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Proving or disproving a statement

Quiz by Oak National Academy: GCSE Maths Higher (2)

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Q 1/6
Score 0
Which of these statements are always true?
30
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Q 2/6
Score 0
Which of these is a general form for any odd number (where $n$ is an integer)?
30
$2n+1$
$4n + 1$
$n + 1$
$2n$
$n$

6 questions

Q.

Which of these statements are always true?

1

30 sec

Q.

Which of these is a general form for any odd number (where

n

is an integer)?

2

30 sec

Q.

The diagram shows a right-angled triangle. Which of these is an expression for the area?

Question Image

3

30 sec

Q.

The diagram shows a right-angled triangle. Which of these formula show the correct relationship between the lengths of the sides?

Question Image

4

30 sec

Q.

Which of these shows the first 4 numbers of the form

9^n

where

n

is a positive integer?

5

30 sec

Q.

Andeep has written out the first 4 numbers of the form

9^n

where

n

is a positive integer. Which of these conjectures hold true for these 4 values?

6

30 sec

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“On this night, we share a roof protecting us from fleets of inequity. Our unification promises a better tomorrow. Those larger than myself, sitting on their marble thrones, sipping blood from cups composed of human skin and singing songs of so-called virtue, grow weaker each moment. Their caravans are revolting. There is hope yet. There is progress! Though tonight may mark a countdown, it is still a celebration. Look at all we have done, not just for Trials but for Palatium Infra as a whole. In four years, when I’m no longer Sovereignty, the Spoiled Purity and his people will continue to strive. So drink! Smoke! Crush up those exotic plants and snort them! We will not falter, weaken, or wane. Our influence is expanding, and somebody new opens their eyes every day. Even the Silbys of Aculeus have reached alarming potentials despite their embittered minds. So long as you relish in tonight, dance, and pray to your “dead” Gods, our revolution shall rise beyond the bounds of class, and when I’m only a commoner, we shall rise again beyond our brainwashed adversaries! Cheers, my people. Cheers!” Followers raised their cups. Some clinked theirs together. Others stood still and screamed breathlessly in agreement. I smiled with courtesy, then stepped off my platform. My voice still rang across the cellar. Speeches before were grander. Those displays were supposed to be emptying, and yet this one left me bloated, swollen tight. I watched as they popped the corks of their bottles and chanted in the name of Purity. Maybe the quality of my words wasn’t what mattered to them anyway, so long as I screamed loud enough. There’s no merit in attacking your people, a voice corrected me. “That’s right,” I said aloud. “Knox, my-my Sovereign!” squealed a nearby devotee, jittering as he stuffed his face with catered pastries. He was one I’d never seen before or had failed to remember. “Look what I’ve found! It’s wine, and not the shoddy Infran kind, either. Earth-made with good fruit! I don’t know how anyone managed to get their hands on this. Maybe some space travel mischief.” He giggled and held up a small glass bottle. “How neat.” “I want you to have it, Sir.” I nodded my head. “Yes, of course. Thank you.” Backing off into the midst of rowdy disciples, I clutched the bottle. What a waste of grapes. It could have been jam instead. Earthly food had a superior taste, ripe with delicate intricacies and nostalgia, but Palatium Infra had mastered the art of alcohol. Why waste your time with a drunkenness so sad and sickening? The booze of trash. Not many more followers approached me. The barren peroration must have upset them. My hands itched to submerge into my suit pockets, and my legs stood suddenly numb, wobbling. Four more years until I’m nothing. But tonight, you are nothing. “Shut up,” I told myself. Tightly packed together in the corner of the dwelling sat the Sibyls. A mound of writhing fabric and tones of skin made up their unified silhouette. I snapped the strap of the nearest gown, balancing on my hands and knees, waving the bottle before them. In their almost rodent nature, narrow noses prodded my way. Their dresses wrinkled and fell to their ankles. Knees dropped, and eyes widened. Many grumbled at me like hungry she-beasts. Those newer ones with faded curtains for hair, sunken eyes, and dirtied nails looked, hid their face, then sobbed. I imagined them in a pack together, fighting wildly against the Spoiled Purity in their rat decorum–biting down with square teeth laced with rabies. “I’ve got you all something,” I said. “Go back off to your pedestal and yap some more. We don’t want it.” A woman rose from the pile and spat. “You don’t even know what it is yet. It's Earth hooch, or more likely a near-flawless replica. I figured you girls would also like a chance to enjoy yourselves tonight.” “Your playmates have been harassing us since the moment you hung the banners and opened the cellar door.” The youngest, with a striking cyan mop upon her head, uncoiled from the mass. What was she now? 20, 21? We celebrated a birthday recently, I thought as she spun around me. “I remember something about a promise. Multiple promises, actually. Are you trying to bribe us into just shutting up and taking it? Because if another sticky, 40-year-old, Earth-born virgin gropes my shoulder, I’m going to have an aneurysm!” the girl continued. “Why not an Infran follower? Do you like it when they touch you?” I returned her accusing tone. “I’m sorry, sweet prophets, that you feel I’ve neglected my duties. I’ll keep a better eye out. Remember, you can always just holler if somebody is bothering you. And Anwen, friend, if I’ve ever tried to bribe you with anything, it was certainly the hair dye. I mean, look at you! Such handsomeness!” I exclaimed. The other Siblys began to encircle her, uttering compliments or even announcements of their envy. Anwen disappeared in a wink with flushed cheeks back into the mound. “I’ll just leave this here.” Smiling, I set down the bottle. ** “141, 143. . .” I counted each step as I trekked the staircase. There was no doubt I lost track somewhere. The ledges kept spawning under my feet, infinitely multiplying until I wasn’t moving at all–swallowing me up in a whirlpool of stone. My tie still hung around my neck, and my blazer remained tied around my hips as a skirt. Streaks of red dribbled off from the cavity in my chest. It was a gorgeous marking, sensual to my fingertips as I traced its edges. Purity, oh, Purity. Purity and his wings of burnt skin. Purity and his many faces. Purity the spoiled. Purity the mutilated. The Silbys did not bother waiting for me. On bare feet, they stormed up the stairs to their room. A trail of red, though in paint unlike mine, streamed after them. None looked remotely near me as they squeaked and gossiped intangibly. I saved them, those Infran broads, enlightened them. As much as they liked to deny it, spit at me, and bask in the thought of their victimhood, in this home, they stood empowered. You’ve done well, my thoughts affirmed, though in the manner of an insincere commentator rather than a hype man. Teeth grace in tile violin goes laundry paper when. It dissolved into an intruding drivel. I rubbed my head and sniveled. “Do you need help, Knox?” called a Silby. Fattened by my coddling, her shadow fell upon me from the doorway steps ahead. I attempted counting again. There must’ve been at least another hundred between me and her. “I’m hallucinating some,” I said, breathing deeply to suppress a burp as I struggled to recall her name. Two syllables. Typically Latin, though sometimes English. Drops of slobber leaked from my mouth. “I’m hallucinating some, Tybal. Do you like your name, Tybal? I would have named you something better. Ty-Tyballinia. No, we’d have to eliminate the ‘ball’ aspect. It sounds too crude.” “One foot in front of the other,” she said. So I walked. Mess greeted me at the doorway. Dirtied culinary obscured the dark wooden countertops, and the sink lay running. I approached the kitchen table, sat, and set my face down upon its cool wooden surface. Assaulting my nose was the smell of neglected flowers, like soil mixed with the kind of sweet cough medicine that would have left me gagging as a child. Open windows whispered songs of the twilight hour through the vessels of busy trolleys and shooting guns. My mouth strained to vomit, but there was nothing in my stomach to regurgitate except the petals of Stulto’s bloom, which came out effortlessly in little sputters. Teetering, I stood up and brushed disgorged plant parts off the tabletop. “Love,” I said as I slogged up yet another staircase. “Are you awake?” She said she’d wait. Somebody’s gotten her. No, she always misses movie night. That sleepyhead, I assured myself. There was a stirring amidst the manor’s cloak of dusk. Portraits of myself, my wife, and my daughter turned to face me as the hallway lights flickered, escaping their quartz frames to penetrate my ears with nonsense. The taxidermied heads of Infran creatures bared their teeth. I stopped to stare at my favorite, an adabactor with daunting spiked tusks poking out from its forehead. Its nose remained black and sharp, and its eyes wide with malice. “Where is my Spes, Adaba-boy? Is she sleepy?” There’s someone in the house. The sounds of the stirring rose along with my blood pressure. Footsteps orbited around me, drawing near and far and then near again, little dancers in the dark. The carpet immersed me in its mass of purples and blues, leaving my skin stained indigo and my vision abstracted. I toiled to reach the master bedroom across the aisle as it stretched out to me with bright lights and celestial howling, like a dove struggling in a pool of oil. Never again with Stulto’s bloom. Never again on what was already a bad night. My hand brushed the doorknob, and the high abruptly faded into only a persistent hum-buzz twirling around my brain. The portraits returned to their typical depression–Spes posing with her ax, Ari’s school photo, and myself in the cap I wore when addressing the military with the Verbis emblem embroidered in its center. All lifeless shots. Who were they for when they captured not the subject’s essence but only some fragment of their identity? They used to feel personal, not advertisements of some supposed characters. Servants, babysitters, and likewise civilian guests, I reminded myself, mustn’t forget whose home they’re in. Yet my body moved independently, taking Ari’s from its hook and laying it backward against the wall to hide her distant grin and tamed posture. It was time for new pictures. Sweet ones, real ones; time was ticking. I approached my own when the stirring began again. Groans and squeals erupted from the vents as if someone had set a pen of pigs loose in my crawlspace. No, not the crawlspace, my bedroom door. I turned the ruby knob. Underneath a blanket wrestled my two squealing piglets, their skins melting together beneath the layer of duvet. Fishnet leggings and manicured nails outstretched and scraped at the sheet beneath them. One raised its head, a salmon-colored man with sweat running down his forehead. Through the crack in the door, we met eyes, his Infran Dr. Sesuss nose flaring its narrow nostrils. No mark of the Spoiled Purity existed carved onto his naked body. My chest felt tight. I stepped back. I was suffocating. Spes emerged from the linens, her hair flowing down her back and her dark skin glistening in front of the bedroom window. She giggled and held the man, the blanket falling and revealing inches of her body I had not seen in months. “Darling,” whispered the rosy-faced man, “look.” He was unfathomably ugly and grotesquely young, with beady, lifeless pupils that dilated when he faced me. The excess flesh on his face sagged while he bit down on his thin lips. My wife faced me, gasped, and strained to cover herself. Suddenly, I was a stranger. A small child who had walked into his parents having sex. I unfurled the door completely. “Get out of my house,” I said. The man stayed in place. “Get out of my house,” I repeated. “Knox,” Spes began. Tears ran down her round cheeks. “Shut up!” I turned to the man, picking up a marble trophy from on top of my dresser. “Get out of my house! I’ll kill you!” “Knox!” Spes sobbed. “God damn it! I hate you! You barely look at me. Every day, there’s less passion. God, God, God, I don’t want to fuck a dead man!” she screamed, “You get out! Get! Get!” My hands wrapped tighter around the statue. That pig of a man was attached to her at the side, his face equipped with a scowl that challenged mine. He thought I was weak; frail like a decaying dementia-ridden senior. I imagined his skull bashed in, his scowl gone, and the feist and confidence in his face beaten into numbness. A new portrait was in order of such brutality, him as a splintered slab of wood, rashed and beaten, a carcass licking my boot. The churning in my brain had come back. Every wall shook. Clock faces came to life and rang in alarm. Indescribable noises caressed my eardrum before breaking into sorrowful weeps. Was it my own? I stared at Spes in motionless frenzy, clenched my teeth, and screamed like a siren. Passionless. What a lie! An excuse, more like. One that erased all my ventures, reducing me to a nobody. But I was not a nobody. I thought of my sect, my campaigns, my endurance through the political brutality of my empty hive-mind world–even my collection of literature, maps, and artifacts. I thought of daring nights alone with Spes when we were young, ravaging each other, two sardonic eggheads suddenly overcome with desire. The veins in my neck throbbed as I gasped for air. It was all I had. I threw the figurine at the man’s head. Eye shut, I heard the thud. A million singing voices of victory flooded out of the cracks in the floorboard. Proving myself a man to the woman I loved in a display of fervent violence was passion. I strained my ears for his cries, though I did not look yet. There had to be a pause, a moment of relief, where I stood tall as a skyscraper and seemingly fought to stay contained in front of my wife and her wounded, quivering paramour. Frantic footsteps rushed off the bed and past my side. I turned and grappled against myself to seize my wife’s shoulder. “Spes!” My eyelids lifted. Escaping was the man with that same numb expression in which I had imagined him. “You’re insane,” he said. I swiveled back towards the bed. With her curly locks flowing over her breasts and her limbs bent at her sides, Spes sat limp pressed against the headboard, her forehead bludgeoned and the statue resting on her stomach. Lips pursed and sweet, my Renaissance beauty reclined there in the guise of a squashed bug. But she was not dead. The desk ornament I flung was only the size of my shoe. Spes, that dramatist, may have been slightly hurt but was far from dead. She only wanted me to think she was to observe me at my most distraught, like a leech feeding on misery. “Get up.” Staggering toward the bed, I said. “You wanted passion? I showed you passion. ‘Shoved it right into your head. Of course, we both know who that gesture was meant for. . .” I fumbled to find my wit. Cold skin met my hands as I stroked her face, unable to resist checking her pulse, even though she was not dead. “I love you, Spes,” I said. Rain pelted against a nearby window. “Spes, please. Please.” No vibration answered my plea. I lifted my hand, sitting next to her now. Tears did not come. There was not any blood on the trophy, but when I picked it up, it felt to be now only a cruel instrument. It depicted a younger me in white marble, with my glasses and collared shirt being the only things painted. Both were in pink. It was a favorable color. I scrambled from the bed to vomit pure digestive bile on the rug. My stomach heaved. I ran my nails along every piece of myself I saw, a dog chasing my tail. As I slammed myself against walls and convulsed, my own heart grew ever louder in my chest. “Dad? I heard–” Ari’s slippered feet hammered across the floor. “Mom? Mom?” I kept my eyes on the storm. Silence fell. “She-She isn’t—your—.” Gasps interrupted every syllable she spoke. “You’re a murderer. Bad. Like they said,” she breathed, “ You beat her!” The words became mush, alphabet soup. Ari ran back down the hall. “My-My mom is dead. . . .Yes. . . Manor of the Trials Sovereignty. . .Ari Sorkin. . . I’m afraid he’s going to hurt me,” she said, presumably over the phone. It was all too fast. I crawled onto the windowsill, opened the glass, and let myself plummet into the alley below. Gusts of wind howled. The lack of motion or sensation informed me I had passed and again lived. Another Palatium Infra, another strange planet in which the celestial endowed rotting men with the opportunity to inhabit. Was this it? Was it all just an impossible limbo of galactic traveling? My surroundings were overwhelmingly gray, an abyss of clouds. Perhaps I had now met the real coming world, and my family and old friends lived here, ready to rush to my sides, lift me up, and jump for joy. Spes would be there. She would be enraged, but at least she’d be there. You are a bad man. You are a bad man. My eyelashes fluttered. There was a tugging sensation in my leg. The fog was wavering along with my ascendance. “No,” I yearned, trying to grip the clouds and stick them in place. “Stay with me.” But the peace was fleeting. I felt the cement under me and the moist garments clinging to my figure. My leg burned. Carefully, I craned my neck, only to observe the promenade as my surroundings. The most underwhelming of filth and danger, individually Infran. Forever my coming world. What a fool I was, having forgotten my blessing. Those idiot Gods could not tell the difference between assassination and self-infliction; a faulty insurance plan. The urge to cry at last set over me, and so I sat and wailed hot salvia into my palm, shielding my mouth to muffle the noise. Thunder echoed my hushed howling. Raindrops turned to pebbles. Under the ambiance of the stormy night, I could have sworn I heard troops stomping, guns cocking, and the chanting of my name. They had all been waiting for this. Billboards came to life, and I could only sit and spectate as the scenery flashed red. I inhaled fear and sobriety through runny nostrils. “Trials Sovereign Vsevolod “Knox” Sorkin is currently at large for the suspected homicide of Spes Sorkin, breaking the first term of the Sovereignty Charter. We now instruct you to report any sightings of the Earth-born, caucasian, roughly 195 centimeters tall, brown-haired, and brown-eyed man to your local Guard post. One can identify the suspected convict specifically by an occult tattoo of Purity’s Coronet on his lower back. No attempted execution or elongated punishment will take place until our Guards conduct an autopsy proving his guilt, per Life’s 1238 commandment. We cannot be sure when or if the Gods will revoke his blessing. Remember, when Gods frown upon strife, opt for a peaceful life. We permit all grieving festivities until Cagidus 4th. Good year!” towering buildings sang out in broadcast, repeating that same convoluted message quicker the instant it ended. Sometimes, the announcer spoke in Latin for the Infran children, other times in Chinese, Hindi, or Spanish to cater to those of irrelevant tongues. You aren’t a bad man. You are a stupid boy. Puddles sloshed. Somebody was approaching. I didn’t dare waste any remaining energy avoiding the Guards and their prodding blades. How did that phrase go? You dug your grave. Now lie in it. And so I embraced the cement. “Knox?” said the Guard. No, her tone was too sincere, and no authority would proceed in such a manner. There wasn’t confirmation on whether or not I was armed, and it wasn’t as if she could shoot me first. She was a partygoer, having just left from the cellar’s backdoor. I shooed her away with my hand. She hovered, and I discerned her shadow hesitating over my body. A man could not rot in peace. “Come on, get up! They’re after you!” Hands reached around my torso, struggling to handle my weight as they urged me onto my feet. That leg, the burning one, my right, trembled and bent unnaturally upon impact with the ground. The partygoer slung my arm over her shoulder, balancing me. My eyes caught a glimpse of a cyan mop. “Anwen?” I rasped, “hu-who let you out?” Keys jangled in her hands–my keys. “I escaped,” she said casually, coercing me to walk beside her. “Quicken your pace. I just heard somebody on your front porch. ‘You see that compost bin down the alley? We’re gonna burrow right down into the depth of that. If they open it and uncover us, I’ll be on top, and I can hide you and act like I’m just a homeless amica trying to take a nap.” With a tightening grip, she led me like livestock to the stinking crate. “I don’t understand, Anwen,” I said. “They’re going to torture and kill you, stupid. You know they’ve been wanting to, and you just handed the opportunity to them!” “I understand that.” It was becoming increasingly challenging to hide the fragility emerging in my voice. “You said you were escaping. Why stop and help your captor?” “What else could I do? Leave you there?” Attempts to shove my wounded body inside its mass of discarded fruits and vegetables began. She yanked down upon my head and submerged me in the fertilizer sea. The evidence grows indisputable, I thought as I stared at the abruptly humane Infran girl, diving in after me, that I belong here. “Damn me to hell! I’ve killed her! My love is dead!” an uncontrollable cry leaped from my mouth. “Shut up! Soon you’ll be, too, if you don’t quiet down.” The actual noise of the Guards darted past us: disorientated marching, guns clanking against each other, cluttered belts rattling, the Latin squawking. One paused to open the bin’s lid, though only rummaged through the surface layer of peat before carrying on. “What are they talking about? I struggle with my Latin,” I whispered. “The search, mainly.” Aggression remained firey in Anwen’s clenched jaw. Though she sat on top of me, there was a monumental distance between our rain-soaked forms. I curled up into a ball, ducked my head between my knees, and dreamt of Spes, ignoring the stench of spoiled food rising from every crevice of my dwelling. The next coming world was due to adopt me again as I forced sleep. I prayed for a canyon of fluffy haze, where I waltzed with pale memories but found nothing but the petrifying stillness of my mind. Killed and ran. Violent as a Guard just to prove a point and watch it backfire. Why would any heaven want to welcome me? I clung to the picture of Spes in my head like it was the last ember of an extinguished flame. “Did you mean to kill her?” Anwen interrogated. “Someone like you would immutably believe yes.” “And who is someone like me? You can’t even treat me like a person for a moment, can you?” grating drama decorated her words. “You know my opinions. I have not seen much of your or your breed’s faces besides that of cruelty and ignorance.” I retorted. “I just saved you! Does that make me cruel and ignorant?” “It makes you an idiot, which is another word for somebody ignorant.” “And why am I an idiot?” She asked. “Because you helping me does no good. Thank you anyhow. Now, do yourself a favor and scram.” As she bent her leg in anticipation, preparing to strike me on the forehead, I sensed an invisible withdrawal widening the gap between us. “You never answered my question,” Anwen took me by the end of my tattered tie suddenly and started her game of shepherd and sheep over again, pulling me back up to the crate’s exit. It appeared as a shining light at the end of a maze of rubbish and mold. “No. Of course not. Spes was my everything,” I sniffled. “I knew it. You couldn’t even bring yourself to hit us, let alone murder your wife. The girls and I always figured you were sensitive.” My heart rate quickened. Today was one of humbling and misery–one to pray a hail spike would fall from the sky as sharp as a needle, pierce into my eyelid, and lobotomize me. I wished I could have merely died or hit my head hard enough not to have to deal with it all. No, I wished I was Anwen with her snarky, careless glow and lack of depth in her eyes. As we emerged from the compost bin together, I fantasized about strangling her until her face turned purple, her weakening spirit no longer categorizing me as “sensitive”, but the thought could only remind me of wielding that trophy and the microscopic traces of my wife’s tender skin tainting it, which turned my guts inside out. “That’s why I think you could use a little help,” Anwen said, “It seems like you can’t walk, either. Your leg is all twisted up.” She undid one of her trim pigtails and handed me the band. “Take off your tie and put up your hair. ‘Will make you less recognizable. Then swallow your pride and stick with me.”
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