Video summary

UPCAT Mock Exam Full Solutions

Main summary

Key takeaways

Educational

Main ideas, concepts, and lessons (with how to solve recurring question types)

Overall strategy for UPCAT/MOCK exam (speed under time pressure)

  • For each problem, prioritize the most efficient method over lengthy concept building.
  • Emphasize mental/fast solving, especially for math.
  • Target speed: about 1 minute per item (often discussed as ~70 seconds).
  • Recommended approach when practicing:
    • Set up the equation quickly
    • Use shortcuts/patterns when they clearly apply
    • Spend time mainly on hard questions, and “accumulate” easy-question time savings.

Algebra / Equations

Q1: Systems of linear equations (2 variables)

Steps

  1. Define variables (e.g., N = notebooks, P = pens).
  2. Write equations from:
    • Total items: N + P = 20
    • Total cost: 35N + 15P = 420
  3. Eliminate one variable:
    • Multiply the first equation by a coefficient (here by 15) to match the other equation and subtract, removing P.
  4. Solve for the remaining variable:
    • After subtraction: 20N = 120 → N = 6
  5. Pick the matching choice.

Q6: Using symmetric identities

Given: x + y = 11 and xy = 24 Steps

  1. Square the sum:
    • (x + y)² = x² + 2xy + y²
  2. Substitute:
    • 11² = x² + 2(24) + y²
  3. Isolate x² + y²:
    • x² + y² = 121 − 48 = 73

Q7: Recognize difference of squares / factoring shortcut

For expressions like (99² − 1²):

  • Recognize as: a² − b² = (a − b)(a + b)
  • Compute quickly: (99−1)(99+1) = 98·100

Q5: Consecutive integers

Given: product of two consecutive integers is 506 Steps

  1. Let integers be x and x + 1
  2. Set up: x(x + 1) = 506 → x² + x − 506 = 0
  3. Shortcut based on answer choices + parity reasoning:
    • Sum of consecutive integers is always odd
  4. Determine the pair whose sum matches required options:
    • Found sum = 45

Q11: Identity expansion to avoid long solving

If (x + 1/x) is given:

  • Use:
    • (x + 1/x)² = x² + 2 + 1/x²
  • Rearrange to get:
    • x² + 1/x² = (given)² − 2

Q1-family recurring theme

  • The method repeatedly stressed: write the two core equations first, then eliminate or use algebraic identities.

Functions / Graph reasoning

Domain & range graph selection

Steps

  • Domain condition: x > 0 and x ≠ 2 (eliminate graphs that don’t match)
  • Range condition: y between −3 and 3 (eliminate graphs that exceed)
  • Choose the graph consistent with both constraints.

Inverse functions (graph of inverse)

  • Key concept: the inverse swaps x and y.
  • If f(x) = 2x:
    • Original: y = 2x
    • Inverse: x = 2y → y = x/2
  • Choose the graph with slope 1/2 (flatter than y = 2x).

Percentage / Discount-type word problems

Combined increase then decrease

  • Increase by 25% → multiply by 1.25
  • Decrease by 20% → multiply by 0.8 (or 4/5)
  • Emphasis:
    • Don’t treat it as “+25 then −20” in raw percent-points.
    • Use multiplicative factors.

Example pattern used

  • Final value relative to original:
    • Original · (1 + a) · (1 − b)

Mental math / Pattern exploitation

  • Use difference of squares and factorization patterns instead of brute multiplying.
  • Use choice-based elimination:
    • If options suggest only one feasible parity/size/result, discard quickly.

Word problems and modeling with “time/distance” or “setup-first”

Boat traveling upstream & downstream

Key modeling idea: Equal quantity in both trips = time (because “at the same time”). Steps

  1. Set time upstream = time downstream.
  2. Use boat speed relative to water:
    • Upstream: boat speed = v − 3 (current opposes)
    • Downstream: boat speed = v + 3 (current helps)
  3. Use time = distance / velocity and solve for v.
  4. Convert back to the requested value (depends on wording).

Age/timeline problems

Steps

  1. Assign ages:
    • Daughter age = x
    • Father age = 5x (from “five times as old”)
  2. Apply timeline shifts (e.g., “8 years ago” and “4 years from now”).
  3. Translate both statements into equations and solve for x.
  4. Convert to present ages.

Geometry highlights

Pythagorean theorem in multiple contexts

  • 2D diagonal (square side s): s√2
  • 3D space diagonal (rectangular prism): √(a² + b² + c²)
  • Rhombus perimeter using diagonals:
    • Diagonals of a rhombus are perpendicular
    • Split into right triangles using half-diagonals
    • Use Pythagorean triple to get side length, then perimeter = 4·side

Circle/sector/area scaling

  • Doubling a dimension:
    • Area (2D) scales by 4
    • Volume (3D) scales by 8
  • Sector area:
    • Sector area = (θ/360)·(πr²)
  • Solve for θ by canceling πr² terms.

Trigonometry & special triangle facts

  • Memorize:
    • Pythagorean triples (e.g., 3-4-5, 5-12-13, etc.)
    • 30-60-90 triangle ratios:
      • x, x√3, 2x
  • Use tangent in “height/base” setups:
    • tan θ = opposite/adjacent
    • height = base·tan θ
    • base = height/tan θ

Trigonometry angle-elevation / depression

Angle elevation

  • Use:
    • tan θ = height / horizontal distance
  • So:
    • height = horizontal distance · tan θ

Elevation change with walking distance

  • Setup uses tangent:
    • height = (horizontal distance)·tan(angle)
  • Express height using different bases, then form an equation from the difference and solve.

Statistics / Probability (last portion)

Mean/average removal & correction

  • Mean relation:
    • mean = (sum of values)/n
  • “One number removed” approach:
    • Use totals:
      • 5-number mean → total of 5
      • remaining 4-number mean → total of 4
    • Difference gives removed number.

Probability by counting / brute force

  • Without replacement:
    • multiply conditional probabilities, updating denominator each draw
  • Independent trials (e.g., coin tosses):
    • brute force outcomes or use counting
    • Example pattern:
      • “At least two heads out of three flips”
      • favorable / total = 4/8 = 1/2
  • “Roll two dice, second > first”:
    • count favorable outcomes by comparison pairs.

Expected/test hypothesis logic

  • For “fewer than” a stated value:
    • use the “less than” (one-tailed) direction.

Odds vs probability (wording pitfall)

  • Odds against event:
    • if P(event)=p, then P(not event)=1−p
    • odds against = (not event cases) : (event cases)

Key “formula memory” points repeatedly stressed

  • Systems of linear equations with elimination
  • Identities:
    • (x + y)² = x² + 2xy + y²
    • a² − b² = (a − b)(a + b)
  • Trig basics:
    • tan θ = opposite/adjacent
    • Special triangles + Pythagorean triples for sin/cos
  • Circle/sector:
    • area scaling and the θ/360 factor
  • Pythagorean triples:
    • memorize common triples to extract side lengths quickly

Detailed bullet list: explicit methodologies/instructional sequences mentioned

  • Approach for math problems under time pressure

    • Identify what is asked (unknown variable/value).
    • Choose the fastest setup:
      • counts/cost equation
      • identity transformation (square/sqrt, difference of squares)
      • geometry formula
      • trig ratio model (tan/sin/cos)
    • Use elimination for 2-variable systems:
      • align coefficients and subtract/add to remove one variable
    • Solve quickly and match the choice.

  • Linear equations (2 variables) setup

    • Define variables (e.g., N and P).
    • Equation 1 from total quantity.
    • Equation 2 from total cost/expression.
    • Eliminate one variable:
      • multiply Equation 1 by a factor so coefficients match Equation 2
      • subtract to remove the chosen variable
    • Solve the remaining variable.

  • Use quadratic/symmetric identity for x² + y²

    • Start with given x + y
    • Square both sides
    • Expand: (x + y)² = x² + 2xy + y²
    • Substitute xy and solve for x² + y².

  • Mental evaluation of nested functions

    • Compute the inner part first (e.g., find f(2)), then substitute layer by layer.
    • Keep numbers small for mental arithmetic.

  • Percentage change

    • Convert each percent change to a multiplicative factor:
      • increase by k% → multiply by (1 + k/100)
      • decrease by k% → multiply by (1 − k/100)
    • Apply sequentially by multiplication (not raw +/− percent points).

  • Consecutive integers product

    • Let first be x, second be x + 1
    • Set x(x + 1) equal to the product
    • Solve or use choice elimination for speed.

  • Upstream/downstream boat

    • Let boat speed in still water be v
    • Upstream: v − current
    • Downstream: v + current
    • “At the same time” → set time upstream = time downstream
    • Use time = distance/velocity and solve.

  • Angle elevation / depression

    • Right triangle model:
      • adjacent = horizontal distance
      • opposite = height difference
    • tan(θ) = opposite/adjacent
    • height = adjacent · tan(θ)

  • 30-60-90 triangle ratio usage

    • If one acute angle is 30°:
      • sides are x (opposite 30), x√3 (opposite 60), 2x (hypotenuse)
    • Use to compute missing side quickly.

  • Circle/sector and area scaling

    • Sector angle:
      • (θ/360)·πr² = sector area → solve for θ
    • Dimension scaling:
      • 1D→2D→3D scaling (square for area, cube for volume)

  • Probability with combinations

    • Without replacement:
      • P(A then B) = P(A)·P(B|A)
    • For “at least k”:
      • count outcomes with k or more successes.

Speakers / sources featured

  • Main speaker: An instructor/host (referred to as “Sir” by viewers; exact name not stated in subtitles).
  • Sources mentioned for materials:
    • “Colejo Updates” (speaker says to check their Facebook page for reviewers/materials)
    • “PINB” (also referenced for exam/reviewer posts)
    • “Scholar of the Town” (Facebook page used for sending a reviewer; speaker repeatedly directs viewers there)
  • Video platform/source: YouTube (context: mock exam “UPCAT Mock Exam Full Solutions”), with no other named external speaker.

Original video