Class 12 Mathematics

Advanced Sample Paper (Olympiad-Oriented)

Time: 3 Hours
Maximum Marks: 100
Difficulty Level: High

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Section A (1 × 10 = 10 Marks)

Answer briefly. Each question carries 1 mark.

1. If $f(x) = |x^2 – 4x + 3|$f(x)=∣x2−4x+3∣, find the number of points where $f(x)$f(x) is non-differentiable.
2. Evaluate:

$$\lim_{x \to 0} \frac{\tan(3x) – 3\tan x}{x^3}$$x→0lim​x3tan(3x)−3tanx​

3. If vectors $\vec{a}, \vec{b}, \vec{c}$a,b,c are coplanar and non-zero, then find the value of

$$\vec{a} \cdot (\vec{b} \times \vec{c})$$a⋅(b×c)

4. Find the order and degree of the differential equation:

$$\left( \frac{d^2y}{dx^2} \right)^3 + \left( \frac{dy}{dx} \right)^2 + y = 0$$(dx2d2y​)3+(dxdy​)2+y=0

5. If $A$A is a 3×3 matrix such that $A^3 = I$A3=I and $A \neq I$A=I, then what is $\det(A)$det(A)?
6. If the random variable X follows binomial distribution with mean 5 and variance 2.5, find n.
7. Evaluate:

$$\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$$∫0π/2​sinx+cosxsinx​dx

8. Find the equation of the tangent to the curve $y = x^x$y=xx at x = 1.
9. If $y = \sin^{-1}(2x\sqrt{1-x^2})$y=sin−1(2×1−x2​), find $\frac{dy}{dx}$dxdy​.
10. Number of real solutions of equation:

$$e^x = x^2$$ex=x2

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Section B (4 × 6 = 24 Marks)

Answer any 6 questions. Each carries 4 marks.

11. Show that the function

$$f(x) = x^3 – 6x^2 + 9x + 1$$f(x)=x3−6×2+9x+1

has exactly two critical points and determine their nature.

12. Solve the differential equation:

$$\frac{dy}{dx} + y\tan x = \sin x$$dxdy​+ytanx=sinx

13. Using vector method, find the shortest distance between the lines:

$$\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{1}$$2x−1​=−1y+1​=1z​ $$\frac{x}{1} = \frac{y-2}{2} = \frac{z+1}{-2}$$1x​=2y−2​=−2z+1​

14. Evaluate:

$$\int \frac{x^2}{\sqrt{a^2 – x^2}} dx$$∫a2−x2​x2​dx

15. If A is a 3×3 matrix and

$$A^2 – 5A + 6I = 0$$A2−5A+6I=0

find $A^{-1}$A−1 in terms of A.

16. A bag contains 5 red and 4 blue balls. Two balls are drawn successively without replacement. Find the probability that both are red given that at least one is red.
17. Prove that:

$$\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{\sqrt{3}}{8}$$sin20∘sin40∘sin80∘=83​​

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Section C (6 × 6 = 36 Marks)

Answer any 6 questions. Each carries 6 marks.

18. Find the area enclosed between the curves:

$$y = x^2 \quad \text{and} \quad y = 2x + 3$$y=x2andy=2x+3

19. Using Lagrange’s Mean Value Theorem, prove that:

$$\ln(1+x) < x \quad \text{for } x > 0$$ln(1+x)<xfor x>0

20. Solve:

$$(x + y)^2 dx + x^2 dy = 0$$(x+y)2dx+x2dy=0

21. If the system of equations:

$$x + y + z = 3$$x+y+z=3 $$2x – y + kz = 1$$2x−y+kz=1 $$x + 2y + 3z = 4$$x+2y+3z=4

has infinitely many solutions, find k.

22. A die is thrown repeatedly until a 6 appears. Find the expected number of throws.
23. Prove that vectors

$$\vec{a} + \vec{b}, \quad \vec{b} + \vec{c}, \quad \vec{c} + \vec{a}$$a+b,b+c,c+a

are coplanar if and only if$$\vec{a} + \vec{b} + \vec{c} = 0$$a+b+c=0

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Section D (10 × 3 = 30 Marks)

Answer any 3 questions. Each carries 10 marks.

24. Investigate the monotonicity, local maxima, local minima, concavity and points of inflection of:

$$f(x) = x^4 – 4x^3 + 6x^2$$f(x)=x4−4×3+6×2

25. Evaluate:

$$\int_0^1 \frac{\ln(1+x)}{1+x^2} dx$$∫01​1+x2ln(1+x)​dx

26. Using eigenvalue method, solve the system:

$$\frac{dx}{dt} = 3x + 4y$$dtdx​=3x+4y $$\frac{dy}{dt} = -4x + 3y$$dtdy​=−4x+3y

Disclaimer: This is an independently created practice paper for educational use only and is not affiliated with any official Olympiad organization.

Answer

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📘 Answer Key (Tabular Format)

Section A

Q. No.	Answer
1	2 points
2	8
3	0
4	Order = 2, Degree = 3
5	1
6	n = 10
7	π/4
8	y = x
9	2
10	2 real solutions

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Section B

Q. No.	Answer
11	Critical points: x = 1 (local max), x = 3 (local min)
12	y = sin x + C cos x
13	Shortest distance = 1 unit
14	$\frac{a^2}{2} \sin^{-1}(x/a) – \frac{x}{2}\sqrt{a^2-x^2} + C$2a2​sin−1(x/a)−2x​a2−x2​+C
15	$A^{-1} = \frac{5I – A}{6}$A−1=65I−A​
16	5/9
17	$\sqrt{3}/8$3​/8

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Section C

Q. No.	Answer
18	Area = 125/6 sq. units
19	Proved using LMVT
20	$x^2 + 2xy = C$x2+2xy=C
21	k = 1
22	6
23	Proved

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Section D

Q. No.	Answer
24	Increasing: (3, ∞); Decreasing: (-∞, 0) ∪ (0,3); Local min at x=0 & x=3; No local max
25	π ln2 / 8
26	$x = e^{3t}(C_1\cos4t + C_2\sin4t)$x=e3t(C1​cos4t+C2​sin4t), $y = e^{3t}(-C_1\sin4t + C_2\cos4t)$y=e3t(−C1​sin4t+C2​cos4t)