Mathmatics model questions class 12

Posted On: Thu Mar 14 2024
Posted By: Bijay Poudel

set-1

Group'A' [ 1 × 11= 11]

Rewrite the correct option in your answer sheet.
1. If 1, ω, ω² are the cube roots of the unit then

ω = ω²
ω²= ω ³
1 + ω + ω²=0
1 + ω = ω²

2. The number of ways that 7 beads of different colors can be strung together so as to form a necklace is

5040
2520
720
360

3. $tan^{-1} \frac{5}{12}$ is equal to

$sin^{-1} \frac{12}{13}$
$cos^{-1} \frac{12}{13}$
$sec^{-1} \frac{12}{13}$
$cosec^{-1} \frac{12}{13}$

4. If 2 cos θ + 1 = 0 is the trigonometric equation of the locus related to the string attached to the wall of a hall then the general value for θ is

$nπ+(-1)^n \frac{2π}{3}$ for n ∈ Z
nπ+ $\frac{2π}{3}$ for n ∈ Z
2nπ ± $\frac{2π}{3}$ for n ∈ Z
2nπ+$\frac{π}{3}$ for n ∈ Z

5. If $\overrightarrow{a} = 2\overrightarrow{i}$ and $\overrightarrow{b}$ = $3\overrightarrow{j}$ where, $\overrightarrow{i}$, $\overrightarrow{j}$, and $\overrightarrow{k}$ unit vectors along X, Y, and Z- axes respectively, then the value $\overrightarrow{b}$× $\overrightarrow{a}$ is equal to

-6$\vec{k}$
6$\overrightarrow{k}$
6$\overrightarrow{i}$
6$\overrightarrow{j}$

6. There is a large grassy area near the president's house in Nepal. The area is the set of all points in a plane. The sum of distances from two fixed places (points) is constant. The conic section represented by the grassy area is…

Circle
Parabola
Hyperbola
Ellipse

7. Four unbiased coins are tossed successively. The mean and variance of the distribution differed by

8. The degree of the differential equation $\frac{d^3y}{dx^2} + 5(\frac{d^2y}{dx^2)^2 + 4(\frac dy}{dx}) + 6=0$ is

9. According to L Hospital's rule the value of $\lim_{x \rightarrow 0}\frac{x^3}{4sinx}$ is equal to

$\frac{3}{4}$
0
$\frac{1}{4}$
∞

10. When the Gauss forward elimination method is used for solving the equation 3x + 4y = 18..........(I) and 3y - x=7....(ii), we apply the operation like...

eqn(i) +4eqn(ii)
eqn(i) +3eqn(ii)
eqn(i)+eqn(ii)
eqn(ii)+3eqn(i)

11. The amount of gravity exerted by the earth on the mass 10Kg(g= $9.8m/s^{2}$) is...

9.8 Joule
9.8 Newton
98 Joule
98 Newton

or, For the quadratic function f(Q) = aQ² + bQ + C for real numbers a,b,c, and a ≠0, the maximum value attained at

($\frac{b}{2a}, \frac{4ac - b^2}{4a})$
($-\frac{b}{2a}, \frac{4ac - b^2}{4a})$
($-\frac{b}{2a}, \frac{ b^2 - 4ac}{4a})$
($-\frac{b}{2a}, \frac{ b^2 - 4ac}{4a})$

Group 'B' [ 5 × 8 =40]

12. The binomial expression for two algebraic terms a and x is given as $(a + x)^n$.

Write the binomial theorem for any positive integer n in expansion form.
Write the general term of the expansion.
Write the single term for C(n,r) + C(n,r) + C(n,r-1).
Write any one property of binomial coefficients
How many terms are there in the expression?

13. Given $n^4 < 10^n$ for a fixed positive integer n≥2, prove that $(n+1)^4< 10^{n +1}$ using principle of mathematical induction.

14. a) Evaluate cos($sin^{-1}\frac{3}{5} + sin^{-1}\frac{5}{13})$.

b) Using the vector method, find the area of the triangle with vertices A(1,4,6), B(-2,5,1), and C(1,-1,1).

15. The information given below relates to the advertisement and sales of a departmental store in lakhs of Nepalese rupees.

	Advertisement Expenditure (X)	sales(y)
Arithmetic Mean	20	100
Standard deviation	3	12
	Correlation coefficient between (X) and (Y) = 0.8

a) Find the two regression equations related to the above data.

b) What should be the advertisement expenditure if the department store wants to attain a sales target of Rs.200 Lakhs?

16. Suman and Nikita are studying the application of derivatives and integration in a class. They ask each other the quiz questions as given below. Based on these
questions answer the following.

f'(x) and g'(x) are derivatives of the function f(x) and g(x). What is the expression equal to $\lim_{x \rightarrow a}\frac{f(x)}{g(x)}$ according to L'Hospital's rule for from ∞/∞?
States Rolle's Theorem.
What is the expression equal to$ \int \frac{1}{x^2 +a^2}\mathrm{d}x$
What does 'C' represent in the expression $\int \frac{dx}{3 sin x + 4 cosx}= \frac{1}{5}$ in $ \left.\right|tan(\frac{x}{2}+ \frac{1}{2}tan^{-1}\frac{4}{3}|) + C?$
Write a difference between derivative and antiderivative.

17. Integrate $\int \frac{1}{x^4 - 1}\mathrm{d}x$ using the concept of partial fraction. Also, give an example of proper rational fraction and improper rational fraction.

18. Use simplex method and maximize: Z(x,y)= x + y subject to constraints 2x + 3y ≥ 22, 2x + y ≥ 14, x ≥ 0, y ≥0.

19. Write any one difference between like parallel forces and unlike parallel forces. A heavy uniform beam whose mass is 60 kg is suspended in a horizontal position by two vertical strings each of which can sustain a tension of 52.5 kg wt. How far from the center of the beam must a body of mass 30 kg be placed so that one of the strings may just break?

If the demand function P= $85 - 4Q - Q^2$, find the consumer's surplus at demand 7 units and price 64 units. Also, make a revenue function for the demand equation $P = 20 + 5Q - Q^2$. Obtain the standard quadratic equation for marginal revenue. Q represents the number of units demanded and represents the price.

Group 'C'

20. A mixture is to be made of three foods, A, B, and C which contain nutrients P, Q, and R as shown in the table below. The quantity of P, Q, and R is 45 units, 54 units, and 45 units respectively.

Food	Units of nutrients per kg of the foods
Food	P	Q	R
A	2	2	4
B	3	5	0
C	4	3	5

Express the information in equation form.
Solve the equations using a matrix.
If the cost per kg of foods A, B, C are Rs. 300, Rs. 240, and Rs. 180 respectively, find the total cost of the mixture by matrix method.

21. A-line makes an angle α, β,γ,δ with the four diagonals of a cube kept in a dining room.

Find the direction ratios of any two diagonals of the cube and express the diagonals in vector form.
Find the angle between any two diagonals of the cube.
Prove that $cos^2α + cos^2β + cos^2γ + cos^2δ = \frac{4}{3}$.

22. A college hostel accommodating 1000 students; one of them came from abroad with an infection of coronavirus, then the hostel was isolated. If the rate at which the virus
spreads is assumed to be proportional to the product of the number 'N' of infected students and the number of non-infected students and the number of infected students is 50 after 4 days.
(a) Express the above information in the form of the differential equation.
(b) Solve the differential equation.
(c) Shows that more than 95% of students will be infected after 10 days.

set -2

Group -A [ 11 × 1 =11 Marks]

Rewrite the correct option in your answer sheet.

1. The polar form of the complex number -1 + $\sqrt{3}i$ is

2(cos 60° + i sin 60°)
2(cos 120° + i sin 120°)
2(cos 60° - i sin 60°)
2(cos 240° + i sin 240°)

2. In an election that will be held on a local level in 2079 there are 6 candidates for secretary and 2 for treasurer. Find in how many ways in which the election may turn out.

3. $sin^{-1} \frac{3}{5} $ is equal to

$tan^{-1}(\frac{4}{5})$
$tan^{-1}(\frac{4}{3})$
$tan^{-1}(\frac{3}{5})$
$tan^{-1}(\frac{3}{4})$

4. The vector product of the two vectors $\overrightarrow{i}$ and $\overrightarrow{k}$, where $\overrightarrow{i}, \overrightarrow{j}$, and $\overrightarrow{k}$ are unit vectors along x,y and z axis respectively is

$\overrightarrow{j}$
-1
0
$- \overrightarrow{j}$

5. The locus of a moving point in a plane such that the difference in the distances between two fixed points is constant is

Ellipse
Parabola
Hyperbola
Circle

6. The distance between the points (4, -1, 5 ) and (-4, 3, 6) is

7. The product of two regression coefficients is

1
<1
≤ 1
>1

8. The integrating factor of the linear differential equation $\frac{dy}{dx} $+ py = Q is

e∫p dx
$e^{∫pdx}$
∫p dx
∫e dx

9. . According to the L’Hospital rule the value of

1
$\frac{1}{2}$
2
$\frac{1}{3}$

10. When the Gauss forward elimination method is used for solving the equations 2x +7y = 3…. (1) and 6x – 5y =-17 ….(2), we apply the operation like….

eqn (1) + 6 eqn (2)
eqn (1) - eqn (2)
3eqn (1) - eqn (2)
3eqn (2) + 2 eqn (1)

11. Find the total revenue of the demand function P = 60- 2Q when the quantity supplied Q is 10.

Group -B [ 8 × 5 = 40 marks]

12. The binomial expression is given by (x + $\frac{1 }{x})^{2n}$

Find the general term in the expansion.
Write the term independent of x in the expansion.
Show that its middle term is $\frac{1.35----(2n - 1)}{n!} (-2)^n$.

13. Prove by the principle of mathematical induction that $3^{2n + 2} - 8n - 9$ is divisible by 64 for any positive integer n.

14. a) If $tan^{-1}a + tan^{-1}b + tan^{-1}c = π$, prove that a + b + c = abc

b) Using vector product, find the area of the parallelogram whose diagonals are represented by the vectors $-3\overrightarrow{i} - 2\overrightarrow{j} + \overrightarrow{k}$ and $\overrightarrow{i} + \overrightarrow{2j} + \overrightarrow{3k}$.

15. The following data represents the information regarding the marks of a student of grade XII in
the second terminal examination. Calculate his marks in English when his mark in mathematics is 80.

	Mathematics	English
Mean of marks	90	75
S.D of marks	3	4

The correlation coefficient between the marks of Mathematics and English is- 0.4.

16. Students of grade XII Sulav and Krish are studying the application of derivatives. They ask each other the questions given below. Based on these questions answer the
following

State Lagrange’s Mean Value theorem.
Show that f(x) =$ x(x-1)^2$ in [0,2] satisfies all the conditions of the theorem.
At what angle does the curve y(1 + x)= x cut the x-axis?

17. Use the concept of integration to evaluate the functions:

$\int \frac{dx}{2 + cosx}$
$\int \frac{x^2dx}{(x^2 +a^2)(x^2 + b^2)}$

18. Solve the following LP problem using the simplex method. Maximize z = 5x + 3y subject to

2x + y ≤40

x + 2y ≤ 50

x,y ≥ 0

19. The demand function for a good is P=60-2Q. The fixed cost for a good is Rs. 192 and the variable cost for each additional unit of good is Rs.20.

Write down the equation for total revenue and total cost in terms of Q.
Find the profit function in terms of Q.
Determine the maximum profit

Group-C (3 × 8 = 24)

20. A company produce three products $p_1, p_2$ and $p_3$. These products require three materials $M_1, M_2$ and $M_3$. For one ton of each of these products, the requirements of the materials are given as:

	$M_1$	$M_2$	$M_3$
$p_1$	2	1	3
$p_2$	0	2	2
$p_3$	3	3	0

The total materials consumed are 24, 28, and 28 tons respectively.

Express the information in equation form.
Solve the equations using the matrix inverse method.
If the cost of the three products are Rs.150, Rs.300 and Rs.200 respectively. Find the total cost of the three products by using a matrix.

21. Prove that the angle between two diagonals of a cube is $cos^{-1}(\frac{1}{3})$.

b) Find the equation of plane through P(a,b,c) and perpendicular to OP.

22.a. Find the differential equation of the family of the lines through the origin. Also, solve the differential equation.

b. Find the general solution of the differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{2x^2}$.

c. What is a linear differential equation? Reduce the equation $\frac{dy}{Pdx} + \frac{1}{x}y = y^2$ to linear form and find its integrating factor also.

Set-3

Group-A [ 11 × 1 =11]

Rewrite the correct option in your answer sheet.

1. The quadratic equation $(3K + 1) x^2 + 2(K+1)x + k=0$ may have the reciprocal roots if K is equal to

1
-1
$\frac{1}{2}$
-$\frac{1}{2}$

2. The consistent and independent system of linear equations in two variables has

no solution
unique solution
two solutions
infinite solutions

3. The solution of $sin^{-1}x = cos^{-1}x$ is

$\frac{1}{2}$
$\frac{1}{\sqrt2}$
-$\frac{1}{\sqrt2}$
±$\frac{1}{\sqrt2}$

4. If $\overrightarrow{a}$.$\overrightarrow{b}$=48, $|\overrightarrow{a}|$= 15 and $|\overrightarrow{b}|$=4, then the value of $|\overrightarrow{a} × \overrightarrow{b}|$ is

$5\sqrt{6}$
$6\sqrt{5}$
$5\sqrt{2}$
$2\sqrt{5}$

5. The length of latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}$ = 1, a>b is

$\frac{2a^2}{b}$
$\frac{a^2}{b}$
$\frac{2b^2}{a}$
$\frac{b^2}{a}$

6. The direction cosines of a line equally inclined to the axes of co-ordinates are

$\frac{1}{3}$,$\frac{1}{3}$,$\frac{1}{3}$
$\frac{1}{\sqrt3}$,-$\frac{1}{\sqrt3}$
$\frac{1}{\sqrt3}$,$\frac{1}{\sqrt3}$,-$\frac{1}{\sqrt3}$
±$\frac{1}{\sqrt3}$, ±$\frac{1}{\sqrt3}$, ±$\frac{1}{\sqrt3}$

7. A die is thrown 144 times successively. Getting an even number is assumed to be a success. The mean and the variance of the distribution differed by

8. The degree of the differential equation xy $\frac{d^2y}{dx^2} + x(\frac{dy}{dx})^2 - y\frac{dy}{dx}$=0 is

0
1
2
undefined

9. According to the L'Hospital rule the value of $\lim_{x \rightarrow 0} \frac{xe^x - 10g(x+1)}{x^2}$

1
$\frac{1}{2}$
$\frac{1}{3}$
$\frac{3}{2}$

10. n Gauss forward elimination method used for solving the system of equations, the coefficient matrix is changed into

Unit Matrix
Upper Triangular Matrix
Symmetric Matrix
Lower Triangular Matrix

11. If the resultant of two like parallel forces acting at a distance of 3 m is 80N at a distance of 75cm from one of the forces, then the force is

or,

If the profit function (π) = $Q^2 - 10Q + 9$, then the breakeven point is

9 or 10
1 or 10
1 or 9
4 or 5

Group-B [ 8 × 5 =40]

12. Let 'x' denote the multiplication operate on the set G={1,-1, i,-i} where 'i' is imaginary unit of complex number.

Construct the operation table for G with multiplication operation.
G is closed under multiplication. Give reason.
Show that the operation is commutative.
Find the identity element and inverse of each element in G.
can you conclude that G forms a group? If yes, explain.

13. a) Rewrte 1 in polar form, By using De Moivre's Theorem, find its complex cube roots.

b) A student has to take 5 courses out of 12 courses in a semester of Higher studies. If two courses are compulsory, find the different number of ways the student can take the course.

14. a) Find the general solution of cos x + cos2x + cos3x=0

b) Find the vertes and the eccentricity of the hyperbola defined by $x^2 - 4y^2 - 4x=0.$

15. From the following data

Age in years(x)	5	15	30	45	50	60
Weight in Kg(y)	10	35	50	65	55	45

Compute the following;

Correlation coefficient by Karl Pearson's method.
Line of regression for estimating X on Y and estimating the most probable age of the weight 37 Kg.

16. Given a function y=f(x), answer the following;

Write the condition for the function f(x) to be differentiable.
If the function is differentiable at x=a, show that it is continuous at that point.
When f(x) = x sin$\frac{1}{x}$, show that it is continuous at x=0 but not differentiable.

17. Integrate $\lmoustache$ $\frac{x^2}{(x^2 + 9)(x^2 +4)} dx$ using the concept of partial fraction. What do you mean by a proper rational fraction? Can it be changed into a proper rational fractional? Give an example.

18. Use the simplex method and maximize;

Z(x,y)= 5x + 5y subject to 2x + y ≤ 20,2x + 3y ≤ 24 and x,y ≥ 0.

19. a) Two, unlike parallel forces, the greater of which is 75N, have a resultant 25N. Find the ratio of the distance of the resultant from the component forces.

b) A projectile thrown from a point in a horizontal plane comes back to the plane in 4 seconds. at a distance of 60m in front of the point of projection. Find the velocity of the projection. (g=$10m/s^2$)

State Hawkins-Simon conditions for the viability of the system. The demand and supply curves for an item are given by $p_d = 20 - 3Q -Q^2$ and $P_5 = Q-1$ respectively. Find the difference between consumer and producer surplus at the equilibrium price.

Group -C [ 3 × 8= 24]

20. Given a series $\frac{1^2}{2!} + \frac{2^2}{3!} + \frac{3^2}{4!} $+ .....,

Write the general term of the series.
Write the expansion of $e^x$ where e is the Euler constant.
Write the series for $\frac{1}{e}$.
Find the sum of the above series in terms of e.

21. P (3,-1,2) and Q(5,-7,4) are two points in space.

find the direction cosines of the line through the points.
find the projection of the line PQ along the X-axis.
write the condition for any two planes to be parallel to each other.
Find the equation of the plane through the points P and Q and perpendicular to another
plane 2𝑥 + 𝑦 + 𝑧 = 5.

20. In a laboratory of KWS, the bacteria is cultured. It is assumed that the rate of growth of bacteria is proportional to the number of bacteria present. The bacteria count is 1,00,000 in the beginning and the number increases by 10% every 2 hours. Answer the following questions:

Express the above information in the form of the differential equation.
Solve the differential equation.
After how many hours the bacteria count will reach 2,00,000?

Set - 4

Group -A [ 11 × 1 =11]

Rewrite the correct option in your answer sheet.

1. In how many ways the letter of the world “RAINBOW” can be arranged such that A and I always come together

120
720
1440
5040

2. Which gives the nth term of the series 1.3 + 2.3 + 3.5 + .....

$n^2 + n$
$n^2 + 2n$
$2n^2 + 2n$
$n + n^2$

3. Which of the following describes the general solution of cosx = k, where -1≤ K ≤ 1?

x= 2𝑛𝜋 ± θ
$x= 𝑛𝜋 ± (-1)^nθ$
x=𝑛𝜋 + θ
x= $(2n + 1) \frac{π}{2}$

4. What is the value of sin($cos^{-1}\frac{2}{3})$ ?

$\frac{2}{3}$
$\frac{\sqrt5}{3}$
0.5
$\frac{1}{\sqrt 2}$

5. If a line has direction rations 1,2,-2, which of the following could be the direction cosines of that line:

$\frac{1}{3},\frac{2}{-3}, \frac{2}{-3}$
$\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$
$\frac{1}{3},\frac{2}{3}, -\frac{2}{3}$
$-\frac{1}{3},\frac{2}{3}, \frac{2}{3}$

6. If $|\overrightarrow{a} × \overrightarrow{b}| = 60$, $|\overrightarrow{a}|=12$ and $|\overrightarrow{b}|=10$, what is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$?

60°
45°
30°
0°

7. In a binomial distribution, if the number of trials is 20 times the probability of failure and the probability of failure is $\frac{1}{2}$, what is the mean?

8. What is the integrating factor of the differential equation: $\frac{dy}{dx} + \frac{1}{x} y = x^2$?

$\frac{1}{x}$
$\frac{1}{x^2}$
x
$x^2$

9. What is the slope of the tangent to the curve 2y = $2 - x^2 $at x= 1?

-2
-1
1
$\frac{1}{2}$

10. If the system of linear equation is inconsistent then the number of solutions to the system is:

11. The pull of the earth on a body is 49N and the acceleration due to gravity is 9.8$m/s^2$. What is the mass of the body?

5Kg
9.8Kg
10Kg
49Kg

If the system of linear equations is inconsistent then the number of solutions to the system is:

Group-B [ 8 × 5 =40]

12. a) If $(1 + x)^n = C_0 + C_1 x + C_2 x^2 + ..... + C_nx^n$ , prove that $C_1 + 2C_2 + 3C_3 + ... + nC_n = n2^{n-1}$

b) Find the term that is free of x, in the expansion of , $(2x - \frac{1}{x})^6.$

13. a) Use the principle of mathematical induction to prove 1 + 2 + 3 + ....n = $\frac{n(n+1)}{2}$

b) Prove that the cube roots of unity from a group under multiplication operation.

14. a) solve cos3x + cos x = cos 2

b) Prove, in any triangle, by vector method that $\frac{sinA}{a} = \frac{sin B}{b} = \frac{sin C}{c}$

15. For two variables X and Y the following information is given ∑x = 15,∑y= 25, $∑x^2 =55, ∑y^2=140$, ∑xy = 78, and n=4. From this information, compute the following

The regression coefficient of X on Y and Y on X
Correlation coefficient of X and Y
Regression equation of X on Y and find X, when Y = 10

16. After completing Derivatives and Antiderivatives John asked you the following question. Please answer them;

State Mean Value Theorem

Use L'Hospital's Rule to evaluate $\lim_{x \rightarrow a}\frac{x^n - a^n}{x-a}$

Evaluate $\lim_{x \rightarrow ∞}\frac{dx}{\sqrt{x^2 - 8x + 18}}$

17. a and (a ≠ b) are any real numbers, evaluate the integral;

$\int\frac{dx}{(x-a)^2 (x-b)^3}$ for a =2 and b=3.

18. Use simplex method and maximize; Z(x,y) = 5x + 12y subject to the constraints 3x + y ≤ 12, x + 2y ≤ 12, x ≥ 0 and y ≥ 0.

19. A stone of mass 1 kg falls from the top of a vertical cliff. After (i) falling for 3 seconds and (ii) descending 800 cm, it reaches the foot of the cliff and penetrates 25 cm into the sand. Find the resistance offered by the sand.,(g= $9.8ms^{-2}$)

The total cost (TVC) = $\frac{1}{5} Q^2 + 2Q$ and the total fixed cost (TFC)=20 where Q is the quantity of goods purchased. If the goods are sold at Rs. 7 per good

Express the total cost and total revenue in terms of Q.
Find the number of goods to be sold that the break–even points will occur.
Find the TR and TC at the break-even points

Group -C [ 3 × 8=24]

20. a) use the row-equivalent matrix method to solve the system,

x-y + 2z =0

x- 2y + 3z =-1

2x - 2y + z=-3

b) If one roots of the equation $ax^2 + bx +c =0$ be four times the other, prove that $4b^2 = 25ac.

21. a) Find the direction cosines of the two lines that satisfy the relation

l + m +n =0 and $ l^2 + m^2 -n^2 =0.$

b) Find the coordinate of the vertex, the eccentricity of the ellipse given by $\frac{x^2}{9} - \frac{y^2}{16} = -1$.

c) Find the equation of the plane through the point (3,-4,5) and parallel to the plane 3x - 5y + 5z =7.

22. Today there are 1000 birds on an island. They breed with a constant continuous growth rate of 10%
per year. From the given information

form a differential equation that describes the situation.
solve the differential equation.
after how many years the population of the birds will double.
find the population of birds after 10 years.

Set - 5

Group-A [ 11 × 1=11]

Rewrite the correct option in your answer sheet.

1. In a hostel there are 6 doors. What is the number of ways that students can enter a hostel and leave the
hostel by a different door?

2. The number of terms in the expansion $(1 + x)^2n$ is

n
2n
2n+1
2n-1

3. What is the value of $x^2 + y^2$, Given that, $cos^{-1}x + cos^{-1}y $= $\frac{π}{2}$

0
1
-1
undefined

4. The equation of the ellipse in standard form with its length of the major axis =8 and eccentricity = $\frac{3}{4}$ is:

$7x^2 - 16y^2 =112$
$7x^2 + 16y^2 =112$
$16x^2 - 7y^2 =112$
$7x^2 - 16y^2 = -112$

5. The distance between points (4,3,-6) and (-2,1,3) is;

6.The area of the parallelogram with diagonals $\overrightarrow{a}$ and $\overrightarrow{b}$ is;

$|\overrightarrow{a} × \overrightarrow{b}|$
$2(\overrightarrow{a} × \overrightarrow{b})$
$\frac{1}{2}|\overrightarrow{a} × \overrightarrow{b}|$
$|-\overrightarrow{a} × \overrightarrow{b}|$

7. The correlation coefficient between two variables lies in between

-1< r < 1
0 < r < 1
-1 ≤ r ≤ 1
0 ≤ r ≤ 1

8. The derivative of $e^{cosx}$ is

$-e^{cosx} sinx$
$sinx e^{cosx}$
$e^{cosx}$
$e^{sinx}$

9. $\int \frac{dx}{x^2 + a^2}$ is;

$\frac{1}{a} tan^{-1} \frac{x}{a}$
$tan^{-1}\frac{x}{a}$
$\frac{1}{a} tan^{-1}x$
$\frac{1}{a} tan^{-1} \frac{a}{x}$

10. The degree of the differential equation $\frac{d^2y}{dx^2} + y$ = sin2x is:

11. While solving the system of linear equations by using Cramer’s rule, if the determinant of the coefficient matrix
D=0, then the system may have

Unique solution
No solution
Infinite solutions
No or infinite solutions

Group -B [ 8 × 5 =40]

12. In how many ways can the letters of the word' COMPUTER' be arranged so that

i) all the vowels are together?

ii) the vowels may occupy only odd positions.

13. Prove by principal of mathematical that $3^{2n} - 1 $ is divisible by 8.

14. Evaluate: cos ($sin^{-1}\frac{4}{5} + tab^{-1} \frac{5}{12})$

b) Find the area of a parallelogram whose diagonals are represented by the vector $-3\overrightarrow{i} -2\overrightarrow{j} + \overrightarrow{k}$ and $\overrightarrow{i} +2\overrightarrow{j} + 3\overrightarrow{k}$

15. Find the regression equation of y on x from the following data;

x	2	4	5	6	8	11
y	18	12	10	8	7	5

Estimate the value of y when x=12.

16. After studying the derivative and integration in a class, Ram and Shyam discussed the following
questions. On the basis of these questions, answer the following:

What is the required condition to apply L Hospital’s Rule?
State the Rolle’s Theorem.
What is the expression equal to $\int\frac{dx}{\sqrt{a^2 - x^2}}$
$\int\frac{dx}{\sqrt{a^2 - x^2}} = log(x + \sqrt{x^2 - a^2} +c)= cosh^{-1}\frac{x}{a}$ +C, what does C represents?
Write the difference between differentiation and anti-differentiation.

17. Integrate; $\int\frac{cos x -sin x}{\sqrt{sin2x}}$ dx. Also, give an example of proper rational fraction and improper rational fraction.

18. Using the simplex method to solve the LP problem:

z= 5x + 7y, subject to 2x + 3y ≤ 13, 3x + 2y ≤ 12, x,y ≥0

19. . Write any one difference between like parallel forces and unlike parallel forces. A heavy uniform beam whose mass is 60 kg is suspended in a horizontal position by two vertical strings each of which can sustain a tension of 52.5 kg wt. How far from the center of the beam must a body of mass 30 kg be placed so that one of the strings may just break?

The demand function for a good is P = 60-2Q. The fixed cost for a good is Rs. 192 and the variable cost for
each additional unit of goods is Rs. 20.

Write down the equations for total revenue and total cost in terms of Q.
Find the TR and TC at the break-evenpoints.

Group- C [ 8 × 3 =24]

20. A mixture is to be made of three foods for dinner, P, Q, and R which contain nutrients X, Y, and Z as shown in the table below. The quantity of X, Y, and Z is 45 units, 54 units, and 45 units respectively.

Food	Units of nutrients per kg of food
Food	X	Y	Z
P	2	2	4
Q	3	5	0
R	4	3	5

Express the information in equation form.
Solve the equations using a matrix.
Calculate the total cost of the mixture by matrix method, when the cost per kg of the foods P, Q, and R are Rs.
300, Rs. 240, and Rs. 180 respectively.

21. What are the direction cosines of a line? If a line makes an angle α,β, γ, δ with the four diagonals of a cube, prove that $cos^2α + cos^2β + cos^2γ + cos^2δ = \frac{4}{3}.$ Also, find the angle between any two diagonals of the cube.

22. A hotel accommodating 2000 tourists; one of them came from abroad with an infection of coronavirus, then the hotel was isolated. If the rate at which the virus spreads is assumed to be proportional to the product of the number 'N' of infected tourists and the number of non-infected and the number of infected tourists is 100 after 8 days.

Express the above information in the form of the differential equation.
Solve the differential equation.
Calculate what percentage of the tourists will be infected after 20 days.

Source: Kathmandu World School, Gundu Bhaktapur

Mathmatics model questions class 12

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