All practice questions

Question 4141Question →

In $\Delta PQR$ , side $p = 4\text{ cm}$ , side $q = 4\sqrt{2}\text{ cm}$ , and $\angle P = 30^\circ$ . If $\angle Q$ is an obtuse angle, what is the measure of $\angle Q$ ?

135^\circ

45^\circ

60^\circ

150^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

The measure of angle

Q

is

135^\circ

.

Applying the Sine Rule gives

\sin Q = \frac{4\sqrt{2} \cdot \sin 30^\circ}{4} = \frac{\sqrt{2}}{2}

. The acute reference angle is

45^\circ

, so the obtuse angle is

180^\circ - 45^\circ = 135^\circ

.

Step-by-Step Solution

1

Apply the Sine Rule to relate sides

p, q

and angles

P, Q

\frac{p}{\sin P} = \frac{q}{\sin Q} \implies \frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin Q}

The Sine Rule connects pairs of opposite sides and angles in any triangle.

2

Solve for

\sin Q

\sin Q = \frac{4\sqrt{2} \cdot \sin 30^\circ}{4} = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}

Substitute

\sin 30^\circ = \frac{1}{2}

and simplify the numerical expression.

3

Determine the obtuse solution for angle

Q

Q = 180^\circ - 45^\circ = 135^\circ

Since

\sin Q = \frac{\sqrt{2}}{2}

, the principal acute angle is

45^\circ

. Because the problem specifies that angle

Q

is obtuse (

90^\circ < Q < 180^\circ

), we take its supplement.

Key Concept

Ambiguous case of the Sine Rule (SSA condition)

Estimated Time:1m 30s

Question 4142Question →

What is the simplified form of the expression $\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}}$ ?

\frac{5 + \sqrt{15}}{2}

\sqrt{5}

5 + \sqrt{15}

\frac{5 - \sqrt{15}}{2}

Show answer & explanation

Answer: $\frac{5 + \sqrt{15}}{2}$

Answer

\frac{5 + \sqrt{15}}{2}

To rationalize the denominator of

\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}}

, multiply both numerator and denominator by the conjugate

\sqrt{5} + \sqrt{3}

. The numerator becomes

\sqrt{5}(\sqrt{5} + \sqrt{3}) = 5 + \sqrt{15}

and the denominator becomes

(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2

, yielding

\frac{5 + \sqrt{15}}{2}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

\sqrt{5} - \sqrt{3}

is

\sqrt{5} + \sqrt{3}

.

Multiplying a binomial surd by its conjugate eliminates the radical terms in the denominator using the difference of two squares identity

(a-b)(a+b) = a^2 - b^2

.

2

Multiply both the numerator and the denominator by the conjugate

\frac{\sqrt{5}(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}

Multiplying the fraction by

\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}

is equivalent to multiplying by

1

, preserving the value of the expression.

3

Expand the numerator and simplify the denominator

Numerator:

\sqrt{5} \times \sqrt{5} + \sqrt{5} \times \sqrt{3} = 5 + \sqrt{15}

. Denominator:

(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2

.

Applying the distributive law to the numerator and difference of squares to the denominator simplifies both parts.

4

Combine the simplified numerator and denominator

\frac{5 + \sqrt{15}}{2}

This is the simplified rationalized form.

Key Concept

Rationalization of Denominators

Question 4143Question →

The table below shows the distribution of scores obtained by $60$ candidates in a competitive aptitude test:

Score Class Interval	Frequency ( $f$ )
$10 - 19$	$5$
$20 - 29$	$10$
$30 - 39$	$18$
$40 - 49$	$15$
$50 - 59$	$12$

Using cumulative frequency estimation (or linear interpolation), what is the interquartile range of the distribution?

18.0

20.0

30.0

15.0

Show answer & explanation

Answer: $18.0$

Answer

The interquartile range of the distribution is

18.0

.

The lower quartile (

Q_1

) corresponds to the

15^{\text{th}}

rank, which is exactly

29.5

. The upper quartile (

Q_3

) corresponds to the

45^{\text{th}}

rank, which interpolates to

47.5

. Subtracting

Q_1

from

Q_3

yields

47.5 - 29.5 = 18.0

.

Step-by-Step Solution

1

Construct the cumulative frequency distribution table with upper class boundaries.

Class intervals, upper class boundaries (

UCB

), frequencies (

f

), and cumulative frequencies (

cf

):
-

10 - 19

:

UCB = 19.5

,

f = 5

,

cf = 5

-

20 - 29

:

UCB = 29.5

,

f = 10

,

cf = 15

-

30 - 39

:

UCB = 39.5

,

f = 18

,

cf = 33

-

40 - 49

:

UCB = 49.5

,

f = 15

,

cf = 48

-

50 - 59

:

UCB = 59.5

,

f = 12

,

cf = 60

Cumulative frequencies and upper class boundaries are necessary to determine quartile ranks and values.

2

Find the lower quartile (

Q_1

).

The position of

Q_1

is

\frac{1}{4} N = \frac{1}{4}(60) = 15^{\text{th}}

candidate score. Since the cumulative frequency reaching

UCB = 29.5

is exactly

15

,

Q_1 = 29.5

.

The 15th value lies precisely at the upper class boundary of the

20 - 29

class interval.

3

Find the upper quartile (

Q_3

) using linear interpolation.

The position of

Q_3

is

\frac{3}{4} N = \frac{3}{4}(60) = 45^{\text{th}}

score.
This falls in the

40 - 49

class interval (

UCB = 49.5

, lower boundary

L = 39.5

,

f = 15

, previous

cf = 33

).

Q_3 = L + \left(\frac{\frac{3N}{4} - cf_{\text{prev}}}{f}\right) \times c = 39.5 + \left(\frac{45 - 33}{15}\right) \times 10 = 39.5 + 8.0 = 47.5

.

Linear interpolation calculates the exact position within the target class interval.

4

Calculate the Interquartile Range (IQR).

\text{IQR} = Q_3 - Q_1 = 47.5 - 29.5 = 18.0

.

The interquartile range is defined as the difference between the upper quartile and the lower quartile.

Key Concept

Interquartile Range estimation from Ogive / Cumulative Frequency Distribution

Estimated Time:1m 30s

Question 4144Question →

If $\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}}$ is expressed in the simplified form $a + b\sqrt{6}$ , where $a$ and $b$ are rational numbers, what is the value of $a + b$ ?

5

7

9

13

Show answer & explanation

Answer: 7

Answer

The correct value of

a + b

is 7.

Multiplying by the conjugate

(3\sqrt{2} + 2\sqrt{3})

reduces the denominator to

18 - 12 = 6

and expands the numerator to

30 + 12\sqrt{6}

. Dividing by 6 yields

5 + 2\sqrt{6}

, giving

a = 5

and

b = 2

, which sums to 7.

Step-by-Step Solution

1

Multiply the numerator and the denominator by the conjugate of the denominator,

(3\sqrt{2} + 2\sqrt{3})

.

\frac{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}{(3\sqrt{2} - 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}

Rationalizing eliminates the surd terms from the denominator using the difference of two squares.

2

Expand both numerator and denominator.

Denominator:

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

. Numerator:

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

.

Apply algebraic identities

(x-y)(x+y) = x^2 - y^2

and

(x+y)^2 = x^2 + 2xy + y^2

.

3

Divide the numerator by the denominator to simplify the expression.

\frac{30 + 12\sqrt{6}}{6} = 5 + 2\sqrt{6}

Both integer and radical coefficients are divisible by 6.

4

Equate

5 + 2\sqrt{6}

to

a + b\sqrt{6}

and calculate

a + b

.

a = 5, b = 2, so a + b = 5 + 2 = 7.

Matching corresponding rational and irrational components.

Key Concept

Rationalization of Binomial Denominators with Surds

Question 4145Question →

The mean of eight consecutive odd numbers is $24$ . What is the median of the first four numbers?

Show answer & explanation

Answer: 20

Answer

The median of the first four numbers is

20

.

Letting the eight consecutive odd numbers be

x, x+2, \dots, x+14

, their sum is

8x + 56

. Dividing by

8

gives a mean of

x + 7 = 24

, so

x = 17

. The first four numbers are

17, 19, 21,

and

23

. The median of these four values is the average of the middle two values (

19

and

21

), which equals

20

.

Step-by-Step Solution

1

Represent the eight consecutive odd numbers algebraically

Let the numbers be

x, x+2, x+4, x+6, x+8, x+10, x+12, x+14

.

Consecutive odd numbers increase by steps of

2

.

2

Set up and solve the mean equation

\frac{(x) + (x+2) + (x+4) + (x+6) + (x+8) + (x+10) + (x+12) + (x+14)}{8} = 24 \implies x + 7 = 24 \implies x = 17

.

The mean of ungrouped data is the sum of all data values divided by the total number of values.

3

Identify the first four numbers in the set

The first four numbers are

17, 19, 21, 23

.

Substitute

x = 17

into

x, x+2, x+4,

and

x+6

.

4

Find the median of the first four numbers

\text{Median} = \frac{19 + 21}{2} = 20

.

For an even count of ordered values (4 items), the median is the arithmetic mean of the two middle terms.

Key Concept

Measures of Central Tendency for Ungrouped Data

Question 4146Question →

Find the maximum integer value of $m$ for which the quadratic inequality $x^2 - mx + 9 > 0$ holds for all real values of $x$ .

Show answer & explanation

Answer: 5

Answer

The maximum integer value of

m

is

5

.

For the quadratic expression

x^2 - mx + 9

to remain strictly positive for all real values of

x

, the quadratic curve must lie completely above the x-axis. Because the coefficient of

x^2

is positive (

1 > 0

), this requires the discriminant to be strictly negative (

D < 0

). Evaluating

b^2 - 4ac < 0

gives

m^2 - 36 < 0

, which simplifies to

-6 < m < 6

. The largest integer strictly less than

6

is

5

.

Step-by-Step Solution

1

Determine the condition for the quadratic expression to be positive for all real values of

x

.

Since the leading coefficient is

1 > 0

, the condition is that the discriminant

D = b^2 - 4ac < 0

.

A parabola opening upward lies entirely above the horizontal axis when it has no real roots.

2

Calculate the discriminant using the coefficients of the quadratic expression.

D = (-m)^2 - 4(1)(9) = m^2 - 36 < 0

.

Here

a = 1

,

b = -m

, and

c = 9

.

3

Solve the quadratic inequality for

m

.

m^2 - 36 < 0 \implies -6 < m < 6

.

The roots of

m^2 - 36 = 0

are

m = -6

and

m = 6

, and the expression is negative strictly between these boundary values.

4

Determine the maximum integer value within the open interval

(-6, 6)

.

The maximum integer value is

5

.

The boundary value

6

is excluded by the strict inequality

m < 6

.

Key Concept

Quadratic Inequalities and Discriminant Conditions for Positive Definiteness

Question 4147Question →

The sum of the first three terms of an arithmetic progression (AP) with a positive common difference $d$ is $21$ . If $2$ is added to the first term, $3$ is added to the second term, and $9$ is added to the third term, the resulting three numbers form consecutive terms of a geometric progression (GP). What is the sum of the first $10$ terms of this arithmetic progression?

Show answer & explanation

Answer: 210

Answer

The sum of the first 10 terms of the arithmetic progression is 210.

Representing the AP terms as

7-d, 7, 7+d

and adding the specified values produces GP terms

9-d, 10, 16+d

. Solving

10^2 = (9-d)(16+d)

yields

d=4

. Consequently, the first term of the AP is

3

. Using

S_{10} = \frac{10}{2}[2(3) + 9(4)]

gives the final answer 210.

Step-by-Step Solution

1

Express AP terms symmetrically and solve for the middle term.

The middle term is

a = 7

, making the terms

7-d

,

7

, and

7+d

.

Choosing terms

a-d, a, a+d

allows the sum equation

3a = 21

to directly isolate the middle term.

2

Set up the geometric progression relation to determine common difference

d

.

The GP terms are

9-d

,

10

, and

16+d

. Solving

10^2 = (9-d)(16+d)

gives

d^2 + 7d - 44 = 0

, yielding

d = 4

.

In any geometric progression, the square of the middle term equals the product of the first and third terms.

3

Determine the first term

a_1

and calculate

S_{10}

.

The first term is

a_1 = 7 - 4 = 3

, and the sum

S_{10} = \frac{10}{2}[2(3) + (10-1)(4)] = 210

.

Applying the AP sum formula

S_n = \frac{n}{2}[2a_1 + (n-1)d]

with

n=10

,

a_1=3

, and

d=4

.

Key Concept

Integrating AP and GP structural relationships to solve for sequence parameters and evaluate finite sums

Question 4148Question →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b + 5$ . What is the identity element of this operation?

Show answer & explanation

Answer: -5

Answer

The identity element of the binary operation is

-5

.

By the definition of an identity element

e

, the relation

a \ast e = a

must hold for all real numbers

a

. Applying the rule

a \ast b = a + b + 5

gives

a + e + 5 = a

. Subtracting

a

from both sides leads to

e + 5 = 0

, which gives

e = -5

.

Step-by-Step Solution

1

Set up the identity element equation

a \ast e = a \implies a + e + 5 = a

By definition of an identity element, operating any element

a

with

e

yields

a

.

2

Solve the equation for

e

e = -5

Subtracting

a

from both sides gives

e + 5 = 0

, which yields

e = -5

.

Key Concept

Identity Element of a Binary Operation

Question 4149Question →

Read the context below and supply the word nearest in meaning to 'unyielding' or 'relentless' that best completes the sentence.

Fill in the blanks below

The special prosecutor displayed an resolve, refusing to halt proceedings or compromise despite intense political pressure.

Show answer & explanation

Answer

The word nearest in meaning to 'unyielding' or 'relentless' in this formal context is 'inexorable'.

In formal legal and administrative contexts, 'inexorable' (or 'implacable') describes an unbending resolve or process that cannot be swayed by external pressure or entreaty, perfectly capturing the prosecutor's steadfast posture.

Step-by-Step Solution

1

Analyze the context of the sentence

The sentence describes a prosecutor who refuses to halt proceedings or yield to intense political pressure, demonstrating firm determination.

Understanding the situational context helps determine the required register and nuance of the target word.

2

Identify the precise contextual synonym for 'unyielding' / 'relentless'

The adjective 'inexorable' signifies a person, resolve, or process that is impossible to stop, alter, or bend by persuasion.

'Inexorable' fits both the semantic requirement of being unyielding and the formal legal/administrative tone of the sentence.

Key Concept

Contextual Synonyms in Formal Register

Question 4150Question →

A straight line passing through the points $(1, 3)$ and $(5, k)$ has a gradient of $2$ . What is the value of $k$ ?

Show answer & explanation

Answer: 11

Answer

The value of

k

is

11

.

The gradient

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting

(1, 3)

,

(5, k)

, and

m = 2

yields

2 = \frac{k - 3}{5 - 1} = \frac{k - 3}{4}

. Multiplying by

4

gives

8 = k - 3

, so

k = 11

.

Step-by-Step Solution

1

Recall the slope/gradient formula for a straight line.

m = \frac{y_2 - y_1}{x_2 - x_1}

The gradient of a straight line passing through two points is the ratio of vertical change to horizontal change.

2

Substitute the known point coordinates and gradient into the formula.

2 = \frac{k - 3}{5 - 1}

We are given

(x_1, y_1) = (1, 3)

,

(x_2, y_2) = (5, k)

, and gradient

m = 2

.

3

Simplify and solve for

k

.

2 = \frac{k - 3}{4} \implies 8 = k - 3 \implies k = 11

Multiplying both sides by

4

clears the fraction, and adding

3

isolates

k

.

Key Concept

Gradient of a straight line passing through two points

Estimated Time:45s

Question 4151Question →

A committee of $3$ members is to be selected from a group of $7$ people. In how many different ways can this committee be formed?

Show answer & explanation

Answer: 35

Answer

35

Selecting a committee of

3

members from

7

people requires calculating the number of combinations, given by

^7C_3 = \frac{7!}{3!4!} = 35

.

Step-by-Step Solution

1

Identify total elements and group size

n = 7

and

r = 3

Since the arrangement or order of members in the committee does not matter, this is a selection problem (combinations).

2

Apply the combinations formula

^nC_r = \frac{n!}{r!(n-r)!}

^7C_3 = \frac{7!}{3!4!}

This formula counts the distinct subsets of size

r

that can be chosen from

n

items.

3

Evaluate the factorial expression

^7C_3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

Expanding

7!

as

7 \times 6 \times 5 \times 4!

allows cancelling

4!

, leaving

\frac{210}{6} = 35

.

Key Concept

Combinations formula

^nC_r = \frac{n!}{r!(n-r)!}

Estimated Time:45s

Question 4152Question →

In a class of $40$ students, $25$ study Mathematics and $18$ study Physics. If $3$ students study neither of the two subjects, how many students study both Mathematics and Physics?

Show answer & explanation

Answer: 6

Answer

6 students study both Mathematics and Physics.

Subtracting the

3

students who study neither subject from the class total of

40

leaves

37

students studying at least one subject. Adding those studying Mathematics (

25

) and Physics (

18

) totals

43

. The excess of

43

over

37

represents the

6

students who study both subjects.

Step-by-Step Solution

1

Subtract the number of students studying neither subject from the total number of students in the class.

n(M \cup P) = 40 - 3 = 37

This gives the number of students who belong to at least one of the two sets.

2

Set up the inclusion-exclusion formula

n(M \cup P) = n(M) + n(P) - n(M \cap P)

.

37 = 25 + 18 - n(M \cap P)

Summing

n(M)

and

n(P)

double-counts the students who study both subjects.

3

Solve for the intersection

n(M \cap P)

.

n(M \cap P) = 43 - 37 = 6

Subtracting the union from the sum of the individual sets isolates the intersection value.

Key Concept

Principle of Inclusion-Exclusion for Two Sets

Question 4153Question →

Given the matrices $A = \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}$ , what is the smaller positive integer value of $k$ for which $\det(A^2 + B) = 4$ ?

5

6

4

8

Show answer & explanation

Answer: 5

Answer

The smaller positive integer value of

k

is 5.

Evaluating

A^2

via matrix multiplication gives

\begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix}

. Adding

B

gives

A^2 + B = \begin{pmatrix} k^2 + 7 & 2k + 1 \\ 3k + 5 & 7 \end{pmatrix}

. Computing

\det(A^2 + B) = (k^2 + 7)(7) - (2k + 1)(3k + 5) = k^2 - 13k + 44

. Setting this to 4 results in

k^2 - 13k + 40 = 0

, which yields roots

k = 5

and

k = 8

. The smaller value is 5.

Step-by-Step Solution

1

Compute matrix

A^2

using matrix multiplication (row by column).

A^2 = \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} k & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix}

Matrix multiplication requires taking the dot product of rows of the first matrix with columns of the second matrix.

2

Add matrix

B

to

A^2

.

A^2 + B = \begin{pmatrix} k^2 + 6 & 2k + 2 \\ 3k + 3 & 7 \end{pmatrix} + \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} k^2 + 7 & 2k + 1 \\ 3k + 5 & 7 \end{pmatrix}

Matrix addition is performed by adding corresponding entries.

3

Calculate the determinant of

A^2 + B

and set it equal to 4.

\det(A^2 + B) = (k^2 + 7)(7) - (2k + 1)(3k + 5) = 7k^2 + 49 - (6k^2 + 13k + 5) = k^2 - 13k + 44 = 4

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is

ad - bc

.

4

Solve the quadratic equation for

k

.

k^2 - 13k + 40 = 0 \implies (k - 5)(k - 8) = 0 \implies k = 5 \text{ or } k = 8

Factoring the quadratic equation gives the two possible integer roots.

Key Concept

Matrix multiplication, matrix addition, determinant of a 2x2 matrix, and quadratic equation solving.

Question 4154Question →

If $5x \equiv 2 \pmod{11}$ , what is the value of $(x^3 - 4x) \pmod{11}$ expressed in standard non-negative remainder form?

7

-4

4

8

Show answer & explanation

Answer: 7

Answer

7

Solving the linear congruence

5x \equiv 2 \pmod{11}

gives

x \equiv 7 \pmod{11}

. Substituting

x = 7

into the expression

(x^3 - 4x)

yields

7^3 - 4(7) = 343 - 28 = 315

. Dividing 315 by 11 gives a quotient of 28 and a remainder of 7. Alternatively, working entirely modulo 11 gives

2 - 6 = -4 \equiv 7 \pmod{11}

. Thus, the canonical non-negative remainder is 7.

Step-by-Step Solution

1

Find the modular inverse of 5 modulo 11

The inverse is 9, since

5 \times 9 = 45 \equiv 1 \pmod{11}

.

To isolate

x

in the linear congruence

5x \equiv 2 \pmod{11}

, multiply both sides by the modular inverse of 5.

2

Solve for x modulo 11

x \equiv 2 \times 9 = 18 \equiv 7 \pmod{11}

.

Simplifying 18 modulo 11 gives the canonical value of

x

.

3

Evaluate

x^3 - 4x

modulo 11

x^3 \equiv 7^3 = 343 \equiv 2 \pmod{11}

and

4x \equiv 4(7) = 28 \equiv 6 \pmod{11}

, so

x^3 - 4x \equiv 2 - 6 = -4 \pmod{11}

.

Substitute

x \equiv 7

into the polynomial expression and reduce each term modulo 11.

4

Convert negative remainder to canonical non-negative form

-4 + 11 = 7 \pmod{11}

.

Modular remainders must be expressed within the standard range

[0, 10]

.

Key Concept

Linear modular congruences and negative remainder canonical reduction

Estimated Time:2m 0s

Question 4155Question →

In $\Delta ABC$ , the side lengths are given as $a = 3\text{ cm}$ , $b = 5\text{ cm}$ , and $c = 7\text{ cm}$ . What is the measure of angle $C$ in degrees?

Show answer & explanation

Answer: 120

Answer

The measure of angle

C

is

120^\circ

.

Using the Cosine Rule

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

, substituting

a=3

,

b=5

, and

c=7

yields

\cos C = \frac{9 + 25 - 49}{30} = -0.5

. Taking the inverse cosine of

-0.5

gives an angle of

120^\circ

.

Step-by-Step Solution

1

Select the appropriate formula for calculating an interior angle given three sides (SSS).

Use the Cosine Rule rearranged for

\cos C

:

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

.

When all three side lengths of a non-right triangle are known, the Cosine Rule is required to find any of its angles.

2

Substitute side lengths

a=3

,

b=5

, and

c=7

into the Cosine Rule equation.

\cos C = \frac{3^2 + 5^2 - 7^2}{2(3)(5)} = \frac{9 + 25 - 49}{30} = -\frac{15}{30} = -0.5.

Simplifying the numerator and denominator determines the exact trigonometric ratio for angle

C

.

3

Find the inverse cosine of

-0.5

in degrees.

C = \arccos(-0.5) = 120^\circ.

A negative cosine value indicates an obtuse angle in the second quadrant (

90^\circ < C < 180^\circ

).

Key Concept

Using the Cosine Rule to determine obtuse angles in SSS triangles

Question 4156Question →

Complete the sentence below by filling in the blanks with the appropriate prepositions.

Fill in the blanks below

Rather than supporting the organization's overarching mission, the project manager's controversial proposal was deemed inimical long-term institutional stability and executed in total defiance established policy.

Show answer & explanation

Answer

The first blank requires 'to' (inimical to) and the second blank requires 'of' (in defiance of).

The correct prepositions are 'to' and 'of'. The formal adjective 'inimical' requires the dependent preposition 'to' (e.g., 'inimical to growth'). The multi-word prepositional idiom indicating open resistance is fixed as 'in defiance of'.

Step-by-Step Solution

1

Identify the dependent preposition required by the adjective 'inimical'.

The adjective 'inimical' strictly takes the preposition 'to' when describing something adverse or harmful.

In standard English grammar, adjectives expressing hostility or opposition take specific prepositional complements ('inimical to').

2

Identify the standard preposition completing the complex prepositional phrase 'in defiance _____'.

The correct fixed prepositional expression is 'in defiance of'.

The noun 'defiance' in this idiomatic phrase collocates specifically with 'of' to signify disregard or resistance.

Key Concept

Dependent Prepositions and Fixed Prepositional Phrases

Question 4157Question →

The set of numbers $k - 2$ , $k$ , $k + 1$ , and $k + 5$ is given, where $k$ is any real constant. What is the variance of this set of numbers?

Show answer & explanation

Answer: 6.5

Answer

The variance of the given set of numbers is 6.5.

The mean of the set is

\bar{x} = k + 1

. Subtracting the mean from each data point gives deviations of

-3

,

-1

,

0

, and

4

. The squares of these deviations are

9

,

1

,

0

, and

16

, which sum to

26

. Dividing this sum by

4

gives a variance of

6.5

. A key statistical property illustrated here is that adding or subtracting a constant

k

from every value in a dataset shifts the mean by

k

but leaves measures of dispersion (such as variance and standard deviation) unchanged.

Step-by-Step Solution

1

Find the mean (\bar{x}) of the given set {k - 2, k, k + 1, k + 5}.

\bar{x} = \frac{(k - 2) + k + (k + 1) + (k + 5)}{4} = \frac{4k + 4}{4} = k + 1

The mean is calculated by summing all values and dividing by the total count of numbers.

2

Determine the deviation of each value from the mean, (x_i - \bar{x}).

(k - 2) - (k + 1) = -3, k - (k + 1) = -1, (k + 1) - (k + 1) = 0, (k + 5) - (k + 1) = 4

Deviations measure how far each data value lies from the mean.

3

Square each individual deviation and sum the results.

(-3)^2 + (-1)^2 + 0^2 + 4^2 = 9 + 1 + 0 + 16 = 26

Squaring converts all deviations into non-negative values.

4

Divide the sum of squared deviations by the total number of observations (N = 4) to find the variance.

\text{Variance} = \frac{26}{4} = 6.5

Variance is defined as the arithmetic mean of the squared deviations from the mean.

Key Concept

Variance and Invariance under Constant Translation

Question 4158Question →

The $2^{\text{nd}}$ , $5^{\text{th}}$ , and $14^{\text{th}}$ terms of a non-constant arithmetic progression (A.P.) form three consecutive terms of a geometric progression (G.P.). If the first term of the A.P. is $3$ , what is the sum of the first $6$ terms of the A.P.?

108

126

99

90

Show answer & explanation

Answer: 108

Answer

108

With first term

a=3

, the terms

T_2 = 3+d

,

T_5 = 3+4d

, and

T_{14} = 3+13d

form a geometric progression. Therefore,

(3+4d)^2 = (3+d)(3+13d)

. Expanding gives

9 + 24d + 16d^2 = 9 + 42d + 13d^2

, which simplifies to

3d^2 - 18d = 0

. Since the sequence is non-constant (

d \neq 0

),

d = 6

. Using the A.P. sum formula

S_n = \frac{n}{2}[2a + (n-1)d]

,

S_6 = 3[2(3) + 5(6)] = 3[6 + 30] = 108

.

Step-by-Step Solution

1

Express the

2^{\text{nd}}

,

5^{\text{th}}

, and

14^{\text{th}}

terms of the A.P. in terms of the first term

a=3

and common difference

d

.

T_2 = 3 + d

,

T_5 = 3 + 4d

, and

T_{14} = 3 + 13d

.

The

n^{\text{th}}

term of an A.P. is given by

T_n = a + (n-1)d

.

2

Set up the condition for consecutive terms of a G.P. and solve for

d

.

(3 + 4d)^2 = (3 + d)(3 + 13d) \implies 9 + 24d + 16d^2 = 9 + 42d + 13d^2 \implies 3d^2 - 18d = 0 \implies d = 6

.

For consecutive terms in a G.P., the middle term squared equals the product of the adjacent terms (

T_5^2 = T_2 \times T_{14}

).

3

Calculate the sum of the first

6

terms of the A.P. using

a=3

and

d=6

.

S_6 = \frac{6}{2}[2(3) + (6-1)(6)] = 3[6 + 30] = 3 \times 36 = 108

.

The sum formula for an A.P. is

S_n = \frac{n}{2}[2a + (n-1)d]

.

Key Concept

Simultaneous conditions connecting A.P. and G.P. terms combined with sequence sum formulas.

Estimated Time:2m 0s

Question 4159Question →

What is the period of the trigonometric function $y = 3\cos(4x)$ ?

90^\circ

120^\circ

180^\circ

1440^\circ

Show answer & explanation

Answer: $90^\circ$

Answer

The period of the given trigonometric function is

90^\circ

.

The period of a function of the form

y = a\cos(bx)

is given by

T = \frac{360^\circ}{b}

. Substituting

b = 4

yields

T = \frac{360^\circ}{4} = 90^\circ

.

Step-by-Step Solution

1

Identify the standard form parameters

For

y = 3\cos(4x)

, amplitude

a = 3

and coefficient of

x

is

b = 4

.

The general form of a cosine function is

y = a\cos(bx + c) + d

.

2

Apply the period formula for trigonometric functions in degrees

Period

T = \frac{360^\circ}{b} = \frac{360^\circ}{4} = 90^\circ

.

The full cycle of a standard cosine graph completes in

360^\circ

, so multiplying the input by

b

compresses the period by a factor of

b

.

Key Concept

Period of a Cosine Function

Estimated Time:45s

Question 4160Question →

Let the universal set be $U = \{x \in \mathbb{Z} : 1 \le x \le 10\}$ . If $P = \{x : x \text{ is a prime number within } U\}$ and $Q = \{x : x \text{ is an even number within } U\}$ , which of the following represents the set $(P \cup Q)'$ ?

\{2\}

\{3, 5, 7\}

\{1, 9\}

\{1, 3, 5, 7, 9\}

Show answer & explanation

Answer: $\{1, 9\}$

Answer

\{1, 9\}

The universal set

U

contains all integers from

1

to

10

. Prime numbers in this range form

P = \{2, 3, 5, 7\}

, and even numbers form

Q = \{2, 4, 6, 8, 10\}

. The union

P \cup Q

combines all elements from both sets:

\{2, 3, 4, 5, 6, 7, 8, 10\}

. Taking the complement of this union relative to

U

leaves the elements

\{1, 9\}

, making the option stating

\{1, 9\}

correct.

Step-by-Step Solution

1

List all elements of the universal set

U

and the subsets

P

and

Q

U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

,

P = \{2, 3, 5, 7\}

, and

Q = \{2, 4, 6, 8, 10\}

Explicit listing helps identify union and complement elements accurately.

2

Find the union of sets

P

and

Q

, denoted as

P \cup Q

P \cup Q = \{2, 3, 4, 5, 6, 7, 8, 10\}

The union includes all elements that belong to set

P

, set

Q

, or both.

3

Determine the complement

(P \cup Q)'

by subtracting

P \cup Q

from the universal set

U

(P \cup Q)' = U \setminus (P \cup Q) = \{1, 9\}

The complement contains all elements of

U

that are not present in

P \cup Q

.

Key Concept

Set Complement and Union Operations

Answer

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