JAMB UTME - Mathematics - 2009

If Q is$\left[\begin{array}{cc}9& -2\\ -7& 4\end{array}\right]$, then |Q| is

Answer Details

To find |Q|, we need to take the determinant of the matrix Q and then take its absolute value. So, we have: |Q| = |9 -2 -7| |-2 0 -4| |-7 -4 0| Expanding along the first row, we get: |Q| = 9(0 - (-4)) - (-2)(-4 - (-7)) - (-7)(-4 - 0) |Q| = 36 - 2(3) + 28 |Q| = 59 Therefore, the correct option is (D) 50.

Each of the interior angles of a regular polygon is 140o. Calculate the sum of all the interior angles of the polygon

Answer Details

In a regular polygon, all the interior angles have the same measure. Let n be the number of sides of the polygon, then we have: sum of all interior angles = (n - 2) x 180 degrees Each interior angle of the polygon is given as 140 degrees. Since the polygon is regular, it follows that all its interior angles are congruent. Hence, we can set up an equation involving the interior angle and solve for the number of sides n. In a regular polygon with n sides, the sum of the interior angles is given by: sum of all interior angles = n x (interior angle) Substituting the given value of the interior angle, we have: sum of all interior angles = n x 140 degrees We can now equate the two expressions for the sum of all interior angles: (n - 2) x 180 degrees = n x 140 degrees Expanding and simplifying the left side, we get: 180n - 360 = 140n Adding 360 to both sides and simplifying, we have: 40n = 360 Therefore, n = 9. Hence, the given polygon has 9 sides. Now, substituting this value of n in the expression for the sum of all interior angles, we have: sum of all interior angles = 9 x 140 degrees = 1260 degrees Therefore, the sum of all the interior angles of the given polygon is 1260 degrees. In conclusion, the answer is option (B) 1260 degrees.

If m * n = n - (m+2) for any real number m and n find he value of 3*(-5)

Answer Details

m * n = n - (m+2)
= -5 - (3+2)
= -5-5
= -10

Determine the value of x for which (x2 - 1) > 0

Answer Details

To find the value of x for which (x^2 - 1) > 0, we need to factorize the expression. (x^2 - 1) can be written as (x - 1)(x + 1). So now we have: (x - 1)(x + 1) > 0 For the product of two factors to be greater than zero, either both factors must be positive or both factors must be negative. If (x - 1) and (x + 1) are both positive, then x > 1. If (x - 1) and (x + 1) are both negative, then x < -1. Therefore, the solution is: x < -1 or x > 1 which means option (A) is the correct answer.

If P $=\left[\begin{array}{cc}x+3& x+2\\ x+1& x-1\end{array}\right]$ evaluate x if |P| = -10

Answer Details

$P=\left[\begin{array}{cc}x+3& x+2\\ x+1& x-1\end{array}\right]$ evaluate x if |P| = -10
(x+3)(x-1) - {(x+1)(x+2)} = -10
x2 - x + 3x - 3 - {x2 + 2x + x + 2} = -10
x2 + 2x - 3 - {x2 + 3x + 2} = -10
-x - 5 = -10
-5 + 10 = x
5 = x
∴x = 5

If x = {n2+1:n is a positive integer and 1 $\le$ n $\le$ 5}, y = {5n:n is a positive integer and 1 $\le$ n $\le$ 5}, find x $\cap$ y.

Answer Details

To find the intersection of two sets, we need to find the elements that are common to both sets. First, let's find the elements of set x: x = {n² + 1 : 1 ≤ n ≤ 5} If we substitute each value of n from 1 to 5 into the formula n² + 1, we get the following values for set x: x = {2, 5, 10, 17, 26} Now, let's find the elements of set y: y = {5n : 1 ≤ n ≤ 5} If we multiply each value of n from 1 to 5 by 5, we get the following values for set y: y = {5, 10, 15, 20, 25} To find the intersection of sets x and y, we need to find the elements that are common to both sets. From the values listed above, we can see that the elements 5 and 10 are in both sets x and y. Therefore, the intersection of x and y is: x ∩ y = {5, 10} Therefore, the correct option is {5, 10}.

If the hypotenuse of a right angle-triangle isosceles triangle is 2cm. What is the area of the triangle?

Answer Details

In an isosceles right triangle, the two legs have the same length, which is equal to √2 times the length of one of the legs. Let's assume that each leg has a length of x, then the hypotenuse (which is 2 cm in this case) can be expressed as: x√2 = 2 Solving for x, we get: x = 2/√2 = √2 The area of an isosceles right triangle can be found by dividing the product of the two legs by 2, so the area in this case would be: (√2 * √2)/2 = 1 cm² Therefore, the correct answer is: - 1 cm²

Which of the following equations represent the graph above?

Answer Details

x = -2 and x = 1/4
x = -2 and 4x = 1
x+2 and 4x-1
(x+2)(4x-1) = 0
4x2 - x + 8x -2 = 0
4x2 + 7x – 2 = 0 but y intercept is positive. Multiply the equation by -1
-4x2 - 7x + 2 = 0
∴y = 2 – 7x – 4x2

What is the mean deviation of x, 2x, x+1 and 3x. If their mean is 2?

Answer Details

The first step to finding the mean deviation is to calculate the mean of the given numbers. We are told that the mean of x, 2x, x+1, and 3x is 2. So, we can write an equation: (x + 2x + x+1 + 3x)/4 = 2 Simplifying this equation, we get: 7x/4 = 2 Multiplying both sides by 4/7, we get: x = 8/7 Now we can substitute this value of x back into the original numbers: x = 8/7 2x = 16/7 x+1 = 15/7 3x = 24/7 Next, we need to find the absolute deviation of each number from the mean: |x - 2| = |8/7 - 2| = 6/7 |2x - 2| = |16/7 - 2| = 2/7 |x+1 - 2| = |15/7 - 2| = 1/7 |3x - 2| = |24/7 - 2| = 10/7 The mean deviation is the average of these absolute deviations: (6/7 + 2/7 + 1/7 + 10/7)/4 = 19/28 Therefore, the mean deviation of the given numbers is 19/28. The answer closest to 19/28 among the options provided is 0.5. Therefore, the answer is (A) 0.5.

The probability of a student passing any examination is 2/3. If the students takes three examination, what is the probability that he will not pass any of them

Answer Details

P(pass) = 2/3
P(not pass) = 1 - 2/3 = 1/3
P(not passing any of the 3 Exams) = 1/3 x 1/3 x 1/3
= 1/27

The distance travelled by a particle from a fixed point is given as s = (t3 - t2 - t + 5)cm. Find the minimum distance that the particle can cover from the fixed point.

Answer Details

The distance travelled by the particle is given by the function s = t^3 - t^2 - t + 5. To find the minimum distance that the particle can cover from the fixed point, we need to find the minimum value of this function. Taking the derivative of s with respect to t, we get s' = 3t^2 - 2t - 1. Setting s' equal to zero and solving for t, we get t = (2 ± √10)/3. To determine whether this critical point is a minimum or maximum, we need to check the second derivative of s. Taking the derivative of s' with respect to t, we get s'' = 6t - 2. When t = (2 + √10)/3, we have s'' > 0, which means that this critical point corresponds to a minimum. Therefore, the minimum distance that the particle can cover from the fixed point is achieved when t = (2 + √10)/3. Substituting this value of t into the function s, we get s_min = s((2 + √10)/3) = (16 - 8√10 + 27√10 - 25)/27 = (-9 + 19√10)/27 ≈ 0.463 cm. Therefore, the correct option is not listed as it is approximately 0.463 cm.

W is directly proportional to U. If w = 5 when U = 3, find when W = 2/7

Answer Details

In the figure above , Ts//xy and XY = TY, ∠SYZ = 34${}^{\circ }$, ∠TXY = 47${}^{\circ }$, find the angle marked n

Answer Details

In Δ TYX
XY = TY ∴y =47${}^{\circ }$ base ∠s of ISCΔ
But y+x+34+47 = 180 interior
opposite ∠s are supplementary
47 + x + 34 + 47 = 180
x + 128= 180
x = 180- 128
x = 52${}^{\circ }$

A polynomial in x whose roots are 4/3 and -3/5 is

Answer Details

If 4/3 and -3/5 are roots of a polynomial
Imply x = 4/3 and - 3/5
3x = 4 and 5x = -3
∴3x-4 = 0 and 5x+3 = 0 are factors
(3x-4)(5x+3) = 0 product of the factors
15x2 + 9x – 20x – 12 = 0 By expansion
15x2 - 11x – 12 = 0

A regular polygon has 150${}^{\circ }$ as the size of each interior angle. How many sides does it have?

Answer Details

A regular polygon has equal interior angles and equal sides. The sum of the interior angles of a polygon with n sides is (n-2) times 180 degrees. So, if a regular polygon has interior angles of 150 degrees, we can use this formula to find the number of sides: (n-2) x 180 = sum of interior angles of the polygon (n-2) x 180 = 150n Simplifying the equation: n - 2 = 150/180 * n n - 2 = 5/6 * n n = 12 Therefore, the regular polygon has 12 sides. Answer: 12.

Simplify $7\frac{1}{12}-4\frac{3}{4}+2\frac{1}{2}$

Answer Details

A binary operation ⊗ defined on the set of integers is such that m⊗n = m + n + mn for all integers m and n. Find the inverse of -5 under this operation, if the identity element is 0

Answer Details

To find the inverse of -5 under this operation, we need to find an integer x such that -5⊗x = 0, where 0 is the identity element. Using the given binary operation, we know that m⊗n = m + n + mn. Substituting -5 for m and x for n, we get: -5⊗x = -5 + x - 5x = -4x - 5 We need to solve -4x - 5 = 0 for x. Adding 5 to both sides gives: -4x = 5 Dividing both sides by -4 gives: x = -5/4 Therefore, the inverse of -5 under this operation is -5/4. So, the correct option is: -5/4

In how many ways can 9 people be seated if 3 chairs are available

Answer Details

If there are 9 people and 3 chairs, the first person can choose any of the 9 chairs. After the first person has chosen their seat, there are only 8 chairs left for the second person to choose from. Once the second person has chosen their seat, there are only 7 chairs left for the third person to choose from. Therefore, the total number of ways in which 3 people can be seated in 9 chairs is: 9 x 8 x 7 = 504 Therefore, the answer is 504, which is option (B).

Find the radius of a sphere whose surface area is 154 cm2

Answer Details

The surface area of a sphere is given by the formula: A = 4πr^2 where A is the surface area and r is the radius of the sphere. In this problem, we are given the surface area of the sphere, which is 154 cm^2. We can use this information to find the radius of the sphere. We start by rearranging the formula to solve for r: r = √(A/4π) Plugging in the value of A = 154 cm^2, we get: r = √(154/4π) ≈ 3.5 cm Therefore, the radius of the sphere is approximately 3.5 cm. So, the correct answer to this problem is option B: 3.50 cm.

In how many ways can a delegation of 3 be chosen from 5 men and 3 women. If at least 1 man and 1 woman must be included?

Answer Details

The histogram above represents the number of candidates that sat for Mathematics examination in a school. How many candidate scored more than 50 marks?

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MarkNo.ofCandidate6040702580159010100595

Find the value of sin 45${}^{\circ }$ - cos 30${}^{\circ }$

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Simplify $\frac{5+\sqrt{7}}{3+\sqrt{7}}$

Answer Details

To simplify the expression $\frac{5+\sqrt{7}}{3+\sqrt{7}}$, we need to rationalize the denominator. To do this, we can multiply the numerator and denominator by the conjugate of the denominator, which is $3-\sqrt{7}$. This gives us: $$\frac{5+\sqrt{7}}{3+\sqrt{7}}\cdot\frac{3-\sqrt{7}}{3-\sqrt{7}}=\frac{(5+\sqrt{7})(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}=\frac{15-5\sqrt{7}+3\sqrt{7}-7}{9-7}$$ Simplifying the numerator and denominator gives: $$\frac{8-2\sqrt{7}}{2}=\boxed{4-\sqrt{7}}$$ Therefore, $\frac{5+\sqrt{7}}{3+\sqrt{7}}$ simplifies to $4-\sqrt{7}$.

Find the value of x in the diagram.

Answer Details

30 + x = 100

x = 100 - 30

= 70o

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

Answer Details

To solve this inequality, we need to work with each inequality separately and then find the values of x that satisfy both. Starting with the first inequality: 3x - 7 ≤ 0 Adding 7 to both sides: 3x ≤ 7 Dividing by 3: x ≤ 7/3 Now, let's work with the second inequality: x + 5 > 0 Subtracting 5 from both sides: x > -5 To satisfy both inequalities, x must be greater than -5 and less than or equal to 7/3. Therefore, the range of values for x is: -5 < x ≤ 7/3

Solve 52(x-1) x 5(x-1) = 0.04

Answer Details

52(x-1) x 5(x-1) = 0.04
52x-2 x 5x-1 = 4/100
52x-2 + x-1 = 1/25
53x-1 = 1/52
53x-1 = 5-2 (Equating the indices)
3x-1 = -2
3x = -2+1
3x = -1
X = -1/3

A chord drawn 5 cm away from the center of a circle of radius 13 cm. Calculate the length of the chord

Answer Details

x2 + 52 = 132
x2 + 25 = 169
x2 = 144

x = √ 144 
= 12
Length of the chord AB = 2x
2*12= 24cm

Evaluate ∫sec2θ dθ

Answer Details

∫sec2θ dθ = ∫ 1/cos2 dθ
∫(cos)-2 dθ,
let u = cos θ
∴∫u-2 = 1/u + c
∫cos θ = sin θ + c
∫sec-2θ = 1/u sin θ + c
= (sinθ / cosθ) + c
= Tan θ + c

Find to infinity, the sum of the sequence $1,\frac{9}{10},{\left(\frac{9}{10}\right)}^2,{\left(\frac{9}{10}\right)}^3,.....$

Answer Details

I.S∩T∩W=S II. S∪T∪W=S

III. T∩W=S

If S⊂T⊂W, which of the above statements are true?

Answer Details

S∩T∩W = S(This means that all the elements of s are in T and also in W.)
S∪T∪W = W (imply W is a universal set for S and T) T∩W = S imply all the elements of S are in T and W)
S ⊂ T ⊂ W
∴ I, II, and III

The distribution above shows the number of days a group of 260 students were absents from school in a particular term. How many students were absent for at least four days in the term

Answer Details

20 + X + 50 + 40 + 2X + 60 = 260
3X + 170 = 260
3X = 260 - 170
3x = 90
x = 30
Absent for at least 4 days

A cliff on the bank of a river is 300 meter high. if the angle of depression of a point on the opposite side of the river is 60${}^{\circ }$, find the width of the river.

Answer Details

We can solve this problem using trigonometry. Let's draw a diagram of the situation:

    A
   /|
  / |
 /  | h = 300 m
/   |
-----------
  x   B

Where point A is the top of the cliff, point B is the unknown point on the opposite side of the river, and x is the width of the river.

We know that the angle of depression from A to B is 60 degrees. This means that the angle of elevation from B to A is also 60 degrees.

Using trigonometry, we can set up the following equation:

tan(60) = h / x

where h is the height of the cliff and x is the width of the river. We can solve for x:

x = h / tan(60)
x = 300 / √3
x = 100√3 meters

Therefore, the width of the river is 100√3 meters. Answer is correct.

5, 8, 6 and k occur with frequency 3, 2, 4, and 1 respectively an d have a mean of 5.7. Find the value of k

Answer Details

The mean of the given numbers is 5.7. Mean = (sum of all numbers) / (total number of numbers) Therefore, (5*3 + 8*2 + 6*4 + k*1) / (3+2+4+1) = 5.7 Simplifying the above equation, we get: (15 + 16 + 24 + k) / 10 = 5.7 55 + k = 57 k = 2 Therefore, the value of k is 2.

The pie chart above represents 400 fruits on display n a grocery store. How many apples are in the store

Answer Details

Evaluate $\frac{81.81+99.44}{20.09+36.16}$ correct to 3 significant figures.

Answer Details

To evaluate the expression: $$\frac{81.81 + 99.44}{20.09 + 36.16}$$ We simply substitute the numbers in the formula and calculate: $$\frac{81.81 + 99.44}{20.09 + 36.16} = \frac{181.25}{56.25} \approx \boxed{3.23}$$ Therefore, the expression evaluates to approximately 3.23.