JAMB UTME - Mathematics - 2001

Pregunta 1 Informe

The distribution of colours of beads in a bowl is given above. What is the probability that a bead selected at random will be blue or white?

7/15

2/5

1/3

1/15

Detalles de la respuesta

Total number of beads
= 1+2+4+5+3
=15
Number of blue beads = 1
P(Blue beads) = 1/15
Numbers of white beads = 4
P(white beads) = 4/15
∴P(Blue of white beads)
= P(Blue) + P(White)
= 1/15 + 4/15
= 5/15
= 1/3

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Pregunta 2 Informe

Find the value of $ \theta $ in the diagram

60

^{\circ}

100

^{\circ}

120

^{\circ}

30

^{\circ}

Detalles de la respuesta

Using cosine formula (t

\sqrt{3}

)2 = t2 + t2 - 2t2 cos

θ

3t2 = 2t2 - 2t2 cos $θ$ = 2t2(1 - cos $θ$ )

1 - cos $θ$ = $\frac{3 t^{2}}{2 t^{2}}$ = $\frac{3}{2}$

cos = 1 - $\frac{3}{2} = - \frac{1}{2}$

$θ$ = cos-1(- $\frac{1}{2}$ ) = 120 $^{\circ}$ and 240 $^{\circ}$

N.B 0 $\geq$ $θ$ 360

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Circles

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Pregunta 3 Informe

Evaluate 21.05347 - 1.6324 x 0.43 to 3 decimal places

20.980

20.351

20.981

20.352

Detalles de la respuesta

Hint: Use BODMAS, in other words, do multiplication of the second and the last first before subtracting value obtained from the first.

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Fractions, Decimals And Approximations

Algebraic Expressions

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Pregunta 4 Informe

Team P and Q are involved in a game of football.What is the probability that the game ends in a draw?

2/3

1/2

1/3

1/4

Detalles de la respuesta

P(Games ends in draw)
This implies that Team P and Q wins
∴ P(P wins) = 1/2
P(Q wins) = 1/2
∴ P(game ends in a draw) = 1/2*1/2 = 1/4

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Probability

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Pregunta 5 Informe

An operation * is defined on the set of real numbers by a*b = a + b + 1. If the identity elements is -1, find the inverse of the element 2 under *.

4

zero

2

-2

Detalles de la respuesta

First, let's find the inverse of the element 2 under the operation * on the set of real numbers. The inverse of an element "a" under the operation "*" is another element "b" such that a*b = b*a = identity element. The identity element under the operation "*" is given as -1, so we have: 2 * b = b * 2 = -1 Substituting the value of the identity element, we get: 2 * b = b * 2 + 1 = -1 Simplifying the equation, we get: 2b = -3 b = -3/2 Therefore, the inverse of the element 2 under the operation * is -3/2. To check if this is correct, we can substitute the values of 2 and -3/2 into the operation * and see if we get the identity element: 2 * (-3/2) = -3/2 * 2 = -1 Since we get the identity element, we can confirm that the inverse of the element 2 under the operation * is -3/2. So, the answer to the question is option (C) -2.

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Pregunta 6 Informe

If y = x sinx, find dy/dx when x = ?/2.

-?/2

-1

1

?/2

Detalles de la respuesta

To find dy/dx, we need to take the derivative of y with respect to x. Using the product rule of differentiation, we have: dy/dx = (sinx) + (x)(cosx) So, when x = ?/2, we can substitute this value into the expression to get: dy/dx = (sin(?/2)) + (?/2)(cos(?/2)) We know that sin(?/2) = 1 and cos(?/2) = 0, so we can simplify the expression: dy/dx = (1) + (?/2)(0) dy/dx = 1 Therefore, the answer is 1.

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Differentiation

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Pregunta 7 Informe

Evaluate (0.142 x 0.275) ÷ 7(0.02) to 3 decimal places.

0.039

0.358

0.033

0.308

Detalles de la respuesta

(0.142 x 0.275) 7(0.02) = (0.14 x 0.14 x 0.275) ÷ (7 x 0.02) = (0.14 x 0.14 x 0.275)/(0.14)
=0.14 x 0.275 = 0.0385 Approx. 0.039

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Fractions, Decimals And Approximations

Percentages

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Pregunta 8 Informe

A car dealer bought a second-hand car for ₦250,000 and spent ₦70,000 refurbishing it. He then sold the car for ₦400,000. What is the percentage gain?

60%

32%

25%

20%

Detalles de la respuesta

The dealer bought the car for ₦250,000 and spent ₦70,000 refurbishing it, for a total cost of: Total cost = ₦250,000 + ₦70,000 = ₦320,000 The dealer then sold the car for ₦400,000. The profit made by the dealer is the difference between the selling price and the cost price: Profit = Selling price - Cost price Profit = ₦400,000 - ₦320,000 = ₦80,000 The percentage gain is the profit expressed as a percentage of the cost price: Percentage gain = (Profit / Cost price) x 100% Substituting the values we get: Percentage gain = (₦80,000 / ₦320,000) x 100% Percentage gain = 0.25 x 100% Percentage gain = 25% Therefore, the percentage gain made by the dealer is 25%, and the answer is.

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Percentages

Fractions, Decimals, Approximations And Percentage

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Pregunta 9 Informe

Find the area bounded by the curves y = 4 - x2 and y = 2x + 1

20(1/3) sq. units

20(2/3) sq. units

10(2/3) sq. units

10(1/3) sq. units

Detalles de la respuesta

Hint:
y = 4 - x2 and y = 2x + 1
=> 4 - x2 = 2x + 1
=> x2 + 2x - 3 = 0
(x+3)(x-1) = 0
thus x = -3 or x = 1.
Integrating x2 + 2x - 3 from (-3, to 1) w.r.t x will give 31/3 = 10(1/3)

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Areas

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Pregunta 10 Informe

Find the number of ways of selecting 8 subjects from 12 subjects for an examination

490

495

496

498

Detalles de la respuesta

This question is asking us to find the number of ways to select 8 subjects out of 12 subjects. This is a combination problem since the order in which the subjects are selected does not matter. The formula for combination is nCr = n!/r!(n-r)!, where n is the total number of items, r is the number of items to be selected, and ! represents the factorial function. Applying this formula to the given problem, we have: 12C8 = 12!/8!(12-8)! = (12x11x10x9)/(4x3x2x1) = 495 Therefore, there are 495 ways of selecting 8 subjects out of 12 subjects. Hence, the correct option is 495.

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Permutation And Combination

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Pregunta 11 Informe

A straight line makes an angle of 30° with the positive x-axis and cuts the y-axis at y = 5. Find the equation of the straight line.

y = (x/10) + 5

y = x + 5

√3y = - x + 5√3

√3y = x + 5√3

Detalles de la respuesta

Cos 30 = 5/x
x cos 30 = 5, => x = 5√3
Coordinates of P = -5, 3, 0
Coordinates of Q = 0, 5
Gradient of PQ = (y2 - y1) (X2 - X1) = (5 - 0)/(0 -5√3)
= 5/5√3 = 1/√3
Equation of PQ = y - y1 = m (x -x1)
y - 0 = 1/√3 (x -(-5√3))
Thus: √3y = x + 5√3

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Pregunta 12 Informe

The chord ST of a circle is equal to the radius, r, of the circle. Find the length of arc ST.

πr/6

πr/2

πr/12

πr/3

Detalles de la respuesta

In a circle, the length of an arc is proportional to the angle it subtends at the center of the circle. Since ST is a chord and its length is equal to the radius of the circle, then angle SOT, where O is the center of the circle, is 60 degrees (because it subtends an equilateral triangle). Therefore, the length of arc ST is 1/6 of the circumference of the circle (because 60 degrees is 1/6 of a full revolution, which is 360 degrees). The circumference of the circle is 2πr, so the length of arc ST is: Length of arc ST = 1/6 * 2πr = πr/3 Hence, the answer is option D: πr/3.

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Pregunta 13 Informe

Triangle SPT is the solution of the linear inequalities

2y - x - 2 ≤ 0, y + 2x + 2 ≥ 0, -2 ≤ x ≤ - 1

-2 ≤ x ≤ 2, y ≥ 0, y + 2x + 2 ≤ 0, x ≤ 0

2y - x - 2 ≤ 0, y + 2x + 2 ≤ 0, y ≥ 0, x ≤ 0

2y - x - 2 ≥ 0, y + 2x + 2 ≤ 0, x ≤ 0

Detalles de la respuesta

Δ SPT is the solution of the inequalities
2y - x - 2 ≤ 0
2y ≤ 2 + x
y ≤ 2/2 + x/2
y ≤ 1 + 1/2x
y + 2x + 2 ≥ 0,
y ≥ -2 -2x
∴ the solution of the inequalities
2y - x - 2 ≤ 0,
y + 2x + 2 ≥ 0
= -2 ≤ x ≤ -1

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Linear Inequalities

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Pregunta 14 Informe

Find the principal which amounts to ₦5,500 at a simple interest in 5 years at 2% per annum.

₦4,900

₦5,000

₦4,700

₦4,800

Detalles de la respuesta

Principal, P = Amount, A - Interest, I.
A = P + I
I = (P.T.R)/100 = (P x 5 x 2)/100 = 10P/100 = P/10
But A = P + I,
=> 5500 = P + (P/10)
=> 55000 = 10P + P
=> 55000 = 11P
Thus P = 55000/11 = ₦5,000

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Sine, Cosine And Tangent Of An Angle

Areas

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Pregunta 15 Informe

Differentiate (2x+5)2(x-4) with respect to x.

4(2x+5)(x-4)

4(2x+5)(4x-3)

(2x+5)(2x-13)

(2x+5)(6x-11)

Differentiation

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Pregunta 16 Informe

Factorize 4x2 - 9y2 + 20x + 25

(2x -3y + 5)(2x - 3y - 5)

(2x - 3y)(2x + 3y)

(2x - 3y +5)(2x + 3y + 5)

(2x + 5)(2x - 9y +5)

Detalles de la respuesta

Given: 4x2 - 9y2 + 20x + 25
Collect like terms: 4x2 + 20x + 25 - 9y2
(2x + 5)(2x + 5) - 9y2
(2x + 5)2 - (3y)2
(2x - 3y +5)(2x + 3y + 5)

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Algebraic Expressions

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Pregunta 17 Informe

The graph shows the cumulative frequency of the distribution of masses of fertilizer for 48 workers in one institution. Which of the following gives the inter-quartile range?

\frac{1}{2} (Q_{3} - Q_{1})

Q3 - Q2

Q3 - Q3

Q3 - Q1

Probability

Linear Inequalities

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Pregunta 18 Informe

If 6Pr = 6, find the value of 6Pr+1

30

33

35

15

Detalles de la respuesta

The notation 6P_r means the number of permutations of r objects selected from a set of 6 distinct objects. Since 6P_r = 6! / (6 - r)!, we know that 6P_r = 6 when r = 1, since 6P₁ = 6! / (6 - 1)! = 6! / 5! = 6. To find the value of 6P_r+1, we can simply multiply 6P_r by (6 - r) since we are selecting one additional object. So, 6P_r+1 = 6P_r * (6 - r) = 6 * (6 - 1) = 30 when r = 1. Therefore, the answer is 30.

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Solution Of Linear Equations

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Pregunta 19 Informe

A cylindrical tank has a capacity of 3080 m3. What is the depth of the tank if the diameter of its base is 14 m?

(Take pi = 22/7)

23 m

25 m

20 m

22 m

Detalles de la respuesta

The formula for the volume of a cylindrical tank is V = πr^2h, where V is the volume, r is the radius, and h is the height or depth of the tank. We are given that the capacity or volume of the tank is 3080 m^3, and we can find the radius by dividing the diameter by 2: d = 14 m r = d/2 = 7 m Substituting the given values in the formula, we get: 3080 = (22/7) x 7^2 x h Simplifying the expression, we get: 3080 = 22 x 7 x 7 x h/7 3080 = 22 x 7 x h 3080 = 154h h = 3080/154 h = 20 Therefore, the depth of the tank is 20 meters. Answer: 20 m

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Pregunta 20 Informe

Find the dimensions of a rectangle of greatest area which has a fixed perimeter p.

square of sides p

square of sides 2p

square of sides (p/2)

square of sides (p/4)

Detalles de la respuesta

Let the rectangle be a square of sides p/4.
So that perimeter of square = 4p
4 x (p/4) = p.

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Pregunta 21 Informe

If $x = \frac{y}{2}$, evaluate $\left(\frac{x^3}{y^3} + \frac{1}{2}\right) \div \left(\frac{1}{2} - \frac{x^2}{y^2}\right)$

5/8

5/2

5/4

5/16

Algebraic Fractions

Quadratic Equations

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Pregunta 22 Informe

Find the variance of 2, 6, 8, 6, 2 and 6

6

5

√6

√5

Detalles de la respuesta

To find the variance of a set of numbers, you need to follow these steps: 1. Find the mean (average) of the set. 2. Subtract the mean from each number in the set. 3. Square each of the differences. 4. Find the mean (average) of the squared differences. Let's apply these steps to the set of numbers given: 2, 6, 8, 6, 2, and 6. 1. The mean of the set is (2+6+8+6+2+6)/6 = 5. 2. Subtracting the mean from each number gives us the following differences: -3, 1, 3, 1, -3, 1. 3. Squaring each difference gives us: 9, 1, 9, 1, 9, 1. 4. Finding the mean of these squared differences gives us (9+1+9+1+9+1)/6 = 5. Therefore, the variance of the set is 5. In summary, variance is a measure of how spread out a set of numbers is. A high variance means that the numbers in the set are far from the mean, while a low variance means that the numbers are closer to the mean.

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Measures Of Dispersion

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Pregunta 23 Informe

⊗	k	l	m
k	l	m	k
l	m	k	l
m	k	l	m

The identity element with respect to the multiplication shown in the table above is

o

m

l

k

Binary Operations

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Pregunta 24 Informe

In the figure above, PQR is a straight line segment, PQ = QT. Triangle PQT is an isosceles triangle, ?SQR is 75$^{\circ}$ and ?QPT is 25$^{\circ}$. Calculate the value of ?RST.

45

^{\circ}

55

^{\circ}

25

^{\circ}

50

^{\circ}

Detalles de la respuesta

In ? PQT,
?PTQ = 25

^{\circ}

(base ?s of isosceles ?)
In ? QSR,
?RQS = ?QPT + ?QTP
(Extr = sum of interior opposite ?s)
?RQS = 25 + 25
= 50

^{\circ}

Also in ? QSR,
75 + ?RQS + ?QSR = 180

^{\circ}

(sum of ?s of ?)
?75 + 50 + ?QSR = 180
125 + ?QSR = 180
?QSR = 180 - 125
?QSR = 55

^{\circ}

But ?QSR and ?RST are the same
?RST = 55

^{\circ}

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Pregunta 25 Informe

Find the number of sides of a regular polygon whose interior angle is twice the exterior angle.

6

2

3

8

Detalles de la respuesta

Let the ext. angle = x
Thus int. angle = 2x
But sum of int + ext = 180 (angle of a straight line).
2x + x = 180
3x = 180
x = 180/3 = 60
Each ext angle = 360/n
=> 60 = 360/n
n = 360/60 = 6

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Triangles And Polygons

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Pregunta 26 Informe

Given the scores: 4, 7, 8, 11, 13, 8 with corresponding frequencies: 3, 5, 2, 7, 2, 1 respectively. The mean score is

7.0

8.7

9.5

11.0

Detalles de la respuesta

To find the mean score, we need to calculate the sum of all the scores, and then divide it by the total number of scores. The scores are given as 4, 7, 8, 11, 13, 8 with corresponding frequencies of 3, 5, 2, 7, 2, 1 respectively. So, the sum of all the scores is: 4 × 3 + 7 × 5 + 8 × 2 + 11 × 7 + 13 × 2 + 8 × 1 = 12 + 35 + 16 + 77 + 26 + 8 = 174 The total number of scores is: 3 + 5 + 2 + 7 + 2 + 1 = 20 Therefore, the mean score is: mean = sum of scores / total number of scores = 174 / 20 = 8.7 So, the answer is.

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Pregunta 27 Informe

The bearings of P and Q from a common point N are 020° and 300° respectively. If P and Q are also equidistant from N, find the bearing of P from Q.

040°

070°

280°

320°

Detalles de la respuesta

Hint: Simply make a sketch of the question and find the bearing to be 070°

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Bearings

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Pregunta 28 Informe

Given an isosceles triangle with length of 2 equal sides t units and opposite side √3t units with angle θ. Find the value of the angle θ opposite to the √3t units.

100°

120°

30°

60°

Detalles de la respuesta

In an isosceles triangle, the two equal sides are opposite to the two equal angles. Therefore, the angle opposite to the side of length √3t would be equal to the other equal angle of the triangle. Let's call this angle x. Then, using the fact that the sum of the angles in a triangle is 180°, we can write: x + x + θ = 180 Simplifying this equation, we get: 2x + θ = 180 Now, we know that the two equal sides of the triangle have length t, so we can use the cosine formula to write: cos(x) = (t/2) / √(3t^2/4) Simplifying this expression, we get: cos(x) = 1/√3 Taking the inverse cosine of both sides, we get: x = 30° Substituting this value of x into the earlier equation, we get: 2(30°) + θ = 180° θ = 120° Therefore, the answer is option (B) 120°.

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Pregunta 29 Informe

Given distribution of color beads: blue, black, yellow, white and brown with frequencies 1, 2, 3, 4, and 5 respectively. Find the probability that a bead picked at random will be blue or white.

7/15

2/5

1/3

1/15

Detalles de la respuesta

There are a total of 1+2+3+4+5 = 15 beads. The probability of picking a blue bead is 1/15, since there is only one blue bead out of the total of 15 beads. The probability of picking a white bead is 4/15, since there are 4 white beads out of the total of 15 beads. To find the probability of picking a blue or white bead, we add the probabilities of picking a blue bead and a white bead: P(blue or white) = P(blue) + P(white) P(blue or white) = 1/15 + 4/15 P(blue or white) = 5/15 P(blue or white) = 1/3 Therefore, the probability of picking a blue or white bead is 1/3.

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Probability

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Pregunta 30 Informe

Find the range of 1/6, 1/3, 3/2, 2/3, 8/9 and 4/3

3/4

5/6

7/6

4/3

Detalles de la respuesta

To find the range of a set of numbers, we need to find the difference between the largest and smallest numbers in the set. The given set of numbers are: 1/6, 1/3, 3/2, 2/3, 8/9, and 4/3. The smallest number in the set is 1/6 and the largest number in the set is 3/2. So, the range of the set is: 3/2 - 1/6 = 9/6 - 1/6 = 8/6 = 4/3 Therefore, the answer is 4/3.

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Number Bases

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Pregunta 31 Informe

The mean score is

7.0

8.7

9.5

11.0

Detalles de la respuesta

Mean ((4*3) + (7*5) + (8*2) + (11*7) + (13*2) + (8*1))/20
(12 + 35 + 16 + 77 + 26 + 8)/20
174/20 = 8.7

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Loci

Measures Of Dispersion

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Pregunta 32 Informe

Triangle SPT is the solution of the linear inequalities

2y - x - 2

\leq

0, y + 2x + 2

\geq

0, x

\geq

0

2y - x - 2

\geq

0, y + 2x + 2

\leq

0, -2

\geq

x

\geq

-1

-2

\geq

x

\geq

2, y

\leq

0, y + 2x + 2

\geq

0, x

\geq

0

2y - x - 2

\geq

0, y + 2x + 2

\geq

0, y

\leq

0, x

\geq

0

Solution Of Linear Equations

Linear Inequalities

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Pregunta 33 Informe

P(-6, 1) and Q(6, 6) are the two ends of the diameter of a given circle. Calculate the radius.

6.5 units

13.0 units

3.5 units

7.0 units

Detalles de la respuesta

The diameter of a circle is the distance between any two points on the circle passing through the center of the circle. In this problem, we are given the two endpoints of the diameter of a circle: P(-6, 1) and Q(6, 6). We can find the distance between these two points using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2) where d is the distance between the two points (P and Q in this case), and (x1, y1) and (x2, y2) are the coordinates of the two points. So, substituting the given values in the distance formula, we get: d = sqrt((6 - (-6))^2 + (6 - 1)^2) d = sqrt(12^2 + 5^2) d = sqrt(144 + 25) d = sqrt(169) d = 13 Therefore, the diameter of the circle is 13 units. The radius of the circle is half the diameter, so: radius = diameter / 2 = 13 / 2 = 6.5 Hence, the radius of the circle is 6.5 units. So, the answer is.

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Circles

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Pregunta 34 Informe

The histogram above shows the distribution of passengers in taxis of a certain motor park. How many taxis have more than 4 passengers

16

17

14

15

Detalles de la respuesta

Looking at the histogram, we can see that the height of the bars represents the number of taxis with a particular number of passengers. We can count the number of taxis with more than 4 passengers by adding up the heights of the bars for 5, 6, 7, and 8 passengers. Counting these bars, we can see that there are a total of 17 taxis with more than 4 passengers. Therefore, the answer is 17.

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Pregunta 35 Informe

The bar chart above shows different colours of passing a particular point of a certain street in two minutes. What fraction of the total number of cars

3/25

2/25

1/5

4/15

Detalles de la respuesta

The total number of cars that passed the point on the street in two minutes is not given in the chart. Therefore, we cannot determine the fraction of the total number of cars from the information provided. The chart only shows the relative frequency or proportion of cars of different colors that passed the point during the two-minute observation period. For example, if the chart shows that 3 out of 25 cars were blue, then the proportion of blue cars passing the point during the two minutes was 3/25. To find the fraction of the total number of cars, we would need to know the total number of cars that passed the point during the two-minute observation period.

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Positive And Negative Integers, Rational Numbers

Sets

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Pregunta 36 Informe

The sixth term of an A.P is half of its twelfth term. The first term of the A.P is equal to

zero

half of the common difference

double the common difference

the common difference

Detalles de la respuesta

Let's assume that the first term of the A.P is 'a', and the common difference is 'd'. We know that for an A.P, the nth term is given by: a + (n-1)d According to the problem, the sixth term is equal to half of the twelfth term. So, we can write: a + 5d = 1/2 (a + 11d) Multiplying both sides by 2 to eliminate the fraction: 2a + 10d = a + 11d Simplifying, we get: a = d Therefore, the first term of the A.P is equal to the common difference. So, the correct answer is "the common difference."

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Sequence And Series

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Pregunta 37 Informe

A point P moves such that it is equidistant from Points Q and R. Find QR when PR = 8cm and angle PRQ = 30°

4√3cm

8cm

8√3cm

4cm

Detalles de la respuesta

Hint: Make a sketch of the moving points such that the hypotenuse is 8cm and the adjacent x cm.
Cos 30 = x/8
x = 8 cos 30 = 8√3cm

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Angles Of Elevation And Depression

Sets

Simple Operations On Algebraic Expressions

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Pregunta 38 Informe

If the gradient of the curve y = 2kx2 + x + 1 at x = 1 is 9, find k.

4

3

2

1

Detalles de la respuesta

The given curve is y = 2kx² + x + 1, and we are asked to find the value of k when the gradient at x = 1 is 9. The gradient of a curve at a particular point is the slope of the tangent to the curve at that point. To find the gradient of this curve at x = 1, we need to differentiate it with respect to x: dy/dx = 4kx + 1 Now we can substitute x = 1 and set the result equal to 9, which gives: 4k(1) + 1 = 9 Simplifying this equation, we get: 4k = 8 Dividing both sides by 4, we obtain: k = 2 Therefore, the value of k that satisfies the given conditions is 2.

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Pregunta 39 Informe

Find the value of P if the line joining (P, 4) and (6, -2) is perpendicular to the line joining (2, P) and (-1, 3).

4

6

3

zero

Detalles de la respuesta

Hint: If two lines a perpendicular, the gradient of one is equal to minus the reciprocal of the other (P, 4) and (6, -2)

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Coordinate Geometry

Simple Operations On Algebraic Expressions

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Pregunta 40 Informe

Solve the equations
m2 + n2 = 29
m + n = 7

(2, 3) and ( 3, 5)

(2, 5) and (5, 2)

(5, 2) and ( 5, 3)

(5, 3) and (3, 5)

Detalles de la respuesta

m2 + n2 = 29 .......(1)
m + n = 7 ............(2)
From (2),
m = 7 - n
but m2 + n2 = 29, substituting;
(7-n)2 + n2 = 29
49 - 14n + n2 + n2 = 29
=> 2n2 -14n + 20 = 0
Thus n2 -7n + 10 = 0
Factorizing;
(n-5)(n-2) = 0
n - 5 = 0, => n = 5
n - 2 = 0, => n = 2.
When n = 5,
m + n = 7, => m = 2,
When n = 2,
m + n = 7, => m = 5.
Thus (m,n) = (5,2) and (2,5)

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Solution Of Linear Equations

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Pregunta 41 Informe

Evaluate ∫2(2x-3)2/3dx

3/5(2x-3)5/3 + k

6/5(2x-3)5/3 + k

2x-3+k

2(2x-3)+k

Detalles de la respuesta

To evaluate the definite integral ∫2(2x-3)^2/3 dx, we can use the following formula: ∫uⁿ du = (uⁿ⁺¹)/(n+1) + C where C is the constant of integration. Using this formula, we can rewrite the integral as: ∫2(2x-3)^2/3 dx = 2 ∫(2x-3)^2/3 dx Let u = 2x - 3, then du/dx = 2 and dx = du/2. Substituting this into the integral, we get: 2 ∫(2x-3)^2/3 dx = 2 ∫u^2/3 (du/2) = ∫u^2/3 du Using the formula, we get: ∫u^2/3 du = (u^5/3)/(5/3) + C Substituting back for u, we get: ∫2(2x-3)^2/3 dx = (2(2x-3)^5/3)/(5/3) + C Simplifying, we get: ∫2(2x-3)^2/3 dx = (6(2x-3)^5/3)/5 + C Therefore, the correct answer is option A: 3/5(2x-3)^5/3 + k.

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Pregunta 42 Informe

If two graphs y = px2 + q and y = 2x2 -1 intersect at x = 2, find the value of p in terms q.

\frac{q - 8}{7}

\frac{7 - q}{4}

\frac{8 - q}{2}

\frac{7 + q}{8}

Detalles de la respuesta

To find the value of p in terms of q, we can use the given information that the two graphs intersect at x = 2. This means that at x = 2, the values of y for both graphs are the same. So, we can substitute x = 2 into both equations and set them equal to each other: px^2 + q = 2x^2 - 1 Substituting x = 2: p(2)^2 + q = 2(2)^2 - 1 4p + q = 7 Solving for p in terms of q, we can isolate p: 4p = 7 - q p = (7 - q) / 4 Therefore, the value of p in terms of q is (7 - q) / 4.

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Quadratic Equations

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Pregunta 43 Informe

What is the rate of change of the volume v of a hemisphere with respect to its radius r when r = 2?

8π

16π

2π

4π

Detalles de la respuesta

To solve this problem, we need to use the formula for the volume of a hemisphere, which is V = (2/3)πr^3. We want to find the rate of change of V with respect to r, which is given by dV/dr = 2πr^2. When r = 2, we can substitute this value into the formula to get dV/dr = 2π(2)^2 = 8π. Therefore, the rate of change of the volume of a hemisphere with respect to its radius when r = 2 is 8π.

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Introductory Calculus

Integration

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Pregunta 44 Informe

Given the scores: 4, 7, 8, 11, 13, 8 with corresponding frequencies: 3, 5, 2, 7, 2, 1 respectively. Find the square of the mode.

49

121

25

64

Detalles de la respuesta

Mode = score with highest frequency = 11.
Square of 11 = 121

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Integration

Linear Inequalities

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Pregunta 45 Informe

Given that $p = 1 + \sqrt{2}$ and $q = 1 - \sqrt{2}$, evaluate $\frac{(p^2 - q^2)}{2pq}$.

2(2+√2)

-2(2+√2)

2√2

-2√2

Detalles de la respuesta

First, let's evaluate p^2 and q^2: p^2 = (1+√2)^2 = 1 + 2√2 + 2 = 3 + 2√2 q^2 = (1-√2)^2 = 1 - 2√2 + 2 = 3 - 2√2 Next, let's substitute the values of p^2 and q^2 into the expression (p^2 - q^2)/(2pq): (p^2 - q^2)/(2pq) = ((3 + 2√2) - (3 - 2√2))/(2(1+√2)(1-√2)) = (4√2)/(2(1-2)) = (4√2)/(-2) = -2√2 Therefore, the answer is -2√2.

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Positive And Negative Integers, Rational Numbers

Modular Arithmetic

Surds (radicals)

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Pregunta 46 Informe

A sector of a circle of radius 7.2cm which subtends an angle of $300^\circ$ at the centre is used to form a cone. What is the radius of the base of the cone?

6cm

7cm

8cm

9cm

5cm

Loci

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Pregunta 47 Informe

The bar chart shows different colours of cars passing a particular point of a certain street in two minutes. What fraction of the cars is yellow

\frac{1}{5}

\frac{2}{25}

\frac{4}{15}

\frac{3}{25}

Detalles de la respuesta

\begin{matrix} colour of cars & Number (frequency) \\ y e l l o w & 3 \\ w h i t e & 4 \\ r e d & 8 \\ g r e e n & 2 \\ b l u e & 6 \\ b l a c k & 2 \\ 25 \end{matrix}

Thus, the fraction of the total numbers that are yellow is $\frac{3}{25}$

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Representation Of Data

Number Bases

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Pregunta 48 Informe

Simplify $ \left(\sqrt[3]{64a^3}\right)^{-1} $

4a

1/8a

8a

1/4a

Number Bases

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Pregunta 49 Informe

Divide: ax3x - 26x2x + 156ax - 216 by a2x - 24ax + 108

ax - 2

ax + 2

ax - 8

ax - 6

Detalles de la respuesta

To divide the given expression, we can use the long division method.

                  ax^2 - 26x + 156a - 216b
     ___________________________________
ax | ax^3 + 3x^2 - 26x^2 + 156ax - 216bx^2 - 24ax + 108
      -ax^3
      -----
           3x^2 - 216bx^2
           3x^2 - 3ax^2
           --------------
                     -216bx^2 + 3ax^2
                     -216bx^2 + 36ax^2
                     ------------------
                                    -33ax^2 + 156a - 216bx^2 - 24ax + 108
                                    -33ax^2 + 33bx^2
                                    ------------------
                                                156a - 33ax^2 - 24ax + 108
                                                156a - 33ax^2 - 156ax + 33ax
                                                -------------------------
                                                             -24ax + 108
                                                             -24ax + 24bx
                                                             -------------
                                                                       108 - 24bx
                                                                       108 - 2ax
                                                                       --------
                                                                                2ax - 24bx + 108

Therefore, the quotient is ax - 2 with a remainder of 2ax - 24bx + 108.

Hence, the answer is ax - 2.

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Algebraic Expressions

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Pregunta 50 Informe

Find the locus of a point which moves such that its distance from the line y = 4 is a constant, k.

y = k + or - 4

y = 4 + or - k

y = 4 + k

y = k - 4

Detalles de la respuesta

The locus of a point which moves such that its distance from the line y = 4 is a constant, k, is the set of all points which are equidistant from the line y = 4. This set of points is a pair of parallel lines, k units above and below the line y = 4. Therefore, the correct answer is: y = k + or - 4.

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Loci

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