Maths MCQ Quiz - Objective Question with Answer for Maths - Download Free PDF

Explanation:

Given:

\(P(A) = P(\frac{A}{B}) = 0.25\)

and \(P(\frac{B}{A}) = 0.5\)

I. \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(A)}\)

⇒ P(A∩B) = P(A) P(B|A)

⇒ P(A∩B) = 0.25 × 0.5 = 0.125

Now

⇒ \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(B)}\)

⇒ \(P(B)= \frac{P(A∩ B)}{P(\frac{A}{B})}\)

⇒ \(P(B) = \frac{0.125}{0.25} = 0.5\)

Now, P(A).P(B) = 0.25 × 0.5 = 0.125 = P(A∩B)

Thus  A and B are independent

II. \(P(\overline A\cup \overline B ) = 1 – P(A ∩ B)\)

= 1 – 0.125 = 0.875

III. \(P(\overline A∩ \overline B ) = 1 – P(A \cup B)\)

= 1 – [P(A) + P(B) – P(A ∩ B)

=  1 – [0.25 + 0.5 – 0.125]

= = 1 – 0.625 = 0.375

So all statements I, II, and III are correct.

∴ Option (d) is correct.

\( \log_{20}(x^2 - 25) < \log_{20} (13x - 55) \)

when \( 13x - 55 > x^2 - 25 \)

\( \Rightarrow x^2 - 13x + 30 < 0 \)

\( \Rightarrow (x - 3)(x - 10) < 0 \)

\( \Rightarrow x \in (3, 10) \)

But \( \log_{20} (x^2 - 25) \) is not defined for \( x \in [-5, 5] \)

\( \Rightarrow \) The solution set of the inequality is \( (5, 10) \).

m = x₂−x₁/ y_2​ − y_1​​

Substituting the points (x1,y1)=(2,18)Unknown node type: span" id="MathJax-Element-8-Frame" role="presentation" style="position: relative;" tabindex="0">(x1,y1)=(2,18)Unknown node type: span(x1,y1)=(2,18)<merror><mtext>Unknown node type: span</mtext></merror> (x1,y1)=(2,18)<span style="font-family: var(--bs-body-font-family); font-size: var(--bs-body-font-size); font-weight: var(--bs-body-font-weight); text-align: var(--bs-body-text-align);"> </span> (x2,y2)=(5,22) (x_2, y_2) = (5, 22)" id="MathJax-Element-9-Frame" role="presentation" style="position: relative;" tabindex="0">(x2,y2)=(5,22)$(x_2, y_2) = (5, 22)$ $(x_2, y_2) = (5, 22)$

 

$$m = \frac{22 - 18}{5 - 2}$$ m=43=1.33

 

The slope m m" id="MathJax-Element-13-Frame" role="presentation" style="position: relative;" tabindex="0">m$m$ $m$  of the best-fit line is 1.33, meaning for every 1 unit increase in X, Y increases by 1.33 units. $$m = \frac{4}{3} = 1.33$$
 

Since the population decreases by a fixed percentage each month, this follows an exponential decay model, making option A (Decreasing exponential) the correct choice.

Solution:

• (A) Incorrect: The data points show that individuals weighing 65 pounds have more offspring than those weighing 50 pounds. So, this statement is false.
• (B) Correct: The number of offspring increases as weight increases, which aligns with what we observe in the graph.
• (C) Incorrect: The number of offspring is not constant—it changes with weight, so this statement is false.
• (D) Incorrect: The lowest weight (40 pounds) corresponds to the lowest number of offspring (5), not the highest.

The correct answer is 27.

It's given that a triangular prism has a volume of 216 cubic centimeters (cm3) and the volume of a triangular prism is equal to Bh, where B is the area of the base and h is the height of the prism.
Therefore, 216 = Bh. It's also given that the triangular prism has a height of 8 cm.
Therefore, h = 8.
Substituting 8 for h in the equation 216 = Bh yields 216 = B(8).
Dividing both sides of this equation by 8 yields 27 = B.
Therefore, the area, in cm2, of the base of the prism is 27 .

Choice C is correct. The volume of right circular cylinder A is given by the expression πr²h, where r is the radius of its circular base and h is its height. The volume of a cylinder with twice the radius and half the height of cylinder A is given by \(\pi(2 r)^2\left(\frac{1}{2}\right) h\), which is equivalent to \(4 \pi r^2\left(\frac{1}{2}\right) h=2 \pi r^2 h\). Therefore, the volume is twice the volume of cylinder A , or 2 × 22 = 44.

Choice A is incorrect and likely results from not multiplying the radius of cylinder A by 2. Choice B is incorrect and likely results from not squaring the 2 in 2 r when applying the volume formula. Choice D is incorrect and likely results from a conceptual error. 

The correct answer is 31.
The circumference of a circle is equal to 2πr centimeters, where r represents the radius, in centimeters, of the circle, and the diameter of the circle is equal to 2r centimeters.
It's given that a circle has a circumference of 31π centimeters.
Therefore, 31π = 2πr.
Dividing both sides of this equation by π yields 31 = 2r.
Since the diameter of the circle is equal to 2r centimeters, it follows that the diameter, in centimeters, of the circle is 31.

Choice C is correct. The first expression (1.5x - 2.4)² can be rewritten as (1.5x - 2.4)(1.5x - 2.4). Applying the distributive property to this product yields (2.25x² - 3.6x - 3.6x + 5.76)-(5.2x² - 6.4). This difference can be rewritten as (2.25x² - 3.6x - 3.6x + 5.76) + (-1)(5.2x² - 6.4). Distributing the factor of -1 through the second expression yields 2.25x² - 3.6x - 3.6x + 5.76 - 5.2x² + 6.4. Regrouping like terms, the expression becomes (2.25x² - 5.2x²) + (-3.6x - 3.6x) + (5.76 + 6.4). Combining like terms yields -2.95x² - 7.2x + 12.16.

Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.

The correct answer is 66. It's given that the right circular cone has a height of 22 centimeters (cm) and a base with a diameter of 6 cm. Since the diameter of the base of the cone is 6 cm, the radius of the base is 3 cm. The volume V, in cm³, of a right circular cone can be found using the formula \(V=\frac{1}{3} \pi r^2 h\), where h is the height, in cm, and r is the radius, in cm , of the base of the cone. Substituting 22 for h and 3 for r in this formula yields \(V=\frac{1}{3} \pi(3)^2(22)\), or V = 66 π. Therefore, the volume of the cone is 66π cm³. It's given that the volume of the cone is nπ cm³. Therefore, the value of n is 66.

Explanation:

Given:

\(P(A) = P(\frac{A}{B}) = 0.25\)

and \(P(\frac{B}{A}) = 0.5\)

I. \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(A)}\)

⇒ P(A∩B) = P(A) P(B|A)

⇒ P(A∩B) = 0.25 × 0.5 = 0.125

Now

⇒ \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(B)}\)

⇒ \(P(B)= \frac{P(A∩ B)}{P(\frac{A}{B})}\)

⇒ \(P(B) = \frac{0.125}{0.25} = 0.5\)

Now, P(A).P(B) = 0.25 × 0.5 = 0.125 = P(A∩B)

Thus  A and B are independent

II. \(P(\overline A\cup \overline B ) = 1 – P(A ∩ B)\)

= 1 – 0.125 = 0.875

III. \(P(\overline A∩ \overline B ) = 1 – P(A \cup B)\)

= 1 – [P(A) + P(B) – P(A ∩ B)

=  1 – [0.25 + 0.5 – 0.125]

= = 1 – 0.625 = 0.375

So all statements I, II, and III are correct.

∴ Option (d) is correct.

The correct answer is Option 1.

Key Points

• Let the probability that A hits the target be P(A)=14" id="MathJax-Element-62-Frame" role="presentation" style="position: relative;" tabindex="0">P(A)=14P(A)=14 $P(A) = \frac{1}{4}$ .
• Let the probability that B hits the target be P(B)=25" id="MathJax-Element-63-Frame" role="presentation" style="position: relative;" tabindex="0">P(B)=25P(B)=25 $P(B) = \frac{2}{5}$ .
• The probability that neither A nor B hits the target is the product of their individual probabilities of missing the target.
• The probability that A misses the target is 1−P(A)=1−14=34" id="MathJax-Element-64-Frame" role="presentation" style="position: relative;" tabindex="0">1−P(A)=1−14=341−P(A)=1−14=34 $1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}$ .
• The probability that B misses the target is 1−P(B)=1−25=35" id="MathJax-Element-65-Frame" role="presentation" style="position: relative;" tabindex="0">1−P(B)=1−25=351−P(B)=1−25=35 $1 - P(B) = 1 - \frac{2}{5} = \frac{3}{5}$ .
• The probability that neither hits the target is (34)×(35)=920" id="MathJax-Element-66-Frame" role="presentation" style="position: relative;" tabindex="0">(34)×(35)=920(34)×(35)=920 $\left(\frac{3}{4}\right) \times \left(\frac{3}{5}\right) = \frac{9}{20}$ .
• The probability that at least one of them hits the target is 1−920=1120" id="MathJax-Element-67-Frame" role="presentation" style="position: relative;" tabindex="0">1−920=11201−920=1120 $1 - \frac{9}{20} = \frac{11}{20}$ .

Additional Information

• This problem involves the concept of complementary probability, which is useful when calculating the probability of at least one event occurring.In probability theory, the complement rule is used to determine the probability of an event not occurring.
• Option 1 is indeed correct as the final probability calculation matches the probability required.

Choice C is correct. The predicted increase in total fat, in grams, for every increase of 1 gram in total protein is represented by the slope of the line of best fit. Any two points on the line can be used to calculate the slope of the line as the change in total fat over the change in total protein. For instance, it can be estimated that the points (20,34) and (30,48) are on the line of best fit, and the slope of the line that passes through them is \(\frac{48-34}{30-20}=\frac{14}{10}\), or 1.4. Of the choices given, 1.5 is the closest to the slope of the line of best fit.
Choices A, B, and D are incorrect and may be the result of incorrectly finding ordered pairs that lie on the line of best fit or of incorrectly calculating the slope.

Choice B is correct. From the shape of the cluster of points, the line of best fit should pass roughly through the points (1,200) and (2.5.350). Therefore, these two points can be used to find an approximate equation for the line of best fit. The slope of this line of best fit is therefore \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{350-200}{2.5-1}\), or 100. The equation for the line of best fit, in slope-intercept form, is y = 100 x + b for some value of b. Using the point (1,200), 1 can be substituted for x and 200 can be substituted for y: 200 = 100(1) + b, or b = 100. Substituting this value into the slope-intercept form of the equation gives y = 100 x + 100. Choice A is incorrect. The line defined by y = 200 x + 100 passes through the points (1,300) and (2,500), both of which are well above the cluster of points, so it cannot be a line of best fit. Choice C is incorrect. The line defined by y = 50 x + 100 passes through the points (1,150) and (2,200), both of which lie at the bottom of the cluster of points, so it cannot be a line of best fit. Choice D is incorrect and may result from correctly calculating the slope of a line of best fit but incorrectly assuming the y-intercept is at (0, 0).

Choice D is correct. The data in the scatterplot roughly fall in the shape of a downward-opening parabola; therefore, the coefficient for the x² term must be negative. Based on the location of the data points, the y- intercept of the parabola should be somewhere between 740 and 760. Therefore, of the equations given, the best model is y = -1.674x2 + 19.76x + 745.73.
Choices A and C are incorrect. The positive coefficient of the x2 term means that these equations each define upward-opening parabolas, whereas a parabola that fits the data in the scatterplot must open downward. Choice B is incorrect because it defines a parabola with a y-intercept that has a negative y-coordinate, whereas a parabola that fits the data in the scatterplot must have a y-intercept with a positive y-coordinate.