Sant Tukaram National Model School, Latur
PRE-BOARD EXAM-I
SUBJECT: MATHEMATICS STANDARD (041)
SUBJECT: MATHEMATICS STANDARD (041)
SECTION - A
This section comprises of 18 multiple choice questions and two questions of assertion and reasoning type of 1 mark each.
Q.1[1 Mark]
The greatest number which divides both 134 and 188, leaving remainders 4 and 6 respectively, is:
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Q.2[1 Mark]
If $f(x) = px^2 + qx + r$, $p \neq 0$ and $p + r = q$, then one of the zeroes of $f(x)$ is:
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Q.3[1 Mark]
Tarun correctly solved a pair of linear equations in two variables and found their only point of intersection as $(5, -1)$. One of the lines was $x - y = 6$. Which of the following could have been the other line?
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Q.4[1 Mark]
If $(1 - p)$ is a root of the quadratic equation $x^2 + px + 1 - p = 0$, then its roots are:
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Q.6[1 Mark]
The perimeters of two similar triangles are $56\text{ cm}$ and $70\text{ cm}$ respectively. If one side of the first triangle is $14\text{ cm}$, then the corresponding side of the second triangle (in cm) is:
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Q.7[1 Mark]
The point which lies on the perpendicular bisector of the line segment joining the points $A(-3, -4)$ and $B(3, 4)$ is:
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Q.8[1 Mark]
If $\tan (A + B) = \sqrt{3}$ and $\tan (A - B) = \frac{1}{\sqrt{3}}$, $0^\circ < A + B \le 90^\circ$, $A > B$, then the value of A and B respectively are:
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Q.9[1 Mark]
T-shirts marked with numbers 4 to 99 are placed in a box. The probability that she gets a T-shirt marked with a number that is either a perfect square or a perfect cube is:
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Q.10[1 Mark]
The diameter of a car wheel is $21\text{ cm}$. The number of complete revolutions it will make in moving $66\text{ km}$ is:
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Q.11[1 Mark]
Two cubes each of volume $64\text{ cm}^3$ are joined end to end to form a solid. The surface area of the resultant cuboid is:
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Q.12[1 Mark]
The mean age of a combined group of men and women is 35 years. If the mean ages of the men and women are 38 years and 30 years respectively, then the percentage of women in the group is:
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Q.13[1 Mark]
Two dice are rolled simultaneously. The probability of getting number less than 4 on each die is:
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Q.14[1 Mark]
$PM$ is a median of $\Delta PQR$ with vertices $P(5, -6)$, $Q(6, 4)$ and $R(0, 0)$. The length of $PM$ is:
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Q.15[1 Mark]
The arc of a circle is of length $6\pi\text{ cm}$ and the sector it bounds has an area of $24\pi\text{ cm}^2$. The radius of the circle is:
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Q.16[1 Mark]
The mean and median of the data are 35.5 and 32 respectively. The value of mode for this data is:
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Q.17[1 Mark]
A flying kite is tied to a point on the ground. The string makes an angle $\theta$ such that $\tan \theta = 12/5$. If length is $52\text{ m}$, the height of the kite is:
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Q.18[1 Mark]
2 cards of diamonds and 4 cards of spades are missing from a pack of 52 cards. The probability of getting a card of heart is:
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Q.19 (A/R)[1 Mark]
Assertion: Parallelogram vertices in order... fourth is (4,1). Reason: Diagonals bisect each other.
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Q.20 (A/R)[1 Mark]
Assertion: Zeroes of $(2k - 1)x^2 + 4x - 3$ are reciprocal, then $k = -1$. Reason: If $a=c$, zeroes are reciprocal.
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SECTION - B
This section comprises of 5 Very Short Answer (VSA) type questions of 2 marks each.
Q.21[2 Marks]
(A) Find the length of the longest unmarked ruler that can exactly measure the dimensions of the hall (9.75m, 6.75m, 5.25m).
OR
(B) Three bells toll at intervals of 5, 6 and 8 minutes. How many more times do they toll together till the park is closed at 7:00 p.m.?
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OR
(B) Three bells toll at intervals of 5, 6 and 8 minutes. How many more times do they toll together till the park is closed at 7:00 p.m.?
Q.22[2 Marks]
(A) $ABCD$ is a trapezium in which $AB$ is parallel to $DC$ and its diagonals intersect each other at the point $O$. Show that $\frac{AO}{BO} = \frac{CO}{DO}$.
OR
(B) In $\Delta PQR$, if $\angle PQR = \angle PSQ$, $PQ = 6\text{ cm}$ and $PS = 3\text{ cm}$, then find the length of $PR$.
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OR
(B) In $\Delta PQR$, if $\angle PQR = \angle PSQ$, $PQ = 6\text{ cm}$ and $PS = 3\text{ cm}$, then find the length of $PR$.
Q.23[2 Marks]
If $\tan\theta = 1/\sqrt{5}$, then find the value of the given trigonometric expression.
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Q.24[2 Marks]
Find the area of a sector of a circle with diameter 28 cm, if the length of the corresponding arc is 22 cm.
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Q.25[2 Marks]
Two dice are thrown simultaneously. Find the probability that the product of the numbers appearing on them is a prime number.
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SECTION - C
This section comprises of 6 Short Answer (SA) type questions of 3 marks each.
Q.26[3 Marks]
Given that $\sqrt{3}$ is an irrational number, prove that $(2 + 3\sqrt{3})$ is an irrational number.
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Q.27[3 Marks]
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $2x^2 - 8x + 5$, then find the value of the algebraic expression.
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Q.28[3 Marks]
(A) Ridhi drew a polygon. Smallest exterior angle is $8^\circ$, subsequent is $4^\circ$ more. Find number of sides.
OR
(B) Find the sum of integers between 1 and 400 that are multiples of 4 as well as of 5.
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OR
(B) Find the sum of integers between 1 and 400 that are multiples of 4 as well as of 5.
Q.29[3 Marks]
Prove that the parallelogram circumscribing a circle is a rhombus.
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Q.30[3 Marks]
In what ratio does the x-axis divide the line segment joining the points $(-4, -6)$ and $(-1, 7)$? Also, find the coordinates.
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Q.31[3 Marks]
(A) Identify the step in which Jyoti has made error(s), if any. Rectify the same and find the correct answer.
OR
(B) Prove the given trigonometric identity involving $\sin x$ and $\cos x$.
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OR
(B) Prove the given trigonometric identity involving $\sin x$ and $\cos x$.
SECTION - D
This section comprises of 4 Long Answer (LA) type questions of 5 marks each.
Q.32[5 Marks]
In a rectangular park measuring $50\text{ m} \times 40\text{ m}$, the gram panchayat plans to construct a swimming pool surrounded by a uniform grassy strip. Find dimensions.
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Q.33[5 Marks]
Prove the Basic Proportionality Theorem, and use it to solve the given Trapezium geometry property.
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Q.34[5 Marks]
(A) A tent is in the shape of a cylinder surmounted by a conical top. Find canvas area and cost.
OR
(B) A solid wooden toy (hemisphere + cone). Find height and painting cost.
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OR
(B) A solid wooden toy (hemisphere + cone). Find height and painting cost.
Q.35[5 Marks]
(A) Heights & Distances: Manish and Mahesh observing a kite from different levels.
OR
(B) Heights & Distances: Angles of depression of two ships from a lighthouse.
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OR
(B) Heights & Distances: Angles of depression of two ships from a lighthouse.
SECTION - E
This section comprises of 3 case-study-based questions of 4 marks each with sub parts (i), (ii), (iii) of marks 1, 1, 2 respectively.
Q.36[4 Marks]
Case Study 1: A student entrepreneur started a company that manufactures sanitizers in two sizes (Linear Equations in Two Variables).
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Q.37[4 Marks]
Case Study 2: Circular meditation garden and designer Arjun measuring tangents (Circles & Trigonometry).
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Q.38[4 Marks]
Case Study 3: Academic coordinator analyzing weekly test scores of Class X (Statistics - Mean, Median, Mode).
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⚠️ IMPORTANT DISCLAIMER:
We do not have any tie-up, affiliation, or official relationship with Sant Tukaram National Model School, Latur, its administration, or any associated educational boards.
This digital solution key and marking scheme is generated independently for educational and learning purposes only. This material has been published only after the examination was officially conducted today (22/10/2026) and the question paper became publicly available.
If school authorities or management has any issue regarding the same, then kindly contact Maheshkumar Sir, the author of these solutions and platform. Its solutions and contents will be modified or removed as per the request.
Students, parents, and teachers are welcome for their suggestions at any time.
We do not have any tie-up, affiliation, or official relationship with Sant Tukaram National Model School, Latur, its administration, or any associated educational boards.
This digital solution key and marking scheme is generated independently for educational and learning purposes only. This material has been published only after the examination was officially conducted today (22/10/2026) and the question paper became publicly available.
If school authorities or management has any issue regarding the same, then kindly contact Maheshkumar Sir, the author of these solutions and platform. Its solutions and contents will be modified or removed as per the request.
Students, parents, and teachers are welcome for their suggestions at any time.
