
    1. 16 + 32x + 48x^2 + 48x^3
    2. 8 + 16x + 24x^2 + 24x^3
    3. 32 + 64x + 96x^2 + 96x^3
    4. 64 + 128x + 192x^2 + 192x^3
  32. Find the sum of the roots of the equation \( 3x^2 - 2 = 1.5x \).
    1. 0.5
    2. 1
    3. 1.5
    4. 2
  33. The sum of the first \( n \) terms of a series is given by the formula \( n^2 \). What is the formula for the \( n \)-th term?
    1. \( 2n - 1 \)
    2. \( n^2 - (n-1)^2 \)
    3. \( n + (n-1) \)
    4. \( n^2 - 1 \)
  34. The sum of the terms in an AP is 110. The last term is 20. Find the number of terms in the AP if its first term is 2.
    1. 10
    2. 11
    3. 12
    4. 13
  35. Find the value of \( \tan(A + B) \) without using a table if \( \sin A = \frac{4}{5} \) and \( \cos B = \frac{\sqrt{2}}{2} \).
    1. 7
    2. 8
    3. 9
    4. 10
  36. In a class of science students, 4 students like only physics, 7 like only chemistry, and 5 like only biology. If 10 students like both Biology and Chemistry, 5 students like Physics and Biology only, and 11 students like both Physics and Chemistry. How many students like the 3 courses if only 39 students out of the 50 students in the class like either of these subjects?
    1. 3
    2. 4
    3. 5
    4. 6
  37. Simplify \( \cos 75^\circ \).
    1. \(\frac{\sqrt{6} - \sqrt{2}}{4}\)
    2. \(\frac{\sqrt{6} + \sqrt{2}}{4}\)
    3. \(\frac{\sqrt{3} - 1}{2}\)
    4. \(\frac{\sqrt{3} + 1}{2}\)
  38. Express \( 45^\circ \) in radians.
    1. \(\frac{\pi}{4}\)
    2. \(\frac{\pi}{2}\)
    3. \(\frac{\pi}{6}\)
    4. \(\frac{\pi}{3}\)
  39. Express \( \cos 4x - \cos 2x \) as a product of two trigonometric ratios.
    1. \(-2 \sin 3x \sin x\)
    2. \(2 \sin 3x \sin x\)
    3. \(-2 \cos 3x \cos x\)
    4. \(2 \cos 3x \cos x\)
  40. Express \( z = 1 + i \) in polar form.
    1. \(\sqrt{2} \text{cis} \frac{\pi}{4}\)
    2. \(\sqrt{2} \text{cis} \frac{\pi}{2}\)
    3. \(2 \text{cis} \frac{\pi}{4}\)
    4. \(2 \text{cis} \frac{\pi}{2}\)
  41. Determine the complex number which satisfies \( z(3 + 5i) = -1 - i \).
    1. \(\frac{-4}{17} + \frac{1}{17}i\)
    2. \(\frac{-4}{17} - \frac{1}{17}i\)
    3. \(\frac{4}{17} + \frac{1}{17}i\)
    4. \(\frac{4}{17} - \frac{1}{17}i\)
  42. Find the value of \( a \) in \( (a + 2x)^7 \) if the coefficient of \( x^4 \) is 35,840.
    1. 4
    2. 5
    3. 6
    4. 7
  43. List the elements of the set \( B = \{ y \mid y^2 = 4 \text{ and } y \text{ is positive} \} \).
    1. \{2\}
    2. \{-2, 2\}
    3. \{4\}
    4. \{-4, 4\}
  44. Find 4 arithmetic means between 6 and 31.
    1. 11, 16, 21, 26
    2. 10, 15, 20, 25
    3. 12, 17, 22, 27
    4. 13, 18, 23, 28
  45. Find 6 geometric means between 2 and 4374.
    1. 6, 18, 54, 162, 486, 1458
    2. 4, 12, 36, 108, 324, 972
    3. 8, 24, 72, 216, 648, 1944
    4. 10, 30, 90, 270, 810, 2430
  46. Express \( \sin 5x \sin 3x \) as a sum of two trigonometric ratios.
    1. \(\frac{1}{2} [\cos 2x - \cos 8x]\)
    2. \(\frac{1}{2} [\sin 8x - \sin 2x]\)
    3. \(\frac{1}{2} [\cos 8x - \cos 2x]\)
    4. \(\frac{1}{2} [\sin 2x + \sin 8x]\)
  47. The sum of the first 3 terms of an AP is 21. If the difference between the third term and the first term is 4, find the sum of the next 4 terms of the sequence.
    1. 56
    2. 60
    3. 64
    4. 68
  48. If \( t = \tan\left(\frac{1}{2}A\right) \), what is the value of \( \cos A \) in terms of \( t \)?
    1. \(\frac{1 - t^2}{1 + t^2}\)
    2. \(\frac{2t}{1 + t^2}\)
    3. \(\frac{1 + t^2}{1 - t^2}\)
    4. \(\frac{t^2 - 1}{t^2 + 1}\)
  49. The first and second term of a sequence is 3 and 5 respectively. If the 3rd term and above is gotten by multiplying 3 to the two terms before and subtracting the term before. Find the sum of the first five terms of such sequence.
    1. 97
    2. 101
    3. 105
    4. 109
  50. Evaluate \( \sum_{r=0}^3 (2r^2 + 4r + 6) \).
    1. 50
    2. 54
    3. 58
    4. 62

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