JAMB UTME - Mathematics - 1997

Question 1 Report

Given that 1/2 log10 P = 1, find the value Of P

104

103

102

101

10-1

Answer Details

1/2log10P
log10P1/2 = 1
P1/2 = 10
p = 102

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Logarithms

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Question 2 Report

Solve the simultaneous equations $ \frac{2}{x} - \frac{2}{x} = 2,\ \frac{4}{x} + \frac{3}{y} = 10 $

x =

\frac{3}{2}

, y =

\frac{3}{2}

x =

\frac{1}{2}

, y =

\frac{3}{2}

x =

\frac{- 1}{2}

, y =

\frac{- 3}{2}

x =

\frac{1}{2}

, y =

\frac{3}{2}

Answer Details

\frac{2}{x} - \frac{2}{x}

= 2.....(1)

\frac{4}{x} + \frac{3}{y}

= 10

\frac{6}{x}

= 12

\to

x =

\frac{6}{12}

x =

\frac{1}{2}

put x =

\frac{1}{2}

in equation (i)
= 4 -

\frac{3}{y}

= 2
= 4 - 2
=

\frac{3}{y}

therefore y =

\frac{3}{2}

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Solution Of Linear Equations

Quadratic Equations

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Question 3 Report

Differentiate $ \frac{6x^3 - 5x^2 + 1}{3x^2} $ with respect to x

\frac{2 + 2}{3 x^{3}}

2 +

\frac{1}{6 x}

2 -

\frac{2}{3 x^{3}}

\frac{1}{5}

Answer Details

\frac{6 x^{3} - 5 x^{2} + 1}{3 x^{2}}

let y = 3x2
y =

\frac{6 x^{3}}{3 x^{2}}

-

\frac{6 x^{2}}{3 x^{2}}

+

\frac{1}{3 x^{2}}

Y = 2x -

\frac{5}{3}

+

\frac{1}{3 x^{2}}

\frac{d y}{d x}

= 2 +

\frac{1}{3}

(-2)x-3
= 2 -

\frac{2}{3 x^{3}}

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Differentiation

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Question 4 Report

In a basket of fruits, there are 6 grapes, 11 bananas and 13 oranges. If one fruit is chosen at random, what is the probability that the fruit is either a grape or a banana?

\frac{17}{30}

\frac{11}{30}

\frac{6}{30}

\frac{7}{3}

Answer Details

There are a total of 6 + 11 + 13 = 30 fruits in the basket. The probability of selecting a grape is 6/30 and the probability of selecting a banana is 11/30. To find the probability of selecting either a grape or a banana, we add these probabilities: 6/30 + 11/30 = 17/30 So the probability of selecting either a grape or a banana is 17/30. Therefore, the answer is: - 17/30

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Probability

Surds (radicals)

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Question 5 Report

Find the difference between the range and the variance of the following set of numbers 4, 9, 6, 3, 2, 8, 10, 5, 6, 7 where $ \sum d^2 = 60 $

2

3

4

6

Measures Of Dispersion

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Question 6 Report

Find the value of k if $ \frac{5+2r}{(r+1)(r-2)} $ expressed in partial fraction is $ \frac{k}{r-2}+\frac{L}{r+1} $ where K and L are constants

3

2

1

-1

Answer Details

5 + 2r = k(r + 1) + L(r - 2)
but r - 2 = 0 and r = 2
9 = 3k
k = 3

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Question 7 Report

Determine x + y if $ \begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 \\ 8 \end{pmatrix} $

3

4

7

12

Answer Details

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Number Bases

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Question 8 Report

Find the range of values of $x$ for which $3x - 7 \le 0$ and $x + 5 > 0$

-5 < x <

\frac{7}{3}

-5

\leq

x

\leq

\frac{7}{3}

-5 < x

\leq

\frac{7}{3}

-5

\leq

x <

\frac{7}{3}

Answer Details

The first inequality 3x - 7 ≤ 0 can be solved as follows: 3x - 7 ≤ 0 3x ≤ 7 x ≤ 7/3 The second inequality x + 5 > 0 can be solved as follows: x + 5 > 0 x > -5 So the range of values of x that satisfies both inequalities is -5 < x ≤ 7/3. The correct answer is -5 < x ≤ 7/3.

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Linear Inequalities

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Question 9 Report

Find the value of (0.006)3 + (0.004)3 in standard form

2.8 x 10-9

2.8 x 10-7

2.8 x 10-8

2.8 x 10-6

Answer Details

To solve this problem, we simply need to evaluate the expressions inside the parentheses first, then add the two results. (0.006)³ = 0.000000216, which can be written in standard form as 2.16 x 10^-7. (0.004)³ = 0.000000064, which can be written in standard form as 6.4 x 10^-8. Adding these two values gives: 2.16 x 10^-7 + 6.4 x 10^-8 = 2.76 x 10^-7 Therefore, the answer is 2.8 x 10-7.

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Number Bases

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Question 10 Report

If U = (s, p, i, e, n, d, o, u, r), X = (s, p, e, n, d) Y = (s, e, n, o, u), Z = (p, n, o, u, r) find X ∩ Y ∪ Z

(p, o, u, r)

(s, p, d, r)

(p, n, d)

(n, d, u)

Algebraic Expressions

Sets

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Question 11 Report

A number is selected at random between 20 and 30, both numbers inclusive. Find the probability that the number is a prime

\frac{2}{11}

\frac{5}{11}

\frac{6}{11}

\frac{8}{11}

Answer Details

Possible outcomes are 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Prime numbers has only two factors
itself and 1
The prime numbers among the group are 23, 29. Probability of choosing a prime number
=

\frac{Number of prime}{No. of total Possible Outcomes}

=

\frac{2}{11}

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Question 12 Report

Evaluate 64.7642 - 35.2362 correct to 3 significant figures

2960

2950

2860

2850

Answer Details

To evaluate the expression 64.7642 - 35.2362, we simply subtract the second number from the first number. 64.7642 - 35.2362 = 29.528 To round this answer to three significant figures, we count three digits starting from the left-most nonzero digit. In this case, the left-most nonzero digit is 2, so we round to three digits after the 2. 29.5 Therefore, the answer is 29.5 to 3 significant figures.

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Fractions, Decimals And Approximations

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Question 13 Report

Sn is the sum of the first n terms of a series given by Sn = n2n - 1. Find the nth term

4n + 1

4n - 1

2n + 1

2n - 1

Answer Details

The given series is: 1 + 3 + 5 + 7 + ... + (2n - 1) Notice that each term in the series is an odd number, and the difference between consecutive terms is 2. To find the nth term, we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d where a1 is the first term, d is the common difference, and n is the term number. In this case, a1 = 1 and d = 2, so we have: an = 1 + (n - 1)2 an = 2n - 1 Therefore, the answer is (D) 2n - 1.

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Sequence And Series

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Question 14 Report

Find the standard derivation of the following data -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

2

3

√10

√11

Answer Details

x =

\frac{\sum x}{N}

=

\frac{0}{11}

= 0

\begin{matrix} x & (x - x) & (x - x)^{2} \\ - 5 & - 5 & 25 \\ - 4 & - 4 & 16 \\ - 3 & - 3 & 9 \\ - 2 & - 2 & 4 \\ - 1 & - 1 & 1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 3 & 9 \\ 4 & 4 & 16 \\ 5 & 5 & 25 \\ 110 \end{matrix}

S.D =

\sqrt{\frac{\sum (x - x)^{2}}{\sum f}}

=

\sqrt{\frac{110}{11}}

=

\sqrt{10}

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Measures Of Dispersion

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Question 15 Report

Find the distance between the point Q (4,3) and the point common to the lines 2x - y = 4 and x + y = 2

3√10

3√5

√26

√13

Answer Details

To find the distance between point Q and the point common to the lines 2x - y = 4 and x + y = 2, we need to first find the coordinates of the point of intersection of the two lines. We can solve the system of equations: 2x - y = 4 x + y = 2 by either substitution or elimination to get the coordinates of the point of intersection, which is (1, 1). Then, we can use the distance formula to find the distance between point Q (4, 3) and (1, 1): d = √[(4 - 1)² + (3 - 1)²] = √[9 + 4] = √13. Therefore, the answer is √13.

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Coordinate Geometry

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Question 16 Report

A chord of a circle of a diameter 42cm subtends an angle of 60o at the centre of the circle. Find the length of the mirror arc

22cm

44cm

110cm

220cm

Circles

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Question 17 Report

The angle of elevation of a building from a measuring instrument placed on the ground is 30o. If the building is 40m high, how far is the instrument from the foot of the building?

\frac{20}{\sqrt 3}

m

\frac{40}{\sqrt 3}

m

20√3m

40√3m

Answer Details

\frac{40}{x}

= tan 30o
x =

\frac{40}{t a n 36}

=

\frac{40}{1 \sqrt{3}}

= 40√3m

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Angles And Intercepts On Parallel Lines

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Question 18 Report

Two binary operations $ \ast $ and $ \oplus $ are defines as $ m \ast n = mn - n - 1 $ and $ m \oplus n = mn + n - 2 $ for all real numbers m, n.

Find the value of $ 3 \oplus (4 \ast 50) $

60

57

54

42

Answer Details

m

*

n = mn - n - 1, m

\oplus

n = mn + n - 2
3

\oplus

(4

*

5) = 3

\oplus

(4 x 5 - 5 - 1) = 3

\oplus

14
3

\oplus

14 = 3 x 14 + 14 - 2
= 54

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Matrices And Determinants

Binary Operations

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Question 19 Report

A survey of 100 students in an institution shows that 80 students speak Hausa and 20 students speak Igbo, while only 9 students speak both language. How many students speak neither Hausa nor Igbo?

orange

9

11

20

Answer Details

In the survey, 80 students speak Hausa, 20 students speak Igbo and 9 students speak both Hausa and Igbo. If we add the number of students speaking only Hausa and the number of students speaking only Igbo, we get 80 + 20 - 9 = 91. Therefore, there are 100 - 91 = 9 students who speak neither Hausa nor Igbo. So, the answer is 9.

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Read lesson note on Ratio, Proportions And Rates (WAEC)

Sets

Ratio, Proportions And Rates

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Question 20 Report

Simplify $ \frac{2\sqrt{3}+3\sqrt{5}}{3\sqrt{5}-2\sqrt{3}} $

\frac{19 + 4 \sqrt{25}}{11}

\frac{19 + 4 \sqrt{15}}{11}

\frac{19 + 2 \sqrt{15}}{11}

\frac{19 + 2 \sqrt{15}}{19}

Answer Details

\frac{(2 \sqrt{3} + 3 \sqrt{5}) (3 \sqrt{5} + 2 \sqrt{3})}{(3 \sqrt{5} - 2 \sqrt{3}) (3 \sqrt{5} - 2 \sqrt{3})}

=

\frac{5 + 12 \sqrt{15}}{33}

=

\frac{19 + 4 \sqrt{15}}{11}

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Read lesson note on Fractions, Decimals, Approximations And Percentage (JAMB)

Read lesson note on Number Bases (JAMB)

Number Bases

Fractions, Decimals, Approximations And Percentage

Number Bases

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Question 21 Report

Each of the base angles of a isosceles triangle is 59o and the verticles of the triangle lie on a circle. Determine rhe angle which the base of the triangle subtends at the centre of the circle.

128o

16o

64o

58o

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Question 22 Report

Age	20	25	30	35	40	45
Number of people	3	5	1	1	2	3

Find the median age of the frequency distribution in the table above.

20

25

30

35

Answer Details

To find the median age, we need to arrange the ages in ascending order and then find the middle value. However, since the table only gives us frequency counts for each age, we first need to construct a cumulative frequency distribution. Age | Number of People | Cumulative Frequency --- | --- | --- 20 | 3 | 3 25 | 5 | 8 30 | 1 | 9 35 | 1 | 10 40 | 2 | 12 45 | 3 | 15 The total number of people is 15, which is an odd number, so the median is simply the value that corresponds to the middle person. Since the cumulative frequency for 8 people is at age 25 and the cumulative frequency for 9 people is at age 30, the median age is 25. Therefore, the answer is: - 25

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Variation

Probability

Statistics

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Question 23 Report

If $X \ast Y = X + Y - XY$, find x when $(x \ast 2) + (x \ast 3) = 68$

24

22

-12

-21

Answer Details

x

*

y = x + y - xy
(x

*

2) + (x

*

3) = 68
= x + 2 - 2x + x + 3 - 3x
= 86
3x = 63
x = -21

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Solution Of Linear Equations

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Question 24 Report

The figure shows circles of radii 3cm and 2cm with centres at X and Y respectively. The circles have a transverse common tangent of length 25cm. Calculate XY.

\sqrt{630}

cm

\sqrt{626}

cm

\sqrt{615}

cm

\sqrt{600}

cm

Circles

Positive And Negative Integers, Rational Numbers

Coordinate Geometry

Surds (radicals)

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Question 25 Report

$ \frac{d}{dx} \cos(3x^2 - 2x) $ is equal to

-sin(6x - 2)

-sin(3x2 - 2x)dx

(6x - 2) sin(3x2 - 2x)

-(6x - 2)sin(3x2 - 2x)

Differentiation

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Question 26 Report

The pie chart shows the income of a civil servant in a month. If his monthly income is N600, find his monthly basic salary

N2000

N2600

N3100

N3450

View Answer

Question 27 Report

Find the volume of the prism.

990cm

880cm3

550cm3

495cm3

Answer Details

To find the volume of a prism, we need to multiply the area of the base by the height of the prism. In this case, the base is a triangle with base 11cm and height 15cm, so its area is 1/2 * base * height = 1/2 * 11cm * 15cm = 82.5cm^2. The height of the prism is given as 6cm. Therefore, the volume of the prism is 82.5cm^2 * 6cm = 495cm^3. Hence, the answer is 495cm^3.

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Question 28 Report

A cone with the sector angle of 45o is cut out of a circle of radius of the cone.

\frac{r}{16}

cm

\frac{r}{6}

cm

\frac{r}{8}

cm

\frac{r}{2}

cm

Circles

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Question 29 Report

Cos 65o has the same value as

Sin 65o

Cos 25o

Cos 115o

Cos 205o

Cos 295o

Trigonometry

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Question 30 Report

The angle between the positive horizontal axis and a given line is 135o. Find the equation of the line if it passes through the point (2,3)

x - y = 1

x + y = 1

x + y = 5

x - y = 5

Answer Details

The problem requires finding the equation of a line that passes through a given point (2,3) and makes an angle of 135 degrees with the positive x-axis. To solve this problem, we need to first find the slope of the line. Since the given angle is measured from the positive x-axis and is 135 degrees, we know that the angle made with the negative x-axis is 45 degrees. Therefore, the slope of the line is the tangent of 45 degrees, which is 1. Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation of the line. The point-slope form is: y - y1 = m(x - x1) where m is the slope and (x1, y1) is a point on the line. Plugging in the values we have, we get: y - 3 = 1(x - 2) Simplifying this equation gives us: y - x + 3 = 0 This equation is in the form of y = mx + b, where m is the slope and b is the y-intercept. We can see that the slope is 1, which we found earlier, and the y-intercept is 3. Therefore, the equation of the line that passes through the point (2,3) and makes an angle of 135 degrees with the positive x-axis is: y - x + 3 = 0 which is equivalent to: y = x - 3 So, the answer is neither (a) nor (b), but is (c) x + y = 5.

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Circles

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Question 31 Report

In the figure, XYZ is a triangle with $XY = 5cm$, $XZ = 2cm$ and XZ is produced to E making the angle < YZE = $150^\circ$. If the angle XYZ = $\theta$, calculate the value of $\sin\theta$

\frac{3}{5}

\frac{1}{2}

\frac{2}{5}

\frac{1}{5}

Answer Details

\frac{5}{\sin 36^{o}} = \frac{2}{\sin θ} = \frac{5}{\frac{1}{2}} = \frac{2}{\sin θ}

10 = $\frac{2}{\sin θ}$

sin $θ$ = $\frac{2}{10}$

= $\frac{1}{5}$

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Sets

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Question 32 Report

If y = x(x4 + x + 1), evaluate $ \int_{0}^{1} y\,dx $

\frac{11}{12}

\frac{11}{16}

\frac{5}{6}

zero

Integration

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Question 33 Report

Find the simple interest rate percent per annum at which ₦1,000 accumulates to ₦1,240 in 3 years

6%

8%

10%

12%

Answer Details

The formula for simple interest is: I = P * r * t Where: I = Interest P = Principal (initial amount borrowed or invested) r = Interest rate per annum (as a decimal) t = Time in years From the given information: P = ₦1,000 I = ₦1,240 - ₦1,000 = ₦240 t = 3 years Substituting these values into the formula, we get: 240 = 1000 * r * 3 Simplifying the equation, we get: r = 240 / (1000 * 3) = 0.08 Converting to a percentage, we get: r = 0.08 * 100% = 8% Therefore, the simple interest rate percent per annum is 8%. Answer: 8%

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Question 34 Report

The nth term of a sequence is given 31 - n , find the sum of the first terms of the sequence.

\frac{13}{9}

1

\frac{1}{3}

\frac{1}{9}

Answer Details

Tn = 31 - n
S3 = 31 - 1 + 31 - 2 + 31 - 3
= 1 +

\frac{1}{3}

+

\frac{1}{9}

=

\frac{13}{9}

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Sequence And Series

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Question 35 Report

If the function f(fx) = x3 + 2x2 + qx - 6 is divisible by x + 1, find q

-5

-2

2

5

Answer Details

To find the value of q, we can use the fact that the given function f(fx) is divisible by x+1. If a polynomial f(x) is divisible by x-a, then f(a) = 0. Using this fact, we can substitute x = -1 in the given function f(fx) = x^3 + 2x^2 + qx - 6, since x+1 is a factor of f(fx): f(f-1) = (-1)^3 + 2(-1)^2 + q(-1) - 6 f(f-1) = -1 + 2 - q - 6 f(f-1) = -5 - q Since f(f-1) is divisible by x+1, we have: f(f-1) = -5 - q = 0 Therefore, q = -5. Hence, the value of q is -5 when f(fx) = x^3 + 2x^2 + qx - 6 is divisible by x+1.

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Trigonometry

Application Of Differentiation

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Question 36 Report

Make F the subject of the formula $ t = \sqrt{\frac{v}{\frac{1}{f}+\frac{1}{g}}} $

\frac{g v - t^{2}}{g t^{2}}

\frac{g t^{2}}{g v - t^{2}}

\frac{v}{\frac{1}{t^{2}} - \frac{1}{g}}

\frac{g v}{t^{2} - g}

Answer Details

t =

\sqrt{\frac{v}{\frac{1}{f} + \frac{1}{g}}}

t2 =

\frac{v}{\frac{1}{f} + \frac{1}{g}}

=

\frac{v f g}{f t g}

\frac{1}{f} + \frac{1}{g}

=

\frac{v}{t^{2}}

= (g + f)t2 = vfg
gt2 = vfg - ft2
gt2 = f(vg - t2)
f =

\frac{g t^{2}}{g v - t^{2}}

View Answer

Question 37 Report

In an examination, the result of a certain school is as shown in the histogram above. How many candidates did the school present?

12

16

18

19

Answer Details

3 + 5 + 8 + 1 + 2 = 19

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Question 38 Report

Find the minimum value of X2 - 3x + 2 for all real values of x

-

\frac{1}{4}

-

\frac{1}{2}

\frac{1}{4}

\frac{1}{2}

Answer Details

y = X2 - 3x + 2,

\frac{d y}{d x}

= 2x - 3
at turning pt,

\frac{d y}{d x}

= 0
∴ 2x - 3 = 0
∴ x =

\frac{3}{2}

\frac{d^{2} y}{d x^{2}}

=

\frac{d}{d x}

(

\frac{d}{d x}

)
= 270
∴ ymin = 2

\frac{3}{2}

- 3

\frac{3}{2}

+ 2
=

\frac{9}{4}

-

\frac{9}{2}

+ 2
= -

\frac{1}{4}

View Answer

Question 39 Report

Find the non-zero positive value of x which satisfies the equation

\[ \left|\begin{matrix} x & 1 & 0 \\ 1 & x & x \\ 0 & 1 & x \end{matrix}\right| = 0 \]

2

4

√3

√2

1

Answer Details

x(x2 - 1) - x = 0
= x3 - 2x = 0
x(x2 - 2) = 0
x = 2

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Positive And Negative Integers, Rational Numbers

Number Bases

Statistics

Logarithms

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Question 40 Report

A point P moves so that is equidistant from point L and M. If LM is 6cm, find the distance of P from LM when P is 10cm from L

12cm

10cm

8cm

6cm

Answer Details

p from LM =

\sqrt{10^{2}}

- 82
=

\sqrt{36}

= 6cm

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Euclidean Geometry

Sets

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Question 41 Report

An arc of a circle subtends an angle $70^\circ$ at the centre. If the radius of the circle is 6cm, calculate the area of the sector subtended by the given angle. $\left(\pi = \frac{22}{7}\right)$

22cm2

44cm2

66cm2

88cm2