lass 11 Mathematics

Advanced / Olympiad-Level Sample Paper

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

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Section A (1 × 10 = 10 Marks)

Answer briefly.

1. If $A = \{x \in \mathbb{R} : |x-2| < 3\}$A={x∈R:∣x−2∣<3}, write A in interval form.
2. Find the domain of

$$f(x) = \frac{\sqrt{5 – x}}{\sqrt{x^2 – 4}}$$f(x)=x2−4​5−x​​

3. Evaluate:

$$\sin 15^\circ \cos 75^\circ + \cos 15^\circ \sin 75^\circ$$sin15∘cos75∘+cos15∘sin75∘

4. If $\log_3(x-1) + \log_3(x+1) = 2$log3​(x−1)+log3​(x+1)=2, find x.
5. Find the number of subsets of a set containing 6 elements.
6. If $^nP_2 = 56$nP2​=56, find n.
7. Solve: $2^{x+1} = 16$2x+1=16
8. If slope of line joining (1,2) and (k,5) is 3, find k.
9. Find coefficient of $x^3$x3 in expansion of $(1 + x)^5$(1+x)5.
10. Number of solutions of equation $\sin x = 1/2$sinx=1/2 in $[0, 2\pi]$[0,2π].

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Section B (4 × 6 = 24 Marks)

Answer any 6 questions.

11. Solve inequality:

$$\frac{x-1}{x+2} > 0$$x+2x−1​>0

12. Prove that:

$$\tan 3\theta = \frac{3\tan\theta – \tan^3\theta}{1 – 3\tan^2\theta}$$tan3θ=1−3tan2θ3tanθ−tan3θ​

13. Find the general term in the expansion of:

$$\left( x^2 + \frac{1}{x} \right)^6$$(x2+x1​)6

14. Solve the system:

$$x + y + z = 6$$x+y+z=6 $$2x – y + 3z = 14$$2x−y+3z=14 $$x + 3y + 2z = 13$$x+3y+2z=13

15. If A and B are two sets such that
    n(A) = 20, n(B) = 15, n(A ∩ B) = 8,
    find n(A ∪ B).
16. Prove that:

$$\sin^4 x + \cos^4 x = 1 – \frac{1}{2}\sin^2 2x$$sin4x+cos4x=1−21​sin22x

17. Find the locus of point P such that sum of distances from (2,0) and (-2,0) is 6.

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Section C (6 × 6 = 36 Marks)

Answer any 6 questions.

18. Solve:

$$|2x – 3| + |x + 1| = 7$$∣2x−3∣+∣x+1∣=7

19. Prove by induction:

$$1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$13+23+33+⋯+n3=(2n(n+1)​)2

20. If $a, b, c$a,b,c are in A.P., prove that:

$$a^2 + b^2 + c^2 = 3b^2 – 2(ac – b^2)$$a2+b2+c2=3b2−2(ac−b2)

21. Find equation of circle passing through (1,1), (2,3), (4,3).
22. Find the term independent of x in:

$$\left( x^2 + \frac{3}{x} \right)^9$$(x2+x3​)9

23. If 5 men and 4 women sit in a row, find number of arrangements such that no two women sit together.

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Section D (10 × 3 = 30 Marks)

Answer any 3 questions.

24. If $z = x + iy$z=x+iy satisfies $|z – 2| = 3$∣z−2∣=3, find its geometrical representation and plot characteristics.
25. Solve:

$$2\sin^2 x – 5\sin x + 2 = 0$$2sin2x−5sinx+2=0

26. Using binomial theorem, approximate:

$$(1.02)^5$$(1.02)5

(correct up to 4 decimal places)

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

Answer

Class 11 Mathematics – Answer Key

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Section A

Q. No.	Answer
1	(-1, 5)
2	(-∞, -2) ∪ [2, 5]
3	1/2
4	x = √10
5	64
6	n = 8
7	x = 3
8	k = 2
9	10
10	2

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Section B

Q. No.	Answer
11	(-∞, -2) ∪ (1, ∞)
12	Identity Proved
13	$T_{r+1} = \binom{6}{r} x^{12-3r}$Tr+1​=(r6​)x12−3r
14	x = 1, y = 2, z = 3
15	27
16	Identity Proved
17	Ellipse: $\frac{x^2}{9} + \frac{y^2}{5} = 1$9×2​+5y2​=1

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Section C

Q. No.	Answer
18	x = 3, x = -3
19	Proved by induction
20	Proved
21	$x^2 + y^2 – 6x – 4y + 6 = 0$x2+y2−6x−4y+6=0
22	84
23	43,200

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Section D

Q. No.	Answer
24	Circle with centre (2,0) and radius 3
25	x = sin⁻¹(1/2), sin⁻¹(2) rejected → Valid solution: x = π/6, 5π/6
26	1.1041 (approx.)