Verified Solution • 2025–26 Session moderate

QJEEM23Mad01c: JEE Main 2023 Mathematics - Class 12 JEEM23Mad01c Solution: Let N denote the number that turns up when a fair die is…

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations

x+y+z=1

2x+Ny+2z=2

3x+3y+Nz=3

has unique solution is k6, then the sum of value of k and all possible values of N is :

Multiple Choice Options

  • A 18
  • B 21
  • C 20
  • D 19

Step-by-Step Verified Solution

For unique solution Δ≠0\Delta \neq 0

i.e. \left|<mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="4" style="font-size: 130.9%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 4.276em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-right: 0.5em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-top: 0.2em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding: 0.2em 0.5em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-top: 0.2em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-right: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-top: 0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="10" style="font-size: 129.3%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 4.276em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-right: 0.5em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-top: 0.2em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding: 0.2em 0.5em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-top: 0.2em; padding-bottom: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-right: 0.5em; padding-top: 0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left: 0.5em; padding-top: 0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="center center center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mi>N</mi></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>3</mn></mtd><mtd><mi>N</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container>\right| \neq 0

⇒(N2−6)−(2N−6)+(6−3N)≠0\Rightarrow\left(N^{2}-6\right)-(2 N-6)+(6-3 N) \neq 0

⇒N2−5N+6≠0\Rightarrow N^{2}-5 N+6 \neq 0

∴N≠2\therefore N \neq 2 and N≠3N \neq 3

∴\therefore Probability of not getting 2 or 3 in a throw of dice =23=\frac{2}{3}

As given 23=k6⇒k=4\frac{2}{3}=\frac{k}{6} \Rightarrow k=4

∴\therefore Required value =1+4+5+6+4=20=1+4+5+6+4=20

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring