## Solve by observation

I came across a question. If x and y are two positive integers such that (3x + 7y) is a multiple of 11, then which of the following will be divisible by 11? The options were: A. 4x+6y B. x+y+4 C. 9x+4y D. 4x-9y First I tried to solve it applying some principles of divisibility and failed. It was a question meant for competitive exam and I had to do it in quick time. Then I did it by observation. As the multiples of 11 are bot

## Geometrical Interpretation of Indefinite Integration

An indefinite integral of a function always gives a family of curves and these curves may not be similar. The family of curve is created due to placement of a constant at the integration. Thus, if integration of F(x) is f(x)+c, then we get a family of curve f(x). Now, if any line parallel to X axis is drawn, it cuts all members of this family at points, say, A, B, C, D, and so on, and the tangents drawn at all such points of intersection are parallel. See the attached PPT vid

{"items":["5f87c3f0cdeaa300175ba102","5f87c3f0cdeaa300175ba103"],"styles":{"galleryType":"Columns","groupSize":1,"showArrows":true,"cubeImages":true,"cubeType":"fill","cubeRatio":1.3333333333333333,"isVertical":true,"gallerySize":30,"collageAmount":0,"collageDensity":0,"groupTypes":"1","oneRow":false,"imageMargin":32,"galleryMargin":0,"scatter":0,"rotatingScatter":"","chooseBestGroup":true,"smartCrop":false,"hasThumbnails":false,"enableScroll":true,"isGrid":true,"isSlider":false,"isColumns":false,"isSlideshow":false,"cropOnlyFill":false,"fixedColumns":1,"enableInfiniteScroll":true,"isRTL":false,"minItemSize":50,"rotatingGroupTypes":"","rotatingCropRatios":"","columnWidths":"","gallerySliderImageRatio":1.7777777777777777,"numberOfImagesPerRow":1,"numberOfImagesPerCol":1,"groupsPerStrip":0,"borderRadius":0,"boxShadow":0,"gridStyle":1,"mobilePanorama":false,"placeGroupsLtr":true,"viewMode":"preview","thumbnailSpacings":4,"galleryThumbnailsAlignment":"bottom","isMasonry":false,"isAutoSlideshow":false,"slideshowLoop":false,"autoSlideshowInterval":4,"bottomInfoHeight":0,"titlePlacement":"SHOW_ON_THE_RIGHT","galleryTextAlign":"center","scrollSnap":false,"itemClick":"nothing","fullscreen":true,"videoPlay":"hover","scrollAnimation":"NO_EFFECT","slideAnimation":"SCROLL","scrollDirection":0,"scrollDuration":400,"overlayAnimation":"FADE_IN","arrowsPosition":0,"arrowsSize":23,"watermarkOpacity":40,"watermarkSize":40,"useWatermark":true,"watermarkDock":{"top":"auto","left":"auto","right":0,"bottom":0,"transform":"translate3d(0,0,0)"},"loadMoreAmount":"all","defaultShowInfoExpand":1,"allowLinkExpand":true,"expandInfoPosition":0,"allowFullscreenExpand":true,"fullscreenLoop":false,"galleryAlignExpand":"left","addToCartBorderWidth":1,"addToCartButtonText":"","slideshowInfoSize":200,"playButtonForAutoSlideShow":false,"allowSlideshowCounter":false,"hoveringBehaviour":"NEVER_SHOW","thumbnailSize":120,"magicLayoutSeed":1,"imageHoverAnimation":"NO_EFFECT","imagePlacementAnimation":"NO_EFFECT","calculateTextBoxWidthMode":"PERCENT","textBoxHeight":0,"textBoxWidth":200,"textBoxWidthPercent":50,"textImageSpace":10,"textBoxBorderRadius":0,"textBoxBorderWidth":0,"loadMoreButtonText":"","loadMoreButtonBorderWidth":1,"loadMoreButtonBorderRadius":0,"imageInfoType":"ATTACHED_BACKGROUND","itemBorderWidth":0,"itemBorderRadius":0,"itemEnableShadow":false,"itemShadowBlur":20,"itemShadowDirection":135,"itemShadowSize":10,"imageLoadingMode":"BLUR","expandAnimation":"NO_EFFECT","imageQuality":90,"usmToggle":false,"usm_a":0,"usm_r":0,"usm_t":0,"videoSound":false,"videoSpeed":"1","videoLoop":true,"jsonStyleParams":"","gallerySizeType":"px","gallerySizePx":670,"allowTitle":true,"allowContextMenu":true,"textsHorizontalPadding":-30,"itemBorderColor":{"themeName":"color_15","value":"rgba(47,46,46,1)"},"showVideoPlayButton":true,"galleryLayout":2,"targetItemSize":670,"selectedLayout":"2|bottom|1|fill|true|0|true","layoutsVersion":2,"selectedLayoutV2":2,"isSlideshowFont":false,"externalInfoHeight":0,"externalInfoWidth":0.5},"container":{"width":670,"galleryWidth":702,"galleryHeight":0,"scrollBase":0,"height":null}}