Secondary Education Examination · Class 10

SEE 2082 Optional Mathematics question paper, Grade Increment

SEE 2082 Optional Mathematics (Grade Increment, GI-2021) has 33 questions worth 75 marks, with 3 hours to answer them. It has 10 of 1 mark, 8 of 2 marks, 11 of 3 marks and 4 of 4 marks.

The SEE examination is conducted by the National Examinations Board (NEB), Nepal. This paper was typed up from the original and checked by the GyanMul team.

Every question is given in Nepali and English, as the paper prints it.

33 questions
75 marks
3 hours
Code GI-2021

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Questions

Answers are shown for the 10 one-mark questions. They are GyanMul's model answers: the NEB does not publish an answer key, so these are not the board's answers.

Question 11 mark
Which function is $f(x) = 5$ ? Write it.
$f(x) = 5$ कुन फलन हो ? लेख्नुहोस् ।
Answer: Constant function. Regardless of the value taken by x, f(x) remains 5 — the output value does not change — therefore this is a constant function.
Question 21 mark
What is the geometric mean between p and q? Write it.
p र q बिचको गुणोत्तर मध्यमा कति हुन्छ ? लेख्नुहोस् ।
Answer: Geometric mean $(G.M.) = \sqrt{pq}$ Because p, G.M., q must be in geometric progression, therefore G.M./p = q/G.M. i.e. $(G.M.)^{2} = pq$ .
Question 31 mark
Write the condition for continuity of a function f(x) at x = a.
फलन f(x) को x = a मा निरन्तरता हुने अवस्था लेख्नुहोस् ।
Answer: $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = f(a)$
Question 41 mark
What is the determinant of matrix [ p q ; r s ]? Write it.
मेट्रिक्स [ p q ; r s ] को डिटरमिन्यान्ट कति हुन्छ ? लेख्नुहोस् ।
Answer: $ps - qr$
Question 51 mark
Write the formula to find angle between a pair of straight lines represented by $ax^{2} + 2$ hxy + $by^{2} = 0$ .
$ax^{2} + 2$ hxy + $by^{2} = 0$ ले प्रतिनिधित्व गर्ने एक जोडा सरल रेखाहरूबिचको कोण पत्ता लगाउने सूत्र लेख्नुहोस् ।
Answer: $\tan \theta = \pm (2\sqrt{h^{2} - ab}) / (a + b)$
Question 61 mark
In which condition a plane surface intersects a cone to form a hyperbola? Write it.
समतलीय सतहले एउटा सोलीलाई कुन अवस्थामा प्रतिच्छेदन गर्दा अति परावलय बन्छ, लेख्नुहोस् ।
Answer: When the plane is parallel to the axis of the cone, so that it cuts both nappes, the section formed is a hyperbola.
Question 71 mark
Express Cos2A in terms of CosA and Sin A.
Cos2A लाई CosA र SinA को रूपमा व्यक्त गर्नुहोस् ।
Answer: $Cos2A = Cos^{2}A - Sin^{2}A$
Question 81 mark
Convert CosM $-$ CosN in product form.
CosM $-$ CosN लाई गुणनफलको रूपमा परिवर्तन गर्नुहोस् ।
Answer: CosM $-$ CosN $= -2 Sin((M + N)/2) \cdot Sin((M - N)/2)$
Question 91 mark
If a⃗ $\cdot \vec{b}$ = |a⃗| |b⃗|, write the relation between a⃗ and b⃗.
यदि a⃗ $\cdot \vec{b}$ = |a⃗| |b⃗| भए a⃗ र b⃗ बिचको सम्बन्ध लेख्नुहोस् ।
Answer: a⃗ and b⃗ are parallel and in the same direction (like parallel vectors). Because a⃗ $\cdot \vec{b}$ = |a⃗||b⃗ $| Cos\theta$ ; the given condition gives $Cos\theta = 1$ , hence $\theta = 0^\circ$ .
Question 101 mark
In the given figure, p' is inverse of the point p. If $op \times op' = 25$ , what is the radius of the inversion circle? Write it.
दिइएको चित्रमा p को विपरित विन्दु p' छ । यदि $op \times op' = 25$ छ भने उत्क्रम वृत्तको अर्धव्यास कति हुन्छ ? लेख्नुहोस् ।
Answer: Radius $r = 5$ units. By the definition of inversion $op \times op' = r^{2}$ . Therefore $r^{2} = 25$ and $r = 5$ .
Question 112 marks
If a polynomial $x^{3} - 3x^{2} + kx + k$ is divided by x + 2, the remainder is k. Find the value of k using remainder theorem.
यदि बहुपदीय $x^{3} - 3x^{2} + kx + k$ लाई x + 2 ले भाग गर्दा शेष k रहन्छ भने शेष साध्यको प्रयोग गरी k को मान पत्ता लगाउनुहोस् ।
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Question 122 marks
Find the equation of line of symmetry of the parabola formed by the equation $y = x^{2} - 4x + 3$ .
समीकरण $y = x^{2} - 4x + 3$ बाट बन्ने पाराबोलाको सममितीय रेखाको समीकरण पत्ता लगाउनुहोस् ।
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Question 132 marks
If [ 2x 7 ; 5 9 ] and [ 9 y ; $-5 4 ]$ are inverse matrices to each other, find the values of x and y.
यदि [ 2x 7 ; 5 9 ] र [ 9 y ; $-5 4 ]$ आपसमा एक अर्काको विपरीत मेट्रिक्सहरू भए x र y का मानहरू पत्ता लगाउनुहोस् ।
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Question 142 marks
If the lines represented by the equation $9x^{2} + 12xy + (k+1)y^{2} = 0$ are coincident, find the value of k.
यदि समीकरण $9x^{2} + 12xy + (k + 1)y^{2} = 0$ ले जनाउने रेखाहरू सम्पाती छन् भने k को मान निकाल्नुहोस् ।
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Question 152 marks
If $Cos(A/3) =$ ½(p + 1/p), find the value of CosA in terms of P.
यदि $Cos(A/3) =$ ½(p + 1/p) भए CosA को मान P को रूपमा पत्ता लगाउनुहोस् ।
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Question 162 marks
Solve: $(0^\circ \le x \le 90^\circ )$ $Cot^{2}x + Cosec^{2}x = 3$
हल गर्नुहोस् : $(0^\circ \le x \le 90^\circ )$ $Cot^{2}x + Cosec^{2}x = 3$
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Question 172 marks
If a⃗ = (4, 3), b⃗ $= (-m, -8)$ and a⃗ $\perp \vec{b}$ , find the value of m. (Both vectors are written in column form.)
यदि a⃗ = (4, 3), b⃗ $= (-m, -8)$ र a⃗ $\perp \vec{b}$ भए m को मान पत्ता लगाउनुहोस् । (दुवै भेक्टर स्तम्भ रूपमा लेखिएका छन् ।)
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Question 182 marks
In a continuous series, if $\sum f|m - M_d| = 220, N = 20$ and median $(M_d) = 44$ , find the coefficient of mean deviation.
यदि एउटा अविच्छिन्न श्रेणीमा $\sum f|m - M_d| = 220, N = 20$ र मध्यिका $(M_d) = 44$ भए मध्यक भिन्नताको गुणाङ्क पत्ता लगाउनुहोस् ।
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Question 193 marks
If $f(x) = 4x - 3$ and g(x) = (x + 2)/5 be two functions, find the value of $f\circ g^{-1}(2)$ .
यदि $f(x) = 4x - 3$ र g(x) = (x + 2)/5 दुईओटा फलनहरू भए $f\circ g^{-1}(2)$ को मान पत्ता लगाउनुहोस् ।
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Question 203 marks
Find the maximum value of objective function $P = 2x + 5y$ under the given constraints. $x + y \le 4, x + 2y \le 6, x \ge 0$ and $y \ge 0$ .
उद्देश्य फलन $P = 2x + 5y$ को दिइएको सर्तहरूको आधारमा अधिकतम् मान पत्ता लगाउनुहोस् । $x + y \le 4, x + 2y \le 6, x \ge 0$ and $y \ge 0$ .
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Question 213 marks
For a real valued function $f(x) = 3x + 5$ , find the values of f(1.99), f(2.01) and f(2). Is the function continuous at $x = 2$ ? Give logic.
वास्तविक फलन $f(x) = 3x + 5$ का लागि f(1.99), f(2.01) र f(2) का मानहरू पत्ता लगाउनुहोस् । के फलन $x = 2$ मा अविच्छिन्न छ ? तर्क दिनुहोस् ।
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Question 223 marks
Solve the given equations by matrix method. $3x - 2y = 4, 2x + 3y = 7$
दिइएका समीकरणहरू मेट्रिक्स विधिद्वारा हल गर्नुहोस् । $3x - 2y = 4, 2x + 3y = 7$
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Question 233 marks
ABCD is a square in which the equation of a diagonal AC is $2x + 3y - 5 = 0$ . If the co-ordinates of vertex B is (5, 3), find the equation of the diagonal BD.
ABCD एउटा वर्ग हो जसमा विकर्ण AC को समीकरण $2x + 3y - 5 = 0$ छ । यदि शीर्षविन्दु B का निर्देशाङ्कहरू (5, 3) छन् भने विकर्ण BD को समीकरण पत्ता लगाउनुहोस् ।
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Question 243 marks
Prove that: $(Sin2\theta / (1 + Cos2\theta )) \cdot (Cot(\theta /2) / (1 + Cos\theta )) = Sec\theta$
प्रमाणित गर्नुहोस् : $(Sin2\theta / (1 + Cos2\theta )) \cdot (Cot(\theta /2) / (1 + Cos\theta )) = Sec\theta$
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Question 253 marks
If $A + B + C = \pi^{c}$ , then prove that: sinA $-$ sinB + sinC $= 4 \sin (A/2) \cdot \cos (B/2) \cdot \sin (C/2)$
यदि $A + B + C = \pi^{c}$ भए प्रमाणित गर्नुहोस् : sinA $-$ sinB + sinC $= 4 \sin (A/2) \cdot \cos (B/2) \cdot \sin (C/2)$
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Question 263 marks
The angles of elevation of the top of a building from two points situated on the same plane and at the same side of the building are $30^\circ$ and $45^\circ$ respectively. If the distance between the two places is 10 m, find the height of the building.
एउटा भवनको छतलाई यसको समतलमा रही एकै तिरका दुई स्थानहरूबाट हेर्दा उन्नतांश कोणहरू क्रमशः $30^\circ$ र $45^\circ$ पाइयो । यदि ती स्थानबिचको दुरी 10 मिटर भए सो भवनको उचाइ पत्ता लगाउनुहोस् ।
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Question 273 marks
$A 2\times 2$ transformation matrix transforms a triangle ABC with vertices A(1, 0), B(2, 4) and $C(5, -1)$ to the $\Delta A'B'C'$ with vertices A'(2, 0), B'(8, 12) and $C'(9, -3)$ . Find the transformation matrix.
एउटा $2 \times 2$ स्थानान्तरण मेट्रिक्सले शीर्षविन्दुहरू A(1, 0), B(2, 4) र $C(5, -1)$ भएका त्रिभुज ABC लाई स्थानान्तरण गर्दा बन्ने प्रतिबिम्ब $\Delta A'B'C'$ का शीर्षविन्दुहरू A'(2, 0), B'(8, 12) र $C'(9, -3)$ छन् भने उक्त स्थानान्तरण मेट्रिक्स पत्ता लगाउनुहोस् ।
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Question 283 marks
Find the quartile deviation from the given data, where the first quartile $(Q_{1}) = 50$ . Marks obtained: $30-40 | 40-50 | 50-60 | 60-70 | 70-80$ No. of Students: 2 | 3 | 6 | 5 | 4
दिइएको तथ्याङ्कबाट चतुर्थांशीय भिन्नता पत्ता लगाउनुहोस्, जहाँ पहिलो चतुर्थांश $(Q_{1}) = 50$ छ : प्राप्ताङ्क (Marks obtained): $30-40 | 40-50 | 50-60 | 60-70 | 70-80$ विद्यार्थीको सङ्ख्या (No. of Students): 2 | 3 | 6 | 5 | 4
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Question 293 marks
Find the standard deviation from the given data. Class Interval: $0-20 | 20-40 | 40-60 | 60-80 | 80-100$ Frequency: 2 | 5 | 6 | 4 | 3
दिइएको तथ्याङ्कबाट स्तरीय भिन्नता पत्ता लगाउनुहोस् : वर्गान्तर (Class Interval): $0-20 | 20-40 | 40-60 | 60-80 | 80-100$ बारम्बारता (Frequency): 2 | 5 | 6 | 4 | 3
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Question 304 marks
The sum of three terms in an arithmetic series is 15. If 1, 4 and 19 are added to them respectively, it forms a geometric series. Find the initial three terms.
एउटा समानान्तरीय श्रेणीमा रहेका तीन पदहरूको योगफल 15 छ । यदि ती पदहरूमा 1, 4 र 19 क्रमशः जोड्दा ती पदहरू ज्यामितीय श्रेणीका बन्नेछन् भने सुरुका तीन पदहरू पत्ता लगाउनुहोस् ।
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Question 314 marks
Find the equation of a circle whose centre lies on the straight line $4x + 5y = 16$ and passes through the points (4, 1) and (6, 5).
विन्दुहरू (4, 1) र (6, 5) भएर जाने र केन्द्र विन्दु सिधा रेखा $4x + 5y = 16$ मा पर्ने वृत्तको समीकरण पत्ता लगाउनुहोस् ।
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Question 324 marks
In a right angled triangle PQR, $\angle$ PQR $= 90^\circ$ . If M is the midpoint of the side PR, prove that PM = MR = QM by using vector method.
एउटा समकोणी त्रिभुज PQR मा $\angle$ PQR $= 90^\circ$ छ । यदि भुजा PR को मध्यविन्दु M भए PM = MR = QM हुन्छ भनी भेक्टर विधि प्रयोग गरी प्रमाणित गर्नुहोस् ।
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Question 334 marks
ABC is a triangle with vertices A(5, 6), B(3, 4) and C(6, 1). If $R_{1}$ represents the reflection on the line y = x and $R_{2}$ represents the rotation about the origin through $+ 90^\circ$ , find the single transformation represented by the combined transformation $R_{1}oR_{2}$ . Transform the $\Delta$ ABC using the single transformation and find the coordinates of the vertices of the image so formed. Also, present $\Delta$ ABC and its image in the same graph paper.
शीर्षविन्दुहरू A(5, 6), B(3, 4) र C(6, 1) भएका त्रिभुज ABC छ । यदि $R_{1}$ ले y = x रेखामा हुने परावर्तन र $R_{2}$ ले उद्गम विन्दुको वरिपरि $+90^\circ$ मा हुने परिक्रमण जनाउँछ भने संयुक्त स्थानान्तरण $R_{1}oR_{2}$ जनाउने एकल स्थानान्तरण पत्ता लगाउनुहोस् । उक्त एकल स्थानान्तरण प्रयोग गरी $\Delta$ ABC लाई स्थानान्तरण गर्नुहोस् र बन्ने प्रतिबिम्बको शीर्षविन्दुहरूका निर्देशाङ्कहरू पत्ता लगाउनुहोस् । साथै $\Delta$ ABC र यसको प्रतिबिम्बलाई एउटै लेखाचित्रमा प्रस्तुत गर्नुहोस् ।
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Questions learners ask

How many questions and marks are in the SEE 2082 Optional Mathematics (Grade Increment, GI-2021) paper?: 33 questions worth 75 marks. There are 10 of 1 mark, 8 of 2 marks, 11 of 3 marks and 4 of 4 marks.
How much time is given for this paper?: 3 hours (180 minutes).
Is this the real board paper?: Yes. It is the SEE 2082 paper printed for Grade Increment, code GI-2021, typed up from the original and checked by the GyanMul team.
Are the answers included?: Answers to the 10 one-mark questions are on this page (GyanMul's model answers; the NEB does not publish an answer key). Worked answers and the marking for every other question are in GyanMul Prep, free with an account.
Can I download this paper as a PDF?: Yes. Press "Print / Save as PDF" at the top of the page and choose "Save as PDF" in your browser's print window.

Source: prep.gyanmul.com/see/past-papers/optional-mathematics/2082/grade-increment · GyanMul Prep

SEE 2082 Optional Mathematics question paper, Grade Increment

SEE 2082 Optional Mathematics (Grade Increment, GI-2021) has 33 questions worth 75 marks, with 3 hours to answer them. It has 10 of 1 mark, 8 of 2 marks, 11 of 3 marks and 4 of 4 marks.

The SEE examination is conducted by the National Examinations Board (NEB), Nepal. This paper was typed up from the original and checked by the GyanMul team.

Every question is given in Nepali and English, as the paper prints it.

33 questions

75 marks

3 hours

Code GI-2021

The account is free. Timer, marking, worked answers and help from Gurubaa, in one place.

Questions

Answers are shown for the 10 one-mark questions. They are GyanMul's model answers: the NEB does not publish an answer key, so these are not the board's answers.

Question 11 mark

Which function is $f(x) = 5$ ? Write it.

$f(x) = 5$ कुन फलन हो ? लेख्नुहोस् ।

Answer: Constant function. Regardless of the value taken by x, f(x) remains 5 — the output value does not change — therefore this is a constant function.

Question 21 mark

What is the geometric mean between p and q? Write it.

p र q बिचको गुणोत्तर मध्यमा कति हुन्छ ? लेख्नुहोस् ।

Answer: Geometric mean

(G.M.) = \sqrt{pq}

Because p, G.M., q must be in geometric progression, therefore G.M./p = q/G.M. i.e.

(G.M.)^{2} = pq

Question 31 mark

Write the condition for continuity of a function f(x) at x = a.

फलन f(x) को x = a मा निरन्तरता हुने अवस्था लेख्नुहोस् ।

Answer:

\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = f(a)

Question 41 mark

What is the determinant of matrix [ p q ; r s ]? Write it.

मेट्रिक्स [ p q ; r s ] को डिटरमिन्यान्ट कति हुन्छ ? लेख्नुहोस् ।

Answer:

ps - qr

Question 51 mark

Write the formula to find angle between a pair of straight lines represented by $ax^{2} + 2$ hxy + $by^{2} = 0$ .

$ax^{2} + 2$ hxy + $by^{2} = 0$ ले प्रतिनिधित्व गर्ने एक जोडा सरल रेखाहरूबिचको कोण पत्ता लगाउने सूत्र लेख्नुहोस् ।

Answer:

\tan \theta = \pm (2\sqrt{h^{2} - ab}) / (a + b)

Question 61 mark

In which condition a plane surface intersects a cone to form a hyperbola? Write it.

समतलीय सतहले एउटा सोलीलाई कुन अवस्थामा प्रतिच्छेदन गर्दा अति परावलय बन्छ, लेख्नुहोस् ।

Answer: When the plane is parallel to the axis of the cone, so that it cuts both nappes, the section formed is a hyperbola.

Question 71 mark

Express Cos2A in terms of CosA and Sin A.

Cos2A लाई CosA र SinA को रूपमा व्यक्त गर्नुहोस् ।

Answer:

Cos2A = Cos^{2}A - Sin^{2}A

Question 81 mark

Convert CosM $-$ CosN in product form.

CosM $-$ CosN लाई गुणनफलको रूपमा परिवर्तन गर्नुहोस् ।

Answer: CosM

-

CosN

= -2 Sin((M + N)/2) \cdot Sin((M - N)/2)

Question 91 mark

If a⃗ $\cdot \vec{b}$ = |a⃗| |b⃗|, write the relation between a⃗ and b⃗.

यदि a⃗ $\cdot \vec{b}$ = |a⃗| |b⃗| भए a⃗ र b⃗ बिचको सम्बन्ध लेख्नुहोस् ।

Answer: a⃗ and b⃗ are parallel and in the same direction (like parallel vectors). Because a⃗

\cdot \vec{b}

= |a⃗||b⃗

| Cos\theta

; the given condition gives

Cos\theta = 1

, hence

\theta = 0^\circ

Question 101 mark

In the given figure, p' is inverse of the point p. If $op \times op' = 25$ , what is the radius of the inversion circle? Write it.

दिइएको चित्रमा p को विपरित विन्दु p' छ । यदि $op \times op' = 25$ छ भने उत्क्रम वृत्तको अर्धव्यास कति हुन्छ ? लेख्नुहोस् ।

Answer: Radius

r = 5

units. By the definition of inversion

op \times op' = r^{2}

. Therefore

r^{2} = 25

and

r = 5

Question 112 marks

If a polynomial $x^{3} - 3x^{2} + kx + k$ is divided by x + 2, the remainder is k. Find the value of k using remainder theorem.

यदि बहुपदीय $x^{3} - 3x^{2} + kx + k$ लाई x + 2 ले भाग गर्दा शेष k रहन्छ भने शेष साध्यको प्रयोग गरी k को मान पत्ता लगाउनुहोस् ।

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Question 122 marks

Find the equation of line of symmetry of the parabola formed by the equation $y = x^{2} - 4x + 3$ .

समीकरण $y = x^{2} - 4x + 3$ बाट बन्ने पाराबोलाको सममितीय रेखाको समीकरण पत्ता लगाउनुहोस् ।

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Question 132 marks

If [ 2x 7 ; 5 9 ] and [ 9 y ; $-5 4 ]$ are inverse matrices to each other, find the values of x and y.

यदि [ 2x 7 ; 5 9 ] र [ 9 y ; $-5 4 ]$ आपसमा एक अर्काको विपरीत मेट्रिक्सहरू भए x र y का मानहरू पत्ता लगाउनुहोस् ।

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Question 142 marks

If the lines represented by the equation $9x^{2} + 12xy + (k+1)y^{2} = 0$ are coincident, find the value of k.

यदि समीकरण $9x^{2} + 12xy + (k + 1)y^{2} = 0$ ले जनाउने रेखाहरू सम्पाती छन् भने k को मान निकाल्नुहोस् ।

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Question 152 marks

If $Cos(A/3) =$ ½(p + 1/p), find the value of CosA in terms of P.

यदि $Cos(A/3) =$ ½(p + 1/p) भए CosA को मान P को रूपमा पत्ता लगाउनुहोस् ।

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Question 162 marks

Solve: $(0^\circ \le x \le 90^\circ )$ $Cot^{2}x + Cosec^{2}x = 3$

हल गर्नुहोस् : $(0^\circ \le x \le 90^\circ )$ $Cot^{2}x + Cosec^{2}x = 3$

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Question 172 marks

If a⃗ = (4, 3), b⃗ $= (-m, -8)$ and a⃗ $\perp \vec{b}$ , find the value of m. (Both vectors are written in column form.)

यदि a⃗ = (4, 3), b⃗ $= (-m, -8)$ र a⃗ $\perp \vec{b}$ भए m को मान पत्ता लगाउनुहोस् । (दुवै भेक्टर स्तम्भ रूपमा लेखिएका छन् ।)

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Question 182 marks

In a continuous series, if $\sum f|m - M_d| = 220, N = 20$ and median $(M_d) = 44$ , find the coefficient of mean deviation.

यदि एउटा अविच्छिन्न श्रेणीमा $\sum f|m - M_d| = 220, N = 20$ र मध्यिका $(M_d) = 44$ भए मध्यक भिन्नताको गुणाङ्क पत्ता लगाउनुहोस् ।

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Question 193 marks

If $f(x) = 4x - 3$ and g(x) = (x + 2)/5 be two functions, find the value of $f\circ g^{-1}(2)$ .

यदि $f(x) = 4x - 3$ र g(x) = (x + 2)/5 दुईओटा फलनहरू भए $f\circ g^{-1}(2)$ को मान पत्ता लगाउनुहोस् ।

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Question 203 marks

Find the maximum value of objective function $P = 2x + 5y$ under the given constraints. $x + y \le 4, x + 2y \le 6, x \ge 0$ and $y \ge 0$ .

उद्देश्य फलन $P = 2x + 5y$ को दिइएको सर्तहरूको आधारमा अधिकतम् मान पत्ता लगाउनुहोस् । $x + y \le 4, x + 2y \le 6, x \ge 0$ and $y \ge 0$ .

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Question 213 marks

For a real valued function $f(x) = 3x + 5$ , find the values of f(1.99), f(2.01) and f(2). Is the function continuous at $x = 2$ ? Give logic.

वास्तविक फलन $f(x) = 3x + 5$ का लागि f(1.99), f(2.01) र f(2) का मानहरू पत्ता लगाउनुहोस् । के फलन $x = 2$ मा अविच्छिन्न छ ? तर्क दिनुहोस् ।

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Question 223 marks

Solve the given equations by matrix method. $3x - 2y = 4, 2x + 3y = 7$

दिइएका समीकरणहरू मेट्रिक्स विधिद्वारा हल गर्नुहोस् । $3x - 2y = 4, 2x + 3y = 7$

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Question 233 marks

ABCD is a square in which the equation of a diagonal AC is $2x + 3y - 5 = 0$ . If the co-ordinates of vertex B is (5, 3), find the equation of the diagonal BD.

ABCD एउटा वर्ग हो जसमा विकर्ण AC को समीकरण $2x + 3y - 5 = 0$ छ । यदि शीर्षविन्दु B का निर्देशाङ्कहरू (5, 3) छन् भने विकर्ण BD को समीकरण पत्ता लगाउनुहोस् ।

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Question 243 marks

Prove that: $(Sin2\theta / (1 + Cos2\theta )) \cdot (Cot(\theta /2) / (1 + Cos\theta )) = Sec\theta$

प्रमाणित गर्नुहोस् : $(Sin2\theta / (1 + Cos2\theta )) \cdot (Cot(\theta /2) / (1 + Cos\theta )) = Sec\theta$

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Question 253 marks

If $A + B + C = \pi^{c}$ , then prove that: sinA $-$ sinB + sinC $= 4 \sin (A/2) \cdot \cos (B/2) \cdot \sin (C/2)$

यदि $A + B + C = \pi^{c}$ भए प्रमाणित गर्नुहोस् : sinA $-$ sinB + sinC $= 4 \sin (A/2) \cdot \cos (B/2) \cdot \sin (C/2)$

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Question 263 marks

The angles of elevation of the top of a building from two points situated on the same plane and at the same side of the building are $30^\circ$ and $45^\circ$ respectively. If the distance between the two places is 10 m, find the height of the building.

एउटा भवनको छतलाई यसको समतलमा रही एकै तिरका दुई स्थानहरूबाट हेर्दा उन्नतांश कोणहरू क्रमशः $30^\circ$ र $45^\circ$ पाइयो । यदि ती स्थानबिचको दुरी 10 मिटर भए सो भवनको उचाइ पत्ता लगाउनुहोस् ।

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Question 273 marks

$A 2\times 2$ transformation matrix transforms a triangle ABC with vertices A(1, 0), B(2, 4) and $C(5, -1)$ to the $\Delta A'B'C'$ with vertices A'(2, 0), B'(8, 12) and $C'(9, -3)$ . Find the transformation matrix.

एउटा $2 \times 2$ स्थानान्तरण मेट्रिक्सले शीर्षविन्दुहरू A(1, 0), B(2, 4) र $C(5, -1)$ भएका त्रिभुज ABC लाई स्थानान्तरण गर्दा बन्ने प्रतिबिम्ब $\Delta A'B'C'$ का शीर्षविन्दुहरू A'(2, 0), B'(8, 12) र $C'(9, -3)$ छन् भने उक्त स्थानान्तरण मेट्रिक्स पत्ता लगाउनुहोस् ।

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Question 283 marks

Find the quartile deviation from the given data, where the first quartile $(Q_{1}) = 50$ . Marks obtained: $30-40 | 40-50 | 50-60 | 60-70 | 70-80$ No. of Students: 2 | 3 | 6 | 5 | 4

दिइएको तथ्याङ्कबाट चतुर्थांशीय भिन्नता पत्ता लगाउनुहोस्, जहाँ पहिलो चतुर्थांश $(Q_{1}) = 50$ छ : प्राप्ताङ्क (Marks obtained): $30-40 | 40-50 | 50-60 | 60-70 | 70-80$ विद्यार्थीको सङ्ख्या (No. of Students): 2 | 3 | 6 | 5 | 4

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Question 293 marks

Find the standard deviation from the given data. Class Interval: $0-20 | 20-40 | 40-60 | 60-80 | 80-100$ Frequency: 2 | 5 | 6 | 4 | 3

दिइएको तथ्याङ्कबाट स्तरीय भिन्नता पत्ता लगाउनुहोस् : वर्गान्तर (Class Interval): $0-20 | 20-40 | 40-60 | 60-80 | 80-100$ बारम्बारता (Frequency): 2 | 5 | 6 | 4 | 3

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Question 304 marks

The sum of three terms in an arithmetic series is 15. If 1, 4 and 19 are added to them respectively, it forms a geometric series. Find the initial three terms.

एउटा समानान्तरीय श्रेणीमा रहेका तीन पदहरूको योगफल 15 छ । यदि ती पदहरूमा 1, 4 र 19 क्रमशः जोड्दा ती पदहरू ज्यामितीय श्रेणीका बन्नेछन् भने सुरुका तीन पदहरू पत्ता लगाउनुहोस् ।

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Question 314 marks

Find the equation of a circle whose centre lies on the straight line $4x + 5y = 16$ and passes through the points (4, 1) and (6, 5).

विन्दुहरू (4, 1) र (6, 5) भएर जाने र केन्द्र विन्दु सिधा रेखा $4x + 5y = 16$ मा पर्ने वृत्तको समीकरण पत्ता लगाउनुहोस् ।

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Question 324 marks

In a right angled triangle PQR, $\angle$ PQR $= 90^\circ$ . If M is the midpoint of the side PR, prove that PM = MR = QM by using vector method.

एउटा समकोणी त्रिभुज PQR मा $\angle$ PQR $= 90^\circ$ छ । यदि भुजा PR को मध्यविन्दु M भए PM = MR = QM हुन्छ भनी भेक्टर विधि प्रयोग गरी प्रमाणित गर्नुहोस् ।

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Question 334 marks

ABC is a triangle with vertices A(5, 6), B(3, 4) and C(6, 1). If $R_{1}$ represents the reflection on the line y = x and $R_{2}$ represents the rotation about the origin through $+ 90^\circ$ , find the single transformation represented by the combined transformation $R_{1}oR_{2}$ . Transform the $\Delta$ ABC using the single transformation and find the coordinates of the vertices of the image so formed. Also, present $\Delta$ ABC and its image in the same graph paper.

शीर्षविन्दुहरू A(5, 6), B(3, 4) र C(6, 1) भएका त्रिभुज ABC छ । यदि $R_{1}$ ले y = x रेखामा हुने परावर्तन र $R_{2}$ ले उद्गम विन्दुको वरिपरि $+90^\circ$ मा हुने परिक्रमण जनाउँछ भने संयुक्त स्थानान्तरण $R_{1}oR_{2}$ जनाउने एकल स्थानान्तरण पत्ता लगाउनुहोस् । उक्त एकल स्थानान्तरण प्रयोग गरी $\Delta$ ABC लाई स्थानान्तरण गर्नुहोस् र बन्ने प्रतिबिम्बको शीर्षविन्दुहरूका निर्देशाङ्कहरू पत्ता लगाउनुहोस् । साथै $\Delta$ ABC र यसको प्रतिबिम्बलाई एउटै लेखाचित्रमा प्रस्तुत गर्नुहोस् ।

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Questions learners ask

How many questions and marks are in the SEE 2082 Optional Mathematics (Grade Increment, GI-2021) paper?

33 questions worth 75 marks. There are 10 of 1 mark, 8 of 2 marks, 11 of 3 marks and 4 of 4 marks.

How much time is given for this paper?

3 hours (180 minutes).

Is this the real board paper?

Yes. It is the SEE 2082 paper printed for Grade Increment, code GI-2021, typed up from the original and checked by the GyanMul team.

Are the answers included?

Answers to the 10 one-mark questions are on this page (GyanMul's model answers; the NEB does not publish an answer key). Worked answers and the marking for every other question are in GyanMul Prep, free with an account.

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