ICSE Class 10 Sample Paper 2026 Maths

ICSE 2026 EXAMINATION

SPECIMEN QUESTION PAPER

MATHEMATICS

Maximum Marks: 80

Time allowed: Three hours

1. Answers to this Paper must be written on the paper provided separately.

2. You will not be allowed to write during first 15 minutes.

3. This time is to be spent in reading the question paper.

4. The time given at the head of this Paper is the time allowed for writing the answers.

5. Attempt all questions from Section A and any four questions from Section B.

6. All working, including rough work, must be clearly shown, and must be done on the same
sheet as the rest of the answer.

7. Omission of essential working will result in loss of marks.

8. The intended marks for questions or parts of questions are given in brackets [ ].

9. Mathematical tables are provided.

Instruction for the Supervising Examiner

Kindly read aloud the Instructions given above to all the candidates present in the Examination
Hall.

T26 511 – SPECIMEN 1 of 13

NOTE:

The Specimen Question Paper in the subject provides a realistic
format of the Board Examination Question Paper and should be used
as a practice tool. The questions for the Board Examination can be set
from any part of the syllabus. However, the format of the Board
Examination Question Paper will remain the same as that of the
Specimen Question Paper.

SECTION A (40 Marks)

(Attempt all questions from this Section.)

Question 1

Choose the correct answers to the questions from the given options. [15]

(Do not copy the question, write the correct answers only.)

(i) (𝑥𝑥 − 2) and (𝑥𝑥 + 2) are the factors of 𝑥𝑥 3 + 𝑥𝑥 2 − 4𝑥𝑥 − 4. The third
factor of the given polynomial is:

(𝑎𝑎) (𝑥𝑥 − 1)

(𝑏𝑏) (𝑥𝑥 − 4)

(𝑐𝑐) (𝑥𝑥 + 1)

(𝑑𝑑) (𝑥𝑥 + 4) [Analyze]

(iii) In the figure given below, AC is a diameter of the circle.

AP = 3 cm and PB = 4 cm and QP ⊥ AB.

If the area of ∆APQ is18 cm2, then the area of shaded portion QPBC is:

(a) 32 cm2

(b) 49 cm2

(c) 80 cm2 [Understanding
(d) 98 cm2 & Analysis]

T26 511 – SPECIMEN 2 of 13

(ii) Radha deposited ₹400 per month in a recurring deposit account for 18
months.

The qualifying sum of money for the calculation of interest is∶

(a) ₹ 3,600

(b) ₹ 7,200

(c) ₹ 68,400

(d) ₹ 1,36,800 [Application]

(iv) In the given diagram, the radius of the circle with centre O is 3 cm. PA
and PB are the tangents to the circle which are at right angle to each other.
The length of OP is:

3
(a) 𝑐𝑐𝑐𝑐
√2

(b) 3 𝑐𝑐𝑐𝑐

(c) 3√2 𝑐𝑐𝑐𝑐
[Analysis &
(d) 6√2 𝑐𝑐𝑐𝑐
Evaluation]

(v) Assertion (A): If 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑏𝑏 then ab =1

Reason (R): 𝑠𝑠𝑠𝑠𝑠𝑠 2 𝜃𝜃 − 𝑡𝑡𝑡𝑡𝑡𝑡2 𝜃𝜃 = 1

(a) (A) is true and (R) is false.

(b) (A) is false and (R) is true.

(c) Both (A) and (R) are true and (R) is the correct explanation of (A).
[Analysis &
(d) Both (A) and (R) are true, but (R) is not the correct explanation of (A).
Evaluation]

T26 511 – SPECIMEN 3 of 13

(vi) A solid sphere is cut into two identical hemispheres.

Assertion (A): The total volume of two hemispheres is equal to the volume
of the original sphere.

Reason (R): The total surface area of two hemispheres together is equal
to the surface area of the original sphere.

(a) (A) is true, (R) is false.

(b) (A) is false, (R) is true. [Analysis]

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

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

(vii) Given that the sum of the squares of the first seven natural numbers is
140, then their mean is:

(a) 20

(b) 70

(c) 280 [Understanding
(d) 980 & Evaluation]

(viii) A bag contains 3 red and 2 blue marbles. A marble is drawn at random.
The probability of drawing a black marble is∶

(a) 0
1
(b)
5
2
(c)
5
3
(d) [Application]
5
(ix) 3
If matrix A=[−1 2] and matrix B=� �, then matrix AB is equal to:
4
(a) [−3]

(b) [8]

(c) [5]

−1 2
(d) � � [Analysis]
3 4

T26 511 – SPECIMEN 4 of 13

(x) A mixture of paint is prepared by mixing 2 parts of red pigments with 5
parts of the base. Using the given information in the following table, find
the values of a, b & c to get the required mixture of paint.

Parts of red pigment 2 4 b 6
Parts of base 5 a 12.5 c

(a) a = 10, b = 10, c = 10

(b) a = 5, b = 2, c = 5

(c) a = 10, b = 5, c = 10 [Application &
(d) a = 10, b = 5, c = 15 Evaluation]

(xi) An article which is marked at ₹ 1,200 is available at a discount of 20%
and the rate of GST is 18%. The amount of SGST is:

(a) ₹ 216.00

(b) ₹ 172.80

(c) ₹ 108.00 [Analysis &
(d) ₹ 86.40 Evaluation]

(xii) The sum of money required to buy 50, ₹ 40 shares at ₹ 38.50 is:

(a) ₹ 1,920

(b) ₹ 1,924

(c) ₹ 1,925

(d) ₹ 1,952 [Application]

(xiii) The roots of quadratic equation x2 – 1 = 0 are:

(a) 0, 0

(b) 1, 1

(c) -1, -1 [Analysis &
(d) +1, -1 Evaluation]

T26 511 – SPECIMEN 5 of 13

(xiv) Which of the following equations represents a line equally inclined to the
axes?

(a) 2x – 3y +7 = 0

(b) x–y=7

(c) x=7 [Analysis &
(d) y = –7 Evaluation]
𝑥𝑥
(xv) Given, 𝑥𝑥 + 2 ≤ 3 + 3 and x is a prime number. The solution set for x is:

(a) ∅

(b) {0}

(c) {1}

(d) {0, 1} [Understanding
& Analysis]

Question 2

(i) While factorizing a given polynomial, using remainder & factor theorem, [4]

a student finds that (2x + 1) is a factor of 2x3 + 7x2 + 2x – 3.

(a) Is the student’s solution correct stating that (2x + 1) is a factor of the
given polynomial?

(b) Give a valid reason for your answer. [Analysis &
Also, factorize the given polynomial completely. Application]

(ii) P is a point on the x- axis which divides the line joining A (- 6, 2) and [4]
B (9, - 4). Find:

(a) the ratio in which P divides the line segment AB.

(b) the coordinates of the point P. [Analysis &
(c) equation of a line parallel to AB and passing through (-3, -2). Evaluation]

T26 511 – SPECIMEN 6 of 13

(iii) In the given figure, AC is the diameter of the circle with centre O. [4]

CD is parallel to BE.

∠AOB = 80⁰ and ∠ACE = 20⁰.

[Analysis &
Evaluation]

Calculate:

(a) ∠ BEC

(b) ∠ BCD

(c) ∠CED

Question 3

(i) -11, -7, -3, ………….,49, 53 are the terms of a progression. [4]

Answer the following:

(a) What is the type of progression?

(b) How many terms are there in all? [Analysis &
Evaluation]
(c) Calculate the value of middle most term.

T26 511 – SPECIMEN 7 of 13

(ii) In the diagram given below, a tilted right circular cylindrical vessel with [4]
base diameter 7 cm contains a liquid. When placed vertically, the height of
the liquid in the vessel is the mean of two heights shown in the diagram.
Find the area of wet surface, when the cylinder is placed vertically on a
horizontal surface.
𝟐𝟐𝟐𝟐
(Use π= 𝟕𝟕 ).

6 cm [Application &
7 cm
Evaluation]
(iii) Use a ruler and compass to answer this question. [5]

(a) Construct a circle of radius 4.5cm and draw a chord AB of length
6.5 cm.

(b) At A, construct ∠CAB=75°, where C lies on the circumference of
the circle.

(c) Construct the locus of all points equidistant from A and B.

(d) Construct the locus of all points equidistant from CA and BA.

(e) Mark the point of intersection of the two loci as P. Measure and [Analysis &

write down the length of CP. Understanding]

SECTION B (40 Marks)
(Attempt any four questions from this Section.)

Question 4
(i) Ms. Kaur invested ₹ 8,000 in buying ₹100 shares of a company paying 6% [3]
dividend at ₹ 80. After a year, she sold these shares at ₹75 each and invested
the proceeds including the dividend received during the first year in buying
₹ 20 shares, paying 15% dividend at ₹ 27 each. Find the:

(a) dividend received by her during the first year. Application &
(b) number of shares purchased by her using the total proceeds. Evaluation]

T26 511 – SPECIMEN 8 of 13

(ii) Solve the following inequation, write the solution set, and represent it on [3]
the real number line.
5𝑥𝑥 3
5x – 21 < 7 – 6 ≤ –37 + x, x ∈ R.
[Evaluation]

(iii) Prove the following trigonometry identity: [4]

(sinθ + cosθ) (cosecθ – secθ) = cosecθ.secθ – 2 tanθ [Application &
Analysis]

Question 5

(i) In the given figure (not drawn to scale) chords AD and BC intersect at P, [3]
where AB = 9 cm, PB = 3 cm and PD = 2 cm.

A

P C

B D

(a) Prove that ∆APB ~ ∆CPD.

(b) Find the length of CD. [Application &
(c) Find area ∆APB : area ∆CPD. Evaluation]

(ii) Mr. Sam has a recurring deposit account and deposits ₹ 600 per month for [3]
2 years. If he gets ₹ 15,600 at the time of maturity, find the rate of interest [Application &
earned by him. Evaluation]

T26 511 – SPECIMEN 9 of 13

(iii) Using step-deviation method, find mean for the following frequency [4]
distribution:

Class 0 – 15 15 – 30 30 – 45 45 – 60 60 – 75 75 – 90

Frequency 3 4 7 6 8 2 [Application &
Evaluation]

Question 6

(i) Find the coordinates of the centroid P of the ∆ABC, whose vertices are [3]
A(–1, 3), B(3, –1) and C(0, 0). Hence, find the equation of a line passing [Analysis &
through P and parallel to AB. Evaluation]

(ii) In the given figure, the parallelogram ABCD circumscribe a circle, touching [3]
circle at P, Q, R and S.

[Analysis &
(a) Prove that: AB = BC Application]

(b) What special name can be given to the parallelogram ABCD?

(iii) The following bill shows the GST rate and the marked price of articles: [4]

Rajdhani Departmental Store

S. No. Item Marked Discount Rate of
Price GST

(a) Dry fruits (1 kg) ₹ 1200 ₹100 12%

(b) Packed Wheat flour (5kg) ₹ 286 Nil 5%

(c) Bakery products ₹ 500 10% 12% [Application &
Evaluation]
Find the total amount to be paid (including GST) for the above bill.

T26 511 – SPECIMEN 10 of 13

Question 7

(i) Draw the necessary diagram for this question. [5]

A man on the top of a lighthouse observes the angle of depression of two
ships on the opposite sides of the lighthouse as 30° and 50° respectively. If
the height of the lighthouse is 80m, find the distance between the two ships.
[Understanding,
Give your answer correct to the nearest meter.
Application &
(Use Mathematical Tables for this Question) Evaluation]

(ii) The marks of 200 students in a test were recorded as follows: [5]

Marks % 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100

No. of
5 7 11 20 40 52 36 15 9 5
students
Using a graph sheet draw ogive for the given data and use it to find the:

(a) median.
[Application,
(b) number of students who obtained more than 65% marks. Analysis &
(c) number of students who did not pass, if the pass percentage was 35. Evaluation]

Question 8

(i) A box containing cards numbered between 0 and 100 are shuffled and a card [3]
is picked at random. Find the probability of getting a card which is:

(a) divisible by 6. [Application &
(b) not divisible by 6. Evaluation]

(ii) If x, y and z are in continued proportion, prove that: [3]
𝑥𝑥 𝑦𝑦 𝑧𝑧 1 1 1 [Application &
+ + = + 3 + 3
𝑦𝑦 2 . 𝑧𝑧 2 𝑧𝑧 2 . 𝑥𝑥 2 𝑥𝑥 2 . 𝑦𝑦 2 𝑥𝑥 3 𝑦𝑦 𝑧𝑧 Analysis]

T26 511 – SPECIMEN 11 of 13

(iii) A manufacturing company prepares spherical ball bearings, each of radius [4]
7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box
can have a maximum of 2156 cm³ of ball bearings. Find the:

(a) maximum number of ball bearings that each box can have.
[Analysis,
(b) mass of each box of ball bearings in kg. Application &
(Use π = 7 )
22 Evaluation]

Question 9

(i) Study the graph given below and answer the following: [3]

y

x

(a) Number of batsmen who scored 500 to 700 runs
[Analysis &
(b) Modal class interval Evaluation]
(c) The value of mode

T26 511 – SPECIMEN 12 of 13

(ii) An Arithmetic Progression (A.P.) has 3 as its first term. The sum of the first [3]
8 terms is twice the sum of the first 5 terms. Find the common difference of [Analysis,
the A.P. Application &
Evaluation]

(iii) The roots of equation (q – r) x2 + (r – p) x + (p - q) = 0 are equal. [4]
[Application &
Prove that: 2q = p + r, that is, p, q & r are in A.P.
Analysis]

Question 10

(i) The sum of the squares of three consecutive even numbers is 596. Find the [3]
numbers. [Analysis,
Application &
Evaluation]

(ii) 1 1 1 0 [3]
Given matrix, X = � � 𝑎𝑎𝑎𝑎𝑎𝑎 𝐼𝐼 = � �, prove that 𝑋𝑋 2 = 4𝑋𝑋 + 5𝐼𝐼.
8 3 0 1
[Application &
Evaluation]

(iii) Use a graph sheet for this question. Take 1 cm = 1 unit along both the x and [4]
y axis. Plot ABCDE, where A (4, 0), B (4, 2), C (2, 2), D (2,4) and E (0,4).

(a) Reflect the points A, B, C and D on the y-axis and name them as F, G,
H and I respectively.

(b) Join the points A, B, C, D, E, I, H, G and F in order. Reflect the figure
ABCDEIHGF on the x-axis and name it as AMNPQRSTF.

(c) Give the geometrical name of the closed figure AEFQ. [Understanding]

T26 511 – SPECIMEN 13 of 13

ICSE 2026 SPECIMEN
DRAFT MARKING SCHEME – MATHEMATICS

Question 1
(i) (c) (x+1) .[15]

(ii) (c) ₹ 68,400

(iii) (c) 80 cm2

(iv) (c) 3√2

(v) (c) Both (A) and (R) is true and (R) is the correct reason for (A).

(vi) (a) (A) is true, (R) is false.

(vii) (a) 20

(viii) (a) 0

(ix) (c) [5]

(x) (d) a = 10, b = 5, c = 15

(xi) (d) ₹ 86.40

(xii) (c) ₹1925

(xiii) (d) +1, -1

(xiv) (b) x – y = 7

(xv) (a) ∅

Question 2
(i) 𝑓𝑓(𝑥𝑥) = 𝟐𝟐𝒙𝒙𝟑𝟑 + 𝟕𝟕𝒙𝒙𝟐𝟐 + 𝟐𝟐𝟐𝟐 − 𝟑𝟑 [4]
1 1 3 1 2 1
𝑓𝑓 �− � = 2 �− � + 7 �− � + 2 �− � − 3 ≠ 0
2 2 2 2
∴ (2𝑥𝑥 + 1) 𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜 𝑓𝑓(𝑥𝑥).
1 1 3 1 2 1
𝑓𝑓 � � = 2 � � + 7 � � + 2 � � − 3 = 0
2 2 2 2
∴ (2𝑥𝑥 − 1) 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜 𝑓𝑓(𝑥𝑥)

T26 511 - SPECIMEN Page 1 of 9

𝑥𝑥 2 + 4𝑥𝑥 + 3
2𝑥𝑥 − 1 𝟐𝟐𝒙𝒙𝟑𝟑 + 𝟕𝟕𝒙𝒙𝟐𝟐 + 𝟐𝟐𝟐𝟐 − 𝟑𝟑
2𝑥𝑥 3 − 𝑥𝑥 2
8𝑥𝑥 2 + 2𝑥𝑥
8𝑥𝑥 2 − 4𝑥𝑥
6𝑥𝑥 − 3
6𝑥𝑥 − 3
××
𝑓𝑓(𝑥𝑥) = (𝟐𝟐𝟐𝟐 − 𝟏𝟏)(𝑥𝑥 2 + 4𝑥𝑥 + 3)
𝑓𝑓(𝑥𝑥) = (𝟐𝟐𝟐𝟐 − 𝟏𝟏)(𝒙𝒙 + 𝟑𝟑)(𝒙𝒙 + 𝟏𝟏)

(ii) (a) 𝑦𝑦 = 0 [4]
−4𝑚𝑚+2𝑛𝑛
= 0, 4𝑚𝑚 = 2𝑛𝑛 → 𝑚𝑚: 𝑛𝑛 = 1: 2
𝑚𝑚+𝑛𝑛
9 × 1 + 2 × (−6)
(b) x= = −1
3
𝑃𝑃(−1, 0)
−4−2 −6 2
(c) 𝑚𝑚𝐴𝐴𝐴𝐴 = 9+6 = 15 = − 5
2
𝑦𝑦 + 2 = − 5 (𝑥𝑥 + 3) → 2𝑥𝑥 + 5𝑦𝑦 = −16

(iii) 1 [4]
(a) ∠𝐵𝐵𝐵𝐵𝐵𝐵 = 180° − 80° = 100° → ∠𝐵𝐵𝐵𝐵𝐵𝐵 = × 100° = 50°
2
(∠ 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡ℎ𝑒𝑒 ∠ 𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)
(b) ∠𝐵𝐵𝐵𝐵𝐵𝐵 = ∠𝐵𝐵𝐵𝐵𝐵𝐵 + ∠𝐴𝐴𝐴𝐴𝐴𝐴 + ∠𝐸𝐸𝐸𝐸𝐸𝐸 = 40° + 20° + 50° = 110°
(c) ∠𝐶𝐶𝐶𝐶𝐶𝐶 = 180° − 110° − 50° = 20°

Question 3
(i) (𝑎𝑎) 𝐴𝐴. 𝑃𝑃. [4]
(𝑏𝑏) 𝑙𝑙 = 53, 𝑎𝑎 + (𝑛𝑛 − 1)𝑑𝑑 = 53
−11 + (𝑛𝑛 − 1)4 = 53 → 𝑛𝑛 = 17
17+1 𝑡𝑡ℎ
(c) Middle term = � 2 � 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 9𝑡𝑡ℎ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

𝑇𝑇9 = 𝑎𝑎 + 8𝑑𝑑 = −11 + 8 × 4 = 21

(ii) 1 7 [4]
ℎ= (1 + 6), 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 → ℎ =
2 2
2
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑜𝑜 𝑤𝑤𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝜋𝜋𝑟𝑟 + 2𝜋𝜋𝜋𝜋ℎ → 𝜋𝜋𝜋𝜋(𝑟𝑟 + 2ℎ)
22 7 7 7
= × � + 2 × � = 115.5 𝑐𝑐𝑐𝑐2
7 2 2 2

T26 511 - SPECIMEN Page 2 of 9

(iii) [5]

(a) 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿ℎ 𝑜𝑜𝑜𝑜 𝐶𝐶𝐶𝐶 = 4.9 𝑐𝑐𝑐𝑐.

SECTION − B
Question 4
(i) (a) 𝑁𝑁𝑁𝑁. 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 =
8000
= 100 [3]
80
6 × 100 × 100
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = = ₹600
100
(b) 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ₹75 × 100 = ₹7500
𝑎𝑎𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ₹8100
8100
𝑁𝑁𝑁𝑁. 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = = 300
27

(ii) 5𝑥𝑥 3 [3]
5𝑥𝑥 − 21 < − 6 ≤ −3 + 𝑥𝑥, 𝑥𝑥 ∈ 𝑅𝑅
7 7
5𝑥𝑥 5𝑥𝑥 3
5𝑥𝑥 − 21 < −6 − 6 ≤ −3 + 𝑥𝑥
7 7 7
5𝑥𝑥 5𝑥𝑥 24
5𝑥𝑥 − < −6 + 21 − 𝑥𝑥 ≤ − + 6
7 7 7
35𝑥𝑥 − 5𝑥𝑥 5𝑥𝑥 − 7𝑥𝑥 −24 + 42
< 15 ≤
7 7 7
30𝑥𝑥 < 105 −2𝑥𝑥 ≤ 18
𝑥𝑥 < 3.5 𝑥𝑥 ≥ −9
7
�𝑥𝑥: − 9 ≤ 𝑥𝑥 < , 𝑥𝑥 ∈ 𝑅𝑅�
2

T26 511 - SPECIMEN Page 3 of 9

(iii) 𝐿𝐿 𝐻𝐻 𝑆𝑆 = (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)(𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) [4]
1 1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
= (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) � − � = (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) � �
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜃𝜃 − 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 1 − 2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃 1 2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜃𝜃
= = = −
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 2𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑅𝑅𝑅𝑅𝑅𝑅
Question 5
(i) (a) 𝐼𝐼𝐼𝐼 ∆𝐴𝐴𝐴𝐴𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝐶𝐶𝐶𝐶𝐶𝐶, ∠𝐵𝐵𝐵𝐵𝐵𝐵 = ∠𝐷𝐷𝐷𝐷𝐷𝐷 (∠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) [3]
∠𝐴𝐴𝐴𝐴𝐴𝐴 = ∠𝐶𝐶𝐶𝐶𝐶𝐶 (∠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)
∴ ∆𝐴𝐴𝐴𝐴𝐴𝐴 ~ ∆𝐶𝐶𝐶𝐶𝐶𝐶 (𝐴𝐴𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎)
𝐴𝐴𝐴𝐴 3
(b) 𝐶𝐶𝐶𝐶
= 2 ∴ 𝐶𝐶𝐶𝐶 = 6𝑐𝑐𝑐𝑐
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (∆𝐴𝐴𝐴𝐴𝐴𝐴) 𝐵𝐵𝐵𝐵 2 9
(c) = 𝐷𝐷𝐷𝐷2 = 4 → 9 ∶ 4
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝐶𝐶𝐶𝐶𝐶𝐶

(ii) 600 × 24 × 25 𝑟𝑟 1 [3]
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = × × = 150 𝑟𝑟
2 100 12
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = ₹15600
600 × 24 + 150𝑟𝑟 = ₹15600
1200
150𝑟𝑟 = ₹15600 − ₹14400 → 𝑟𝑟 = = 8%
150

(iii) 𝐶𝐶lass 𝑥𝑥 𝑢𝑢 = 𝑑𝑑/𝑖𝑖 𝑓𝑓 𝑓𝑓𝑓𝑓 [4]
0 – 15 7.5 -3 3 -9
15 – 30 22.5 -2 4 -8
30 – 45 37.5 -1 7 -7
45 – 60 52.5 0 6 0
60 – 75 67.5 1 8 8
75 – 90 82.5 2 2 4
30 -12
∑ 𝑓𝑓𝑓𝑓 −12
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 𝐴𝐴 + × 𝑖𝑖 = 52.5 + × 15 = 52.5 − 6 = 46.50
∑ 𝑓𝑓 30

Question 6
(i) (a) P �
−1+3+0 3+(−1)+0
,
2 2
� = 𝑃𝑃 �3 , 3� [3]
3 3
−1−(3) −4
(b) 𝑚𝑚𝐴𝐴𝐴𝐴 = = 4 = −1 𝑚𝑚𝐶𝐶𝐶𝐶 = −1
3−(−1)

2 2
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒, 𝑦𝑦 − = −1 �𝑥𝑥 − � → 3𝑥𝑥 + 3𝑦𝑦 = 4
3 3

T26 511 - SPECIMEN Page 4 of 9

(ii) (𝑎𝑎) 𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴, 𝐵𝐵𝐵𝐵 = 𝐵𝐵𝐵𝐵, 𝐷𝐷𝐷𝐷 = 𝐷𝐷𝐷𝐷 𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶𝐶𝐶 = 𝐶𝐶𝐶𝐶 [3]
(𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡 𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒)
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, (𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵) + (𝐷𝐷𝐷𝐷 + 𝐶𝐶𝐶𝐶) = (𝐴𝐴𝐴𝐴 + 𝐷𝐷𝐷𝐷) + (𝐵𝐵𝐵𝐵 + 𝐶𝐶𝐶𝐶)
𝐴𝐴𝐴𝐴 + 𝐷𝐷𝐷𝐷 = 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 → 2 𝐴𝐴𝐴𝐴 = 2 𝐵𝐵𝐵𝐵 ∴ 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵
(b) Rhombus

(iii) 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 [4]

S. No. 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐺𝐺𝐺𝐺𝐺𝐺 𝑇𝑇𝑇𝑇𝑇𝑇
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
1. 𝐷𝐷𝐷𝐷𝐷𝐷 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ₹ 1200 ₹ 1100 12% 12 × 1100
= 132
(1𝑘𝑘𝑘𝑘) 100
2. 𝑊𝑊ℎ𝑒𝑒𝑒𝑒𝑒𝑒 ₹ 286 ₹ 286 5% 5 × 286
= 14.30
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 100
3. 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 ₹ 500 ₹ 450 12% 12 × 450
= 54
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 100
Total ₹1836 ₹ 200.30

Grand total ₹ 2036.30

Question 7
(i) [5]
A
30° 50°

80 𝑚𝑚

30° 50°
C B D
𝐴𝐴𝐴𝐴
𝐼𝐼𝐼𝐼 ∆𝐴𝐴𝐴𝐴𝐴𝐴, = tan 30°
𝐵𝐵𝐵𝐵
80 1
= → 𝐵𝐵𝐵𝐵 = 80√3 = 80 × 1.7321 𝑚𝑚
𝐵𝐵𝐵𝐵 √3
𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵
𝐼𝐼𝐼𝐼 ∆𝐴𝐴𝐴𝐴𝐴𝐴, = tan 50° 𝑜𝑜𝑜𝑜 = 𝑡𝑡𝑡𝑡𝑡𝑡40°
𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴
𝐵𝐵𝐵𝐵
= 0.8391 → 𝐵𝐵𝐵𝐵 = 80 × 0.8391 𝑚𝑚
80
𝐶𝐶𝐶𝐶 = 80 × 1.732 𝑚𝑚 + 80 × 0.839 𝑚𝑚 = 80(1.7321 + 0.8391) 𝑚𝑚
80(2.5712) = 205.696 𝑚𝑚 = 206 𝑚𝑚

T26 511 - SPECIMEN Page 5 of 9

(ii) Marks (%) 𝑓𝑓 𝑐𝑐𝑐𝑐 [5]
0 – 10 5 5 (a) 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 53 ± 1
10 – 20 7 12 (b) 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 65% = 46 ± 2
(c) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷’𝑡𝑡 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 31 ± 2
20 – 30 11 23
30 – 40 20 43
40 – 50 40 83
50 – 60 52 135
60 – 70 36 171
70 – 80 15 186
80 – 90 09 195
90 – 100 05 200

Scale: x-axis, 2cm = 10% marks

y-axis, 2cm= 20 students

Question 8
(i) (a) {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96} [3]
16
𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 6) =
99
16 83
(b) 𝑃𝑃(𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 6) = 1 − 99 = 99

(ii) 𝑥𝑥 𝑦𝑦 [3]
= → 𝑦𝑦 2 = 𝑥𝑥𝑥𝑥
𝑦𝑦 𝑧𝑧

T26 511 - SPECIMEN Page 6 of 9

𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 3 + 𝑦𝑦 3 + 𝑧𝑧 3
𝐿𝐿 𝐻𝐻 𝑆𝑆 = + + =
𝑦𝑦 2 . 𝑧𝑧 2 𝑧𝑧 2 . 𝑥𝑥 2 𝑥𝑥 2 . 𝑦𝑦 2 𝑥𝑥 2 . 𝑦𝑦 2 𝑧𝑧 2
𝑥𝑥 3 + 𝑦𝑦 3 + 𝑧𝑧 3 𝑥𝑥 3 𝑦𝑦 3 𝑧𝑧 3
= 3 3+ 3 3+ 3 3
𝑥𝑥 3 𝑧𝑧 3 𝑥𝑥 𝑧𝑧 𝑥𝑥 𝑧𝑧 𝑥𝑥 𝑧𝑧
3
1 𝑦𝑦 1 1 1 1
= 3 + 6 + 3 = 3 + 3 + 3 = 𝑅𝑅 𝐻𝐻 𝑆𝑆
𝑧𝑧 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑦𝑦 𝑥𝑥

(iii) 2156 2156 [4]
(𝑎𝑎) 𝑁𝑁𝑁𝑁. 𝑜𝑜𝑜𝑜 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = =
4 3 4 22 7 3
3 × 𝜋𝜋 × 𝑟𝑟 3 × 7 × �10�
2156 × 3 × 7 × 10 × 10 × 10
= = 1500
4 × 22 × 7 × 7 × 7
(b) 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑏𝑏𝑏𝑏𝑏𝑏 = 4 𝑔𝑔𝑔𝑔 × 1500 = 6 𝑘𝑘𝑘𝑘

Question 9
(i) (𝑎𝑎) 5 [3]
(b) 400 – 500
(c) Mode = 430 runs
y

x

(ii) 8 5 [3]
𝑎𝑎 = 3, 𝑆𝑆8 = 2 𝑆𝑆5 → [2 × 3 + (8 − 1)𝑑𝑑] = 2 � [2 × 3 + (5 − 1)𝑑𝑑]�
2 2
3
4[6 + 7𝑑𝑑] = 5[6 + 4𝑑𝑑] → 24 + 28𝑑𝑑 = 30 + 20𝑑𝑑 → 𝑑𝑑 =
4

T26 511 - SPECIMEN Page 7 of 9

(iii) 𝑎𝑎 = 𝑞𝑞 − 𝑟𝑟, 𝑏𝑏 = 𝑟𝑟 − 𝑝𝑝 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 𝑝𝑝 − 𝑞𝑞 [4]
𝑓𝑓𝑓𝑓𝑓𝑓 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟, 𝑏𝑏 2 = 4𝑎𝑎𝑎𝑎 → (𝑟𝑟 − 𝑝𝑝)2 = 4 (𝑞𝑞 − 𝑟𝑟)(𝑝𝑝 − 𝑞𝑞)
𝑟𝑟 2 + 𝑝𝑝2 − 2𝑝𝑝𝑝𝑝 = 4[𝑝𝑝𝑝𝑝 − 𝑞𝑞 2 − 𝑝𝑝𝑝𝑝 + 𝑞𝑞𝑞𝑞)
𝑟𝑟 2 + 𝑝𝑝2 − 2𝑝𝑝𝑝𝑝 + 4𝑝𝑝𝑝𝑝 = 4[𝑝𝑝𝑝𝑝 − 𝑞𝑞 2 + 𝑞𝑞𝑞𝑞]
(𝑝𝑝 + 𝑟𝑟)2 = 4[𝑞𝑞(𝑝𝑝 + 𝑟𝑟) − 𝑞𝑞 2 ]
(𝑝𝑝 + 𝑟𝑟)2 − 4𝑞𝑞(𝑝𝑝 + 𝑟𝑟) + 4𝑞𝑞 2 = 0
𝑙𝑙𝑙𝑙𝑙𝑙 (𝑝𝑝 + 𝑟𝑟) = 𝑦𝑦
𝑦𝑦 2 − 4𝑞𝑞𝑞𝑞 + 4𝑞𝑞 2 = 0
(𝑦𝑦 − 2𝑞𝑞)2 = 0
𝑦𝑦 − 2𝑞𝑞 = 0
𝑜𝑜𝑜𝑜 𝑝𝑝 + 𝑟𝑟 = 2𝑞𝑞 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

Question 10
(i) 𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡ℎ𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 (𝑥𝑥 − 2), 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 (𝑥𝑥 + 2) [3]
(𝑥𝑥 − 2)2 + 𝑥𝑥 2 + (𝑥𝑥 + 2)2 = 596 → 3𝑥𝑥 2 = 588 → 𝑥𝑥 2 = 196 ∴ 𝑥𝑥 = 14
The required numbers are 12, 14 & 16

(ii) 1 1 1 1 [3]
𝑋𝑋 2 = � �� �
8 3 8 3
1 × 1 + (1) × (8) 1 × (1) + (1) × 3
=� �
(8) × 1 + 3 × (8) (8) × (1) + 3 × 3

1+8 1+3
=� �
8 + 24 8 + 9
9 4
∴ 𝑋𝑋 2 = � �
32 17
1 1 4 4
𝑎𝑎𝑎𝑎𝑎𝑎 4𝑋𝑋 = 4 � �=� �
8 3 32 12
4 4 5 0 9 4
4𝑋𝑋 + 5𝐼𝐼 = � �+� �=� �
32 12 0 5 32 17
∴ 𝑋𝑋 2 = 4𝑋𝑋 + 5𝐼𝐼, 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

T26 511 - SPECIMEN Page 8 of 9

(iii) (a) Square [4]

[

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