Mathematics Questions & Answers Practice

August 26, 2019

Mathematics Questions & Answers Practice

1. A group of market women sell at least one of yam, plantain and maize. 12 of them sell maize, 10 sell yam and 14 sell plantain. 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only while 3 sell all the three items. How many women are in the group?

A. 25
B. 19
C. 18
D. 17

Correct Answer. Option A

Explanation

Let the three items be M, Y and P.
n{M ∩ Y} only = 4-3 = 1
n{M ∩ P) only = 5-3 = 2
n{ Y ∩ P} only = 2
n{M} only = 12-(1+3+2) = 6
n{Y} only = 10-(1+2+3) = 4
n{P} only = 14-(2+3+2) = 7
n{M∩P∩Y} = 3

Number of women in the group = 6+4+7+(1+2+2+3) as above =25 women

2. If log 10 to base 8 = X, evaluate log 5 to base 8 in terms of X.

A. $\frac{1}{2}$ X
B. X- $\frac{1}{4}$
C. X- $\frac{1}{3}$
D. X- $\frac{1}{2}$

Explanation

l o g_{8} 10

l o g_{8} 2 x 5

l o g_{8} 2

l o g_{8} 5

2^{3}

l o g_{8} 2 = y

2 = 8^{y}

y = \frac{1}{3}

\frac{1}{3} = l o g_{8} 2

\frac{1}{3}

l o g_{8} 5 = X - \frac{1}{3}

Correct Answer. Option C

3. Find the value of X if

\frac{\sqrt{2}}{x + \sqrt{2}} = \frac{1}{x - \sqrt{2}}

\frac{\sqrt{2}}{x + \sqrt{2}} = \frac{1}{x - \sqrt{2}}

A. 3√2+4
B. 3√2-4
C. 3-2√2
D. 4+2√2

Correct Answer: Option A

Explanation

Start your solution by cross-multiplying, then collect like terms and factorize accordingly to get the unknown.

4. If {(a²b^-3c)^3/4/a^-1b⁴c⁵} = a^pb^qc^r; what is the value of p+2q?

A. (5/2)
B. -(5/4)
C. -(25/4)
D. -10

Correct Answer: Option D

Explanation

Hint: Use BODMAS and algebra to arrive at the values of P = 5/2, q = -25/4 and r = -9/2.
Then substitute the values of p and q into p+2q to get -10.

5. A trader bought 100 oranges at 5 for N1.20, 20 oranges got spoilt and the remaining were sold at 4 for N1.50. Find the percentage gain or loss.

A. 30% gain
B. 25% gain
C. 30% loss
D. 25% loss

Correct Answer: Option B

Explanation

Cost price (cp) = (100/5) x N1.20 =N24.00
Selling price (sp) = 100-20 = 80
(80/4) x N1.50 = N30.00

Gain = SP-CP = N30.00-N24.00 = N6.00
Gain% = (gain/CP) x 100 = 25%

6. What is the answer when 24346 is divided by 426?

A. 23₆
B. 35₆
C. 52₆
D. 55₆

Correct Answer: Option B

Explanation

Hint: Two methods can be used;
1. Direct division (if you know division in number bases)
2. Convert both sides to base 10, divide and convert your answer back to base 6. Your answer should be 356

7. If

\frac{(a^{2} b^{- 3} c)^{\frac{3}{4}}}{a^{- 1} b^{4} c^{5}} = a^{p} b^{q} c^{r}

What is the value of p+2q?

\frac{(a^{2} b^{- 3} c)^{\frac{3}{4}}}{a^{- 1} b^{4} c^{5}} = a^{p} b^{q} c^{r}

Explanation

Hint: apply basic mathematics rules beginning from BODMAS to algebra, and follow solution carefully to arrive at p =(5/2), q = -(25/4) and r = -(9/2).

Then p+2q will give you

\frac{5}{2} + 2 (\frac{- 25}{4}) = - 10

\frac{(a^{2} b^{- 3} c)^{\frac{3}{4}}}{a^{- 1} b^{4} c^{5}} = a^{p} b^{q} c^{r}

8. if 29 x (3Y)9 = 35 x (3Y)5, find the value of Y.

If 2₉ x (3Y)₉ = 3₅ x (3Y)₅, find the value of Y.

A. 4
B. 3
C. 2
D. 1

Correct Answer: Option D

Explanation

The correct answer is 1.888888
PROVE==> (2x9⁰) (3x9⁰ + Yx9¹) = (3x5⁰)(3x5⁰ + Yx5¹).
You multiply to get 2(3+9Y)=3(3+5Y).
You further multiply to get 6+18Y=9+15Y. ==>
Collect like terms 18Y-15Y=9-6.
3Y/3=3/3.
Y=1

9. If

m * n = (\frac{m}{n} - \frac{n}{m}

) for m, n belong to R, evaluate -3*4

A. -25/12 $\frac{- 25}{12}$

\frac{- 7}{12}

\frac{7}{12}

\frac{25}{12}

Correct Answer: Option C

Explanation

m * n = \frac{m}{n} - \frac{n}{m} = \frac{- 3}{4} - \frac{4}{- 3} = \frac{- 3}{4} + \frac{4}{3} = \frac{(- 9 + 16)}{12} = \frac{7}{12}

\sqrt{\frac{(0.0023 * 750)}{(0.00345 * 1.25)}}

10.

\sqrt{\frac{(0.0023 * 750)}{(0.00345 * 1.25)}}

\sqrt{\frac{(0.0023 * 750)}{(0.00345 * 1.25)}}

Simplify

\sqrt{\frac{(0.0023 * 750)}{(0.00345 * 1.25)}}

\sqrt{\frac{(0.0023 * 750)}{(0.00345 * 1.25)}}

\sqrt{\frac{(0.0023 * 750)}{(0.00345 * 1.25)}}

A. 15
B. 20
C. 40
D. 75

Correct Answer: Option B

Explanation

Multiply appropriately to remove decimals on both numerator and denominators. Such that you have

$\sqrt{\frac{(2300 x 750)}{(345 x 125)}} D i v i d i n g, => \sqrt{4} 00 = 20$ $\sqrt{\frac{(2300 x 750)}{(345 x 125)}} D i v i d i n g, => \sqrt{4} 00 = 20$

$\sqrt{\frac{(2300 x 750)}{(345 x 125)}} D i v i d i n g, => \sqrt{4} 00 = 20$

11. The sum of two numbers is twice their difference. If the difference of the numbers is P, find the larger of the two numbers

A. p/2
B. 3p/2
C. 5p/2
D. 3p

Correct Answer: Option B

Explanation

Let the numbers be x and y
x+y = 2p.....(i)
x-y = p......(ii)
2x = 3p
x = 3p/2

12. A binary operation * is defined by a*b = ab+a+b for any real number a and b. if the identity element is zero, find the inverse of 2 under this operation.

A. 2/3
B. 1/2
C. -1/2
D. -2/3

Correct Answer: Option D

Explanation

a * a^{- 1} = a a^{- 1} + a + a^{- 1} = e

if e = 0

{2.2}^{- 1} + 2 + 2^{- 1} = 0

collecting like terms, we have:

{3.2}^{- 1} + 2 = 0

2^{- 1}

= -

\frac{2}{3}

13. Factorize completely X²+2XY+Y²+3X+3Y-18

A. (x+y+6)(x+y-3)
B. (x-y-6)(x-y+3)
C. (x-y+6)(x-y-3)
D. (x+y-6)(x+y+3)

Correct Answer: Option A

14. Tope bought X oranges at N5.00 each and some mangoes at N4.00 each. if she bought twice as many mangoes as oranges and spent at least N65.00 and at most N130.00, find the range of values of X.

A. 4≤X≤5
B. 5≤X≤8
C. 5≤X≤10
D. 8≤X≤10

Correct Answer: Option C

{2.2}^{- 1} + 2 + 2^{- 1} = 0

15. Three consecutive positive integers k, l and m are such that l² = 3(k+m). Find the value of m

A. 4
B. 5
C. 6
D. 7

Correct Answer: Option A

Explanation

6² = 3(k+m)
6² = 3(8+4)
36 = 3 x 12
36 = 36

thus, m = 4.

16. Express

\frac{1}{X^{3} - 1}

in partial fractions

A. 1/3{(1/x-1)-(x-2/x²-x+1)}
B. 1/3{(1/x-1)-(x+2/x²+x+1)}
C. 1/3{(1/x-1)-(x-2/x²+x+1)}
D. 1/3{(1/x-1)-(x-2/x²-x-1)}

Correct Answer: Option B

17. The first term of a geometric progression is twice its common ratio. Find the sum of the first two terms of the G.P if its sum to infinity is 8.

A. 8/5
B. 8/3
C. 72/25
D. 56/9

Correct Answer: Option C

Explanation

Le the common ratio be r so that the first term is 2r.

Sum, s = a/(1-r)
ie. 8 = 2r/(1-r)

8(1-r) = 2r, r = 8/5.

Sn = a(1-rⁿ)/(1-r)
Solve further to get 72/25

18. Divide 4x3-3x+1 by 2x-1

A. 2x²-x+1
B. 2x²-x-1
C. 2x²+x+1
D. 2x²+x-1

Correct Answer: Option D

19 . Find a positive value of ã if the coordinate of the centre of a circle X²+y²-2ãx+4y-ã = 0 is (ã,-2) and the radius is 4 units.

A. 1
B. 2
C. 3
D. 4

Correct Answer: Option C

20. A man 1.7m tall observes a bird on top of a tree at an angle...

A man 1.7m tall observes a bird on top of a tree at an angle of 30⁰. if the distance between the man's head and the bird is 25m, what is the height of the tree?

A. 26.7m
B. 14.2m
C. $1.7 + (25 \frac{\sqrt{3}}{3} m$

1.7 + (25 \frac{\sqrt{2}}{2} m

Correct Answer: Option C

21. Evaluate:

\int_{0}^{z} (s i n x - c o s x) d x W h e r e l e t t e r z = \frac{π}{4} . (π = p i)

A. $\sqrt{2 + 1}$

\sqrt{2} - 1

- \sqrt{2} - 1

1 - \sqrt{2}

Correct Answer: Option B

22. Find the volume of solid generated when the area enclosed by y = 0, y = 2x, and x = 3 is rotated about the x-axis.

A. 81 π cubic units
B. 36 π cubic units
C. 18 π cubic units
D. 9 π cubic units

Correct Answer: Option B

Explanation

y = 2 x V = \int π^{2} d y b u t y = 2 x V = \int π 4 x^{2} d x V = \frac{4 (3)^{3} π}{3} - \frac{4 (3)^{3} π}{3} V = \frac{4 * 27 π}{3} = 36 π c u b i c u n i t s

23. What is the derivative of t² sin (3t - 5) with respect to t?

y = 2 x V = \int π^{2} d y b u t y = 2 x V = \int π 4 x^{2} d x V = \frac{4 (3)^{3} π}{3} - \frac{4 (3)^{3} π}{3} V = \frac{4 * 27 π}{3} = 36 π c u b i c u n i t s

A. 6t cos (3t - 5)
B. 2t sin (3t - 5) - 3t² cos (3t - 5)
C. 2t sin (3t - 5) + 3t² cos (3t - 5)
D. 2t sin (3t - 5) + t² cos 3t

Correct Answer: Option C

Explanation

t² sin (3t - 5) = 2t sin ( 3t - 5) + t² x 3 cos (3t - 5) = 2t sin (3t - 5) + 3t² cos (3t - 5)

24.

\int_{- 2}^{1} (x - 1)^{2} d x

Evaluate

\int_{- 2}^{1} (x - 1)^{2} d x

\int_{- 2}^{1} (x - 1)^{2} d x

y = 2 x V = \int π^{2} d y b u t y = 2 x V = \int π 4 x^{2} d x V = \frac{4 (3)^{3} π}{3} - \frac{4 (3)^{3} π}{3} V = \frac{4 * 27 π}{3} = 36 π c u b i c u n i t s

y = 2 x V = \int π^{2} d y b u t y = 2 x V = \int π 4 x^{2} d x V = \frac{4 (3)^{3} π}{3} - \frac{4 (3)^{3} π}{3} V = \frac{4 * 27 π}{3} = 36 π c u b i c u n i t s

A. $\frac{- 10}{3}$

B. 7
C. 9
D. 11

25. In ∆MNO, MN = 6 units, MO = 4 units and NO = 12 units. If the bisector of and M meets NO at P, calculate NP.

A. 4.8 units
B. 7.2 units
C. 8.0 units
D. 18.0 unit Correct Answer: Option C

y = 2 x V = \int π^{2} d y b u t y = 2 x V = \int π 4 x^{2} d x V = \frac{4 (3)^{3} π}{3} - \frac{4 (3)^{3} π}{3} V = \frac{4 * 27 π}{3} = 36 π c u b i c u n i t s

26. Find the tangent to the acute angle between the lines 2x+y = 3 and 3x-2y = 5.

A. -7/4
B. 7/8
C. 7/4
D. 7/2

Correct Answer: Option C

27. Find the tangent to the acute angle between the lines 2x+y = 3 and 3x-2y = 5.

A. -7/4
B. 7/8
C. 7/4
D. 7/2

Correct Answer: Option C

28. From a point P, the bearings of two points Q and R are N67⁰W and N23⁰E respectively. If the bearing of R from Q is N68⁰E and PQ = 150m, calculate PR

A. 120m
B. 140m
C. 150m
D. 160m

Correct Answer: Option C

29. Find the equation of the locus of a point P(x,y) such that PV = PW, where V = (1,1) and W = (3,5)

A. 2x + 2y = 9
B. 2x + 3y = 8
C. 2x + y = 9
D. x + 2y = 8

Correct Answer: Option D

Explanation

The locus of a point P(x,y) such that PV = PW where V = (1,1) and W = (3,5). This means that the point P moves so that its distance from V and W are equidistance.

PV = PW

\sqrt{(x - 1)^{2} + (y - 1)^{2}} = \sqrt{(x - 3)^{2} + (y - 5)^{2}}

.

Squaring both sides of the equation,
(x-1)² + (y-1)² = (x-3)² + (y-5)².

x²-2x+1+y²-2y+1 = x²-6x+9+y²-10y+25

Collecting like terms and solving, x + 2y = 8.

30. Find the area bounded by the curve y = x(2-x). The x-axis, x = 0 and x = 2.

A. 4 sq units
B. 2 sq units
C. $\frac{4}{3} s q u n i t s$

D. $\frac{1}{3} s q u n i t s$

Correct Answer: Option C

31. If the minimum value of y = 1 + hx - 3x² is 13, find h.

A. 13
B. 12
C. 11
D. 10

Correct Answer: Option B

32. Let P = {1, 2, u, v, w, x}; Q = {2, 3, u, v, w, 5, 6, y} and R = {2, 3, 4, v, x, y}.

Determine (P-Q) ∩ R

A. {1, x}
B. {x y}
C. {x}
D. ɸ

Correct Answer: Option C

33.If the population of a town was 240,000 in January 1998 and it increased by 2% each year, what would be the population of the town in January, 2000?

A. 480,000
B. 249,696
C. 249,600
D. 244,800

Correct Answer: Option B

Explanation

1st year, Population = 240,000 x (2/100) = 4800.
Being the 2nd year population = 240,000 x 4800 = 244800.
Increase in Pop. in 2nd year = 244800 x (2/100) = 4896
Jan 2000, Pop. = 244800 + 4896 = 249,696

34. If

\frac{(2 \sqrt{3} - \sqrt{2})}{(\sqrt{3} + 2 \sqrt{2})} = m + n \sqrt{6}

, find the values of m and n respectively.

A. 1, -2
B. -2, 1
C. $\frac{- 2}{5}$ , 1
D. 2, 3/5

\frac{(2 \sqrt{3} - \sqrt{2})}{(\sqrt{3} + 2 \sqrt{2})} = m + n \sqrt{6}

, find the values of m and n respectively.

A. 1, -2
B. -2, 1
C. $\frac{- 2}{5}$ , 1
D. 2, 3/5

Correct Answer: Option B

Explanation

Rationalize

\frac{(2 \sqrt{3} - \sqrt{2})}{(\sqrt{3} + 2 \sqrt{2})}

and equate to

m + n \sqrt{6}

. Such that m = -2, and n = 1.

35. In a youth club with 94 members, 60 like modern music, and 50 like traditional music. The number of members who like both traditional and modern music is three times those who do not like any type of music. How many members like only one type of music?

A. 8
B. 24
C. 62
D. 86

Correct Answer: Option C

Explanation

Use a venn diagram:

60-3x+3x+50-3x = 94-x.
110-3x+x = 94
-2x = 94-110

=>-2x = -16, this x = -8.

Members that like only one game:
= 60 - 3x + 50 - 3x
= 60 - 3x8 + 50 - 3x8
= 60 - 24 + 50 - 24
= 36 + 26 = 62

36, Evaluate (2.813 x 10^-3 x 1.063) (5.637 x 10^-2)

A. 0.056
B. 0.055
C. 0.054
D. 0.54

Correct Answer: Option B

37. A man wishes to keep his money in a savings deposit at 25% compound interest so that after three years he can buy a car for N150,000. How much does he need to deposit?

A. N112,000.50
B. N96,000.00
C. N85,714.28
D. N76,800.00

Correct Answer: Option D

Explanation

Amount A = P(1+r)ⁿ;
A = N150,000, r = 25%, n = 3.
150,000 = P(1+0.25)³ = P(1.25)³

P = 150,000/1.25³ =N76,800.00

38. If 314₁₀ - 256₇ = 340_x, find x.

A. 7
B. 8
C. 9
D. 10

Correct Answer: Option A

Explanation

314₁₀ - 256₇ = 340_x,
Convert 256₇ and 340_x to base 10, such that:
314 - 139 = 3x² + 4x
=> 3x² + 4x - 175 = 0 (quadratic)
Factorising, (x - 7) (3x + 25) = 0,
either x = 7 or x = -25/3 ( but x cannot be negative)

Therefore, x = 7

39. Simplify 3(2ⁿ⁺¹) - 4(2^n-1) 2ⁿ⁺¹ - 2ⁿ

A. 2ⁿ⁺¹
B. 2^n-1
C. 4
D. 1/4

Correct Answer: Option C

Explanation

Start by expanding 3(2ⁿ⁺¹) - 4(2^n-1) 2ⁿ⁺¹ - 2ⁿ:

3 x 2ⁿ x 2¹ - 2² x 2ⁿ x 2^-1 2ⁿ x 2 -2ⁿ

Solving the equation above gives;
2ⁿ x 2¹ x 2 2ⁿ = 2 x 2 = 4

40. If P344₆ - 23P2₆ = 2PP2₆, find the value of the digit P.

A. 2
B. 3
C. 4
D. 5

Correct Answer: Option D

Explanation

Covert everything to base 10 and collect like terms, such that:
210P - 42P = 434 + 406
168P = 840
P = 840/168 = 5

41. A binary operation * is defined by a*b = a^b. If a*2 = 2-a, find the possible values of a.

A. 1, -1
B. 1, 2
C. 2, -2
D. 1, -2

Correct Answer: Option D

Explanation

a * b = a^b
a * 2 = a² = 2 - a
a² + 2a - a - 2 = 0
a(a+2) - 1(a+2) = 0
(a-1)(a+2) = 0
a = 1, -2

42. The 3rd term of an A.P is 4x - 2y and the 9th term is 10x - 8y. Find the common difference.

A. 19x - 17y
B. 8x - 4y
C. x - y
D. 2x

Correct Answer: Option C

Explanation

n = 3, U₃ = 4x - 2y,
U_n = a + (n+1)d,
4x - 2y = a + 2d (1)

U₉ = 10x - 8y,
10x - 8y = a + 8d (2)

Solving (1) and (2), d = 6(x-y)/6
d = x-y.

43. Find the inverse of p under the binary operation * defined by p*q = p + q - pq, where p and q are real numbers and zero is the identity

A. p
B. p -1
C. p/(p-1)
D. p/(p+1)

Correct Answer: Option C

Explanation

If P-1 is the inverse of P and O is the identity, Then P-1 * P = P * P-1 = 0
i.e. P-1 + P - P-1.P = 0
P-1 - P-1.P = -P
P-1(1 - P) = -P
P-1 = -P/(P-1)
= P/(P-1)

44. Evaluate (1/2 - 1/4 - 1/8 - 1/16 + ...) - 1

A. 2/3
B. zero
C. -2/3
D. -1

Correct Answer: Option C

Explanation

S = a/(1+r), where a = 1/2, r = 1/2.

S = 1/2 3/2 = 1/2 x 3/2 = 1/3.
1/3 - 1 = -2/3

45. if (x - 1), (x + 1) and (x - 2) are factors of the polynomial ax³ + bx² + cx - 1, find a, b, c in that order.

A. -1/2, 1., 1/2
B. 1/2, 1, 1/2
C. 1/2, 1, -1/2
D. 1/2, -1, 1/2

Correct Answer: Option A

Explanation

This is a polynomial of the 3rd order, thus x should have three answers. Use the factors given to get values of x as 1, -1 and -2.

Form three equations, and carry out elimination and subsequent substitution to get a = -1/2, b = 1, and c = 1/2

46. A trader realizes 10x - x² naira profit from the sale of x bags on corn. How many bags will give him the desired profit?

A. 4
B. 5
C. 6
D. 7

Explanation
y = 10x - x²
dy/dx = 10 -x
As dy/dx = 0,
10 - 2x = 0
2x = 10
x = 5

47. Solve the inequality 2 - x > x².

A. x < -2 or x > 1
B. x > 2 or x< -1
C. -1 < x < 2
D. -2 < x < 1

Solve the inequality 2 - x > x².

A. x < -2 or x > 1
B. x > 2 or x< -1
C. -1 < x < 2
D. -2 < x < 1

Correct Answer: Option D

Explanation

2 - x > x²
2 - x - x² > 0

(x+2)(x-1) < 0

48. If α and β are the roots of the equation 3x² + 5x - 2 = 0, find the value of 1/α + 1/β

A. -5/3
B. -2/3
C. 1/2
D. 5/2

Correct Answer: Option D

Explanation

1/α + 1/β = β+α/αβ
3x² + 5x - 2 = 0
x² + 5x/3 - 2/3 = 0
αβ = -2/3
β+α = -5/3

Thus; β+α/αβ = -2/3 -2/3 = -5/2

49. A frustrum of pyramid with square base has its upper and lower sections as squares of sizes 2m and 5m respectively and the distance between them 6m. Find the height of the pyramid from which the frustrum was obtained.

A. 8.0 m
B. 8.4 m
C. 9.0 m
D. 10.0 m

Correct Answer: Option D

50. P is a point on one side of the straight line UV and P moves in the same direction as UV. If the straight line ST is on the locus of P and angle VUS = 60°, find angle UST.

A. 310°
B. 130°
C. 80°
D. 50°

Correct Answer: Option B

Explanation

Make a good sketch of the question, and follow this steps.
Since ST is parallel to UV => angle USR = 50°(alternate to VUS.
Angle UST = 180° - 50° = 130° (angle on a straight line)