CBSE Class 12 Maths Sample Paper 2026, with Solutions PDF Download

Thissectioncomprisesofmultiplechoicequestions(MCQs)of1mark each.

Selectthecorrectoption(Question1-Question18)

1.

(A)sin⁻¹𝑥                  (B) sin⁻¹(2𝑥)                      (C)sin⁻¹𝑥)                     (D)2sin⁻¹𝑥

(

2

ForVisuallyImpaired:

InverseTrigonometricFunction,whose domain is[−1,1,is…

]

33

(A)    cos⁻¹𝑥                                                       (B)cos⁻¹𝑥)

(

3

(C)     cos⁻¹(3𝑥)                                                 (D)3cos⁻¹𝑥

𝑚×4                                      𝑛×3                                                  𝑝×𝑞

𝐴𝐶botharedefinedandaresquarematricesofsameorder,thenvalueof𝑚,𝑛,𝑝

and𝑞are:

(A)𝑚=𝑞=3𝑎𝑛𝑑 𝑛=𝑝=4                                      (B)𝑚=2,𝑞=3𝑎𝑛𝑑 𝑛=𝑝=4

(C)𝑚=𝑞=4𝑎𝑛𝑑𝑛=𝑝=3                                       (D)𝑚=4,𝑝=2𝑎𝑛𝑑𝑛=𝑞=3

Ifthematrix𝐴=[3             𝑝        𝑡]isskew-symmetric,thenvalueof𝑞+𝑡is….

𝑝+𝑟

𝑞    −4      0

(A)−2                      (B)0                                 (C) 1                           (D)2

(A)3                        (B) 9                                (C)3 𝐼                          (D) 9 𝐼

Theinverseof thematrix [0        2    0]is…

0    0    5

¹     0    0

0    0    3                                                 ⎡3                          ⎤

(A) [0    2    0]                                         (B)0       1          0

2

5    0    0                                                 ⎢0    0     1⎥

[             ₅]

−1             0       0

⎡    3                                      ⎤                                        −3      0       0

(C)     0      −1             0                                  (D)[0         −2      0]

2

⎢0         0      −1⎥                                          0       0      −5

[                       ₅]

Value of the determinant |                              |is sin23^𝑜                cos23^𝑜

(A)0                          (B)¹                                 (C)^√3                        (D)1

2                                                                              2

cos𝑥     ,𝑥>𝜋

is continuousat𝑥=𝜋,then thevalueof𝑘is

(A)        𝜋                (B)      ⁻¹                         (C)   0                    (D)⁻²

𝜋                                                                                                                                     𝜋

(A)^𝜋     −1                                 (B)^𝜋+1                                                    (C)−𝜋              −1                      (D)−𝜋             +1

4                  2                                           4                2                                                                4                  2                                       4                   2

(A)(−∞,−1                             (B)(−∞,1                                        (C)[−1,∞)                (D)[−1,1

]                            )                                                                 ]

4                                                        4                                                            4                                                              44

(A)straight lines      (B)parabolas           (C)Circles              (D)Ellipses

𝑎

(A)𝑎+𝑏∫𝑏𝑓(𝑏−𝑥)𝑑𝑥                                        (B)𝑎+𝑏∫𝑏𝑓(𝑎−𝑥)𝑑𝑥

2           𝑎                                                                                                                                2          𝑎

(C)    𝑏−𝑎∫𝑏𝑓(𝑥)𝑑𝑥                                        (D)𝑎+𝑏∫𝑏𝑓(𝑥)𝑑𝑥

2           𝑎                                                                                                                             2          𝑎

(A)−1                                                 (B)        ¹                     (C)        ¹                (D)        ¹

10                                                           20                                                                    4                                                           5

(A)⁶units                 (B)    ¹⁴units                (C)    ⁸units            (D)⁵units

√14                                                            √6                                                                √14                                                     √6

(A)   4units             (B)    √34units           (C)5units              (D)5√2units

(A)𝑎⃗                       (B)𝑏⃗⃗+ 𝑐⃗                  (C)𝑎⃗− 𝑏⃗⃗              (D)𝑐⃗

Maximise𝑍=3𝑥+ 2𝑦+1

Subjectto the constraints𝑥≥0,𝑦≥0,3𝑥+4𝑦≤12,

He/ShenoticesthebelowshowngraphforthesaidLPPproblem,andfindsthat a constraint is missing in it:

Helphim/herchoosetherequiredconstraintfromthe graph.

Themissingconstraintis

(A)𝑥+ 2𝑦≤2                                                                           (B)2𝑥+ 𝑦≥2

(C)2𝑥+𝑦≤2                                                                            (D)𝑥+2𝑦≥2

If 𝑍=𝑎𝑥 +𝑏𝑦 +𝑐, where 𝑎,𝑏,𝑐 > 0, attains its maximum value at two of its corner points (4,0) and (0,3) of the feasible region determined by the system of linear inequalities, then

(A)4𝑎=3𝑏                  (B)3𝑎=4𝑏             (C)4𝑎+𝑐=3𝑏              (D)3𝑎+𝑐=4𝑏

Thenoneof theotherpossiblepointsat which the objectivefunctionattainsits minimum value is

(A)(2,9)                  (B)   (6,6)                (C)(4,7)                  (D)(0,0)

ForVisuallyImpaired:

The graphoftheinequality3𝑥+5𝑦<10is the

(A)         Entire𝑋𝑌 −plane

(B)         OpenHalf planethatdoesn’tcontainorigin

(C)        OpenHalfplanethat containsorigin,butnotthepoints oftheline3𝑥 + 5𝑦 = 10

(D)          Halfplane that containsorigin andthepointsoftheline3𝑥+5𝑦=10

(A)      1                              (B)     1                               (C)1                                        (D)1

81                                        100                                        90

(Question numbers 19 and 20 are Assertion-Reason based questions carrying1markeach.Twostatementsaregiven,onelabelledAssertion(A) and the other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below.)

(A)   Both(A)and(R) aretrueand (R) isthecorrect explanationof (A).

(B)   Both(A)and(R) are truebut(R) is notthecorrectexplanationof(A).

(C)   (A)is truebut(R)is false.

(D)   (A)is falsebut(R) istrue.

)

2                                                                                                 4

Reason(R):Principalvaluebranchofsin⁻¹𝑥is[−𝜋,𝜋]andthatofs𝑒𝑐⁻¹𝑥

22

is[0,𝜋] −{𝜋}.

2

Reason(R):Thevector(𝑎⃗+𝑏⃗⃗)isperpendiculartotheplane of𝑎⃗and𝑏⃗⃗

Thissectioncomprisesof5veryshortanswer(VSA)typequestionsof2markseach.

21B

3

OR

Findthedomainofcos⁻¹(3𝑥−2)

(

4           2                                                                          𝒅𝒙

23B

(𝑥−1)3

OR

Findout the areaof shaded region intheenclosed figure.

Find out the area of the region enclosed by the curve𝑦²= 𝑥, 𝑥=3and 𝑥-axis in the first quadrant.

definitionofderivatives,find𝑓^′(5).

𝑂𝑃 is a median. Find the area of the parallelogram formed by the two adjacent sides as 𝑂𝐴 and 𝑂𝑃.

Thissectioncomprisesof6shortanswer(SA)typequestionsof3marks each.

26B.

𝑑𝑥            (log(𝑥𝑒))2

OR

2

If𝑥=𝑎(𝜃−sin𝜃),𝑦=𝑎(1 − cos𝜃)find^𝑑𝑦.

𝑑𝑥2

28B

28A

28B

−4

integralrepresentonthe graph?

OR

Using integrationfindtheareaoftheregion{(𝑥,𝑦)∶𝑥²− 4𝑦≤0,𝑦−𝑥≤0}

ForVisuallyImpaired:

Definethefunction𝑦=|𝑥+ 1|.Evaluate∫2|𝑥+1|𝑑𝑥.Whatdoesthevalueof

−4

thisintegralrepresent?

OR

Using integrationfind theareaenclosedwithinthecurve:25𝑥²+ 16𝑦²=400

29B

𝑟⃗=(2𝚤̂−𝚥̂+2𝑘̂)+𝜇(3𝚤̂+6𝚥̂+2𝑘̂)

measuredparalleltothez-axis.

OR

Find the point of intersection of the line 𝑟⃗ = (3𝚤̂+𝑘̂)+𝜇(𝚤̂+𝚥̂+𝑘̂)and the line through (2,−1,1) parallel to the z-axis. How far is this point from the z-axis?

30

Maximise𝑍=2𝑥+𝑦subjectto

𝑥+ 𝑦≤1200

𝑥+ 𝑦≥600

𝑦≤𝑥

2

𝑥≥0,𝑦≥0.

ForVisuallyImpaired:

The objective function 𝑍 =3𝑥+2𝑦of a linear programming problem under some constraints is to be maximized and minimized. The corner points of the feasible region are 𝐴(600,0),𝐵(1200,0),𝐶(800,400)and 𝐷(400,200). Find the pointat which𝑍 ismaximum andthepoint atwhich 𝑍 isminimum.Also,find the corresponding maximum and minimum values of 𝑍.)

Thissectioncomprisesof4longanswer(LA)typequestionsof5marks each

Fortwomatrices𝐴=[2             −5    −1]and𝐵=[0        −1    −1],findtheproduct𝐴𝐵

−2     4       1                      2      0       3

andhencesolve thesystemofequations:

3𝑥− 6𝑦−𝑧=3

2𝑥− 5𝑦− 𝑧+ 2=0

−2𝑥+ 4𝑦+𝑧=5

33B

0         1+𝑥2

OR

Find∫(3sin𝜃−2)cos𝜃𝑑𝜃

5−𝑐𝑜𝑠2𝜃−4sin𝜃

34B

𝑑𝑥

OR

Findtheparticularsolutionofthedifferentialequation:

𝑥𝑦                                                           𝑥⁄𝑦

2𝑦𝑒^⁄𝑑𝑥+ (𝑦− 2𝑥𝑒                )𝑑𝑦=0giventhat𝑦(0)=1

3                                                                                               2                 2

whosey-coordinateis1.Findtheco-ordinatesoftheirpointofintersection.Find the vector equation of a line perpendicular to both the given lines and passing through this point of intersection.

This section comprises of 3 case-study/passage-based questions of 4 marks each with subparts.Thefirsttwocasestudyquestionshavethreesubparts(I),(II),(III)ofmarks1, 1, 2 respectively. The third case study question has two subparts of 2 marks each

A city’s traffic management department is planning to optimize traffic flow by analyzingtheconnectivitybetweenvarioustrafficsignals.Thecityhasfivemajor spots labelled 𝐴, 𝐵, 𝐶, 𝐷, 𝑎𝑛𝑑 𝐸.

Thedepartmenthascollectedthefollowingdata regardingone-waytrafficflow between spots:

1.   Trafficflowsfrom𝐴 to𝐵,𝐴to𝐶,and𝐴 to𝐷.

2.   Trafficflowsfrom 𝐵to𝐶and𝐵to𝐸.

3.   Trafficflowsfrom𝐶to𝐸.

4.   Trafficflowsfrom 𝐷to𝐸and𝐷to𝐶.

Thedepartmentwantsto representandanalyzethisdatausingrelationsand functions. Use the given data to answer the following questions:

I.             Isthetrafficflowreflexive?Justify.                                                                   [1]

II.             Is the traffic flow transitive? Justify.                                                             [1] IIIA.Representtherelationdescribingthetrafficflowasasetoforderedpairs.

Alsostate thedomainand range of therelation.

OR

IIIB.Doesthetrafficflowrepresentafunction?Justifyyour answer.                       [2]

LEDbulbsareenergy-efficientbecausetheyusesignificantlylesselectricitythan traditional bulbs while producing the same amount of light. They convert more energyintolightratherthanheat,reducingwaste.Additionally,theirlonglifespan means fewer replacements, saving resources and money over time.

Acompany manufactures a new type of energy-efficient LED bulb. The cost of production and the revenue generated by selling x bulbs (in an hour) are modelled as

𝐶(𝑥)=0.5𝑥²−10𝑥+150and𝑅(𝑥)=−0.3𝑥²+20𝑥respectively,where

𝐶(𝑥)and𝑅(𝑥)arebothin₹.

To maximize the profit, the company needs to analyze these functions using calculus. Use the given models to answer the following questions:

I.             Derivetheprofitfunction𝑃(𝑥)                                                                  [1]

II.             Findthecriticalpointsof𝑃(𝑥).                                                                   [1]

IIIA.Determine whetherthecriticalpointscorrespondtoamaximum ora minimum profit by using the second derivative test.

OR

IIIB.Identifythepossiblepracticalvalueof𝑥 (i.e.,thenumberofbulbsthatcan realistically be produced and sold) that can maximize the profit, if the resources available and the expenditure on machines allows to produce minimum 10 but not more than 18 bulbs per hour. Also calculate the maximum profit.                                                                                        [2]

Excessiveuseofscreenscanresultinvisionproblems,obesity,sleepdisorders, anxiety, low retention problems and can impede social and emotional comprehension and expression. It is essential to be mindful of the amount of time we spend on screens and to reduce our screen-time by taking regular breaks, setting time limits, and engaging in non-screen-based activities.

In a class of students of the age group 14 to 17, the students were categorised into three groups according to a feedback form filled by them. The first group constituted of the students who spent more than 4 hours per day on the mobile screen or the gaming screens, while the second group spent 2 to 4 hours /day onthesameactivities.Thethirdgroupspentlessthan2hours/dayonthesame. Thefirstgroupwith thehigh screentime is60%of allthestudents,whereasthe second group with moderate screen time is 30% and the third group with low screen time is only 10% of the total number of students. It was observed that 80% students of first group faced severe anxiety and low retention issues, with 70% of second group, and 30% of third group having the same symptoms.

I.             What is the total percentage of students who suffer from anxiety and low retention issues in the class?                                                                  [2]

II.             A student is selected at random, and he is found to suffer from anxiety and low retention issues. What is the probability that he/she spends screen time more than 4 hours per day?                                               [2]