JAMB UTME - Mathematics - 2015

The area of a circle of radius 4cm is equal to (Take π = 3.142 )

Answer Details

Area of a circle (A) = πr2
Radius = 4cm
π = 3.142
A = 3.142 × (4)2
= 3.142 × 16
= 50.272
A = 50.3cm2 (1dp)

Given that tan x = $\frac{2}{3}$, where 0o d" x d" 90o, Find the value of 2sinx.

Answer Details

tan x = $\frac{2}{3}$(given), is illustrated in a right-angled $\Delta$
thus m2 = 22 + 32
= 4 + 9 = 13
m = $\sqrt{13}$
Hence, 2sin x = 2 x $\frac{2}{m}$
2 x$\frac{2}{\sqrt{13}}$
= $\frac{4}{\sqrt{13}}$
= $\frac{4}{\sqrt{13}}=\frac{\sqrt{13}}{\sqrt{13}}$
= $\frac{4\sqrt{13}}{13}$

The area of an ellipse is 132cm2. The length of its major axis is 14cm. Find the length of it minor axis.

Answer Details

The area of an ellipse (e) = $\frac{\pi ab}{4}$
Let a rep. the length of its major axis = 14cm
Let b rep. the length of its minor axis = ?
π = 3.142
Area of an ellipse = 132cm2
132 = (π14 × b) ÷ 4
132 = (3.142 × 14b) ÷ 4
3.142 × 14b =132 × 4
b = (1324 × 4) ÷ (3.142 × 14)
= 528/43.988
= 12cm
The length of its minor axis (b) = 12cm

A public car dealer marked up the cost of a car at 30% in an attempt to make 20% gross profit. Due to the value of dollar, he now placed 20% discount on the car. What profit or loss will he make?

Answer Details

Let assume the cost price is 100%
Marked up price + cost price = 20 + 100 = 120%
Discount at 20% = 20/100 × 120 % of cost price
Selling price = cost price − gain
= (120 − 24)% of cost price
= 96% of cost price
Loss = (100− 96)% of cost price
= 4% of cost price
∴ He will make a 4% loss.

Calculate the value of x and y if 27x ÷ 81x+2y = 9, x + 4y = 0

Answer Details

${27}^x÷{81}^{(x+2y)}=9$
$\left(27\right)x=9\times {81}^{(x+2y)}$
$(3^3{)}^x=3^2\times 3^{4(x+2y)}$
$=3^{(2+4x+8y)}$
$3^{3x}=3^{(2+4x+8y)}$
$3x=2+4x+8y$
$3x-4x-8y=2........\left(1\right)$
$x+4y=0........\left(2\right)$
$-4y=2$
$y=(-2)÷4=-\frac{1}{2}$
$y=-\frac{1}{2}$
Substitute the value of y into equation (2)
i.e $x+4y=0$
$x+4(-\frac{1}{2})=0$
$x-2=0$
$x=2$
$\therefore x=2,y=-\frac{1}{2}$
Method II
${27}^x÷{31}^{(x+2y)}=9$
$3^{3x}\times 3^{(-4x-8y)}=3^2$
$3^{(3x-8y)}=3^2$
$-x-8y=2........\left(1\right)$
$x+4y=0........\left(2\right)$
$-4y=2$
$y=-\frac{2}{4}=-\frac{1}{2}$
$y=-\frac{1}{2}$
Substitute the value of y into equation 2
$x+4y=0$
$x+4(-1)÷2)=0$
$x-2=0$
$x=2$
$x=2,y=-\frac{1}{2}$

A man sells his new brand car for ₦420,000 at a gain of 15%. What did it cost him?

Answer Details

If the man sold the car for ₦420,000 at a gain of 15%, then we can find the cost price of the car using the following formula: Selling price = Cost price + Profit We know the selling price is ₦420,000 and the profit is 15% of the cost price. Let's call the cost price "x." Profit = 15% of x = 0.15x Substituting the values in the formula: ₦420,000 = x + 0.15x ₦420,000 = 1.15x To solve for x, we need to divide both sides by 1.15: x = ₦420,000 / 1.15 x = ₦365,217.39 Therefore, the cost price of the car was ₦365,217.39. So, option B - ₦365,217 is the correct answer.

Add 11012,101112 and 1112

Answer Details

11012 + 101112 + 1112 = 101011

If duty is levied at 25%, find the duty to be added to a bill of ₦80.

Answer Details

The correct answer is ₦20. To find the duty to be added to a bill of ₦80, we need to calculate 25% of ₦80. 25% means 25 out of 100, so we can calculate 25% of ₦80 by multiplying ₦80 by 25/100. ₦80 * 25/100 = ₦20 So the duty to be added to a bill of ₦80 is ₦20.

The probability of an event A is 1/5. The probability of B is 1/3 . The probability both A and B is 1/15. What is the probability of either event A or B or both?

Answer Details

The probability of either event A or B or both can be calculated using the formula for the union of two events: P(A or B) = P(A) + P(B) - P(A and B) where P(A) is the probability of event A, P(B) is the probability of event B, and P(A and B) is the probability of both events happening at the same time. Plugging in the given probabilities, we get: P(A or B) = 1/5 + 1/3 - 1/15 = 7/15 So, the probability of either event A or B or both is 7/15.

The extension of a stretched string is directly proportional to its tension. If the extension produced by a tension of 8 Newtons is 2cm, find the extension produced by a tension of 12 Newtons.

Answer Details

Since the extension of a stretched string is directly proportional to its tension, we can use the formula:

extension = k * tension

where k is a constant of proportionality. We can find k by using the known values of tension and extension:

k = extension / tension
k = 2 cm / 8 N = 0.25 cm/N

Now that we have found k, we can use it to find the extension produced by a tension of 12 Newtons:

extension = k * tension
extension = 0.25 cm/N * 12 N = 3 cm

So, the extension produced by a tension of 12 Newtons is 3 cm.

Find the value of x if $[1÷{64}^{(x+2)}]=[4^{(x?3)}÷{16}^x]$

Answer Details

$[1÷{64}^{(x+2)}]=[4^{(x-3)}÷{16}^x]$
${64}^{-(x+2)}=\left[4^{(x-3)}\right]÷\left[{16}^x\right]$
Break down 4, 16 and 64 into smaller index numbers:
$2^{-6(x+2)}=2^{2(x-3)}÷2^{4\left(x\right)}$
$2^{-6x-12}=2^{2x-4x-6}$
$2^{-6x-12}=2^{-2x-6}$
− 6x − 12 = − 2x − 6
Collect the like terms:
−6x + 2x = −6 + 12
−4x =6
x = $\frac{6}{4}$
x = $\frac{-3}{2}$

If y = $(x^2+3x?1)÷(3x+4).$ Find dy/dx

Answer Details

dy/dx $(4x^3+3x^2+2x+1)$
= $12x^2+6x+2$
⇒ $12x^2+6+2$
Divide both sides by 2
$\frac{12x^2}{2}-\frac{6x}{2}-\frac{2}{2}$
= $6x^2+3x+1$
dy/dx
= $6x^2+3x+1$

Simplify (0.09)2 and give your answer correct to 4 significant figures

Answer Details

(0.09)2 = 0.09 × 0.09
= 0.0081
= 0.008100 to 4 significant figures
Please, start counting first from the non-zero digits i.e. 8

The sum of two numbers is 5; their product is 14. Find the numbers.

Answer Details

Let x represent the first number;
Then, the other is (5 − x) , since their sum is 5 and their product is 14
x(5 − x) = 14
5x − x2 = 14
x2 − 5x − 14 = 0
(x2 − 7x) + (2x − 14) = 0
x(x − 7) + 2(x − 7) = 0
(x − 7)(x + 2) = 0
Either (x − 7) = 0 or (x + 2) = 0
x = 7 or x = − 2
The two numbers are 7 and − 2

Find the range of the value of x satisfying the inequalities 5 + x $\le$ 8 and 13 + x $\ge$ 7

Answer Details

The given inequalities are: 5 + x ≤ 8 13 + x ≥ 7 Simplifying them, we get: x ≤ 3 x ≥ -6 So the range of values of x that satisfy both the inequalities is: -6 ≤ x ≤ 3 Therefore, option (C) is the correct answer. To arrive at this answer, we first solved each inequality separately to get an idea of the possible range of values for x. Then we looked for the overlap between these ranges, which gives us the range of values that satisfy both inequalities.

If an investor invest ₦450,000 in a certain organization in order to yield X as a return of ₦25,000. Find the return on an investment of ₦700,000 by Y in the same organization.

Answer Details

To solve the problem, we need to use the concept of proportionality. We can set up a proportion between the return and the investment amount: Return/Investment = X/₦450,000 We can rearrange this to solve for X: X = (Return * ₦450,000) / Investment Now, we can use this equation to find the return Y on an investment of ₦700,000: Y = (X * ₦700,000) / ₦450,000 Substituting the given values, we get: X = (₦25,000 * ₦450,000) / ₦450,000 = ₦25,000 Y = (₦25,000 * ₦700,000) / ₦450,000 = ₦38,888.89 (rounded to ₦38,888.90K) Therefore, the answer is option D: ₦38,888.90K.

Factorize $x^2-2x-15$

Answer Details

The expression x^2 - 2x - 15 can be factored into the product of two binomials. To find these binomials, we can use the "ac method," where we try to find two numbers that multiply to -15 and add up to -2. These numbers are -5 and 3. We can then write the expression as:

(x - 5)(x + 3)

So, the expression x^2 - 2x - 15 can be factored as (x - 5)(x + 3).

In the venn diagram, the shaded region is

Answer Details

A man bought a car for ₦800 and sold it for ₦520. Find his loss per cent

Answer Details

The loss percentage can be calculated as follows: Loss percentage = (Loss / Cost price) x 100 Here, the cost price of the car is ₦800, and the selling price is ₦520. Since the selling price is less than the cost price, the man has incurred a loss. Loss = Cost price - Selling price Loss = ₦800 - ₦520 Loss = ₦280 Substituting the values in the formula: Loss percentage = (280 / 800) x 100 Loss percentage = 0.35 x 100 Loss percentage = 35% Therefore, the man incurred a loss of 35%.

If A = (−3,5) and B = (4,−1) find the co-ordinate of the mid point

Answer Details

A pipe made of metal 10cm thick has an external radius of 11cm. find the area of metal in making 2.4cm of pipe

Answer Details

Simplify √30 × √40

Answer Details

√30 × √40
= √(30 × 40)
= √(3 × 10 × 4 × 10)
= √(400 × 3)
= √400 × √3
= 20 × √3
= 20√3

Factorize $a^2-b^2-4a+4$

Answer Details

The trinomial = $a^2-4a+4$
$a^2-b^2-4a+4=(a^2-4a+4)-b^2$
$(a^2-2a-2a+4)-b^2$
$a(a-2)-2(a-2)-b^2$
$(a-2{)}^2-b^2$
$(a-2+b)(a-2-b)$

Determine the third term of a geometrical progression whose first and second term are 2 and 14 respectively

Answer Details

1st G.P. = a =2
2nd G.P. = ar − 1 = 54
2(r) = 54
r = 54/2 = 27
r = 27
3rd term = ar2 = (2) (27)2
2 × 27 × 27
= 1458

Simplify $6\frac{1}{12}-2¾+1½$

Answer Details

6½ − 2¾ + 1½
73/12 − 11/4 + 3/2
[(73 − 33 + 18) ÷ (12)]
58/12
29/6
$4\frac{5}{6}$

The probability of an outcome A is 1/6. The probability of the B outcome is 1/4. If the probability of A or B or both is 1/12, what is the probability of both outcomes A and B?

Answer Details

P(A outcome) = $\frac{1}{6}$, P(B outcome) = ¼
P(A ∪ B) = 1/12, P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∪ B)
1/6 + 1/4 − 1/12
= (2 + 3 − 1) ÷ 12
= 4/12
= 1/3
P(A ∩ B) = 1/3

Given that Z = {1,2,4,5} what is the power of set Z?

Answer Details

The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself. The formula for finding the power set of a set is 2^n, where n is the number of elements in the original set. In this case, Z has four elements, so the number of subsets is 2^4 = 16. Therefore, the power set of Z has 16 elements, which is the answer.

Rationalize 5 ÷ (2 − √3)

Answer Details

To rationalize the expression 5 ÷ (2 − √3), we need to eliminate the square root in the denominator. We can do this by multiplying both the numerator and denominator by the conjugate of the denominator, which is 2 + √3. This is because when we multiply the conjugate of the denominator by the original denominator, we get a difference of squares that simplifies the expression: 5 ÷ (2 − √3) * (2 + √3) / (2 + √3) Simplifying the numerator by multiplying the terms inside the brackets, we get: 5(2 + √3) / (2^2 - (√3)^2) Simplifying further, we get: 5(2 + √3) / (4 - 3) 5(2 + √3) / 1 Therefore, the rationalized form of the expression is 5(2 + √3), which is equivalent to option C.

How many sides has a regular polygon whose interior angles are 120° each?

Answer Details

A regular polygon is a polygon with equal sides and equal angles. The formula to find the interior angles of a regular polygon is: Interior Angle = (n - 2) x 180 / n where n is the number of sides of the polygon. We are given that the interior angles of the regular polygon are 120°. Substituting this value into the formula, we get: 120 = (n - 2) x 180 / n Simplifying this equation, we get: 120n = 180n - 360 360 = 60n n = 6 Therefore, the regular polygon has 6 sides.

Simplify $\frac{1}{(x+1)}+\frac{1}{(x-1)}$

Answer Details

[1 ÷ (x+1)] + [1 ÷ (x − 1)]
= ((x − 1) + [(x + 1)) ÷ (x+1)(x − 1)]
Using the L.C.M.
= (x − 1 + x + 1) ÷ (x + 1)(x − 1)
= (x + 2 − 1 + 1) ÷ (x + 1)(x − 1)
= 2x ÷ (x + 1)(x − 1) =2x ÷ (x + 1)(x − 1)

Simplify 4 √[(390695T− 8)½)]

Answer Details

If (159.75)10 = (y)6. Find x

Answer Details

(159.75)10 = (y)6
$\begin{array}{cc}6& 159\\ 6& 26\text{R 3}\\ 6& 4\text{R 2}\\ & 0\text{R 4}\end{array}\uparrow$
15910 = 4236
Then Convert 0.75 to base 6
0.75 x 6 =4.5 i.e 4.5 - 3.00 = 1.5
0.5 x 6 = 3.00 423 + 1.5 =424.5
159.75 = (424.5)6
(159)10 = (424.5)6

Find the simple interest on ₦325 in 5years at 3% per annum.

Answer Details

The formula for finding the simple interest on a principal amount P for a period of t years at an interest rate of r per annum is:

simple interest = P * r * t / 100

In this case, P = 325, t = 5, and r = 3, so we can plug in the values into the formula:

simple interest = 325 * 3 * 5 / 100
simple interest = 48.75

So, the simple interest on ₦325 in 5 years at 3% per annum is ₦48.75.

Integral ∫$(5x^3+7x^2-2x+5)$dx