KEAM 2025 April 23 Question Paper (Available) :Download Solutions with Answer Key

Evaluate the integral: \[ \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx \]

• (A) \( \frac{2}{3} x^3 + 2x + C \)
• (B) \( \frac{1}{3} x^3 + 3x + C \)
• (C) \( \frac{1}{3} x^3 + x + C \)
• (D) \( \frac{2}{3} x^3 + 3x + C \)

Solve for \( a \) and \( b \) given the equations: \[ \sin x + \sin y = a, \quad \cos x + \cos y = b, \quad x + y = \frac{2\pi}{3} \]

• (A) \( a = \frac{1}{2}, \quad b = \frac{1}{2} \)
• (B) \( a = \frac{1}{2}, \quad b = \frac{1}{\sqrt{2}} \)
• (C) \( a = 0, \quad b = 1 \)
• (D) \( a = 1, \quad b = 1 \)

Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).

• (A) \( \left[ 0, \frac{\pi}{2} \right] \)
• (B) \( (-\infty, \infty) \)
• (C) \( \left(0, \infty \right) \)
• (D) \( \left( \cos^{-1} \left( \frac{1}{5} \right), \infty \right) \)

Given the function \( h(x) = f(g(x)) \), where \( f(x) = f'(x) = 3 \), and \( g(x) = 9 \), find \( g'(3) \), \( f'(3) \), and \( h'(3) \).

• (A) \( g'(3) = 6, \quad f'(3) = 9, \quad h'(3) = 3 \)
• (B) \( g'(3) = 9, \quad f'(3) = 6, \quad h'(3) = 6 \)
• (C) \( g'(3) = 3, \quad f'(3) = 9, \quad h'(3) = 9 \)
• (D) \( g'(3) = 9, \quad f'(3) = 6, \quad h'(3) = 9 \)

If \( -5 < x \leq -1 \), then \( -21 \leq 5x + 4 \leq b \). Find \( b \).

• (A) \( b = -11 \)
• (B) \( b = -16 \)
• (C) \( b = -12 \)
• (D) \( b = -13 \)

An unbiased die is tossed until a sum \( S \) is obtained. If \( X \) denotes the number of times tossed, find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).

• (A) \( \frac{1}{4} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{1}{3} \)
• (D) \( \frac{1}{5} \)

Evaluate the integral: \[ \int e^x \sec(x) \left( \tan(x) + 1 \right) \, dx \]

• (A) \( e^x \sec(x) + C \)
• (B) \( e^x \sec(x) \left( \tan(x) + 1 \right) + C \)
• (C) \( e^x \sec(x) \tan(x) + C \)
• (D) \( e^x \sec(x) \tan(x) + e^x \sec(x) + C \)

Given that \( \vec{a} \parallel \vec{b} \), \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), and \( |\vec{a}| = 7 \), find \( |\vec{b}| \).

• (A) \( |\vec{b}| = 7 \)
• (B) \( |\vec{b}| = 14 \)
• (C) \( |\vec{b}| = \frac{49}{7} \)
• (D) \( |\vec{b}| = \frac{49}{14} \)

If \( f(x) = \sqrt{x - 3} + 4 \sqrt{5 - x} \), find the domain of \( f(x) \).

• (A) \( [3, 5] \)
• (B) \( [3, 5) \)
• (C) \( (3, 5] \)
• (D) \( (3, 5) \)

Given the function \( F(x) = |\sin(3x)| - \cos(3x) \), for \( \frac{\pi}{6} \leq x \leq \frac{\pi}{3} \), find \( f' \left( \frac{\pi}{4} \right) \).

• (A) \( -\frac{3}{2} \)
• (B) \( \frac{3}{2} \)
• (C) \( -\frac{1}{2} \)
• (D) \( \frac{1}{2} \)

If \( f(x) = \cos x \), find the following expression: \[ \frac{1}{2} \left[ f(x + y) + f(y - x) - f(x) \cdot f(y) \right] \]

• (A) \( \cos(x + y) + \cos(y - x) - \cos x \cdot \cos y \)
• (B) \( \cos(x + y) - \cos(y - x) - \cos x \cdot \cos y \)
• (C) \( 2 \cos(x + y) - \cos x \cdot \cos y \)
• (D) \( \cos(x + y) + \cos(y - x) + \cos x \cdot \cos y \)

Given: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \quad and \quad \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \]
\text{Find the value of \( \alpha - \beta \).

• (A) \( 0 \)
• (B) \( 1 \)
• (C) \( 2 \)
• (D) \( 3 \)

Given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \), find \( C \).

• (A) \( C = -26 \)
• (B) \( C = 28 \)
• (C) \( C = -24 \)
• (D) \( C = 16 \)

Given the following information: \[ n(A \times B) = 160, \quad n(B \times C) = 80, \quad n(A \times C) = 240 \]
\text{Find \( n(A) \).

• (A) \( n(A) = 16 \)
• (B) \( n(A) = 20 \)
• (C) \( n(A) = 24 \)
• (D) \( n(A) = 30 \)

Find the equation of the circle touching the x-axis at \( (9, 0) \) and the line \( y = 14 \).

• (A) \( (x - 9)^2 + y^2 = 14^2 \)
• (B) \( (x - 9)^2 + (y - 7)^2 = 7^2 \)
• (C) \( (x + 9)^2 + (y - 14)^2 = 14^2 \)
• (D) \( (x - 9)^2 + (y - 7)^2 = 14^2 \)

Evaluate the limit: \[ \lim_{x \to \infty} \frac{\sqrt{\cos^2 x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \]

• (A) \( 0 \)
• (B) \( 1 \)
• (C) \( -1 \)
• (D) Undefined

The velocity is given as \( \mathbf{v} = 3\hat{i} + 3\hat{j} \). Find the acceleration \( \mathbf{a} \).

• (A) \( \mathbf{a} = 3\hat{i} + 3\hat{j} \)
• (B) \( \mathbf{a} = 0 \)
• (C) \( \mathbf{a} = 6\hat{i} + 6\hat{j} \)
• (D) \( \mathbf{a} = 3\hat{i} + 6\hat{j} \)

If \( \sin \alpha = \frac{12}{13} \), and \( \frac{\pi}{6} \leq \alpha \leq \frac{3\pi}{2} \), find \( \tan \alpha \).

• (A) \( \frac{5}{12} \)
• (B) \( \frac{5}{13} \)
• (C) \( \frac{12}{5} \)
• (D) \( \frac{13}{5} \)

Find \( \tan 15^\circ + \tan 45^\circ \).

• (A) \( 1 + \tan 15^\circ \)
• (B) \( \tan 60^\circ \)
• (C) \( \tan 60^\circ + 1 \)
• (D) \( \tan 15^\circ + 1 \)

Evaluate the integral: \[ \int_{\frac{\pi}{10}}^{\frac{2\pi}{5}} \frac{\cot^3 x}{1 + \cot^3 x} \, dx \]

• (A) \( \frac{1}{2} \)
• (B) \( \frac{1}{3} \)
• (C) \( 0 \)
• (D) \( \frac{1}{4} \)

Evaluate the integral: \[ \int \frac{\sin^1 x}{\sqrt{1 - x^2}} \, dx \]

• (A) \( \frac{1}{2} \)
• (B) \( \frac{\sin^2 x}{2} \)
• (C) \( \frac{\cos^2 x}{2} \)
• (D) \( \frac{\sin x}{2} \)

Given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).

• (A) \( 7 \)
• (B) \( 4 \)
• (C) \( 14 \)
• (D) \( 3 \)

Find the focus of the parabola \( x^2 - 4x + 8y + 4 = 0 \).

• (A) \( (2, 1) \)
• (B) \( (1, 2) \)
• (C) \( (-2, 1) \)
• (D) \( (2, -1) \)

Given the system of equations: \[ Z + \bar{Z} = 4 \quad and \quad Z - \bar{Z} = 6 \]
\text{Find \( |Z| \).

• (A) 13
• (B) 7
• (C) 10
• (D) 6

A planet revolves around the sun with a time period 27 times that of planet B. Planet A is at \( x \) times the distance of planet B from the sun. Find the value of \( x \).

• (A) \( 13 \)
• (B) \( 12 \)
• (C) \( 10 \)
• (D) \( 15 \)

Relations between the speed of X-ray, gamma-ray, and UV rays when they travel in a vacuum.

• (A) X-rays travel faster than UV rays and gamma rays.
• (B) Gamma-rays travel faster than UV rays and X-rays.
• (C) UV rays travel faster than X-rays and gamma rays.
• (D) All of them travel at the same speed.

If the frequency of the cyclotron is doubled, then the radius becomes?

• (A) Doubled
• (B) Halved
• (C) Quadrupled
• (D) Unchanged

Electrostatic force is maximum when charge \( Q \) is placed at ------?

• (A) At the center of the sphere
• (B) At the surface of the sphere
• (C) At the corner of the cube
• (D) At infinity

In an equilateral triangle with each side having resistance \( R \), what is the effective resistance between two sides?

• (A) \( \frac{R}{3} \)
• (B) \( \frac{2R}{3} \)
• (C) \( R \)
• (D) \( \frac{R}{2} \)

A cylindrical vessel contains 16 kg at 1 atm. A certain amount of substance is taken out so that pressure becomes 0.7 atm. Find the amount taken out (in kg).

• (A) 2.5 kg
• (B) 3.5 kg
• (C) 4 kg
• (D) 5 kg

What phenomenon is explained by the wave nature of electromagnetic radiation?

• (A) Diffraction
• (B) Reflection
• (C) Refraction
• (D) Polarization

Fehling's solution is a mixture of:

• (A) Copper sulfate and sodium hydroxide
• (B) Sodium hydroxide and potassium cyanide
• (C) Copper sulfate and potassium cyanide
• (D) Copper sulfate and sodium tartrate

Which of the following is the set of neutral oxides?
a) \( \text{Al_2O_3, Cl_2O_7 \), b) \( N_2O, CO \)

• (A) \( Al_2O_3, Cl_2O_7 \)
• (B) \( N_2O, CO \)
• (C) \( N_2O, SO_2 \)
• (D) \( SO_2, CO_2 \)

Initial concentration of a reaction is \( 1.68 \times 10^{-2} \) and after 10 minutes concentration becomes \( 0.84 \times 10^{-2} \). Then the rate of concentration in minutes is:

• (A) \( 0.084 \)
• (B) \( 0.042 \)
• (C) \( 0.014 \)
• (D) \( 0.021 \)

Number of sigma and pi bonds in methyl but-1-ene is:

• (A) 10 sigma bonds and 3 pi bonds
• (B) 9 sigma bonds and 4 pi bonds
• (C) 8 sigma bonds and 4 pi bonds
• (D) 8 sigma bonds and 3 pi bonds

If \( Z = \frac{2 - i}{\alpha + i} \) and \( 4 \, Re(Z) = 3 \, Im( \overline{Z} ) \), find \( \alpha \).

• (A) \( \alpha = -2 \)
• (B) \( \alpha = 3 \)
• (C) \( \alpha = -3 \)
• (D) \( \alpha = 2 \)

Find the vertex of the parabola \( 4y = x^2 - 6x + 17 \).

• (A) \( (3, 7) \)
• (B) \( (1, 4) \)
• (C) \( (3, 4) \)
• (D) \( (1, 7) \)

Solve the system of equations: \[ \begin{bmatrix} 4 & 9
12 & -3
8 & -2 \end{bmatrix} \begin{bmatrix} 7
9 \end{bmatrix} = \begin{bmatrix} \alpha
\beta \end{bmatrix} \]

• (A) \( \alpha = 166, \beta = 54 \)
• (B) \( \alpha = 153, \beta = 49 \)
• (C) \( \alpha = 155, \beta = 50 \)
• (D) \( \alpha = 160, \beta = 56 \)

The cubic polynomial \( 2x^3 - 3x^2 - 36x + 28 \) is increasing in the range of \( x \). Find the interval where the function is increasing.

• (A) \( x > 2 \)
• (B) \( x < -2 \)
• (C) \( -2 < x < 2 \)
• (D) \( x < 0 \)

If \( \cot^{-1} \left( \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \right) \), then find \( \sec^2 \theta \).

• (A) \( 1 \)
• (B) \( 2 \)
• (C) \( 3 \)
• (D) \( 4 \)

What is the order of the SN2 reaction for the compounds: \[ 2-methyl-2-bromo-butene, \quad 2-bromo-butene, \quad 1-bromo-butane \]

• (A) 1, 2, 3
• (B) 3, 2, 1
• (C) 2, 1, 3
• (D) 3, 1, 2

Which of the following bond enthalpies is the least: \[ C = C, C = O, C \equiv N, C = N \]

• (A) \( C = C \)
• (B) \( C = O \)
• (C) \( C \equiv N \)
• (D) \( C = N \)

The reaction shown is: \[ Phenol + CHCl_3 + NaOH \rightarrow Salicylaldehyde \]
What is the name of the above reaction?

• (A) Aldol condensation
• (B) Reimer-Tiemann reaction
• (C) Friedel-Crafts reaction
• (D) Kolbe-Schmidt reaction

If the half-life of \( D \) is 1500 years and \( B \) is 2000 years, what is the mean lifetime?

• (A) 1750 years
• (B) 1800 years
• (C) 1900 years
• (D) 1850 years

Common oxidation state of Cr?

• (A) +3
• (B) +2
• (C) +6
• (D) +1

What is the formula of lanthanoids with sulfur?

• (A) \( La_2S_3 \)
• (B) \( LaS_2 \)
• (C) \( La_3S_4 \)
• (D) \( La_4S_3 \)

Among the following, which one is incorrect? \[ BrF_5 \rightarrow Trigonal bipyramidal, \quad SF_4 \rightarrow See saw, \quad NH_3 \rightarrow Pyramidal, \quad XeF_4 \rightarrow Square planar \]

• (A) BrF\(_5\) \quad Trigonal bipyramidal
• (B) SF\(_4\) \quad See saw
• (C) NH\(_3\) \quad Pyramidal
• (D) XeF\(_4\) \quad Square planar

Which among the following has the highest molar elevation constant?

• (A) CHCl\(_3\)
• (B) CCl\(_4\)
• (C) CH\(_3\)COOH

Find the rate constant at 310K if the initial concentration is 0.72 mol L\(^{-1}\) and the final concentration is 1.44 mol L\(^{-1}\) at 10 minutes.

• (A) 0.05
• (B) 0.0693
• (C) 0.091
• (D) 0.13

Arrange in the order of dipole moment: \[ NH_3, \, NF_3, \, H_2S, \, CHCl_2 \]

• (A) NH\(_3\) \(>\) NF\(_3\) \(>\) H\(_2\)S \(>\) CHCl\(_2\)
• (B) NF\(_3\) \(>\) NH\(_3\) \(>\) CHCl\(_2\) \(>\) H\(_2\)S
• (C) NH\(_3\) \(>\) H\(_2\)S \(>\) CHCl\(_2\) \(>\) NF\(_3\)
• (D) CHCl\(_2\) \(>\) H\(_2\)S \(>\) NF\(_3\) \(>\) NH\(_3\)

Arrange in the order of conductivity: \[ Na, Ag, Fe, Cu \]

• (A) Ag \(>\) Cu \(>\) Na \(>\) Fe
• (B) Na \(>\) Cu \(>\) Fe \(>\) Ag
• (C) Cu \(>\) Fe \(>\) Ag \(>\) Na
• (D) Fe \(>\) Na \(>\) Cu \(>\) Ag

Find the pH of the solution if \( [H^+] = 2 \times 10^{-4} \, mol/L \).

• (A) 3.0
• (B) 3.7
• (C) 4.5
• (D) 5.0

Which among the following acts as a Lewis acid? \[ A) AlCl_3, \quad B) NH_3, \quad C) OH^-, \quad D) H_2O \]

• (A) AlCl\(_3\)
• (B) NH\(_3\)
• (C) OH\(^-\)
• (D) H\(_2\)O

When toluene is treated with chromium oxide and acetic anhydride, followed by hydrolysis, what is the product formed?

• (A) Toluene-2,4-diol
• (B) Acetyl toluene
• (C) Benzyl alcohol
• (D) Benzoic acid

Layer's test is used for what purpose?

• (A) To detect aldehydes
• (B) To detect ketones
• (C) To detect carboxylic acids
• (D) To detect aromatic amines

Which of the following has the highest \( K_b \) value?

• (A) CHCl\(_3\)
• (B) CCl\(_4\)
• (C) CH\(_3\)COOH
• (D) NH\(_3\)

What technique is used to separate chloroform from aniline?

• (A) Fractional distillation
• (B) Simple distillation
• (C) Filtration
• (D) Evaporation

What is the IUPAC name of phenyl isopentylether?

• (A) 1-Phenylpentan-2-yl ether
• (B) Phenylmethylether
• (C) 2-Phenylethyl ether
• (D) Phenyl isopropylether

The disease is caused by a deficiency of riboflavin.

• (A) Scurvy
• (B) Pellagra
• (C) Beriberi
• (D) Rickets

In which of the following cases does manganese have a +7 oxidation state?

• (A) KMnO\(_4\)
• (B) MnO\(_2\)
• (C) MnCl\(_2\)
• (D) Mn\(_2\)O\(_3\)

A body of mass 0.8kg moves with velocity \( v = 2x^2 + 2 \, m/s \). What is the work done during its motion from \( x = 0 \) to \( x = 2 \, m \)?

• (A) 4.8 J
• (B) 3.2 J
• (C) 6.4 J
• (D) 2.4 J

A body of mass 0.8kg moves with velocity \( v = 2x^2 + 2 \, m/s \). What is the work done during its motion from \( x = 0 \) to \( x = 2 \, m \)?

• (A) 4.8 J
• (B) 3.2 J
• (C) 6.4 J
• (D) 2.4 J

A solid sphere, hollow sphere, and solid cylinder start sliding from an inclined plane without rolling. Then the ratio of time taken by them to reach the ground is?

• (A) 1:2:3
• (B) 1:1:1
• (C) 1:3:2
• (D) 3:2:1

A liquid of density \( d \) is moving down in a vessel of height \( h \) with an acceleration \( a < g \). Then the pressure at the bottom is?

• (A) \( hdg \)
• (B) \( hd(g - a) \)
• (C) \( hd(g + a) \)
• (D) \( \frac{hdg}{a} \)

De Broglie wavelength of a quantum photon having energy \( E \)?

• (A) \( \lambda = \frac{h}{E} \)
• (B) \( \lambda = \frac{h}{\sqrt{E}} \)
• (C) \( \lambda = \frac{E}{h} \)
• (D) \( \lambda = \frac{h}{\sqrt{2E}} \)

Work done on splitting a spherical drop into 8 droplets?

• (A) \( 7 \times W \)
• (B) \( 8 \times W \)
• (C) \( 6 \times W \)
• (D) \( 4 \times W \)

The fundamental frequency of a string of length \( l \) is \( n \). The string is cut into 3 parts \( l_1, l_2, \) and \( l_3 \), each having fundamental frequency \( n_1, n_2, n_3 \). Then, what is the relationship between \( n_1, n_2, n_3 \)?

• (A) \( n_1 = n_2 = n_3 \)
• (B) \( n_1 < n_2 < n_3 \)
• (C) \( n_1 > n_2 > n_3 \)
• (D) \( n_1 = 2n_2 = 3n_3 \)

Tension of a rope when a man of mass \( m \) climbs up or down with an acceleration \( a < g \)?

• (A) \( T = mg \)
• (B) \( T = m(g - a) \)
• (C) \( T = m(g + a) \)
• (D) \( T = \frac{mg}{a} \)

Wave nature of light can be used to explain which phenomenon?

• (A) Reflection
• (B) Refraction
• (C) Diffraction
• (D) Absorption

What is the maximum wavelength to excite a hydrogen atom?

• (A) \( \lambda = \frac{1}{13.6 \, eV} \)
• (B) \( \lambda = \frac{c}{R} \)
• (C) \( \lambda = \frac{h}{E} \)
• (D) \( \lambda = \frac{c}{E} \)

How can we decrease the effective capacitance of a parallel plate capacitor?

• (A) Increase the distance between the plates
• (B) Decrease the distance between the plates
• (C) Increase the area of the plates
• (D) Increase the dielectric constant

\( M^0 L T^{-1} \) is the dimension of ___________?

• (A) Power
• (B) Work
• (C) Energy
• (D) Momentum

The inward electric flux through a closed surface is \( 6 \times 10^{-5} \) and the outward flux is \( 3 \times 10^{-5} \). Then the total charge enclosed is?

• (A) \( 9 \times 10^{-5} \, C \)
• (B) \( 3 \times 10^{-5} \, C \)
• (C) \( 6 \times 10^{-5} \, C \)
• (D) \( 0 \, C \)

Relation between the drift velocity and an electric field is ----?

• (A) \( v_d = \frac{E}{\mu} \)
• (B) \( v_d = E \times \mu \)
• (C) \( v_d = \frac{\mu}{E} \)
• (D) \( v_d = \mu \times E^2 \)

Find the resistance required to be connected to a galvanometer of resistance \(100 \, \Omega\) with a full scale deflection of 1mA into a voltmeter of range 1V.

• (A) \( 1000 \, \Omega \)
• (B) \( 100 \, \Omega \)
• (C) \( 900 \, \Omega \)
• (D) \( 200 \, \Omega \)

Brewster's angle should lie between?

• (A) \( 0^\circ \) to \( 45^\circ \)
• (B) \( 45^\circ \) to \( 90^\circ \)
• (C) \( 0^\circ \) to \( 90^\circ \)
• (D) \( 0^\circ \) to \( 60^\circ \)

The ratio of the angular speed of the minute hand to the second hand of a watch is?

• (A) \( 1 : 60 \)
• (B) \( 1 : 3600 \)
• (C) \( 1 : 360 \)
• (D) \( 1 : 120 \)

Number of degrees of freedom for the monoatomic gas molecule is?

• (A) 3
• (B) 2
• (C) 5
• (D) 6

After a collision, two particles move together, then the collision is?

• (A) Elastic
• (B) Inelastic
• (C) Perfectly elastic
• (D) Super-elastic

Which of the following statement is true?

• (A) Saturation current depends on intensity
• (B) Saturation current does not depend on intensity
• (C) K.E depends on intensity
• (D) Photocurrent depends on frequency

In a Carnot engine, efficiency is dependent on ________?

• (A) Temperature of the hot reservoir
• (B) Temperature of the cold reservoir
• (C) Both hot and cold reservoir temperatures
• (D) Heat input

The product of the first five terms of a GP is 32. What is the 3rd term?

• (A) 4
• (B) 2
• (C) 8
• (D) 16

What is the value of n when \( t_n = \frac{n(n+6)}{n+4} \) and \( t_n = 5 \)?

• (A) \( 3 \)
• (B) \( 4 \)
• (C) \( 5 \)
• (D) \( 6 \)

If \(x + z = 2y\) and \(y = \frac{\pi}{4}\), what is \( \tan x \cdot \tan y \cdot \tan z \)?

• (A) 1
• (B) 0
• (C) \( \sqrt{2} \)
• (D) 2

If \( 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \), then \( \alpha = ? \)

• (A) \( \frac{1}{2} \)
• (B) \( \frac{\sqrt{2}}{2} \)
• (C) \( \frac{1}{\sqrt{2}} \)
• (D) \( 1 \)

If \( A \) is a \( 3 \times 3 \) matrix and \( |B| = 3|A| \) and \( |A| = 5 \), then find \( \left| \frac{adj B}{|A|} \right| \).

• (A) 3
• (B) 9
• (C) 1
• (D) 5

Find the value of \( Z^2 \) if \( Z = \left( 1 + \frac{1}{i} \right) \).

• (A) 4
• (B) 5
• (C) 6
• (D) 3

Find the standard deviation of the numbers: -3, 0, 3, 8.