Vinculum Round 1

Theorems

100

a²+b²=c²

Pythagoras Theorem

200

If f(a)=0, then (x−a) is a factor of f(x).

Factor Theorem

300

If f(x) is divided by (x−a), the remainder is f(a)  

Remainder Theorem

400

(a + b)ⁿ = ∑ (nCr) * a^(n-r) * b^r (where the summation is from r = 0 to n)

Bionmal Theorem

Double Points

500

If F(x) is the integral of f(x), then:

d/dx [∫ f(x) dx] = f(x)

Fundamental Theorem of Calculus

Trigonometry

100

What is cos⁡(60^∘)?

1/2

200

In a right-angled triangle, the opposite side is 3 and the hypotenuse is 5. What is sin⁡(θ)?

3/5

300

In a right triangle, perpendicular is 5 and the hypotenuse is 13. What is cos⁡(θ)?

12/13

Steal Question

400

In a right triangle, the adjacent side is 4 and the hypotenuse is 5. What is cos⁡(θ)? 

4/5

500

In a right-angled triangle, angle θ is such that tan⁡(θ)= 3/4
Find the exact values of sin⁡(θ) and cos(θ)

sin(θ)= 3/5 and cos(θ)= 4/5

Linear Equations

100

Solve: x−7= −12

-5

200

Solve: 5x−3 = 2x+6

x = 3

Shield

300

Solve: 3(x+2) = 2(x+5)

x = 4

400

Solve: 2x + 3/4 = 5

x= 17/8

500

Solve: x/3 + 2 = 2x - 1/4

27/20 

Integration

100

∫ x dx

x²/2 + C

200

∫ 3x² dx

x³ + C

300

∫ (2x+5) dx

x² + 5x + C

400

∫ x⁻² dx

− 1/x + C

500

∫ x^1/2 dx

2/3x^3/2 + C

Switch Question

Functions and Graphs

100

If f(x)= 2x + 3, find f(4).

11

200

If f(x)= x²− 1 and f(x)= 8, find x.

±3

Steal Points

300

If f(x)= x + 2 and g(x)= 3x, find f[g(2)].

8

400

The graph of y=x² is shifted 3 units up. What is the new equation?

y= x² + 3

Hint

500

Let f(x) = x² - 5x + 6
Find the value of f(4) - f(2)

2