Class 11 IMO — Full Syllabus Mock Test

Q1 Logical Reasoning

The equation sin x = 2 has:

The range of sin x is from −1 to 1.

Q2 Logical Reasoning

What is the focal distance of a point (x₁, y₁) on the parabola x² = -4ay?

For x² = -4ay, focus is (0, -a) and directrix is y = a. By definition, focal distance equals distance to directrix. Distance from (x₁, y₁) to y = a is |y₁ - a|.

Q3 Logical Reasoning

Which measure is most affected by extreme values?

Mean uses every observation and is sensitive to outliers.

Q4 Mathematical Reasoning

The value of 10P3 is:

10×9×8 = 720.

Q5 Logical Reasoning

Two sets are given: A = {a, b} and B = {c, d}. Is A × B equal to B × A?

The Cartesian product is generally non-commutative. A × B ≠ B × A unless the sets A and B are identical.

Q6 Mathematical Reasoning

A line cuts x-axis at 6 and y-axis at 3. Its equation is:

x/6 + y/3 =1 ⇒ x+2y=6.

Q7 Mathematical Reasoning

The midpoint of (−4, −2, 6) and (4, 2, −6) is:

Average of corresponding coordinates gives (0,0,0).

Q8 Achievers Section

If lim(x→1) [x⁴ − 1] / [x − 1] = lim(x→k) [x³ − k³] / [x² − k²], find the value of k.

LHS: 4(1)³ = 4. RHS: Divide numerator and denominator by (x−k) to get [3k²] / [2k] = (3/2)k. Equating them: 4 = (3/2)k, so k = 8/3.

Q9 Achievers Section

If f(x)=5x and g(x)=2x, then (f−g)(x) equals:

5x−2x=3x.

Q10 Everyday Mathematics

Distance of point (2,−3) from y-axis is:

Distance from y-axis equals |x|.

Q11 Everyday Mathematics

The length of the longest pole that can be kept in a rectangular warehouse of dimensions 12 m × 9 m × 8 m is:

The longest pole aligns with the main diagonal of the cuboid. Length = √(l² + b² + h²) = √(12² + 9² + 8²) = √(144 + 81 + 64) = √289 = 17 m.

Q12 Mathematical Reasoning

The Arithmetic Mean (AM) between 13 and 19 is:

The AM between two numbers a and b is (a + b)/2. So, (13 + 19)/2 = 32/2 = 16.

Q13 Logical Reasoning

The number of solutions of cos 2x = 1 in the interval [0, 2π] is:

cos 2x = 1 means 2x = 2nπ → x = nπ. For x ∈ [0, 2π], the solutions are 0, π, and 2π. Thus, there are 3 solutions.

Q14 Mathematical Reasoning

Find the general solution of cos(x/2) = 0 (where n ∈ Z).

cos θ = 0 implies θ = (2n + 1)π/2. So x/2 = (2n + 1)π/2 → x = (2n + 1)π.

Q15 Mathematical Reasoning

If R = {(1,a),(2,b),(3,a)}, then the range of R is:

Range consists of actual second components used.

Q16 Everyday Mathematics

An electrician in Chennai defines complex power as S = V × I̅ (where I̅ is the conjugate of current). If voltage V = 10+5i and current I = 2+i, calculate S.

I̅ = 2 - i. S = (10+5i)(2-i) = 20 - 10i + 10i - 5i² = 20 - 5(-1) = 25.

Q17 Mathematical Reasoning

If the sides of a triangle are 3, 5, and 7, what is the measure of its greatest angle?

The greatest angle is opposite the longest side (7). Let this angle be C. cosC = (3² + 5² − 7²)/(2 × 3 × 5) = (9 + 25 − 49)/30 = −15/30 = −1/2. Therefore, C = 120°.

Q18 Mathematical Reasoning

Find the equations of the tangents to the circle x² + y² = 25 that are parallel to the line 3x − 4y = 0.

The given line has slope m = 3/4. Tangents parallel to it have equations y = (3/4)x ± a√(1 + m²). Here a = 5. So, y = 3/4 x ± 5√(1 + 9/16) = 3/4 x ± 5(5/4). Multiplying by 4 gives 4y = 3x ± 25, or 3x − 4y = ± 25.

Q19 Achievers Section

The number of ways to arrange the letters of SUCCESS is:

SUCCESS has 7 letters with S repeated 3 times and C repeated 2 times, so the number of arrangements is 7!/(3!·2!) = 5040/12 = 420.

Q20 Logical Reasoning

Find the left-hand limit of f(x) = |x − 3| / (x − 3) as x approaches 3.

For x < 3, |x − 3| = −(x − 3). So the expression is −(x − 3) / (x − 3) = −1. The left-hand limit is −1.

Q21 Mathematical Reasoning

cos 90° equals:

Standard value.

Q22 Everyday Mathematics

A bicycle wheel is modelled by x²+y²=196. Its radius is:

Radius=√196=14.

Q23 Logical Reasoning

Which of the following is a statement?

A statement is a sentence that is definitely true or false. '2 + 3 = 5' is true and hence a statement.

Q24 Achievers Section

What is the range of the function y = sec²x + cosec²x?

sec²x + cosec²x = 1/cos²x + 1/sin²x = (sin²x + cos²x) / (sin²x cos²x) = 1 / (1/4 sin²2x) = 4 / sin²2x. Since the maximum value of sin²2x is 1, the minimum value of the expression is 4. Range is [4, ∞).

Q25 Everyday Mathematics

In a whispering gallery in a Delhi museum shaped like a semi-ellipse, the length is 100 m and height is 30 m. How far are the foci from the center?

2a=100 => a=50. b=30. c = √(a² - b²) = √(2500 - 900) = √1600 = 40. Distance is 40 m.

Q26 Logical Reasoning

What is the maximum distance from the point (10, 7) to the circle x² + y² − 4x − 2y − 20 = 0?

Centre is (2, 1) and r = √(4 + 1 + 20) = 5. Distance of point to centre d = √((10 − 2)² + (7 − 1)²) = √(64 + 36) = 10. Max distance = d + r = 10 + 5 = 15.

Q27 Logical Reasoning

What is the total number of mutually perpendicular coordinate planes defined within a standard 3D Cartesian space?

There are exactly 3 fundamental mutually perpendicular coordinate planes in standard 3D geometry: the xy-plane, the yz-plane, and the zx-plane.

Q28 Mathematical Reasoning

The derivative of tan x is:

d/dx(tan x)=sec²x.

Q29 Mathematical Reasoning

Find the negation of: 'For every real number x, x² + 1 > 0'.

The negation of 'For every x, P(x)' is 'There exists an x such that ~P(x)'. The negation of '>' is '≤'.

Q30 Logical Reasoning

A truth table for a compound statement involving 3 distinct logical variables (p, q, r) will have how many rows?

Each variable can be True or False (2 possibilities). For n variables, the number of rows is 2ⁿ. For 3 variables, 2³ = 8 rows.

Q31 Mathematical Reasoning

Which of the following is a valid property of the modulus for two complex numbers z₁ and z₂?

The modulus of a product is the product of the moduli. The other options fail the triangle inequality generally.

Q32 Mathematical Reasoning

The standard deviation of the first 10 natural numbers is roughly:

Variance of first n natural numbers is (n² − 1)/12. For n=10, Variance = (100 − 1)/12 = 99/12 = 8.25. Standard deviation = √8.25 ≈ 2.87.

Q33 Everyday Mathematics

The distance of point (3,4) from x-axis is:

Distance from x-axis equals absolute y-coordinate.

Q34 Mathematical Reasoning

The 7th term of the GP 1, 3, 9, 27, ... is:

T₇ = 3⁶ = 729.

Q35 Logical Reasoning

A coin is biased so that a Head is twice as likely to occur as a Tail. What is the probability of getting a Head?

Let P(Tail) = x. Then P(Head) = 2x. Since total probability is 1, x + 2x = 1, so 3x = 1, meaning x = 1/3. P(Head) = 2/3.

Q36 Mathematical Reasoning

The amplitude of y = 3sin x is:

Amplitude is absolute coefficient of sin x.

Q37 Logical Reasoning

Which measure of central tendency is most severely affected by extreme outliers in the data?

The mean uses every single value in its calculation, so a very large or very small extreme value disproportionately pulls the mean in that direction.

Q38 Mathematical Reasoning

The focal distance of a point P(x, y) on the parabola y² = 12x is given by:

For y² = 4ax, the focal distance of a point (x, y) is |x + a|. Here 4a = 12, so a = 3. The focal distance is x + 3 (since x ≥ 0 for this parabola).

Q39 Logical Reasoning

Find the 10th term of the series 3, 7, 13, 21, 31, ... using the method of differences.

Differences: 4, 6, 8, 10 (an AP). tⁿ = a + sum of (n-1) differences = 3 + (n-1)/2[2(4) + (n-2)2] = n² + n + 1. For n=10, 100 + 10 + 1 = 111.

Q40 Mathematical Reasoning

A number is chosen from 1 to 30. Probability it is a prime number is:

There are 10 primes up to 30. Hence 10/30=1/3.

Q41 Everyday Mathematics

A survey of 100 people found 60 use UPI, 45 use cards and 20 use both. How many use at least one payment method?

60+45−20=85.

Q42 Achievers Section

The endpoint of latus rectum of x² = 20y in the first quadrant is:

a = 5. Endpoints are (±10, 5).

Q43 Logical Reasoning

Find all pairs of consecutive even positive integers, both of which are larger than 5, such that their sum is strictly less than 23.

x > 5 and x + x + 2 < 23 → 2x < 21 → x < 10.5. Valid even x: 6, 8, 10. The pairs are (6, 8), (8, 10), and (10, 12).

Q44 Mathematical Reasoning

In ∆ABC, if sides a = 16, b = 24, and c = 20, what is the value of cosB?

By the Cosine Rule, cosB = (a² + c² − b²)/(2ac) = (16² + 20² − 24²)/(2 × 16 × 20) = (256 + 400 − 576)/640 = 80/640 = 1/8.

Q45 Logical Reasoning

If sin θ + cosec θ = 2, then what is the value of sin²θ + cosec²θ?

Squaring both sides of sin θ + cosec θ = 2 gives sin²θ + cosec²θ + 2(sin θ × cosec θ) = 4. Since sin θ × cosec θ = 1, we get sin²θ + cosec²θ + 2 = 4, so sin²θ + cosec²θ = 2.

Q46 Mathematical Reasoning

The coefficient of x⁶ in (1 + 2x)⁷ is:

Coefficient = ⁷C₆×2⁶=7×64=448.

Q47 Everyday Mathematics

The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter is at least 61 cm, find the minimum length of the shortest side.

Let shortest side be x. Longest = 3x. Third = 3x − 2. x + 3x + 3x − 2 ≥ 61 → 7x − 2 ≥ 61 → 7x ≥ 63 → x ≥ 9.

Q48 Everyday Mathematics

A viral message is sent to 4 people. Each of those sends it to 4 new people, and so on. How many people receive the message by the 5th level?

This is a GP: 4 + 16 + 64 + 256 + 1024. Sum S5 = 4(4⁵ − 1)/(4 − 1) = 4(1023)/3 = 4(341) = 1364.

Q49 Everyday Mathematics

The population of bacteria in a lab in Mumbai follows a curve where the instantaneous rate is given by lim(h→0) (2^(3+h) − 8) / h. What is this value?

Rewriting 2^(3+h) as 2³ × 2ʰ = 8 × 2ʰ. The expression is 8(2ʰ − 1)/h. As h approaches 0, (2ʰ − 1)/h approaches log(2). So the result is 8 log(2).

Q50 Mathematical Reasoning

Calculate the distance between the foci of x²/25 + y²/9 = 1.

a=5, e = √(1 - 9/25) = 4/5. Distance = 2ae = 2(5)(4/5) = 8.