Class 11 IMO — Full Syllabus Mock Test

Q1 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.

Q2 Mathematical Reasoning

The line with x-intercept 4 and y-intercept 2 is:

x/4 + y/2 =1 ⇒ x+2y=4.

Q3 Everyday Mathematics

A bus fare function f(x)=2x+10 depends on distance x. The fare as distance approaches 20 km is:

Substitute x=20. Fare = 2×20+10 = ₹50.

Q4 Achievers Section

The period of y = cos 3x is:

Period = 2π/3.

Q5 Everyday Mathematics

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

Distance from y-axis equals |x|.

Q6 Mathematical Reasoning

The arithmetic mean of 7 and 19 is:

(7+19)/2 = 13.

Q7 Logical Reasoning

According to De Morgan’s Laws, the complement of the union of two sets, (A ∪ B)', is equal to:

De Morgan's first law states that the complement of a union is the intersection of their complements: (A ∪ B)' = A' ∩ B'.

Q8 Everyday Mathematics

A student using E-learning Hub Yudgam takes two quizzes. The probability of passing Math is 0.7, passing Physics is 0.8, and passing both is 0.6. What is the probability of passing exactly one quiz?

P(exactly one) = P(Math only) + P(Physics only) = [P(Math) − P(Both)] + [P(Physics) − P(Both)] = [0.7 − 0.6] + [0.8 − 0.6] = 0.1 + 0.2 = 0.3.

Q9 Logical Reasoning

If a function f(x) has a derivative f'(x) > 0 for all x in (a, b), then in this interval the function f(x) is strictly:

A positive derivative means the slope of the tangent is positive, indicating that the function values are increasing as x increases.

Q10 Mathematical Reasoning

According to Pascal's rule, 10C4 + 10C5 is equal to:

Pascal's rule states nCr + nC(r−1) = (n+1)Cr. Therefore, 10C5 + 10C4 = 11C5.

Q11 Mathematical Reasoning

The line y = mx + c touches the ellipse x²/a² + y²/b² = 1 if:

Standard condition for tangency to an ellipse is c = ±√(a²m² + b²), hence c² = a²m² + b².

Q12 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.

Q13 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.

Q14 Logical Reasoning

Which among the strictly following defined spatial points sits nearest to the absolute spatial origin (0, 0, 0)?

Compare the squared absolute distances: A is 4+4+1 = 9. B is 9. C is 1+1+1 = 3. D is 0+4+4 = 8. The smallest squared distance is 3, proving (1, 1, 1) is absolutely closest.

Q15 Achievers Section

The solution set of all real numbers x satisfying x < 1 or x > 4 is:

Values less than 1 or greater than 4 form two separate intervals.

Q16 Achievers Section

A light ray passing through (1, 2) reflects on the x-axis at point A. The reflected ray passes through (5, 3). Find the x-coordinate of A.

Image of (1,2) across x-axis is (1,−2). The reflected ray lies on the line joining (1,−2) and (5,3). Equation: y+2 = (5/4)(x−1) → 4y+8 = 5x−5. At x-axis, y=0 → 8 = 5x−5 → 5x=13 → x=13/5.

Q17 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).

Q18 Everyday Mathematics

In a class, 18 students like Maths, 12 like Science and 5 like both. If one student is selected from 25 students, probability that the student likes Maths or Science is:

18+12−5=25 students like at least one subject.

Q19 Mathematical Reasoning

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

T₇ = 3⁶ = 729.

Q20 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.

Q21 Mathematical Reasoning

The number of permutations of 5 distinct objects taken 3 at a time is:

5P3 = 5!/(2!) = 60.

Q22 Logical Reasoning

The equation sin x = 2 has:

The range of sin x is from −1 to 1.

Q23 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.

Q24 Everyday Mathematics

The median of ₹120, ₹140, ₹150, ₹170, ₹200 is:

Middle value is ₹150.

Q25 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.

Q26 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.

Q27 Logical Reasoning

If f(x)=5 for all x≠2, then lim(x→2) f(x) equals:

A constant function has limit equal to the constant value.

Q28 Logical Reasoning

If |x + 2| ≤ 9 and x is an integer, how many solutions exist?

−9 ≤ x + 2 ≤ 9 → −11 ≤ x ≤ 7. Number of integers = 7 − (−11) + 1 = 19.

Q29 Mathematical Reasoning

Express cos 2x in terms of tan x.

cos 2x = cos²x − sin²x. Rewriting as (cos²x − sin²x) / (cos²x + sin²x). Dividing numerator and denominator by cos²x gives (1 − tan²x) / (1 + tan²x).

Q30 Mathematical Reasoning

The slope of the line passing through points (2, 3) and (6, 11) is:

m = (11−3)/(6−2) = 8/4 = 2.

Q31 Mathematical Reasoning

Evaluate lim(x→0) (tan4x)/(tan2x).

Using tan(kx) ≈ kx near 0, limit = 4/2 = 2.

Q32 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.

Q33 Everyday Mathematics

In a group of 50 students, 28 study Hindi, 22 study Sanskrit and 8 study both. How many study at least one language?

28+22−8=42.

Q34 Mathematical Reasoning

The foci of x²/16 + y²/64 = 1 are:

c² = 64 − 16 = 48, so c = 4√3.

Q35 Everyday Mathematics

A wheel of radius 14 cm rotates through π/2 radians. Distance travelled by a point on rim is:

Arc length = rθ =14×π/2=7π.

Q36 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.

Q37 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.

Q38 Everyday Mathematics

In how many ways can 7 beads of different colours be strung to form a necklace?

For a necklace, clockwise and anticlockwise arrangements are indistinguishable. The formula is (n - 1)! / 2. So, 6! / 2 = 720 / 2 = 360.

Q39 Mathematical Reasoning

A letter is chosen from the word PROBABILITY. Probability that it is a vowel is:

Vowels are O,A,I,I,I = 5 out of 11 letters.

Q40 Logical Reasoning

Which function is even?

cos(−x)=cosx.

Q41 Everyday Mathematics

Ravi hits a cricket ball whose path is modeled by y = 2x - 0.1x² (x and y in meters). Find the maximum height reached by the ball.

The path is y = -0.1(x² - 20x). Completing the square gives y = -0.1(x - 10)² + 10. The vertex is at (10, 10). The maximum height is the y-coordinate of the vertex, which is 10 m.

Q42 Mathematical Reasoning

For a triangle ABC, if a = 5, b = 6, and c = 7, the area is 6√6. What is the length of the circumradius R?

The area ∆ can be expressed as ∆ = abc / 4R. Therefore, R = abc / 4∆. Substituting the given values: R = (5 × 6 × 7) / (4 × 6√6) = 210 / (24√6) = 35 / (4√6).

Q43 Mathematical Reasoning

The general solution of the equation sin 2x = 0 is (where n ∈ Z):

sin θ = 0 means θ = nπ. Hence, 2x = nπ → x = nπ/2.

Q44 Logical Reasoning

If |x + 2| ≤ 9 and x is an integer, how many solutions exist?

−9 ≤ x + 2 ≤ 9 → −11 ≤ x ≤ 7. Number of integers = 7 − (−11) + 1 = 19.

Q45 Logical Reasoning

If A ⊂ B, then what is A ∪ B?

If A is a subset of B, all elements of A are already in B. Taking the union adds no new elements to B, so A ∪ B = B.

Q46 Mathematical Reasoning

Find the equations of the directrices for x²/16 + y²/9 = 1.

a=4, e=√7/4. Directrices are x = ±a/e = ±4 / (√7/4) = ±16/√7.

Q47 Mathematical Reasoning

Distance between (4,4,4) and (1,0,0) equals:

Distance = √(3²+4²+4²)=√41.

Q48 Achievers Section

The foci of x²/50 + y²/18 = 1 are:

√32 = 4√2, so both forms are correct.

Q49 Mathematical Reasoning

The probability of an impossible event is:

An impossible event contains no favorable outcomes, making its probability 0 / n(S) = 0.

Q50 Logical Reasoning

The equation cos x = −2 has:

cos x cannot be less than −1.