Class 10 Compulsory Mathematicsकक्षा १० अनिवार्य गणित — SEE notes & solutions

450 जना मानिसहरूमा गरिएको एउटा सर्वेक्षणमा 200 जनाले चिया, 250 जनाले कफि मन पराएको पाइयो । 50 जनाले यी दुईमध्ये कुनै पनि मन पराएनन् ।

In a survey of 450 people, 200 people liked tea and 250 people liked coffee. But 50 people did not like any of these two drinks.

(a) यदि T र C ले क्रमश: चिया र कफि मन पराउने मानिसहरूको समूहलाई जनाउँछ भने \(n(\overline{T \cup C})\) को गणनात्मकता लेख्नुहोस् ।

If T and C denote the set of people who like tea and coffee respectively, write the cardinality of \(n(\overline{T \cup C})\).

(b) माथिको जानकारीलाई भेनचित्रमा प्रस्तुत गर्नुहोस् ।

Present the above information in a Venn diagram.

(c) चिया मात्र मन पराउने मानिसहरूको सङ्ख्या पत्ता लगाउनुहोस् ।

Find the number of people who liked tea only.

(d) चिया र कफि दुवै मन पराउने मानिसको सङ्ख्या र कफि मात्र मन पराउने मानिसको सङ्ख्याबिच तुलना गर्नुहोस् ।

Compare the number of people who like both tea and coffee with the number of people who like coffee only.

एक जना किसानले 2 वर्षका लागि एउटा सहकारीमा रु.50,000 वार्षिक चक्रिय व्याज \(8\%\) प्रति वर्षका दरले पाउने गरी जम्मा गरेछ ।

A farmer deposited Rs. 50,000 in a co-operative for 2 years to get the annual compound interest at the rate of \(8\%\) per annum.

(a) त्रैमासिक चक्रिय व्याजमा एक वर्षमा कति पटक व्याज गणना गरिन्छ ? लेख्नुहोस् ।

How many times the interest is calculated in the quarterly compound interest in one year? Write it.

(b) किसानले 2 वर्षको अन्त्यमा कति वार्षिक चक्रिय व्याज प्राप्त गर्दछ ? पत्ता लगाउनुहोस् ।

How much annual compound interest will the farmer receive at the end of 2 years? Find it.

(c) सोही रकमको उही व्याजदर र अवधिको अर्धवार्षिक चक्रिय व्याज वार्षिक चक्रिय व्याजभन्दा कतिले बढी हुन्छ ? पत्ता लगाउनुहोस् ।

By how much the semi-annual compound interest is more than the annual compound interest of the same sum at the same rate and for the same period of time? Find it.

एउटा विद्युतीय बस रु.45,00,000 मा किनियो । 2 वर्षमा बसको प्रयोगबाट रु.12,00,000 आम्दानी भयो । बसको मूल्यमा वार्षिक \(10\%\) को दरले ह्रास आउँछ ।

An electric Bus is purchased for Rs.45,00,000. Using the bus for 2 years Rs.12,00,000 is earned. The value of the bus depreciates at the rate of \(10\%\) per annum.

(a) यदि बसको सुरुको मूल्य \(V_0\), वार्षिक ह्रासदर R र बसको T वर्षपछिको मूल्य \(V_T\) भए \(V_T\) लाई \(V_0\), R र T को रूपमा लेख्नुहोस् ।

If the initial price of bus is \(V_0\), annual rate of depreciation is R and price of the bus after T years is \(V_T\) then express \(V_T\) in terms of \(V_0\), R and T.

(b) पहिलो वर्षमा बसको मूल्य कतिले घटेछ ? पत्ता लगाउनुहोस् ।

How much the price of the bus is depreciated in first year? Find it.

(c) यदि 2 वर्षपछि सो बस बिक्की गरियो भने कति प्रतिशत नाफा वा नोक्सान हुन्छ ? पत्ता लगाउनुहोस् ।

If the bus is sold after 2 years, what will be the percentage of profit or loss? Find it.

नबिन विदेश जानको लागि अमेरिकी डलर साट्न बैङ्क गएछ । सो दिन 1 डलर ($) को खरिद दर रु.138.23 र बिक्री दर रु.138.83 थियो ।

Nabin went to bank to exchange American dollars to visit abroad. In that day the buying rate of 1 dollar ($) was Rs. 138.23 and selling rate was Rs.138.83.

(a) खरिद दरभन्दा बिक्री दर कतिले बढी छ ? पत्ता लगाउनुहोस् ।

By how much the selling rate is more than the buying rate? Find it.

(b) अमेरिकि डलर 500 सँग कति नेपाली रुपैया साट्न सकिन्छ ? पत्ता लगाउनुहोस् ।

How much Nepali rupees can be exchanged with American dollar 500? Find it.

(c) केही दिनपछि 1 डलरको बिक्रीदर रु.139.80 हुन्छ भने नेपाली मुद्रा कति प्रतिशतले अवमूल्यन भएछ ? पत्ता लगाउनुहोस् ।

After some days the selling rate of dollar 1 becomes Rs.139.80 then by what percent the Nepali currency was devaluated? Find it.

वर्ग आधार भएको एउटा पिरामिडको ठाडो उचाई 12 से.मि. र आधार भुजा 10 से.मि. छन् ।

The vertical height of a square based pyramid is 12 cm and its base side is 10 cm.

[Figure: Square based pyramid with vertical height 12cm and base side 10cm]

(a) वर्ग आधार भएको पिरामिडमा कतिवटा त्रिभुजाकार सतहहरू हुन्छन् ? लेख्नुहोस् ।

How many triangular surfaces are there in a square based pyramid? Write it.

(b) सो पिरामिडको आयतन पत्ता लगाउनुहोस् ।

Find the volume of the pyramid.

(c) सो पिरामिडको पुरा सतहको क्षेत्रफल पत्ता लगाउनुहोस् ।

Find the total surface area of the pyramid.

दिइएको चित्रमा, एउटा ठोस वस्तु समान अर्धव्यास भएका वेलना र सोली मिली बनेको छ । उक्त ठोस वस्तुमा वेलनाकार भागको लम्बाइ 28 से.मी. र सोली भागको छड्के उचाइ 17 से.मी. छ । वेलनाकार भागको आयतन 5632 घन से.मी. छ ।

In the given figure, a combined solid object is formed with the combination of cylinder and cone having same radius. In the solid object, the length of cylindrical part is 28 cm and the slant height of conical part is 17 cm. The volume of the cylindrical part is 5632 cubic cm.

[Figure: Combined solid object: cylinder of length 28cm and cone with slant height 17cm, radius shown as 17cm]

(a) उक्त ठोस वस्तुको आधार कस्तो आकारको छ ? लेख्नुहोस् ।

What type is the shape of the base of solid object? Write it.

(b) सोली भागको उचाइ र वेलनाकार भागको लम्बाइ तुलना गर्नुहोस् ।

Compare the height of conical part and the length of cylindrical part.

(c) के वेलनाकार भागको आयतन सोली भागको आयतनको पाँच गुणा छ ? गणना गरी पुष्टि गर्नुहोस् ।

Is the volume of cylindrical part five times the volume of conical part? Justify with calculation.

एउटा आयतकार कोठाको लम्बाइ, चौडाइ र उचाइ क्रमश: 12 मि., 8 मि. र 3 मि. छन् । सो कोठामा 2 मि. किनारा भएका दुईओटा वर्गाकार भयालहरू छन् र एउटा \(1.5\) मि. \(\times 1\) मि. को ढोका छ ।

The length, breadth and height of a rectangular room are 12 m, 8 m and 3 m respectively. There are two square windows with edges 2 m and a door of size \(1.5\) m \(\times 1\) m in the room.

(a) सो कोठाको भुईँको क्षेत्रफल पत्ता लगाउनुहोस् ।

Find the area of the floor of the room.

(b) भयाल र ढोकाले ओगटेको क्षेत्रफल बाहेक सो कोठाको चार भित्ता र सिलिङमा प्रतिवर्ग मिटर रु.15 को दरले रङ लगाउँदा कति खर्च लाग्छ ? निकाल्नुहोस् ।

How much does it cost to coloring the four walls and ceiling of the room excluding the area occupied by the windows and door at the rate of Rs.15 per square meter? Calculate it.

3 र 27 को बिचमा 7 वटा समानान्तरीय मध्यमाहरू छन् ।

There are 7 arithmetic means between 3 and 27.

(a) \(a\) र \(b\) बिचको समानान्तरीय मध्यमा निकाल्ने सूत्र लेख्नुहोस् ।

Write the formula to calculate arithmetic mean between \(a\) and \(b\).

(b) दिइएको अनुक्रमको 5 औँ मध्यमा कति हुन्छ ? पत्ता लगाउनुहोस् ।

What is the 5th mean of the given sequence? Find it.

(c) 3 र 27 बीचको समानान्तरीय मध्यमा र गुणोत्तर मध्यमा कुन कतिले ठूलो छ ? तुलना गर्नुहोस् ।

Which one is greater by how much in arithmetic mean and geometric mean between 3 and 27? Compare it.

एउटा आयताकार चौरको परिमिति र क्षेत्रफल क्रमश: 44 मिटर र 120 वर्ग मिटर छन् ।

The perimeter and area of a rectangular ground are 44 meter and 120 square meters respectively.

(a) वर्ग समीकरण \(ax^2 + bx + c = 0, a \neq 0\) हल गर्ने सूत्र लेख्नुहोस् ।

Write the formula to solve the quadratic equation \(ax^2 + bx + c = 0, a \neq 0\).

(b) चौरको लम्बाइ र चौडाइ पत्ता लगाउनुहोस् ।

Find the length and breadth of the ground.

(c) चौरको लम्बाइलाई घटाएर वर्गाकार बनाउँदा नयाँ चौरको क्षेत्रफल कति प्रतिशतले बढी वा कम हुन्छ ? पत्ता लगाउनुहोस् ।

If the ground is made a square by reducing the length side, by what percent the area will be increased or decreased? Find it.

(a) सरल गर्नुहोस् (Simplify): \(\dfrac{a}{a-b} + \dfrac{b}{b-a}\)

(b) हल गर्नुहोस् (Solve): \(2^x + \dfrac{1}{2^x} = 2\dfrac{1}{2}\)

दिइएको चित्रमा \(EC \parallel AB\), \(DA \parallel CB\) र \(DF \perp BC\) छन् ।

In the given figure, \(EC \parallel AB\), \(DA \parallel CB\) and \(DF \perp BC\).

[Figure: Figure showing EC parallel AB, DA parallel CB, DF perpendicular BC with points E, D, C, F, A, B labeled]

(a) एउटै आधार र उही समानान्तर रेखाहरू बिच रहेका त्रिभुज र समानान्तर चतुर्भुजको क्षेत्रफल बिचको सम्बन्ध लेख्नुहोस् ।

Write the relation between the areas of triangle and parallelogram standing on the same base and between same parallel lines.

(b) यदि \(BC = 6\) से.मि. र \(DF = 8\) से.मि. भए \(\triangle ABE\) को क्षेत्रफल पत्ता लगाउनुहोस् ।

If \(BC = 6\) cm and \(DF = 8\) cm, find the area of \(\triangle ABE\).

(c) दिइएको चित्रमा यदि \(\triangle AOB\) को क्षेत्रफल र \(\triangle COD\) को क्षेत्रफल बराबर छन् भने \(AD \parallel BC\) हुन्छन् भनी प्रमाणित गर्नुहोस् ।

In the given figure, if area of \(\triangle AOB\) and area of \(\triangle COD\) are equal, then prove that \(AD \parallel BC\).

[Figure: Figure with points A, D on top and B, C on bottom, O at intersection of diagonals]

दिइएको चित्रमा O वृत्तको केन्द्रविन्दु र PQRS एउटा चाक्रिय चतुर्भुज हो ।

In the given diagram, O is the centre of the circle and PQRS is a cyclic quadrilateral.

[Figure: Circle with centre O and cyclic quadrilateral PQRS inscribed, with P, Q, R, S on the circle]

(a) \(\angle QRS\) र वृहत \(\angle QOS\) बिचको सम्बन्ध लेख्नुहोस् ।

Write the relationship between \(\angle QRS\) and reflex \(\angle QOS\).

(b) यदि \(\angle QPS = 46^\circ\) भए \(\angle QOS\) को मान पत्ता लगाउनुहोस् ।

If \(\angle QPS = 46^\circ\), find the value of \(\angle QOS\).

(c) \(\angle QPS + \angle QRS = 180^\circ\) हुन्छ भनी प्रयोगद्वारा प्रमाणित गर्नुहोस् ।

(3 से.मि. भन्दा बढी अर्धव्यास भएका दुईवटा वृत्तहरू आवश्यक छन् ।)

Verify experimentally that: \(\angle QPS + \angle QRS = 180^\circ\).

(Two circles having radii more than 3 cm are necessary.)

(a) \(MN = 8\) से.मी., \(NO = 7\) से.मि. र \(\angle MNO = 60^\circ\) भएको \(\triangle MNO\) को रचना गर्नुहोस् । साथै उक्त \(\triangle MNO\) को क्षेत्रफलसँग बराबर हुने एउटा आयत NPQR को पनि रचना गर्नुहोस् ।

Construct a \(\triangle MNO\) in which \(MN = 8\) cm, \(NO = 7\) cm and \(\angle MNO = 60^\circ\). Also construct a rectangle NPQR equal in area to \(\triangle MNO\).

(b) \(\triangle MNO\) र आयत NPQR को क्षेत्रफल किन बराबर हुन्छन् ? कारण दिनुहोस् ।

Why the areas of \(\triangle MNO\) and the rectangle NPQR are equal? Give the reason.

दिइएको चित्रमा, 30 मिटर अग्लो एउटा स्तम्भ (AB) को टुप्पोबाट एउटा घर (CD) को छानाको अवनति कोण (\(\angle HAC\)) \(30^\circ\) छ । स्तम्भ र घरबिचको दूरी \((BD) = 10\sqrt{3}\) मिटर छ ।

In the given figure, from the top of a tower (AB) 30 meter high, the angle of depression (\(\angle HAC\)) of the roof of a house (CD) is \(30^\circ\). The distance between tower and house \((BD) = 10\sqrt{3}\) meter.

[Figure: Tower AB 30m high, house CD, angle of depression HAC = 30 degrees, BD = 10 root 3 m, with points A, H at top, E, C at house roof level, B and D at ground]

(a) उन्नतांश कोणको परिभाषा लेख्नुहोस् ।

Write the definition of the angle of elevation.

(b) घरको छानोबाट स्तम्भको टुप्पोको उन्नतांश कोण कति हुन्छ ? लेख्नुहोस् ।

What is the angle of elevation of the top of the tower from the roof of the house? Write it.

(c) घरको उचाइ पत्ता लगाउनुहोस् ।

Find the height of the house.

(d) \(AE = EC\) भएको अवस्थामा \(\angle CAH\) को मान कति हुन्छ ? कारण दिनुहोस् ।

What is the value of \(\angle CAH\) when \(AE = EC\)? Give reason.

तालिकामा 15 जना विद्यार्थीहरूले 50 पूर्णाङ्कको एउटा परीक्षामा प्राप्त गरेको प्राप्ताङ्क दिइएको छ ।

The marks obtained by 15 students in an examination with full mark 50 are given in table.

(a) निरन्तर श्रेणीको पहिलो चर्तुयांश निकाल्ने सूत्र लेख्नुहोस् ।

Write the formula to calculate the first quartile of continuous series.

(b) दिइएको तथ्याङ्कको पहिलो चतुर्थांश पत्ता लगाउनुहोस् ।

Find the first quartile of the given data.

(c) दिइएको तथ्याङ्कको मध्यक निकाल्नुहोस् ।

Calculate the mean of the given data.

(d) मध्यकभन्दा बढी अंक प्राप्त गर्ने अधिकतम् विद्यार्थीहरूको सङ्ख्या कति हुन सक्दछन् ? पत्ता लगाउनुहोस् ।

What could be the maximum number of students who obtained more marks than mean? Find it.

एउटा भोलामा 7 ओटा काला र 4 ओटा राता उस्तै र उत्रै बलहरू छन् । दुई ओटा बलहरू एकपछि अर्को गरी पुनः नराखी भिकिएको छ ।

A bag contains 7 black and 4 red balls of same shape and size. Two balls are drawn randomly one after another without replacement.

(a) यदि B र R दुई अनाश्रित घटनाहरू भए \(P(B \cap R)\) को सूत्र लेख्नुहोस् ।

If B and R be two independent events then write the formula of \(P(B \cap R)\).

(b) सबै सम्भावित परिणामहरूको सम्भाव्यतालाई वृक्ष चित्रमा देखाउनुहोस् ।

Show the probability of all possible outcomes in a tree diagram.

(c) दुवै बल कालो नै पर्ने सम्भाव्यता पत्ता लगाउनुहोस् ।

Find the probability of getting both black balls.

(d) दुवै बल रातो पर्ने सम्भाव्यता, दुवै बल कालो पर्ने सम्भाव्यता भन्दा कति बढी वा घटी छ ? पत्ता लगाउनुहोस् ।

By how much the probability of getting both red balls is more or less than the probability of getting both black balls? Find it.