Class 11 Sample Paper 2026 Math

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Practice Paper
(2025-26)

Class – XI

Mathematics (Code: 041)
Time: 3 hours​ ​ ​ ​ ​ ​ ​ ​ ​ Maximum Marks: 80
​ ​ ​ ​ ​ ​ ​
1.​ This Question paper contains 38 questions divided into five sections A,B,C,D,E.
Each section is compulsory. However, there are internal choices in some questions.
2.​ Section A has question number (1-18) as MCQ’s and Question number (19-20 )
Assertion-Reason based questions of 1 mark each.
3.​ Section B has Question number (21-25) of Very Short Answer (VSA)-type
questions of 2 marks each.
4.​ Section C has Question number (26-31) of Short Answer (SA)-type questions of 3
marks each.
5.​ Section D has Question number (32-35 ) of Long Answer (LA)-type questions of 5
marks each.
6.​ Section E has Question number (36-38) of Source based/Case based/passage
based/integrated units of assessment questions (4 marks each) with sub parts.
7.​ There is no overall choice however an internal choice have been provided in 2
questions in Section -B , 3 questions in Section- C and 2 questions in Section- D

Section – A
Question Number 1-18 are of MCQ type question one mark each.
1. 𝑐𝑜𝑠30˚ + 𝑐𝑜𝑠90˚ + 𝑐𝑜𝑠150˚ + 𝑐𝑜𝑠210˚ = 1

(a) 1 (b ) 0

− 3 1
(c) (d) 2
2

2 Which of the following is not equal to cos 2x ? 1

2 2 2
(a) 𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑥 (b ) 1 − 2𝑠𝑖𝑛 𝑥
2 2
1−𝑡𝑎𝑛 𝑥
(c) 1 − 2𝑐𝑜𝑠 𝑥 (d) 2
1+𝑡𝑎𝑛 𝑥

3 1
2
Let X= {x: x is an integer and 𝑥 < 10} which of the following correctly list all the
elements of set X?

(a) {1, 2, 3} (b ) {0, 1, 2, 3}

(c) {− 1, − 2, − 3, 1, 2, 3} (d) {− 1, − 2, − 3, 0, 1, 2, 3}

4 The domain of the function f given by f(x)= 2
𝑥−4
is equal to : 1
𝑥 −16

(a) R-{4} (b ) R-{-4}

(c) R (d) R - {4,-4}

5. 1
If [.] denote the greatest integer function, then which of the following statement is true ?

(a) [x] = x for all real x (b) [x] = x , only when x is an integer

(c) [x] = x+1, if x is not an integer (d) [x] = x-1, if x is an integer

6 1
Let 𝑐𝑜𝑠𝑥 = 2 , where x is in radians & lies in third quadrant. If x =
𝑎π
with a and b 1
𝑏
coprime positive integers, then a+b= ?

(a) 2 (b) 3

(c) 5 (d) 7

7 Which of the following complex number is equal to 𝑍 = 𝑖
1+2+3+.........+2025 1

(a) 1+0i (b) -1+0i

(c) 0-i (d) 0+i

8 10 1
Total number of terms in the expansion of (𝑥 − 2 𝑥 + 1) is
(a) 10 (b ) 11

(c) 20 (d) 21

1
(5+2𝑖)(4−𝑖)
If 𝑧 = 1+3𝑖
, then Im (𝑧. 𝑧¯)equals to :

9
(a) 0 (b) 2

(c) 4 (d) 6

10 4
If 𝑥+2 < 0 then x belongs to : 1

(a) (− ∞, − 2) (b) (− 2, ∞)

(c) (− ∞, − 2) ∪ (− 2, ∞) (d) 𝑥≠ − 2

1
7−𝑧
11 If 𝑓(𝑧) = 2 , where 𝑧 = 1 + 2𝑖 , then |𝑓(𝑧)| is :
1−𝑧

(a) 2
|𝑧| (b)|𝑧|

(c)2|𝑧| (d) None of these

12 If the focus of the parabola is (4,0) and its directrix is x=3 then its equation is : 1

2 2
(a) 𝑦 = 16𝑥 (b ) 𝑦 =− 16𝑥
2 2
(c) 𝑥 = 16𝑦 (d) 𝑥 =− 16𝑦

13 The reflection of the point (2, − 5, 3) in xz -plane is : 1
(a) (2, 5, 3) (b ) (2, − 5, 3)

(c) (− 2, − 5, 3) (d) (2, − 5, 3)

14 𝑐𝑜𝑠4𝑥
If 𝑄 = 𝑐𝑜𝑠2𝑥 , then (5Q+2) equals : 1

(a) 5cot2x+3 (b) 5tan⁡2x+3

(c) 3−5tan2x (d) 3+5sec2x

15 Equation of the line passing through the point (1,2) and perpendicular to the line x+y+1=0 is 1
:
(a) y-x+1=0 (b) y-x-1=0

(c) y-x+2=0 (d) y-x-2=0

16 If P(C)=0.4 , P(D)=0.5 and P(𝐶∩𝐷)=0.2 Then P(𝐶∪𝐷) equal to : 1

(a) 0.7 (b ) 0.8

(c) 0.9 (d) 1.1

17 If A ,B and C are three mutually exclusive and exhaustive events of an experiment such 1
that 𝑃(𝐴) = 0. 2, 𝑃(𝐵) = 0. 3 then 𝑃(𝐶) equals to :
(a) 0.2 (b ) 0.3

(c) 0.5 (d) 0.7

18 Mean and standard deviation of the numbers 2,4,6,8,10 are : 1
(a) Mean = 6, Standard deviation = 2.83 (b ) Mean = 5, Standard deviation = 2

(c) Mean = 6, Standard deviation = 3.16 (d Mean = 5, Standard deviation = 2.24

(ASSERTION-REASON BASED QUESTIONS )

In the following questions, a statement of assertion (A) is followed by a statement of
Reason (R). Choose the correct answer out of the following choices.

(a) Both A and R are true and R is the correct explanation of A.

(b) Both A and R are true but R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

19 Assertion (A): The inequality −2x>6 is equivalent to x<−3 1

Reason (R): When both sides of an inequality are divided or multiplied by a negative number, the
inequality sign reverses.

20 Assertion (A) :If cos a +cos b +cos c =3 then sin a +sin b +sin c =0 1

Reason (R ) : The cosine of any real angle lies between -1 and 1

(Section B)
This section contains 5 Very Short Answer (VSA) type questions of
2 marks each.
Q 21 2
How many chords can be drawn through 21 points on a circle?

Q 22 Prove that: 1 x 1! + 2 x 2! +3 x 3! +……...+n x n! = ( n+1)! – 1 2

Q 23 3 2
𝑢
If(𝑥 + 𝑖𝑦) = 𝑢 + 𝑖𝑣 then show that 𝑥 + 𝑦 = 4 𝑥 − 𝑦
𝑣
(2 2
).
OR
4 4
Evaluate (1 + 𝑖) + (1 − 𝑖)
Q 24 2
𝑐𝑜𝑠 𝑥 2
Find the derivative of 1+𝑠𝑖𝑛𝑥 w. r. t. x.
Q 25 Prove that: sin(n+1)x.sin (n+2)x+ cos(n+1)x.cos (n+2)x =cosx 2

OR
If tan x + tan y = a and tan x tan y=b, then prove that
𝑎
𝑡𝑎𝑛(𝑥 + 𝑦) = 1−𝑏

Section C
This section contains 6 Short Answer (SA)-type questions of 3 marks each.

Q26 In what ratio is the line joining the points (2, 3) and (4, -5) divided by the line passing 3
through the points (6, 8) and (-3, -2).

Q27 Solve for real x :
(2𝑥−1) (3𝑥−2)
⩾ 4 −
(2−𝑥) 3
3 5

Q28 If tany =2,
π 𝑦 𝑦 𝑦
< 𝑦 < π then find the value of 𝑠𝑖𝑛 2 , 𝑐𝑜𝑠 2 & 𝑡𝑎𝑛 2 3
2

OR
1
Find the value of 𝑠𝑖𝑛22 2 ˚

Q29 If A={𝑥: 𝑥ϵ𝑁, 1 < 𝑥≤5} and 3

B={𝑥: 𝑥ϵ𝑁, 3≤𝑥 < 7}
then find the value of
(a)𝐴∪𝐵 (b)𝐴∩𝐵
Q30 Find the equation of the circle which passes through the point (2,3) and whose center lies at 3
the point of intersection of the lines x−y=1 and x+y=5
OR
If the length of the major axis of an ellipse is 20 and the length of the minor axis is 16, find
the equation of the ellipse with center at the origin and major axis along the x-axis.

Q. 31 𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥 𝑑𝑦
If 𝑦 = 𝑠𝑖𝑛𝑥−𝑐𝑜𝑠𝑥 then find 𝑑𝑥 at x=0 3

OR
1−𝑐𝑜𝑠𝑚𝑥
Find the value of 1−𝑐𝑜𝑠𝑛𝑥

(SECTION D )

This section contains four Long Answer (LA)-type questions of 5marks each.
Q 32 5
3 5 4
Find the coefficient of 𝑥 in the product of (2 − 3𝑥) (1 + 𝑥) using binomial theorem.
●​ OR
8 8 8 8
Find (𝑥 + 2) + (𝑥 − 2) hence or otherwise evaluate ( 5 + 2) + ( 5 − 2)

Q 33 If the first term of AP is 2 and the sum of first five terms is equal to one fourth of the sum of 5
next five terms , then show that 20 th term is -112.

Q 34 (a) Find the sum of the series 4+44+444+-------+n terms -. 5
1 1 1 1

(b) Prove that 2 . 4 . 8 . 16
2 8 24 64
−−−−−−−−−− ∞ = 2

OR
2 2
If p and q are roots of 𝑥 − 4𝑥 + 𝑎 = 0 and r,s are roots of 𝑥 − 8𝑥 + 𝑏 = 0 where p,q,r
𝑎+𝑏 10
,s form a Geometrical Progression Prove that 𝑏−𝑎 = 6
Q35 The mean and variance of eight observations are 9 and 9.25 respectively . If six of 5
observations are 6,7,10,12,12 and 13, then find the remaining two observations .

Section E
Source based/Case based/passage based/integrated units of assessment
Questions
Q36 Geometric mathematics is deeply intertwined with art, particularly in patterns involving
circles such as mandalas, rangoli, and ancient mosaic designs. Consider a pattern formed by
drawing a circle of radius 60 cm. Inside this circle, another circle is drawn whose radius is
half that of the previous circle, and this process continues infinitely so that each new circle
has half the radius of the previous one. Such recursive, concentric circle patterns are widely
used in traditional and modern art forms.
Suppose the radius of the first (outermost) circle is 60 cm.

Answer the following:
(i) What is the radius of the third circle?​ 1
(ii) Find the area of the second inner circle.​ 1
(iii) What is the perimeter (circumference) of the fourth circle?​ 2
OR​
(iii) Find the total sum of areas of all circles if the process continues infinitely.

Q37 A school is organizing an inter-school event where 5 different competitions (Quiz, Debate,
Elocution, Science Exhibition, and Sports) will take place on the same day. There are 6
student organizers selected from Class XI. Each competition requires a unique student
organizer to lead it; no student can lead more than one event.

During the planning, the following sub-questions arise:
(i) In how many different ways can the 6 organizers be assigned so that 5 of them are 1
selected to lead the 5 competitions, while the sixth organizer acts as a supervisor?
(ii) If two of the competitions (Quiz and Science Exhibition) must be led by female students
and among the 6 organizers, only 2 are females, in how many ways can the assignments for 1
all competitions be made?
(iii) Suppose the organizers have to be seated in a row for a photo and the supervisor must 2
be seated at one end. In how many ways can they be arranged?
OR
(iii) If the event is conducted annually and the 6 organizers are to be assigned to the 5 events
and supervisor role in such a way that no student repeats the same role in two consecutive
years, what is the minimum number of years before all possible assignments are exhausted
without any repetition for any individual in any given role?

Q 38 A satellite dish has a shape called a paraboloid, where each cross section is parabola. Since 2+2
radio signals (parallel to axis) will bounce off the surface of the dish to the focus, the
receiver should be placed at the focus. The dish is 12 ft across, and 4.5 ft deep at the vertex.
Based on above situation answer the following:
(I)Write the equation of the parabola representing the cross section of the dish with the
vertex at the origin.
(ii)In the parabola representing the satellite dish, find the distance from the vertex to the
focus where the receiver should be placed.

​