Day 22 — Gate II

Updated

July 30, 2026

Day 22 — Gate II

Stage II · gate day
Goal: Closed-book fluency across Days 11–21: expressions, linear equations/inequalities, \(2\times 2\) systems (including matrix form), polynomials, factoring, rational expressions, functions, linear/quadratic graphs, exponentials, and \(\sum\) notation with closed forms.

Why this matters

Stage III begins logic and proofs. That stage assumes you can manipulate algebra without it consuming all working memory. Gate II certifies that variables, equations, functions, and sums are tools — not obstacles.

Theory

Coverage map

Days Skills
11 Expressions, distribute, substitute, identities
12 Linear equations; identity vs contradiction
13 Inequalities; intervals; flip rule; absolute value
14 \(2\times 2\) systems; \(\det\); \(Ax=b\); Cramer; inverse
15–16 Polynomials; Horner; factoring; zero-product
17 Rational expressions; domain; partial fractions setup
18 Functions; domain/range; composition; inverse idea
19 Slope, vertex, quadratic formula, complete square
20 Exponential growth/decay; solve \(b^{x}=c\)
21 \(\sum\), linearity, \(\sum i\), geometric, telescoping, double sums

How to sit the gate

  1. Write warm-up definitions closed book (below).
  2. Sections A–D timed if you like (\(100\)–\(140\) min total).
  3. Section proofs require justifications.
  4. Log misses by day number; repair within a week.

Warm-up definitions (blank paper)

  • Zero-product property
  • Domain of \(\dfrac{p(x)}{q(x)}\)
  • Slope formula; vertex \(x=-\dfrac{b}{2a}\)
  • Function \(f:A\to B\); injective / inverse idea
  • \(\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}\)
  • Geometric sum \(\sum_{i=0}^{n} r^{i}=\dfrac{1-r^{n+1}}{1-r}\) (\(r\neq 1\))
  • \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\)
  • Solution cases for a \(2\times 2\) linear system
  • Flip rule for inequalities

Algebra hygiene checklist

  • Domain before solving rationals
  • Flip inequality when multiplying by negative
  • Check system solutions in both equations
  • Factor completely before canceling
  • Expand to verify factoring
  • Off-by-one on sum limits
  • \(\det=0\) needs consistency check, not automatic “no solution”
  • Inverse exists iff \(\det\neq 0\)

Section map

Section Focus Suggested time
A Expressions, linear eq/ineq, absolute value \(25\) min
B Systems \(2\times 2\), matrices, polynomials, factoring \(30\) min
C Rational, functions, linear/quadratic, exponential \(30\) min
D Summation + mixed proofs \(30\)–\(40\) min

Worked examples (calibration)

Example 1 — Equation + inequality

Solve \(3(x-2)=2x+1\) → \(3x-6=2x+1\) → \(x=7\).
Solve \(-2x+4\le 10\) → \(-2x\le 6\) → \(x\ge -3\), interval \([-3,\infty)\).

Example 2 — System + matrix

\(\begin{cases}x+y=5\\ 2x-y=4\end{cases}\) → add: \(3x=9\), \(x=3\), \(y=2\).
\(A=\begin{pmatrix}1&1\\2&-1\end{pmatrix}\), \(\det=-1-2=-3\neq 0\).
Cramer: \(x=\frac{\det\begin{pmatrix}5&1\\4&-1\end{pmatrix}}{-3}=\frac{-5-4}{-3}=3\).

Example 3 — Inverse

\(A^{-1}=-\frac{1}{3}\begin{pmatrix}-1&-1\\ -2&1\end{pmatrix}=\frac{1}{3}\begin{pmatrix}1&1\\ 2&-1\end{pmatrix}\).
Check \(A A^{-1}=I\).

Example 4 — Factor & rational

\(\dfrac{x^{2}-9}{x^{2}-x-6}=\dfrac{(x-3)(x+3)}{(x-3)(x+2)}=\dfrac{x+3}{x+2}\), \(x\neq 3,-2\).

Example 5 — Quadratic

\(f(x)=x^{2}-4x-5=(x-5)(x+1)\); vertex at \(x=2\), \(f(2)=-9\).

Example 6 — Exponential + sum

\(2^{x}=32\Rightarrow x=5\).
\(\sum_{i=1}^{n}(2i)=2\cdot\frac{n(n+1)}{2}=n(n+1)\).

Example 7 — Function

\(f(x)=\frac{1}{x-1}\), domain \(x\neq 1\); \((f\circ f)(x)=\frac{x-1}{1-(x-1) wait}\): \(f(f(x))=1/(\frac{1}{x-1}-1)=\frac{x-1}{1-(x-1)}=\frac{x-1}{2-x}\).

Example 8 — Geometric sum

\(\sum_{i=0}^{n} 2^{i}=2^{n+1}-1\).

Exercises — Section A (Expressions, linear, inequalities)

Starred items have answer sketches below.

A1.* Simplify \(3(2x-1)-2(x+4)+x\).

A2. Expand \((x-3)(x+5)-x(x-1)\).

A3.* Solve \(4(x-1)=2x+10\); check.

A4. Classify \(3(x+2)=3x+6\) and \(3(x+2)=3x+5\).

A5. Solve \(\dfrac{x}{2}-\dfrac{x}{5}=3\).

A6.* Solve \(-3\le 2x+1<7\); interval notation.

A7. Solve \(|2x-3|\ge 5\); write as union of intervals.

A8. Solve \(|x+1|<0.5\).

A9. Word: three consecutive integers sum to \(75\). Find them.

A10. Rearrange \(A=\frac{1}{2}h(b_1+b_2)\) for \(b_1\).

Exercises — Section B (Systems, poly, factor)

B1.* Solve \(\begin{cases}2x+y=7\\ x-2y=-4\end{cases}\) by elimination; write \(Ax=b\) and \(\det A\).

B2. Use Cramer’s rule on B1 (verify).

B3.* Find \(A^{-1}\) for \(A=\begin{pmatrix}2&1\\ 1&1\end{pmatrix}\); solve \(Ax=\begin{pmatrix}1\\ 0\end{pmatrix}\).

B4. Classify: \(\begin{cases}x+2y=3\\ 2x+4y=6\end{cases}\) and \(\begin{cases}x+2y=3\\ 2x+4y=7\end{cases}\).

B5. Multiply \((x^{2}+1)(x-2)\); degree of product.

B6.* Horner-evaluate \(2x^{3}-3x+1\) at \(x=2\).

B7. Factor \(6x^{2}-7x-3\); solve \(6x^{2}-7x-3=0\).

B8. Factor \(x^{3}-8\); solve \(x^{3}=8\).

B9. Factor completely over reals: \(x^{4}-1\).

B10. Divide \(x^{3}-2x^{2}-x+2\) by \(x-1\) (long division); write factorization if remainder \(0\).

Exercises — Section C (Rational, functions, quad, exp)

C1.* Domain and simplify \(\dfrac{x^{2}-x-6}{x^{2}-9}\).

C2. Compute \(\dfrac{2}{x-1}-\dfrac{1}{x}\); single fraction.

C3.* Partial fractions: \(\dfrac{3x+5}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}\); find \(A,B\).

C4. Solve \(\dfrac{x}{x-2}=3\); check domain.

C5. \(f(x)=x^{2}+1\), \(g(x)=x-1\): compute \((f\circ g)(x)\) and \((g\circ f)(x)\).

C6.* Inverse of \(f(x)=5x-2\) on \(\mathbb{R}\).

C7. Vertex form of \(f(x)=x^{2}-6x+4\) by completing the square; vertex coordinates.

C8. Discriminant and roots of \(2x^{2}-3x-2=0\).

C9. Line through \((1,2)\) perpendicular to \(y=2x+3\).

C10.* \(A(t)=100\cdot (1.05)^{t}\); solve \(A(t)=200\) for \(t\) in log form. Half-life style: if decay \(A=80\cdot(1/2)^{t/10}\), find \(A(30)\).

Exercises — Section D (Summation + proofs)

D1.* Closed form: \(\sum_{i=1}^{n}(3i+1)\).

D2. Evaluate \(\sum_{i=0}^{6} 2^{i}\).

D3.* Telescoping: \(\sum_{i=1}^{n}\bigl(\frac{1}{i}-\frac{1}{i+1}\bigr)\).

D4. Double sum: \(\sum_{i=1}^{n}\sum_{j=1}^{i} 1\).

D5. Nested loop iterations: outer \(i=1..n\), inner \(j=1..n\) vs inner \(j=1..i\).

D6.* Prove \(\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\) by induction.

D7. Prove: product of two odds is odd.

D8. Prove flip: if \(c<0\) and \(a<b\) then \(ac>bc\).

D9. Prove \(\det\neq 0\) \(\Rightarrow\) unique solution idea for \(2\times 2\) (via inverse or Cramer).

D10. Geometric: prove \(\sum_{i=0}^{n} r^{i}=\frac{1-r^{n+1}}{1-r}\) for \(r\neq 1\).

Answer key sketches (starred + selected half)

A1. \(6x-3-2x-8+x=5x-11\).
A3. \(4x-4=2x+10\), \(2x=14\), \(x=7\).
A6. \(-4\le 2x<6\), \(-2\le x<3\), \([-2,3)\).

B1. From second \(x=2y-4\); \(2(2y-4)+y=7\), \(5y-8=7\), \(y=3\), \(x=2\). \(A=\begin{pmatrix}2&1\\1&-2\end{pmatrix}\), \(\det=-4-1=-5\).
B3. \(\det=1\), \(A^{-1}=\begin{pmatrix}1&-1\\ -1&2\end{pmatrix}\). \(x=A^{-1}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\ -1\end{pmatrix}\).
B6. Horner coeffs \(2,0,-3,1\): \(2\); \(4\); \(8-3=5\); \(10+1=11\).

C1. \(\dfrac{(x-3)(x+2)}{(x-3)(x+3)}=\dfrac{x+2}{x+3}\), \(x\neq\pm 3\). (Also \(x\neq 3\) from cancel.)
C3. \(3x+5=A(x+2)+B(x+1)\). \(x=-1\): \(2=A(1)\), \(A=2\). \(x=-2\): \(-1=B(-1)\), \(B=1\).
C6. \(f^{-1}(x)=\frac{x+2}{5}\).
C10. \(t=\log_{1.05}2=\frac{\ln 2}{\ln 1.05}\); \(A(30)=80\cdot\frac{1}{8}=10\).

D1. \(3\cdot\frac{n(n+1)}{2}+n=\frac{3n(n+1)+2n}{2}=\frac{n(3n+5)}{2}\).
D3. \(1-\frac{1}{n+1}=\frac{n}{n+1}\).
D6. Standard induction as Day 21.

Additional half-key:
A4. Identity; contradiction.
A7. \(2x-3\le -5\) or \(2x-3\ge 5\) → \(x\le -1\) or \(x\ge 4\).
B4. Infinite (dependent); empty (inconsistent).
B7. \((3x+1)(2x-3)\); \(x=-\frac13\) or \(x=\frac32\).
C7. \((x-3)^{2}-5\); vertex \((3,-5)\).
C8. \(\Delta=9+16=25\); \(x=2\), \(x=-\frac12\).
D2. \(2^{7}-1=127\).
D4. \(\frac{n(n+1)}{2}\).
D5. \(n^{2}\) vs \(n(n+1)/2\).

Optional rapid-fire

RF1. Combine \(2(x-3)-4(1-x)\).
RF2. Solve \(|x-2|\le 3\); interval.
RF3. \(\det\begin{pmatrix}4&1\\ 2&3\end{pmatrix}\) and inverse factor \(1/\det\).
RF4. Factor \(x^{2}-9x+20\); roots.
RF5. Domain of \(\frac{x}{x^{2}-1}\).
RF6. Vertex of \(y=x^{2}+2x-3\).
RF7. \(3^{x}=81\).
RF8. \(\sum_{i=1}^{n} 2=\;?\)
RF9. \(\sum_{i=0}^{4} 3^{i}\).
RF10. Is \(f(x)=x^{3}\) invertible on \(\mathbb{R}\)?

RF answers: \(6x-10\); \([-1,5]\); \(\det=10\), \(A^{-1}=\frac{1}{10}\begin{pmatrix}3&-1\\ -2&4\end{pmatrix}\); \((x-4)(x-5)\); \(x\neq\pm 1\); \((-1,-4)\); \(x=4\); \(2n\); \(\frac{3^{5}-1}{2}=121\); yes, \(f^{-1}=\sqrt[3]{\cdot}\).

Must-pass items (certify Stage II)

You should be able to, without notes:

  1. Solve a \(2\times 2\) system two ways and via \(A^{-1}\) or Cramer once.
  2. Produce \(\sum_{i=1}^{n} i\) and a geometric sum closed form with proof sketch.
  3. Complete the square for one quadratic.
  4. Simplify one rational expression with domain.
  5. Flip an inequality correctly under negative multiplication.

Repair schedule template

Weak day Symptom Redo plan Date

Self-score rubric

Score band Meaning
\(\ge 85\%\) Stage III ready; spot-repair only
\(70\)–\(84\%\) Repair weak days with targeted rework
\(50\)–\(69\%\) Re-work failed sections with notes, then retest
\(<50\%\) Restart Stage II foundations (Days 11–14 first)

CS connection

Gate II underwrites asymptotic sums, solving for parameters in models, matrix transforms in 2D, and function composition pipelines. Stage III will quantify statements about these objects.

Common pitfalls (exam mode)

Pitfall What to do instead
Flip forget Mark every \(\times\) by negative
Cancel terms not factors Factor first
Extraneous rational roots Check domain
\(\det=0\) auto empty Test consistency
Sum off-by-one Count terms \(b-a+1\)
Inverse vs reciprocal \(f^{-1}\) undoes \(f\)
Losing root dividing by \(x\) Factor + zero-product

Checkpoint

  • Warm-up definitions written closed-book
  • Sections A–D attempted
  • At least one \(2\times 2\) matrix system solved (B1–B3)
  • At least one \(\sum\) closed form derived (D1–D4)
  • Starred answers checked; miss log by day number
  • Repair plan dated

Write two takeaways and your three weakest Stage II topics.

Deep dive — mixed Gate II drills with sketches

G1 — Expression → equation → inequality.
Simplify \(3(2x-1)-2(x+4)+x=5x-11\).
Solve \(4(x-1)=2x+10\Rightarrow x=7\).
Solve \(-3\le 2x+1<7\Rightarrow -2\le x<3\), interval \([-2,3)\).
Absolute value: \(|2x-3|\ge 5\Rightarrow x\le -1\) or \(x\ge 4\).

G2 — System + matrix trifecta.
\(\begin{cases}2x+y=7\\ x-2y=-4\end{cases}\): \(x=2\), \(y=3\) (elimination).
\(A=\begin{pmatrix}2&1\\1&-2\end{pmatrix}\), \(\det=-5\).
\(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\) has \(\det=1\), \(A^{-1}=\begin{pmatrix}1&-1\\ -1&2\end{pmatrix}\); \(A^{-1}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\ -1\end{pmatrix}\).

G3 — Factor, rational, domain.
\(\dfrac{x^{2}-x-6}{x^{2}-9}=\dfrac{(x-3)(x+2)}{(x-3)(x+3)}=\dfrac{x+2}{x+3}\) for \(x\neq\pm 3\).
Partial fractions setup: \(\dfrac{3x+5}{(x+1)(x+2)}=\dfrac{2}{x+1}+\dfrac{1}{x+2}\).

G4 — Quadratic + exponential.
\(x^{2}-6x+4=(x-3)^{2}-5\), vertex \((3,-5)\).
\(2x^{2}-3x-2=0\): \(\Delta=25\), roots \(2\) and \(-\frac12\).
\(A(t)=100\cdot(1.05)^{t}=200\Rightarrow t=\log_{1.05}2\).

G5 — Summation must-pass.
\(\sum_{i=1}^{n}(3i+1)=\frac{n(3n+5)}{2}\).
\(\sum_{i=0}^{6}2^{i}=127\).
Telescoping \(\sum_{i=1}^{n}\bigl(\frac1i-\frac1{i+1}\bigr)=\frac{n}{n+1}\).
Double \(\sum_{i=1}^{n}\sum_{j=1}^{i}1=\frac{n(n+1)}{2}\).

G6 — Proof fragments to rewrite cold.
- Induction for \(\sum i\).
- Geometric \(S-rS\).
- Flip rule: \(c<0\), \(a<b\Rightarrow ac>bc\).
- \(\det\neq 0\Rightarrow\) unique \(2\times 2\) solution via inverse/Cramer.

Extra rapid-fire (second pass)

RF11. Solve \(|x-2|\le 3\); interval.
RF12. Factor \(x^{2}-9x+20\); roots.
RF13. Domain of \(\frac{x}{x^{2}-1}\).
RF14. \(\sum_{i=0}^{4}3^{i}\); \(\sum_{i=1}^{n}2\).
RF15. Inverse of \(f(x)=5x-2\); is \(x^{3}\) invertible on \(\mathbb{R}\)?

RF answers (11–15): \([-1,5]\); \((x-4)(x-5)\), roots \(4,5\); \(x\neq\pm 1\); \(\frac{3^{5}-1}{2}=121\), \(2n\); \(\frac{x+2}{5}\), yes with cube root.

Timed second-pass strategy

  1. One full \(2\times 2\) solved two ways + \(\det\) check.
  2. One rational simplify with domain line.
  3. One \(\sum\) induction or geometric proof.
  4. One inequality with a negative multiply (mark the flip).
  5. Update repair table by day number (11–21), not by vague “algebra.”

Tomorrow

Day 23 — Propositions & truth tables. Stage III begins: what is a proposition; connectives; full truth tables; tautology and contradiction.