Following is the model question paper issued by the National Examination Board for Grade XI (11) students of Nepal after the revised and updated curriculum has been adopted in the teaching system.

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GRADE XI
Mathematics
Model questions
Time: 3 hrs
Full marks: 75

Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. 

 Attempt all questions

Group A

Rewrite the correct option in your answer sheet.

1. Which one of the following is a statement?

  1. The fishes are beautiful.
  2. Study mathematics.
  3. x is a capital of country y.
  4. Water is essential for health.
Answer: d (because the statement is true and not false at the same time).

2. The value of $\sqrt{-16} * \sqrt{-25}$ is
  1. -20
  2. -20 i
  3. 20 i
  4. 20
Answer: a

3. If $< C = 60^O$, b = 5cm, and a = 4 cm of $\triangle$ ABC, what is the value of c?
  1. 3.58 cm
  2. 4.58 cm
  3. 4.89 cm
  4. 4.56
4. In a triangle ABC, B = 120$^O$, a = 1, c = 1 then the other angles and sides are
  1. 35, 45, $\sqrt{2}$
  2. 10, 50, $\sqrt{3}$
  3. 20, 40, 2
  4. 30, 30, $\sqrt{3}$
5. The cosine of the angle between the vectors $\vec{a} = \vec{i} - 2\vec{j} + 3 \vec{k}$ and $\vec{b}= \vec{i} + 3 \vec{j} + 3 \vec{k}$ is
  1. $\dfrac{1}{14}$
  2. 14
  3. $\sqrt{14}$
  4. 196
6. The equation of parabola with the vertex at the origin and the directrix y - 2= 0 is
  1. $x^2 - 8 y = 0$
  2. $y^2 + 8y = 0$
  3. $x^2 + 8y = 0$
  4. $y^2 - 8y = 0$
Answer: a

7. A mathematical problems is given to three students Sumit, Sujan and Rakesh whose chance of solving it are $\dfrac{1}{2}, \dfrac{1}{3} \ and \ \dfrac{1}{a}$ respectively. The probability that the problem is solved is $\dfrac{3}{4} ? The possible values of a are:
  1. $\dfrac{9}{2}$
  2. 4
  3. $\dfrac{1}{4}$
  4. $\dfrac{1}{8}$
8. $\displaylines{ lim \\ \theta \to 0} \dfrac{sin \theta}{ \theta}$ is equal to
  1. 0
  2. $\infty$
  3. 1
  4. $\dfrac{0}{0}$
9. The derivates of $\dfrac{4x^2 + 3}{3x^2 - 2}$ is
  1. $\dfrac{-34x}{(3x^2 - 2)^2}$
  2. $\dfrac{30x^2}{3x^2 - 2}$
  3. $\dfrac{-32x}{(3x^2 - 2)^3}$
  4. $\dfrac{-31x}{(3x-2)^2}$
10. By Newton's Raphson, the positive root of $x^3 - 18 = 0$ in (2,3) is
  1. 2.666
  2. 2.621
  3. 2.620
  4. 2.622
11. Two forces acting at an angle of 45$^O$ have a resultant equal to $\sqrt{10}$N, if one of the forces be $\sqrt{2}$N, what is the other force
  1. 1N
  2. 2N
  3. 3N
  4. 4N
OR

The total cost function of a producer is given as C = 500 + 30Q + $\dfrac{1}{2}Q^2$. What is the marginal cost (MC) at Q = 4 is
  1. Rs 38
  2. Rs 34
  3. Rs 30
  4. Rs 28

Group B (5 * 8 = 40)

12. A function $f(x) = x^2$ is given. Answer the following questions for the function f(x).
  1. What is the algebraic nature of the function?
  2. Write the name of the locus of the curve.
  3. Write the vertex of the function.
  4. Write any one property for sketching the curve.
  5. Write the domain of the function.
13. Compare the sum of n terms of the series: $1 + 2a + 3a^2 +4a^3 + ...$ and $a + 2a + 3a + 4a+...$ up to n terms.

14. a) In any triangle, prove that: (b+c) sin $\dfrac{A}{2}$ = a sin $\left ( \dfrac{A}{2} +B \right )$. (3)

b) Express $\vec{r}$ = (4,7) as the linear combination of $\vec{a}$ = (5,-4) and $\vec{b}$ = (-2,5). (2)

15. Calculate the appropriate measure of Skewness for the data below.


16. Define different types of discontinuity of a function. Also write the conditon for increasing, decreasing and concavity of function. (2+3)

17. Evaluate $\int \dfrac{x^2 dx}{\sqrt{a^2 - x^2}}$

18. Define Trapezoidal rule. Evaluate using Trapezoidal rule for $\int^1_0 \dfrac{dx}{1 + x} n = 4$.

19. State sine law and use it to prove Lami's theorem.

OR

A decline in the price of good X by Rs. 5 causes an increase in its demand by 20 units to 50 units. The new price is X is 15. 

(i) Calculate elasticity of demand.

(ii) The elasticity of demand is negative, what does it mean?

Group C

20 a) The factor of expression $\omega ^3 - 1$ are $\omega - 1$ and $\omega^2 + \omega + 1$. If $\omega^3 -1  = 0$

(i) Find the possible values of $\omega$ and write the real and imaginary roots of $\omega$. (2)

(ii) Prove that $ \left | \displaylines{ 1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1 } \right | = 0.$ Where n is positive integer. (4)

(b) Verify that: $| x + y | \leq |x| + |y|$ with x = 2 and y = -3. (2)

21 a) The single equation of pair of lines $2x^2 + 3xy + y^2 +5x + 2y - 3 =0$.

(i) Find the equation of pair of straight lines represented by the single equation. (4)

(ii) Are the pair of lines represented by the equation pass through origin? Write with reason. (1)

(iii) Find the points of intersection of the pair of lines. (2)

(b) If three vector $\vec{a}, \vec{b} \ and \ \vec{c}$ are mutually perpedincudar units vectors in space then write a relation between them. (1)

22 (i) Distinguish between derivative and anti-derivative of a function. Write their physical meanings and illustrate with example in your context. Find, the differntial coefficient of $\text{log} \sin x$ with respect to x. (1+2+2)

(ii) Find the area bounded by the y-axis, the curve $x^2 = 4 (y-2)$ and the line $y=11$. (3)



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