Grade 12 Mathematics Model Paper Set-1

Grade 12 Mathematics Model Paper Set-1

As per the new syllabus, NEB Grade 12 Mathematics paper is divided into three groups: Group A which has 11 objective questions with 1 mark weightage of each question, Group B which has 8 long questions with 5 marks each, and Group C having 3 very long questions with weightage of 8 marks for each question.
Download the paper from the link below. Check out our notes here.
Total time- 3 hrs
Full marks: 75

Group ‘A’ (1 × 11 = 11)

Rewrite the correct option in your answer sheet

1. The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by:
(a) 6! x 5!
(b) 6 x 5
(c) 5! x 4!
(d) 7! x 5!

2. If one root of the equation 2 x − px + q = 0 is twice the other, then
(a) 2p = 2q
(b) 4p² = q
(c) 9p² = 2q
(d) 2p² = 9q

3. The value of

sin\left [ \frac{\pi}{3} - sin^{-1}\left (-\frac{1}{2} \right )\right ]

(a) 0
(b) – 1
(c) 1
(d) -1/2

4. If cos pθ = cosqθ , p ≠ q , then
(a) θ = 2nπ
(b) θ = 2nπ/(p-q)
(c) θ = 2nπ/(p+q)
(d) (b) θ = 2nπ/(p±q)

5. The equation to the ellipse whose foci are (±2,0)

\begin{align*} a.\ \frac{x^{2}}{12}+\frac{y^{2}}{16}=1\ \ \ b.\ \frac{x^{2}}{16}+\frac{y^{2}}{12}=1\\ c.\ \frac{x^{2}}{16}+\frac{y^{2}}{8}=1\ \ \ d.\ \frac{x^{2}}{8}+\frac{y^{2}}{16}=1\end{align*}

6. If i, j,k are the three mutually orthogonal unit vectors then i x ( j x k) + j x(k x i) + k x(i x j) equals
(a) 0
(b) 1
(c) – 1
(d) none of these

7. In a certain town, 40% of the people have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. If a person selected at random from the town, has brown hair, the probability that he also has brown eyes is
(a) 1/5
(b) 3/8
(c) 1/3
(d) 2/3

8. If tan^-1(2x/1-x²) then dy/dx =

\begin{align*} a.\ \frac{2}{1+x^{2}}\ \ \ b.\ \frac{2}{1-x^{2}}\\ c.\ \frac{2x}{1+x^{2}}\ \ \ d.\ \frac{2x}{1-x^{2}}\end{align*}

9. The value of the given integral is equal to

\int\frac{1}{\sqrt{a^{2}-x^{2}}} 

\begin{align*} a.\ \frac{1}{a}tan^{-1}\frac{x}{a}+c\ \ \ b.\ tan^{-1}\frac{x}{a}+c\\ c.\ \frac{1}{a}sin^{-1}\frac{x}{a}+c\ \ \ d.\ sin^{-1}\frac{x}{a}+c \end{align*}

10. Gauss Elimination method is used for solving
(a) algebraic equations
(b) exponential equations
(c) trigonometric equations
(d) linear simultaneous equations

11. The least velocity with which a cricket ball can be thrown 10 m horizontally is
(a) 10 ms^-1
(b) 20 ms^-1
(c) 100 ms^-1
(d) 200 ms^-1

OR

According to the principle of dynamics of market price, the rate of change of price is
(a) directly proportional to the excess demand
(b) inversely proportional to the excess demand
(c) directly proportional to the excess supply
(d) none of the above

Group ‘B’ (5 × 8 = 40)

12. If (1+x)ⁿ = C₀ + C₁x + C₂x² + …. + C_nxⁿ, prove that C₀C_n + C₁C_n-1 + … + C_nC₀ = (2n!)/(n! n!).

13. State De-Moivre’s theorem and use it to find the cube roots of unity. Verify the sum of the three roots of unity is zero.

14. (a) If tan^-1x + tan^-1y + tan^-1z = π then show that: x + y + z = xyz
(b) Find the unit vector perpendicular to the vectors i + j −3k and −i −2 j −3k.

15. Define correlation. Find Karl Pearson’s coefficient of correlation of the marks of the following distribution.

16. Find, from the first principle, the derivative of log(tan x).

17. Evaluate:

\int\frac{dx}{1+sinx+cosx} 

18. Using Simplex method, maximize z = 5x +3y subject to 2x + y ≤ 40, x + 2y ≤ 50, x, y ≥ 0.

19. Find the velocity and the direction of projection of a shot that passes in a horizontal direction just over the top of a wall which is 250 m off and 125 m high. (g = 9.8 ms-2)

OR

The demand and supply functions are given by Q_d = -6P_t + 20 and Q_s = 3P_t-1 -16 respectively. Assuming equilibrium, find P_t when P₀ =15.

Group C [8 x 3= 24]

20. (a) Prove by mathematical induction: 1+ 3+ 4 +…+ (2n −1) = n²
    (b) Using Cramer’s rule, solve: x − 2y = −7 and 3x + 7y = 5
    (c) Define abelian group. If (G, *) is an abelian group, prove that (a*b)^-1 = a^-1 * b^-1 for all a,b ∈ G.

21. (a) The direction ratios of two lines are a1 ,b1 ,c1 and a2 ,b2 ,c2 respectively.
    (i) Find the angle between two lines.
    (ii) Find the condition under which two lines are perpendicular.
    (iii) Find the condition under which two lines are parallel.
    (b) Define vector product of two vectors.

22. (a) Solve:

\frac{dy}{dx}+\frac{1+cos2y}{1-cos2x}=0

b. State Mean Value Theorem. Verify it for the function f(x) = 2x² −10x + 29 in [2, 7].