JEE B.Arch cheat sheet

Orthographic projection rule: a solid's front view shows length and height; the top view shows length and depth; the side view shows depth and height. Missing-line questions are solved by projecting each visible edge into the other two views.

Counting a cube's unit cells: a cuboid a×b×c units cut into unit cubes yields a·b·c cubes total; cubes with exactly two painted faces = 4(a−2)+4(b−2)+4(c−2) when all dimensions exceed 2.

Three-face painted cubes in an a×b×c painted block number exactly 8 (the corners) whenever a, b, c ≥ 2; cubes with no paint = (a−2)(b−2)(c−2).

Mirror images: a vertical mirror reverses left–right but preserves up–down; a horizontal (water) mirror reverses up–down but preserves left–right. Letters asymmetric about the mirror axis change; symmetric ones (A, H, I, M, O, T, U, V, W, X, Y) do not.

Paper folding and punching: unfold step by step in reverse order, reflecting each hole across the fold line used in that step; the number of holes doubles with each fold unless a hole lies on the crease.

Series and analogy reasoning: check differences, then ratios, then alternating sub-series, then squares/cubes; for figure series check rotation (typically 90° or 45° steps), element addition/subtraction, and shading alternation.

Number-series and coding tests: letter positions A=1…Z=26; opposite-letter pairs sum to 27 (A–Z, B–Y, C–X). In coding–decoding, test shift, reversal, and positional-value patterns before assuming a keyword substitution.

Architectural awareness: know the designer and city of widely taught works - Le Corbusier's Chandigarh Capitol Complex, Louis Kahn's IIM Ahmedabad, Charles Correa's Jawahar Kala Kendra (Jaipur), B.V. Doshi's IIM Bangalore, and the Lotus Temple by Fariborz Sahba (Delhi).

Spatial development: a cube net has 6 squares and 11 distinct valid nets; opposite faces on a net are never adjacent in the folded cube, so faces separated by one square in a straight strip are opposite.

Quadratic ax²+bx+c=0: roots (−b±√(b²−4ac))/(2a); sum = −b/a, product = c/a; discriminant D>0 distinct real, D=0 equal real, D<0 complex conjugates.

Conic sections in standard form: circle x²+y²=r²; parabola y²=4ax with focus (a,0) and directrix x=−a; ellipse x²/a²+y²/b²=1 with e=√(1−b²/a²); hyperbola x²/a²−y²/b²=1 with e=√(1+b²/a²).

Straight lines: distance from (x₁,y₁) to ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²); angle between lines tanθ=|(m₁−m₂)/(1+m₁m₂)|; parallel lines have equal slopes, perpendicular slopes multiply to −1.

Limits and continuity: lim(x→0) sinx/x = 1, lim(x→0)(1−cosx)/x² = 1/2, lim(x→0)(eˣ−1)/x = 1, lim(x→0)ln(1+x)/x = 1; L'Hôpital applies only to 0/0 or ∞/∞ indeterminate forms.

Derivatives: d/dx(xⁿ)=nxⁿ⁻¹, d/dx(sinx)=cosx, d/dx(cosx)=−sinx, d/dx(tanx)=sec²x, d/dx(lnx)=1/x, d/dx(eˣ)=eˣ; product rule (uv)′=u′v+uv′, quotient rule (u/v)′=(u′v−uv′)/v².

Definite integrals: ∫₀^a f(x)dx = ∫₀^a f(a−x)dx; ∫₀^(2a) f(x)dx = 2∫₀^a f(x)dx when f(2a−x)=f(x); area between curves = ∫|f(x)−g(x)|dx over the intersection interval.

Vectors: a·b = |a||b|cosθ; a×b has magnitude |a||b|sinθ and direction normal to both; scalar triple product a·(b×c) gives the parallelepiped volume; coplanar vectors satisfy a·(b×c)=0.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B); Bayes' theorem P(Aᵢ|B)=P(B|Aᵢ)P(Aᵢ)/ΣP(B|Aⱼ)P(Aⱼ).

Matrices and determinants: det(AB)=detA·detB; A⁻¹=adj(A)/det(A) exists only when det(A)≠0; for a 3×3 matrix |A|=a₁(b₂c₃−b₃c₂)−a₂(b₁c₃−b₃c₁)+a₃(b₁c₂−b₂c₁).

Line hierarchy: outlines are the heaviest, visible edges medium, hidden edges dashed, and construction/centre lines lightest; consistent line weight is the single largest scoring factor in freehand architectural sketching.

Human figure scale: an average adult is about 7.5 head-heights tall; door height ≈ 2.1 m, ceiling ≈ 3.0 m, eye level ≈ 1.6 m, chair seat ≈ 0.45 m - use these to keep drawn figures and openings believable.

One-point perspective: all receding edges converge to a single vanishing point on the horizon at eye level; front faces stay true shape. Two-point perspective uses two vanishing points on the horizon, and verticals remain vertical.

Vanishing-point placement: a vanishing point placed far outside the picture frame reduces distortion; objects below the horizon show their top faces, objects above show their undersides.

Isometric drawing uses three axes at 120° with true lengths along each; oblique projection keeps the front face true and draws depth at 30–45° with half or full scale.

Composition rule of thirds: place the principal mass or focal element near a third-line intersection rather than dead centre; leave sky/foreground margins so the subject is not crowded by the frame.

Shading and rendering: hatch lines follow the form's surface direction; light source is fixed for the whole drawing, giving a highlight, mid-tone, core shadow, and reflected light on curved forms.

Memory drawing of everyday objects (chair, table, staircase, hand tools) is judged on correct proportion, plausible structure, and clean line quality rather than artistic flourish.