JAMB UTME - Mathematics - 2007

A binary operation Δ is defined by aΔb = a + b + 1 for any numbers a and b. Find the inverse of the real number 7 under the operation Δ, if the identity element is -1

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The identity element for the operation Δ is -1, which means that for any real number a, aΔ(-1) = (-1)Δa = a. To find the inverse of 7 under the operation Δ, we need to find a number x such that 7Δx = xΔ7 = -1, which is the identity element. So, we can start by setting up the equation: 7Δx = 7 + x + 1 = xΔ7 = x + 7 + 1 = -1 Simplifying each side of the equation, we get: x + 8 = -1 and 7 + x + 1 = -1 Solving for x in the first equation, we get: x = -1 - 8 = -9 Therefore, -9 is the inverse of 7 under the operation Δ, because 7Δ(-9) = (-9)Δ7 = -1, which is the identity element.

Simplify $\frac{3}{5}÷(\frac{2}{7}\times \frac{4}{3}÷\frac{4}{9})$

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If y = x cos x, find dy/dx

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To find dy/dx of y = x cos x, we can use the product rule of differentiation, which states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. In this case, we have: y = x cos x Using the product rule, we get: dy/dx = cos x - x sin x Therefore, the correct option is: cos x - x sin x. To explain it in simple terms, the derivative of x cos x is equal to cos x minus x times the derivative of cos x, which is -sin x. This gives us cos x - x sin x as the answer.

Find the value of $\frac{tan{60}^o-tan{30}^o}{tan{60}^o+tan{30}^o}$

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Evaluate 101122 - 10122

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101122 - 10122 = (1x23 + 0x22 + 1x21 + 1x20)2 - (1x22 + 0x21 + 1x22)2
(1x8 + 0x4 + 1x2 + 1x1) 2 - (1x4 + 0x2 + 1x1) 2)
= (8 + 0 + 2 + 1) 2 - (4 + 0 + 1) 2
= 112 - 52
= 16 x 6 = 96
9610 to base 2
2/96 = 48 R 0
2/48 = 24 R 0
2/24 = 12 R 0
2/12 = 6 R 0
2/6 = 3 R 0
2/3 = 1 R 1
2/1 = 0 R 1
11000002

If 5, 8, 6 and 2 occur with frequencies 3, 2, 4 and 1 respectively, find the product of the modal and medial number.

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To find the modal and medial numbers from the given frequency distribution, we need to first determine the mode and median of the dataset. The mode is the number that occurs most frequently, and the median is the middle number when the data is arranged in order. In this case, the mode is 6, which occurs with a frequency of 4. The median can be found by arranging the numbers in order: 2, 5, 5, 5, 6, 6, 6, 6, 8, 8 The median is the middle number, which is also 6. Therefore, the modal number is 6, and the medial number is also 6. The product of the modal and medial numbers is: 6 × 6 = 36 Therefore, the answer is 36.

Evaluate $\frac{(05652?04375{)}^2}{0.04}$ correct to three significant figures

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A senatorial candidate had planned to visit seven cities prior to a primary election. However, he could only visit four of the cities. How many different itineraries could be considered?

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Number of itineraries = 7P4
=$\frac{7!}{(7-4)!}=\frac{7!}{3!}=\frac{7\times 6\times 5\times 4\times 3!}{3!}=840$

W ∝ L2 and W = 6 when L = 4. If L = √17 find W

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From the given relation, we have W ∝ L^2. This means that W is directly proportional to L^2. We can write this as W = kL^2, where k is the constant of proportionality. To find the value of k, we can use the given values of W and L. We have W = 6 when L = 4. Substituting these values in the equation above, we get: 6 = k(4^2) 6 = 16k k = 6/16 k = 3/8 Now, we can use this value of k to find W when L = √17. Substituting these values in the equation W = kL^2, we get: W = (3/8)(√17)^2 W = (3/8)(17) W = 51/8 W = 6 3/8 Therefore, the answer is 6 3/8.

Determine the value of ${?}_0^{\frac{?}{2}}(?2cosx)dx$

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The graph above is represented by

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The roots of the graph are -2, -1 and 1

y = (x + 2)(x + 1)(x - 1) = (x + 2)(x2 - 1)

= x3 + 2x2 - x - 2

Find the value of x for which the function f(x) = 2x3 - x2 - 4x + 4 has a maximum value

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f(x) = 2x3 - x2 - 4x – 4
f’(x) = 6x2 - 2x – 4
As f’(x) = 0
Implies 6x2 - 2x – 4 = 0
3x – x – 2 = 0 (By dividing by 2)
(3x – 2)(x + 1) = 0
3x – 2 = 0 implies x = -2/3
Or x + 1 = 0 implies x = -1
f’(x) = 6x2 - 2x – 4
f’’(x) = 12x – 2
At max point f’’(x) < 0

∴f’’(x) = 12x – 2 at x = -1
= 12(-1) – 2
= -12 – 2 = -14
∴Max at x = 1

Find y, if $\sqrt{12}-\sqrt{147}+y\sqrt{3}=0$

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We are given an equation: $$ \sqrt{12} - \sqrt{147} + y\sqrt{3} = 0 $$ To solve for $y$, we can isolate the $\sqrt{3}$ term on one side of the equation: \begin{align*} \sqrt{12} - \sqrt{147} + y\sqrt{3} &= 0 \\ y\sqrt{3} &= \sqrt{147} - \sqrt{12} \\ y &= \frac{\sqrt{147} - \sqrt{12}}{\sqrt{3}} \\ \end{align*} To simplify the expression, we can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$: \begin{align*} y &= \frac{(\sqrt{147} - \sqrt{12})\sqrt{3}}{\sqrt{3}\sqrt{3}} \\ y &= \frac{\sqrt{441} - \sqrt{36}}{3} \\ y &= \frac{21 - 6}{3} \\ y &= \boxed{5} \end{align*} Therefore, $y=5$.

The table above shows the number of pupils in each age group in a class. What is the probability that a pupil chosen at random is at least 1 years old?

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P(At east 11 yrs) = P(11yrs) + P(12yrs)
= 27/40 + 7/40
= 34/40
= 17/20

A man made a profit of 5% when he sold an article for N60,000.00. How much would he have sell the article to make a profit of 26%

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5% profit = 100 + 5 = 105%
26% profit = 100 + 26 = 126%
∴ 105% → N60,000
1% → 60000/15
126% = 1000/105 x 126/1
=N72,000

Find the sum to infinity of the series $2+\frac{3}{2}+\frac{9}{8}+\frac{27}{32}+......$

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The volume of a hemispherical bowl is $718\frac{2}{3}$. Find its radius .

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A binary operation ⊕ on real numbers is defined by x⊕y = xy + x + y for any two real numbers x and y. The value of (-3/4)⊕6 is

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The given binary operation ⊕ on real numbers is defined as x⊕y = xy + x + y for any two real numbers x and y. Substituting the values x = -3/4 and y = 6 in the given expression, we get: (-3/4)⊕6 = (-3/4)×(6) + (-3/4) + (6) = (-9/2) + (21/4) = (-18/4) + (21/4) = 3/4 Therefore, the value of (-3/4)⊕6 is 3/4. Hence, option (A) is the correct answer.

Calculate the length of an arc of a circle diameter 14 cm, which substends an angle of 90${}^{\circ }$ at the center of the circle

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The length of an arc of a circle can be calculated using the formula L = rθ, where L is the length of the arc, r is the radius of the circle, and θ is the central angle subtended by the arc (in radians). In this case, the diameter of the circle is 14 cm, so the radius is half of that, which is 7 cm. The central angle subtended by the arc is 90 degrees, or π/2 radians. Plugging in the values, we get L = 7 × π/2 = 7π/2 cm. Therefore, the length of the arc is 7π/2 cm. So, the correct option is  7π/2 cm.

In the diagram P, Q, R, S are points on the circle RQS = 30o. PRS = 50o and PSQ = 20o. What is the value of xo + yo?

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Draw a line from P to Q

< PQS = < PRS (angle in the sam segment)

< PQS = 50o

Also, < QSR = < QPR(angles in the segment)

< QPR = xo

x + y + 5= = 180(angles in a triangle)

x + y = 180 - 50

x + y = 130o

In a basket, 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?

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There are 6+11+13=30 fruits in the basket. The probability of choosing a grape is 6/30 and the probability of choosing a banana is 11/30. The probability of choosing either a grape or a banana is the sum of these probabilities: 6/30 + 11/30 = 17/30. Therefore, the answer is 17/30.

If X = {all the perfect squares less than 40}

Y = {all the odd numbers fro, 1 to 15}. Find X ∩ Y.

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All the perfect squares < 40

X = {1, 4, 9, 16, 25, 36}
All the odd numbers from 1 to 15
Y = {1, 3, 5, 7, 9, 11, 13, 15}
X ∩ Y = {1, 9}

Given

P = {1, 3, 5, 7, 9, 11}

And Q = {2, 4, 6, 8, 1, 12}. Determine the relationship between P and Q

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The given sets P and Q are two different sets containing some numbers. To determine the relationship between the sets, we need to compare the elements of the sets. By comparing the elements of P and Q, we can see that they have only one element in common, which is 1. The other elements in the sets are unique and do not appear in both sets. Therefore, we can say that P and Q have some elements that are different and some that are the same. Hence, the correct answer is P ∩ Q ≠ ∅.

The area of a square is 144 sq cm. Find the length of its diagonal

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BD = $\sqrt{x^2+x^2}$
= $\sqrt{{12}^2+{12}^2}$
= $\sqrt{144+144}$
= 2(144)
= 12$\sqrt{2}$cm

The solution of the quadratic inequality (x3 + x - 12) ≥ 0 is

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(x3 + x - 12) ≥ 0
(x + 4)(x - 3) ≥ 0
Either x + 4 ≥ 0 implies x ≥ -4
Or x - 3 ≥ 0 implies x ≥ 3
∴ x ≥ 3 or x ≥ -4

The histogram above represents the weights of students who travelled out to their school for an examination. How many people made the trip.

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The pie chart above illustrate the amount of private time a student spends in a week studying various subjects. Find the value of k

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To find the value of k, we need to use the information given in the pie chart. The sum of the angles in a circle is 360 degrees. In this case, the pie chart represents the amount of private time a student spends in a week studying various subjects, so the sum of the angles in the chart should be 360 degrees. We can start by finding the angles of the sectors for the subjects that are mentioned in the chart: Math, Science, English, History, and Others. The angles are: Math: 90 degrees Science: 60 degrees English: 60 degrees History: 60 degrees Others: k degrees We know that the sum of these angles should be 360 degrees. Therefore, we can write the equation: 90 + 60 + 60 + 60 + k = 360 Simplifying this equation, we get: 330 + k = 360 Subtracting 330 from both sides, we get: k = 30 Therefore, the value of k is 30 degrees. To explain it simply, we can use the fact that the sum of the angles in a circle is 360 degrees to find the value of k. We can find the angles of the sectors for the mentioned subjects and use the sum of these angles to set up an equation. By solving the equation, we can find that the value of k is 30 degrees.

The nth term of the sequence 3/2, 3, 7, 16, 35, 74 ..... is

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The nth term of the sequence is 5 . 2n-2 - n. To understand why, we can look at how the sequence is generated. The first term is 3/2, the second term is 3, and each subsequent term is generated by doubling the previous term and subtracting its position in the sequence. For example, to get the third term, we double the second term (which is 3) to get 6, and then subtract the position of the term (which is 3) to get 3+3=6. Similarly, to get the fourth term, we double the third term (which is 6) to get 12, and then subtract the position of the term (which is 4) to get 12-4=8. Using this pattern, we can derive the general formula for the nth term: 5 . 2n-2 - n.

Integrate $\frac{x^2-\sqrt{x}}{x}$ with respect to x

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To integrate x² - √x/x with respect to x, we can start by factoring the expression as follows: x² - √x/x = x - 1/√x Then we can integrate each term separately: ∫(x - 1/√x) dx = ∫x dx - ∫(1/√x) dx The first integral is straightforward: ∫x dx = 1/2 x² + C1 For the second integral, we can use the substitution u = √x, du/dx = 1/(2√x), dx = 2√x du: ∫(1/√x) dx = ∫2 du = 2u + C2 = 2√x + C2 Substituting back u = √x, we get: ∫(1/√x) dx = 2√x + C2 Putting everything together, we have: ∫(x² - √x/x) dx = ∫x dx - ∫(1/√x) dx = (1/2 x² + C1) - (2√x + C2) = 1/2 x² - 2√x + C where C = C1 - C2 is the constant of integration. Therefore, the correct option is x²/2 - 2√x + K, where K = C is the constant of integration.

If the lines 3y = 4x - 1 and qy = x + 3 are parallel to each other, the value of q is

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To determine the value of q that makes the two lines 3y = 4x - 1 and qy = x + 3 parallel to each other, we need to remember that parallel lines have the same slope. The slope of the line 3y = 4x - 1 can be found by rearranging the equation into slope-intercept form, y = (4/3)x - 1/3, where the slope is 4/3. Similarly, the slope of the line qy = x + 3 is 1/q. For these two lines to be parallel, their slopes must be equal. Therefore, we can set 4/3 equal to 1/q and solve for q: 4/3 = 1/q q = 3/4 Therefore, the value of q that makes the two lines parallel is 3/4.

x10 = 12145 find x.

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x10 = 1214 5, 1 x 53 + 2 x 52 + 1 x 51 + 4 x 5o
= 125 + 50 + 5 + 4 18410
x = 184

$\begin{array}{cccc}\text{Age in years}& 10& 11& 12\\ \text{Number of pupils}& 6& 27& 7\end{array}$

The table above shows the number of pupils in each age group in a class. What is the probability that a pupil chosen at random is at least 11 years old?

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A particle P moves between points S and T such that angles SPT is always constant of ST constant. Find the locus off P

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What is the value of k if the mid-point of the line joining (1 - k, - 4) and (2, k + 1) is (-k , k)?

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(1-k+2) / 2 = - k and -4 + k + 1 = k
3-k = -2k and -3 + k = 2k
K = -3 and k = -3

The graph above is represented by

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x = -2, x = -1 and x = 1
then the factors; x+2, x+1 and x-1
Product of the factors; (x+2)(x+1)(x-1)
= y = (x + 2)(x2 - x + x - 1)
= y = (x+2)(x2-1)
x3 - x + 2x2 - 2 = y
x3 + 2x2 - x - 2 = y

The area of a square is 144 sqcm. Find the length of the diagonal.

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To find the length of the diagonal of a square, we need to use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). In this case, the two legs of the right triangle are the sides of the square, and the hypotenuse is the length of the diagonal. Let's call the length of one side of the square "x". Then, we know that the area of the square is given by: x^2 = 144 sqcm Taking the square root of both sides, we get: x = 12 cm Now, using the Pythagorean theorem, we can find the length of the diagonal: d^2 = x^2 + x^2 = 2x^2 d = sqrt(2x^2) = x * sqrt(2) = 12 * sqrt(2) cm Therefore, the length of the diagonal is 12√2 cm.

Find the sum to infinity to the following series 3 + 2 + $\frac{4}{3}$ + $\frac{8}{9}$ + $\frac{16}{17}$ + .....

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To find the sum to infinity of this series, we need to determine if it is a converging or diverging series. We can do this by finding the common ratio between each term. The common ratio between the second and first term is 2/3. The common ratio between the third and second term is 4/3. The common ratio between the fourth and third term is 8/9, and so on. We can see that the common ratio is less than 1, so the series is converging. Therefore, we can use the formula for the sum of an infinite geometric series: S = a/(1 - r) where S is the sum, a is the first term, and r is the common ratio. In this case, the first term is 3 and the common ratio is 2/3. So, plugging these values into the formula, we get: S = 3/(1 - 2/3) = 3/(1/3) = 9 Therefore, the sum to infinity of this series is 9. So, the answer to the question is option (D) 9.

Make L the subjects of the formula if $\sqrt{\frac{42w}{5l}}$

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$\sqrt{$ \sqrt{\frac{42w}{5l}}$$ d^2={\left(\sqrt{\frac{42W}{5l}}\right)}^2{d}^2=\frac{42W}{5l}5l{d}^2=42Wl=\frac{42W}{5d^2}$}$

If x10 = 12145 find x

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x10 = 12145
= x10 = 1 x 53 + 2 x 52 + 1 * 51 + 4 x 50
= 1 x 125 + 2 x 25 + 1 x 5 + 4 x 1
= 125 + 50 + 5 + 4
= 184

Each of the interior angles of a regular polygon is 140o. How many sides has the polygon?

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The sum of the interior angles of a polygon can be found using the formula: S = (n - 2) × 180o where S is the sum of the interior angles, and n is the number of sides in the polygon. For a regular polygon, all interior angles have the same measure. In this case, the interior angle of the polygon is given as 140o. Therefore, we can use the formula: S = n × 140o Substituting this into the formula for the sum of interior angles, we get: n × 140o = (n - 2) × 180o Simplifying this equation, we get: 140n = 180n - 360 Solving for n, we get: 40n = 360 n = 9 Therefore, the polygon has 9 sides. The answer is 9.

If log102 = x, express log1012.5 in terms of x

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We know that log base 10 of 2 is x, which means that 10 to the power of x is equal to 2. To express log base 10 of 12.5 in terms of x, we need to find a way to write 12.5 in terms of 2 and x. We can write 12.5 as 10 to the power of 1.09691 (approximately) using a calculator. Now, we can use the laws of logarithms to simplify the expression: log base 10 of 12.5 = log base 10 of (10^1.09691) = 1.09691 * log base 10 of 10 = 1.09691 Therefore, we want to find an expression among the given options that equals 1.09691 when x is substituted into it. We can check each option by substituting x into it and simplifying: : 2(1 + x) = 2 + 2x Substituting x = log base 10 of 2 gives 2 + 2(log base 10 of 2), which does not equal 1.09691. : 2 + 3x Substituting x = log base 10 of 2 gives 2 + 3(log base 10 of 2), which also does not equal 1.09691. : 2(1 - x) = 2 - 2x Substituting x = log base 10 of 2 gives 2 - 2(log base 10 of 2), which also does not equal 1.09691. : 2 - 3x Substituting x = log base 10 of 2 gives 2 - 3(log base 10 of 2), which equals 1.09691. Therefore, the correct answer is: 2 - 3x.

If the lines 2y - kx + 2 = 0 and y + x - k/2 = 0 Intersect at (1, -2), find the value of k

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The problem gives two equations of two lines and a point of intersection between them. We need to find the value of "k" in one of the equations. The point of intersection (1, -2) lies on both lines, so it must satisfy both equations. Substituting x=1 and y=-2 in the first equation 2y - kx + 2 = 0 gives: 2(-2) - k(1) + 2 = 0 Simplifying this equation: -4 - k + 2 = 0 -2 - k = 0 k = -2 Therefore, the value of k is -2. Option (C) is the correct answer.

Factorize 2t2 + t - 15

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To factorize 2t² + t - 15, we need to find two binomials that multiply to give us 2t² + t - 15. To do this, we can use the factoring method called "AC method." First, we need to find two numbers whose product is 2(-15) = -30 and whose sum is 1. These numbers are 6 and -5. Next, we replace the middle term t with 6t - 5t: 2t² + 6t - 5t - 15 Then we group the terms: (2t² + 6t) - (5t + 15) We factor out the greatest common factor from each group: 2t(t + 3) - 5(t + 3) We notice that we have a common binomial factor of (t + 3), so we can factor it out: (t + 3)(2t - 5) Therefore, the factored form of 2t² + t - 15 is (t + 3)(2t - 5). So, the correct option is: (t + 3)(2t - 5).

What is the mean deviation of 3, 5, 8, 11, 12 and 21?

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The mean deviation is a measure of the spread of a dataset. It is calculated by finding the average of the absolute deviations of the values from the mean. First, we need to find the mean of the given values: mean = (3+5+8+11+12+21)/6 = 60/6 = 10 Next, we find the absolute deviation of each value from the mean: |3-10| = 7 |5-10| = 5 |8-10| = 2 |11-10| = 1 |12-10| = 2 |21-10| = 11 The sum of these absolute deviations is: 7+5+2+1+2+11 = 28 Finally, we find the mean deviation by dividing the sum of absolute deviations by the number of values: mean deviation = 28/6 = 4.7 Therefore, the mean deviation of the given values is 4.7.

In a basket, 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

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To find the probability that a fruit chosen at random is either a grape or a banana, we need to first determine the total number of fruits in the basket, which is the sum of grapes and bananas, since we are only interested in these two types of fruits. Total number of grapes and bananas = 6 + 11 = 17 Therefore, the probability of choosing either a grape or a banana is: P(grape or banana) = (number of grapes + number of bananas) / total number of fruits = 17 / (6 + 11 + 13) = 17 / 30 Hence, the probability that the fruit chosen is either a grape or a banana is 17/30. Therefore, the answer is not any of the options provided.

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To solve this problem, we need to use the formula for the mean of a frequency distribution: mean = (sum of (value × frequency))/total frequency From the table, we can see that the total frequency is 25, and the sum of (value × frequency) is: 5 × 4 + 6 × 6 + 7 × 7 + 8 × 5 + 9 × 3 = 20 + 36 + 49 + 40 + 27 = 172 So, the mean mark is: mean = 172/25 = 6.88 We also know that the total mark scored is 200, so we can set up an equation: total mark = mean × total frequency + y Substituting in the values we know, we get: 200 = 6.88 × 25 + y Solving for y, we get: y = 200 - 6.88 × 25 = 11 Therefore, the value of y is 11. Answer: 11.

A man 40 m from the foot of a tower observes the angle of elevation of the tower to be 30${}^{\circ }$. Determine the height of the tower.

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The problem involves finding the height of a tower, given the distance of a person from the foot of the tower and the angle of elevation of the tower from the person. In this case, the person is 40 meters away from the foot of the tower, and observes the angle of elevation to be 30 degrees. To solve for the height of the tower, we can use the tangent function, which relates the opposite (height) and adjacent (distance) sides of a right triangle to the tangent of an angle. Let h be the height of the tower. Then, we have: tangent(30 degrees) = opposite/adjacent tangent(30 degrees) = h/40 Using a calculator, we can evaluate the tangent of 30 degrees to be approximately 0.577. Substituting this value into the equation above, we get: 0.577 = h/40 To solve for h, we can multiply both sides by 40: 0.577 x 40 = h h = 23.08 Therefore, the height of the tower is approximately 23.08 meters. The closest option is 20m, but it's not the correct answer. The correct answer is not given in the options, but it is approximately 23.08 meters, which is between the options 1 and 4.

In how many ways can 6 subjects be selected from 10 subjects for an examination

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To find the number of ways to select 6 subjects from 10 subjects, we can use the formula for combinations, which is: nCr = n! / (r! * (n-r)!) where n is the total number of subjects and r is the number of subjects to be selected. In this case, we have n = 10 and r = 6, so we can plug these values into the formula: 10C6 = 10! / (6! * (10-6)!) Simplifying this expression gives: 10C6 = (10*9*8*7*6*5) / (6*5*4*3*2*1) Canceling out the common factors, we get: 10C6 = 10*9*8*7 / 4*3*2*1 10C6 = 210 Therefore, there are 210 ways to select 6 subjects from 10 subjects for an examination. Therefore, the correct options are (a) and (d).

Find the locus of point equidistant from two straight lines y - 5 = 0 and y - 3 = 0

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Locus of point P equidistant from y - 5 = 0 and y - 3 = 0 is y = 4 i.e y - 4 = 0

Find the value of x for which 2(32x-1) = 162

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