Mathematics: Calculus and Coordinate Geometry
CG PET Mathematics tests Class 11–12 algebra, trigonometry, coordinate geometry, calculus, vectors/3D geometry and probability. The paper rewards recognising standard forms quickly and checking domain restrictions before simplifying.
Calculus rules
- d/dx(x^n) = nx^(n−1); use chain, product and quotient rules. d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(e^x)=e^x, d/dx(ln x)=1/x.
- For maxima/minima on a closed interval [a,b]: solve f′(x)=0, then evaluate f at all critical points and at a and b. The largest value is the maximum, the smallest is the minimum.
- ∫x^n dx = x^(n+1)/(n+1)+C (n≠−1); a definite integral equals F(b)−F(a).
- f is increasing where f′>0 and decreasing where f′<0; f″>0 at a critical point indicates a local minimum.
Coordinate geometry
- Distance = √[(x2−x1)²+(y2−y1)²]; midpoint = ((x1+x2)/2,(y1+y2)/2).
- Slope m = (y2−y1)/(x2−x1); parallel lines have equal slopes, perpendicular lines have m1m2 = −1.
- Circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).
Worked example
Find the maximum value of f(x)=x³−3x on [−2,2]. f′(x)=3x²−3=3(x−1)(x+1), so critical points are x=±1. Evaluate: f(−2)=−8+6=−2, f(−1)=−1+3=2, f(1)=1−3=−2, f(2)=8−6=2. The maximum is 2, attained at x=−1 and x=2.
Traps
Skipping endpoint checks on a closed interval; cancelling factors without noting where they are zero; using degree mode in trigonometric limits; misapplying the section formula (internal vs external); confusing radius with diameter; forgetting the ± when solving equations with squares.