Math Quiz on Third Roots

Questions and Answers

What is the range of the function $f(x) = sin^{-1}x + sec^{-1}x + tan^{-1}x$?

( – 1, 1)

{ – 1, 1}

[ – 1, 1] (correct)

( – ∞ , ∞ )

Which of the following represents the range of the function $f(x) = log_2 [3x - [x + [x + [x]]]]$?

[0, 1) (correct)

(0, 1]

[0, 1]

{0, 1}

Given the relation $f(x) f'(x) = f(x) + f'(x)$ for $x eq 0$, if $f(4) = -63$, what could be the value of $f(4)$?

-26

-511 (correct)

-124

-7

If $f(x) = cosx + sinx$ and $g(x) = x^2 - 1$, what does the composition $f(g(x))$ yield?

$cos(x^2 - 1) + sin(x^2 - 1)$ Signup and view all the answers

In the polynomial function $f(x)$ fulfilling the condition $f(x)f'(x) = f(x) + f'(x)$, what can be inferred about its degree?

It is quadratic. Signup and view all the answers

What are the maximum and minimum values of $( ext{sin}^{-1}(x))^3 + ( ext{cos}^{-1}(x))^3$?

$rac{ ext{π}}{3}, rac{7 ext{π}}{3}$ Signup and view all the answers

For the condition $sin^{-1} (sin(x)) < rac{ ext{π}}{3}$, which range for x is valid?

x ∈ (0, 1) Signup and view all the answers

Which condition must be fulfilled for the inequality $sin^{-1} (sin(x)) < rac{ ext{π}}{3}$?

$1 + x^2 > 0$ Signup and view all the answers

What is the result of the expression $( ext{sin}^{-1}(x))^3 + ( ext{cos}^{-1}(x))^3$ for any valid x?

$ ext{π}$ Signup and view all the answers

Which of the following ranges for x gives valid results for the inequality $sin^{-1} (sin(x)) < rac{ ext{π}}{3}$?

x ∈ (0, 1) Signup and view all the answers

What incorrect assumption might lead to misinterpreting the values for $( ext{sin}^{-1}(x))^3 + ( ext{cos}^{-1}(x))^3$?

Believing both terms can exceed 1 Signup and view all the answers

If $x$ is in the range of $(-1, 1)$, which of the following must be true regarding $( ext{sin}^{-1}(x))^3 + ( ext{cos}^{-1}(x))^3$?

Remains within certain bounds Signup and view all the answers

If $sin^{-1}(sin(x))$ maps to which values for $x$ in question deals?

Values can be both negative and positive depending on range Signup and view all the answers

If the roots are denoted by γ1 and γ2, what is the value of |γ1 + γ2|?

10 Signup and view all the answers

For which interval does the value of 'a' lie if the given expression is always negative for all real 'x'?

(-∞, -1/2) Signup and view all the answers

In the quadratic equation given, what is the correct form of the expression for it to be always negative?

ax^2 + bx + c with b^2 - 4ac < 0 Signup and view all the answers

What can be said about the second term of the polynomial ax^2 + 2(a + 1)x + (9a + 4)?

It is linear in 'x' Signup and view all the answers

If the roots of the equation λ(x^2 - x) + x + 5 = 0 are α and β, what do they relate to?

They are always real numbers Signup and view all the answers

What value does 'c' take when ensuring the expression remains negative?

4 Signup and view all the answers

If the expression ax^2 + 2(a + 1)x + (9a + 4) has one root, which condition must hold?

The discriminant is zero Signup and view all the answers

Which of the following values would make ax^2 + 2(a + 1)x + (9a + 4) positive for large values of 'x'?

a > 0 Signup and view all the answers

What is the radius of curvature of the convex mirror if the distance between the object and the plane mirror is 30 cm and no parallax is observed between the images?

25 cm Signup and view all the answers

When a plano-convex lens is silvered on its plane surface and illuminated from left to right, what focal length does it achieve?

28 cm Signup and view all the answers

What focal length is achieved when the same plano-convex lens is silvered on its curved surface and illuminated from the opposite side?

10 cm Signup and view all the answers

If the refractive index of the material of the lens is to be found, which value corresponds to the lens that achieves a 10 cm focal length when its curved surface is silvered?

1.51 Signup and view all the answers

In optical systems, what effect does the placement of a thin rod along the principal axis have on the focal point?

No effect on the focal length Signup and view all the answers

What is the primary characteristic that defines an optical system created by silvering the curved surface of a lens?

It provides a fixed focal length Signup and view all the answers

What happens to the distance between the object and the plane mirror when there is no parallax observed between the images?

It should equal 30 cm Signup and view all the answers

What is the molecular formula of caffeine?

C7H5NO2 Signup and view all the answers

Which of the following combinations of atomic masses for elements X and Y would satisfy the given mole and weight conditions?

40, 30 Signup and view all the answers

If caffeine has a molecular weight of 194 and contains 29% nitrogen by mass, how many nitrogen atoms are in one molecule of caffeine?

3 Signup and view all the answers

How does the number of atoms in 560g of iron compare to that in 70g of nitrogen?

Twice that of 70g N Signup and view all the answers

Given a plano-convex lens, what effect does polishing its surfaces have on its focal length?

No effect on the focal length Signup and view all the answers

Which of the following statements about plano-convex lenses is true?

They can change focal length based on surface treatment Signup and view all the answers

Which property of the lens material needs to be determined if the lens exhibits a focal length of 28 cm when silvered?

Refractive index of the material Signup and view all the answers

What is the value of w if a metal displacing 0.7 liters of hydrogen gas at STP has an equivalent weight of 28?

14g Signup and view all the answers

What mass corresponds to 0.05 moles of the compound XY2 if its weight is given as 5g?

5g Signup and view all the answers

Which compound represents a higher molecular complexity, XY2 or X2Y3?

X2Y3 Signup and view all the answers

Determine the number of molecules in 3.011 x 10^23 molecules of X2Y3.

3.011 x 10^23 Signup and view all the answers

If a compound has a molecular formula C7H5N2O, what element is absent from this compound?

Oxygen Signup and view all the answers

If the atomic weight of the given metal is 28, how would you calculate the grams that displace 0.7 liters of hydrogen?

Use molar volume relationships Signup and view all the answers

Study Notes

Roots and Expressions

Third roots of two equal roots are represented as γ₁ and γ₂.
The absolute value of their sum |γ₁ + γ₂| is provided with options: (A) 10, (B) 12, (C) 13, (D) 42.

Inequalities and Intervals

Quadratic expression ax² + 2(a + 1)x + (9a + 4) must be negative for all real x.
Determine the intervals for 'a' under consideration of given choices:
- (A) (−∞, −1/2), (B) (1/4, ∞), (C) (1/2, 2), (D) None.

Roots of Equations

For roots α and β of λ(x² - x) + x + 5 = 0, the maximum and minimum values of the expression (sin⁻¹x)³ + (cos⁻¹x)³ can be determined with options:
- (A) −π/2, π/2, (B) −π/8, π/8, (C) π/3, 7π/3, (D) −π/32, π/32.

Trigonometric Conditions

For the condition sin⁻¹(sin(x)) < π - 3, possible intervals for 'x' can be:
- (A) x ∈ (−1, 0), (B) x ∈ (0, 1), (C) x ∈ (−1, 1), (D) x ∈ φ.

Image Formation in Mirrors

An object distance of 30 cm between the object and a plane mirror results in no parallax between the images.
Calculate the radius of curvature for a convex mirror with options:
- (A) 12.5 cm, (B) 25 cm, (C) 75 cm, (D) 100 cm.

Optical Systems with Lenses

A plano-convex lens silvered on its plane surface has a focal length of 28 cm when illuminated from one side.
If the same lens is silvered on the curved surface and illuminated from the opposite side, it behaves as an optical system with a focal length of 10 cm.
Determine the refractive index of the lens material with options:
- (A) 1.75, (B) 1.51, (C) 1.55, (D) 1.50.

Atomic Mass Calculations

For compounds XY₂ and X₂Y₃, identify the atomic masses of elements X and Y based on the given weights of moles.
Possible atomic masses include:
- (A) 20, 30, (B) 30, 40, (C) 40, 30, (D) 80, 60.

Number of Atoms Representation

In 560 g of Fe (atomic mass 56 g/mol), the quantity of atoms is compared to that in 70 g of N and 20 g of H, asking if each is accurate through given options.

Molecular Weight in Caffeine

Caffeine, with a molecular weight of 194 g, contains 29% nitrogen by mass. Determine the number of nitrogen atoms in one caffeine molecule with potential answers:
- (A) 2, (B) 6, (C) 4, (D) 3.

Displacement of Hydrogen Gas

A metal with equivalent weight 28 displaces 0.7 liters of hydrogen gas at STP, calculating the value of 'w' for the displaced gas.

Function Range Analysis

The range of the function f(x) = log₂[3x - [x + [x + [x]]] is given, where [·] represents the greatest integer function, with possible ranges:
- (A) [0, 1], (B) {0, 1}, (C) (0, 1], (D) [0, 1).

Polynomial Functions

Given the polynomial condition f(x)f'(1/x) = f(x) + f'(1/x) for x ≠ 0 and that f(4) = −63, find the corresponding value of f with choices:
- (A) −511, (B) −26, (C) −124, (D) −7.

Combining Functions

Given f(x) = cosx + sinx and g(x) = x² - 1, explore implications or relationships within these function definitions.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your understanding of the properties of third roots with this quiz. Determine the value of |γ1 + γ2| and choose the correct answer from the provided options. This quiz is crucial for students preparing for competitive exams.