बहुपद \( f(x) = 5x^4 - 3xy + 8x + 6 \) को घात कुन हो? What is the degree of the polynomial \( f(x) = 5x^4 - 3xy + 8x + 6 \)?
6
4
xy
3
[1A]
यदि सम्बन्ध \( R: A \to B \) लाई \( R = \{(1,3), (2,4), (3,4), (3,6), (5,7)\} \) ले परिभाषित गरिएको छ भने यसको डोमेन के हो? If \( R: A \to B \) is a relation defined as \( R = \{(1,3), (2,4), (3,4), (3,6), (5,7)\} \), then what is its domain?
\(\{1,2,3,4,5,6,7\}\)
\(\{3,4,6,7\}\)
\(\{1,2,3,4,5\}\)
\(\{1,2,3,5\}\)
[1B]
\( \sin^2\theta \) बराबर कुन त्रिकोणमितिय अनुपात हो? Which trigonometric ratio is equal to \( \sin^2\theta \)?
\( \sqrt{1 - \cos^2\theta} \)
\( \dfrac{1}{\cos ec\theta} \)
\( 1 - \cos^2\theta \)
\( \dfrac{1}{\cos^2\theta} \)
[1C]
\( 9^\circ \) मा कति सेन्टेसिमल मिनेट हुन्छन्? How many centesimal minutes are there in \( 9^\circ \)?
1000'
540'
900'
10'
[1D]
\( p^2 = h^2 - b^2 \) पाइथागोरस सम्बन्धको एक रूप हो। यदि यसको दुवै तर्फलाई \( p^2 \) ले भाग गरिन्छ भने तलका मध्ये कुन सम्बन्ध प्राप्त हुन्छ? \( p^2 = h^2 - b^2 \) is one of the Pythagoras relations. If you divide its both sides by \( p^2 \), then which of the following relation is obtained?
\( \sin^2\theta + \cos^2\theta = 1 \)
\( \csc^2\theta - \cot^2\theta = 1 \)
\( \sec^2\theta - \tan^2\theta = 1 \)
\( \sec^2\theta \cdot \cos^2\theta = 1 \)
[1E]
बिन्दुहरू \( A(x_1, y_1) \) र \( B(x_2, y_2) \) लाई जोड्ने रेखाको ढाल के हो? What is the slope of a line joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?
बिन्दुहरू \( P(-4, 0) \) र \( Q(0, -3) \) बीचको दुरी कति हो? Which of the following is the distance between the points \( P(-4, 0) \) and \( Q(0, -3) \)?
7 units
1 units
5 units
-7 units
[1G]
यदि \( \Sigma x = 45 \) र \( \bar{x} = 9 \) भए \( \Sigma f \) को मान कति हो? If \( \Sigma x = 45 \) and \( \bar{x} = 9 \), then what is the value of \( \Sigma f \)?
36
54
5
405
[1H]
बहुपद \( f(x) = 5x^4 - 3xy + 8x + 6 \) को घात कुन हो? What is the degree of the polynomial \( f(x) = 5x^4 - 3xy + 8x + 6 \)?
4
5
2
1
[1A]
यदि सम्बन्ध \( R: A \to B \) लाई \( R = \{(1,3), (2,4), (3,4), (3,6), (5,7)\} \) ले परिभाषित गरिएको छ भने यसको डोमेन के हो? If the relation \( R: A \to B \) is defined as \( R = \{(1,3), (2,4), (3,4), (3,6), (5,7)\} \), what is its domain?
\(\{3, 4, 6, 7\}\)
\(\{1, 2, 3, 5\}\)
\(\{1, 2, 4, 5\}\)
\(\{1, 2, 3, 4, 5, 6, 7\}\)
[1B]
\( \sin^2\theta \) बराबर कुन त्रिकोणमितिय अनुपात हो? Which trigonometric ratio is equal to \( \sin^2\theta \)?
\( 1 + \cos^2\theta \)
\( 1 - \cos^2\theta \)
\( \tan^2\theta \)
\( \sec^2\theta - 1 \)
[1C]
\( 9^\circ \) मा कति सेन्टेसिमल मिनेट हुन्छन्? How many centesimal minutes are there in \( 9^\circ \)?
1000'
900'
100'
540'
[1D]
\( p^2 = h^2 - b^2 \) पाइथागोरस सम्बन्धको एक रूप हो। यदि यसको दुवै तर्फलाई \( p^2 \) ले भाग गरिन्छ भने तलका मध्ये कुन सम्बन्ध प्राप्त हुन्छ? \( p^2 = h^2 - b^2 \) is one form of the Pythagoras relation. If both sides are divided by \( p^2 \), which of the following relations is obtained?
बिन्दुहरू \( A(x_1, y_1) \) र \( B(x_2, y_2) \) लाई जोड्ने रेखाको ढाल के हो? What is the slope of a line joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?
\( \dfrac{y_2 - y_1}{x_2 - x_1} \)
\( \dfrac{x_2 - x_1}{y_2 - y_1} \)
\( \dfrac{y_1 + y_2}{x_1 + x_2} \)
\( \dfrac{x_1 - x_2}{y_1 - y_2} \)
[1F]
बिन्दुहरू \( P(-4, 0) \) र \( Q(0, -3) \) बीचको दुरी कति हो? What is the distance between the points \( P(-4, 0) \) and \( Q(0, -3) \)?
5 units
7 units
\( \sqrt{7} \) units
\( \sqrt{5} \) units
[1G]
यदि \( \Sigma x = 45 \) र \( \bar{x} = 9 \) भए \( \Sigma f \) को मान कति हो? If \( \Sigma x = 45 \) and \( \bar{x} = 9 \), then what is the value of \( \Sigma f \)?
5
9
45
405
[1H]
Group 'B'
Let \( A = \{1, 3, 4\} \) and \( B = \{9, 16\} \) be two sets. Hari defined a relation \( R: A \to B \) satisfying the condition \( y = x^2 \).
Define ordered pair.
Find \( A \times B \).
Find the relation \( R \) in ordered pair form.
Write the relation between range of the relation \( (R) \) and domain of the inverse relation \( (R^{-1}) \) with reason.
An ordered pair \((x, y)\) is a pair of elements in definite order, where order of elements matters.
\( A \times B\) is \(\{(1, 9), (1, 16), (3, 9), (3, 16), (4, 9), (4, 16)\} \)
Find the relation \( R \) in ordered pair form is \( R = \{(x, y) \in A \times B \mid y = x^2\} \)
or\( R = \{(3, 9), (4, 16)\} \)
Relation between range of \( R \) and domain of \( R^{-1} \) is
Here Range of \( R \): \( \{9, 16\} \)
Again Domain of \( R^{-1} \): \( \{9, 16\} \)
Hence
The range of \( R \) equals the domain of \( R^{-1} \)
The elements of matrix \( A \) of order \( 2 \times 2 \) is given by \( a_{ij} = 2i - j \).
Construct the given matrix \( A \) according to given condition.
Write the type of matrix \( A \).
To construct the 2 x 2 matrix \( A \), we compute \( A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \)
Here
For \( a_{11} \), we take \( i = 1, j = 1 \), so the value of \( a_{11} \) is \( a_{11} =2i-j= 2(1) - 1 = 1 \)
For \( a_{12} \), we take \( i = 1, j = 2 \), so the value of \( a_{12} \) is \( a_{12} =2i-j= 2(1) - 2 =0 \)
For \( a_{21} \), we take \( i = 2, j = 1 \), so the value of \( a_{21} \) is \( a_{21} =2i-j= 2(2) - 1 = 3 \)
For \( a_{22} \), we take \( i = 2, j = 2 \), so the value of \( a_{22} \) is \( a_{22} =2i-j= 2(2) - 2 = 2 \)
Hence , the matrix is \( A = \begin{pmatrix} 1 & 0 \\ 3 & 2 \end{pmatrix} \)
The matrix \( A \) is square matrix.
Two angles of a triangle are \( 50^\circ \) and \( 75^\circ \).
How many degrees are there in \( \pi^c \)? Write it.
Find the remaining angle of the triangle in degree.
The degrees in \( \pi^c \) is \( \pi^c = \pi \cdot \frac{180}{\pi} = 180^\circ \)
Remaining angle of the triangle in degrees are
Given angles are \( \measuredangle A=50^\circ\) \( \measuredangle B=70^\circ\)
Remaining angle is \( 180^\circ - 50^\circ-70^\circ = 55^\circ \)
Study the given right angled triangle and answer the following questions.
Find the length of perpendicular \( (p) \) from the given right angled triangle.
Find the value of: \( \sin\theta + \cos\theta - \tan\theta \).
The length of perpendicular \( (p) \) is \(p=\sqrt{h^2-b^2} \)
or\(p=\sqrt{10^2-6^2} \)
or\(p=\sqrt{64} \)
or\(p=8 \)
The value is \( \sin\theta + \cos\theta - \tan\theta \)
or\(\frac{p}{h}+\frac{b}{h}-\frac{p}{b}\)
or\(\frac{8}{10}+\frac{6}{10}-\frac{8}{6}\)
or\(\frac{4}{5}+\frac{3}{5}-\frac{4}{3}\)
or\(\frac{4 \times 3+3 \times 3-4 \times 5}{5 \times 3}\) or\(\frac{12+9-20}{15}\)
or\(\frac{1}{15}\)
If \( \sin^4 A - \cos^4 A = 1 - 2\cos^2 A \) is a trigonometrical identity then:
Write the two factors of \( \sin^4 A - \cos^4 A \).
Prove that: \( \sin^4 A - \cos^4 A = 1 - 2\cos^2 A \).
The two factors of \( \sin^4 A - \cos^4 A \) are \( \sin^4 A - \cos^4 A \)
or\( (\sin^2 A)^2 - (\cos^2 A)^2 \)
or\( (\sin^2 A+\cos^2 A)(\sin^2 A-\cos^2 A) \)
The LHS is \( \sin^4 A - \cos^4 A \)
or\( (\sin^2 A+\cos^2 A)(\sin^2 A-\cos^2 A) \)
or\( 1(\sin^2 A-\cos^2 A) \)
or\( \sin^2 A-\cos^2 A \)
or\( 1-\cos^2 A-\cos^2 A \)
or\( 1-2\cos^2 A \)
\( P(-1, 5) \) and \( Q(2, -1) \) are the end points of line segment \( PQ \).
What is the image of a point \( A(x, y) \) under the reflection on y-axis? Write it.
Find the co-ordinates of the end points of image line \( P'Q' \) under the reflection on y-axis.
The image of a point \( A(x, y) \) under the reflection on y-axis is \( (x, y) \to (-x, y) \)
The co-ordinates of the end points of image line \( P'Q' \) under the reflection on y axis are \( P(-1, 5) \to (-5, 1) \) \( Q(2, -1) \to (1, -2) \)
\( P(-1, 5) \) र \( Q(2, -1) \) ले रेखाखण्ड \( PQ \) का अन्त्य बिन्दुहरू हुन्। \( P(-1, 5) \) and \( Q(2, -1) \) are the end points of line segment \( PQ \).
Y-अक्षमा प्रतिबिम्बन गर्दा बिन्दु \( A(x, y) \) को प्रतिबिम्ब के हुन्छ? लेख्नुहोस्। What is the image of a point \( A(x, y) \) under the reflection on y-axis? Write it.
\( y = -x \) मा प्रतिबिम्बन गर्दा प्रतिबिम्ब रेखाखण्ड \( P'Q' \) का अन्त्य बिन्दुहरूको निर्देशांक फेला पार्नुहोस्। Find the co-ordinates of the end points of image line \( P'Q' \) under the reflection on \( y = -x \).
the image of a point \( A(x, y) \) under the reflection on y-axis is \(A'(-x, y)\)
The image of the point \( P(-1, 5) \) is \(P'(-5, 1)\)
The image of the point \( Q(2, -1) \) is \(Q'(-2, 1)\)
If \( (2, 2) \) is the mid-point of the line segment joining the points \( M(-1, 2) \) and \( N(5, y) \), then:
Find the value of \( y \).
Find the co-ordinates of a point which divides \( MN \) externally in the ratio \( 1:3 \).
If \( T(2,6) \) is another point then prove that: \( MT = NT \).
Given midpoint is \( (2, 2) \), so y-coordinate \(= \frac{y_1 + y_2}{2} \)
or\(2= \frac{2 + y}{2} \)
or\( 2 + y = 4 \)
or\( y = 2 \)
Coordinates of the point are \( M(-1, 2) \), \( N(5, 2) \)
External section formula is \( \left( \frac{m x_2 - n x_1}{m - n}, \frac{m y_2 - n y_1}{m - n} \right) \)
Here x-coordinate: \( \frac{1 \cdot 5 - 3 \cdot (-1)}{1 - 3} = \frac{5 + 3}{-2} = \frac{8}{-2} = -4 \)
Again y-coordinate: \( \frac{1 \cdot 2 - 3 \cdot 2}{1 - 3} = \frac{2 - 6}{-2} = \frac{-4}{-2} = 2 \)
Hence, the point is \( (-4, 2) \)
We use distance formula to get \( MT = \sqrt{(2 - (-1))^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Again \( NT = \sqrt{(5 - 2)^2 + (2 - 6)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Hence \( MT = NT \)
विद्यार्थीहरूले पहिलो त्रैमासिक परीक्षामा प्राप्त गरेका अंकहरू तलको बाट दिइएको छ: The marks obtained by the students in first terminal examination are given below:
Marks
15
20
25
30
35
No. of students
4
7
4
3
1
डिस्क्रिट डेटाको मध्यक फेला पार्ने सूत्र लेख्नुहोस्। Write the formula to find mean in discrete data.
दिइएको डेटाको मध्यिका पत्ता लगाउनुहोस। Find the median of the given data.
Formula to find mean in discrete data is \(\text{Mean } (\bar{X}) = \frac{\sum fx}{\sum f} \)
The median of the given data is
Marks (\(x\))
No. of students (\(f\))
Cumulative Frequency (\(c.f.\))
15
4
4
20
7
4 + 7 = 11
25
4
11 + 4 = 15
30
3
15 + 3 = 18
35
1
18 + 1 = 19
Now, position of Median is MD=\(\left( \frac{N + 1}{2} \right)^\text{th} \text{ term} \) MD=\(\left( \frac{19 + 1}{2} \right)^\text{th} \text{ term}\) MD=\(\left( \frac{20}{2} \right)^\text{th} \text{ term}\) MD=\(10^\text{th} \text{ term}\)
Hence, the orresponding value, which is median, is Median= 20
Group 'C'
यदि \( P = \{1, 6\} \) र \( Q = \{4\} \) दुई समुहहरू हुन् भने, If \( P = \{1, 6\} \) and \( Q = \{4\} \) are two sets, then,
\( Q \times P \) लाई क्रम जोडा रूपमा फेला पार्नुहोस् र यसलाई मिलान चित्रमा प्रस्तुत गर्नुहोस्। Find \( Q \times P \) in ordered pair form and represent it in a mapping diagram.
\( Q \times P \) का बिन्दुहरूलाई x-अक्षमा प्रतिबिम्बित गर्नुहोस्। Reflect the elements of \( Q \times P \) on x-axis.
\( Q \times P \) का बिन्दुवहरू जोड्ने रेखाखण्डलाई आन्तरिक रूपमा 2:3 को अनुपातमा विभाजन गर्ने बिन्दुको निर्देशांक फेला पार्नुहोस्। Find the co-ordinates of a point which divides the line segment joining the elements of \( Q \times P \) internally in the ratio 2:3.
Given sets are \( P = \{1, 6\} \), \( Q = \{4\} \)
The Cartesian product \( Q \times P \)is \( Q \times P = \{(4, 1), (4, 6)\} \)
The mapping diagram is
The reflection of the elements of \( Q \times P \) on the x-axis are
or\( (4, 1) \to (4, -1) \)
or\( (4, 6) \to (4, -6) \)
Coordinates of the point dividing the line segment joining \( (4, 1) \) and \( (4, 6) \) internally in the ratio 2:3 is \(M= \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right) \)
Here, \( m = 2 , n = 3 , x_1=4, y_1=1, x_2=4, y_2=6 \)
Hence \(M= \left( \frac{2 \cdot 4 + 3 \cdot 4}{2 + 3}, \frac{2 \cdot 6 + 3 \cdot 1}{2 + 3} \right )\)
or\(M= \left( \frac{20}{5}, \frac{15}{5} \right )\)
or\( M=(4, 3) \)
∆ABC मा, ∠A=70°, ∠B=50° र ∠C=60° भए, In ∆ABC, ∠A=70°, ∠B=50° and ∠C=60°, then,
दिइएको ∆ABC को मध्यिका कोण फेला पार्नुहोस्। Find the median angle of the given ∆ABC.
∠A र ∠B को योगफललाई रेडियनमा परिवर्तन गर्नुहोस्। Convert the sum of ∠A and ∠B in to radian.
डोल्मा ले भनिन् "∆ABC एक बिषमबाहु त्रिभुज हो" र पासाङ ले भनिन् "∆ABC एक न्यून कोण त्रिभुज हो।" कसको भनाई सही छ? तपाईंको तर्कसहित उत्तर दिनुहोस्। Dolma said “∆ABC is a scalene triangle” and Pasang said “∆ABC is an acute angled triangle.” Whose saying is correct? Give your logical answer.
∠A र ∠B बाट बनाउन सकिने सबै सम्भावित क्रम जोडाहरू निर्माण गर्नुहोस्। Construct all possible ordered pairs formed by ∠A and ∠B.
Median is the middle value when ordered data, so \( 50^\circ, 60^\circ, 70^\circ \)
The median angle of \( \triangle ABC \) is Median angle \( 60^\circ \)
The sum of \( \angle A \) and \( \angle B \) into radians is \( \angle A + \angle B\)
or\(70^\circ + 50^\circ \)
or\(120^\circ \)
or\(120 \cdot \frac{\pi}{180} = \frac{2\pi}{3} \)
All angles different, so all sides are different. Thus, Dolma said “∆ABC is a scalene triangle” , is correct.
Next, the triangle has all angles \( 90^\circ \)
Thus, Pasang said “∆ABC is an acute angled triangle", is correct.
Both Dolma and Pasang are correct.
All possible ordered pairs formed by \( \angle A \) and \( \angle B \) is \( (70^\circ, 50^\circ), (50^\circ, 70^\circ) \)
यदि \( A = \begin{bmatrix} \sin^2\theta & -\cot^2\theta \\ \cos^2\theta & \csc^2\theta \end{bmatrix}_{2\times2} \) एक म्याट्रिक्स हो भने, If \( A = \begin{bmatrix} \sin^2\theta & -\cot^2\theta \\ \cos^2\theta & \csc^2\theta \end{bmatrix}_{2\times2} \) is a matrix, then,
\( a_{11} \times a_{22} \) को मान फेला पार्नुहोस्। Find the value of \( a_{11} \times a_{22} \).
दिइएको म्याट्रिक्सको सदस्यहरूको माध्यक \( \frac{1}{2} \) हो भन्ने प्रमाणित गर्नुहोस्। Prove that the mean of the elements of given matrix is \( \frac{1}{2} \).
the value of \( a_{11} \times a_{22} \) is \( a_{11} \times a_{22} \)
or\( \sin^2\theta \times \csc^2\theta \)
or\( \cancel{\sin^2\theta} \times \frac{1}{\cancel {\sin^2\theta}} \)
or\( 1\)
The mean of the elements of given matrix is \(\dfrac{a_{11}+a_{12}+a_{21}+a_{22}}{4} \)
or \(\dfrac{\sin^2\theta + \csc^2\theta+\cos^2\theta - \cot^2\theta}{4} \)
or \(\dfrac{(\sin^2\theta + \cos^2\theta)+ (\csc^2\theta- \cot^2\theta)}{4} \)
or \(\dfrac{1+1}{4} \)
or \(\dfrac{1}{2} \)
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