Class 11 IMO — Full Syllabus Mock Test

Q1 Logical Reasoning

The equation cos x = −2 has:

cos x cannot be less than −1.

Q2 Everyday Mathematics

A rubber ball is dropped from a height of 10 metres. Each time it bounces, it rises to 3/4 of its previous height. The total distance covered by the ball before coming to rest is:

Distance = 10 + 2 × (10 × 3/4) + 2 × (10 × (3/4)²) + ... = 10 + 15 / (1 − 3/4) = 10 + 15 / (1/4) = 10 + 60 = 70 m.

Q3 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).

Q4 Logical Reasoning

Which function is odd?

sin(−x)=−sinx.

Q5 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.

Q6 Mathematical Reasoning

The coordinates of the midpoint of A(−2, 3, 5) and B(4, 1, 7) are:

Midpoint = ((−2+4)/2, (3+1)/2, (5+7)/2) = (1, 2, 6).

Q7 Everyday Mathematics

A road is represented by line y = 2x + 1. Another road parallel to it passes through (0,5). Its equation is:

Parallel lines have same slope 2.

Q8 Mathematical Reasoning

The term independent of x in the expansion of (x + 1/x)¹⁰ is:

T_(r+1) = 10Cr x^(10−r) (x⁻¹)ʳ = 10Cr x^(10−2r). For x⁰, 10−2r = 0 → r = 5. Term = 10C5 = 252.

Q9 Mathematical Reasoning

In ∆ABC, the value of a(cosB cosC + sinB sinC) evaluates to:

The term (cosB cosC + sinB sinC) is the trigonometric expansion for cos(B−C). Thus, the expression simply translates to a cos(B−C).

Q10 Mathematical Reasoning

cos 90° equals:

Standard value.

Q11 Mathematical Reasoning

A parabola has directrix y = -6 and vertex at origin. Its focus is:

Focus is (0, a) where a = 6.

Q12 Mathematical Reasoning

Which point is a vertex of x²/64 + y²/49 = 1?

a = 8, so vertices are (±8,0).

Q13 Logical Reasoning

If f(x)=x² and h=0.01, then the average rate of change from x=1 to x=1+h is closest to:

[(1.01)²−1]/0.01=2.01.

Q14 Everyday Mathematics

Aarav is flying a kite in Jaipur. The string makes an angle θ with the ground such that 2sin²θ = 1. If θ is acute, what is the angle?

2sin²θ = 1 → sin²θ = 1/2. For an acute angle, sin θ = 1/√2, which corresponds to 45°.

Q15 Mathematical Reasoning

If z = 2 − 2i, then arg(z) is:

Point is in fourth quadrant.

Q16 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.

Q17 Logical Reasoning

What is the principal argument of any positive real number?

Positive real numbers lie on the positive x-axis, which corresponds to an angle of 0 radians.

Q18 Mathematical Reasoning

The length of the latus rectum of x² = 24y is:

x² = 4ay gives a = 6. Length of latus rectum = 4a = 24.

Q19 Everyday Mathematics

A bus in Delhi travels a displacement represented by z₁ = 3+4i km. Another bus travels z₂ = 6-2i km. What is the straight-line distance between their final positions?

Distance = |z₁ - z₂| = |(3+4i) - (6-2i)| = |-3 + 6i| = √( (-3)² + 6² ) = √(9 + 36) = √45 = 3√5 km.

Q20 Achievers Section

If P(A)=3/5, then odds in favour of A are:

Odds in favour = P(A):P(A') = 3/5:2/5 = 3:2.

Q21 Mathematical Reasoning

Calculate lim(x→0) (sin 2x + sin 3x) / (2x).

Separate the fractions: [ (sin 2x)/(2x) ] + [ (sin 3x)/(2x) ]. First limit is 1. Second can be written as (3/2) × [ (sin 3x)/(3x) ] = (3/2) × 1. Sum = 1 + 3/2 = 5/2.

Q22 Logical Reasoning

The compound statement (p → q) ∧ (q → p) is logically equivalent to:

The biconditional statement 'p if and only if q' strictly means 'if p then q' AND 'if q then p'.

Q23 Everyday Mathematics

A population grows by 1% annually. Using approximation, (1.01)¹⁰ is about:

(1+x)¹⁰≈1+10x=1.10.

Q24 Logical Reasoning

Find the foci of 25x² + 9y² = 225.

x²/9 + y²/25 = 1. Vertical ellipse. c = √(25-9) = 4. Foci are (0, ±4).

Q25 Mathematical Reasoning

What is the area of an equilateral triangle inscribed in the circle x² + y² = a²?

The circumradius of the triangle is R = a. For an equilateral triangle, side length s = R√3 = a√3. Area = (√3/4)s² = (√3/4)(3a²) = (3√3/4)a².

Q26 Mathematical Reasoning

The centre of the circle x² + y² − 12x + 14y + 24 = 0 is:

Centre=(−g,−f)=(6,−7).

Q27 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.

Q28 Achievers Section

The standard deviation of data 1, 3, 5 is:

Standard deviation = √(variance) = √(8/3).

Q29 Everyday Mathematics

A clock pendulum in a Kolkata museum swings creating an angle θ from the vertical satisfying 2cos²θ − √3 cos θ = 0. Find the maximum acute angle of swing.

cos θ(2cos θ − √3) = 0. cos θ = 0 (θ = 90°) or cos θ = √3/2 (θ = 30°). The pendulum doesn't swing to 90° typically; the acute swing angle here is 30°.

Q30 Everyday Mathematics

A pizza of radius 21 cm has a sector of angle 60°. Its area is:

Area=(60/360)×π×21²=231 cm² using π=22/7.

Q31 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.

Q32 Logical Reasoning

Which of the following is true for the Cartesian product of empty sets?

By definition, the Cartesian product of any set with an empty set is the empty set. There are no ordered pairs to form.

Q33 Mathematical Reasoning

Solve 7x + 3 < 5x + 9, and represent the solution in interval notation for x ∈ R.

2x < 6 → x < 3. The interval is (−∞, 3).

Q34 Mathematical Reasoning

The negation of the statement 'All equilateral triangles are isosceles' is:

The negation of a universal statement 'For all x, P(x)' is an existential statement 'There exists an x such that not P(x)'.

Q35 Mathematical Reasoning

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

m = (11−3)/(6−2) = 8/4 = 2.

Q36 Logical Reasoning

The axis of symmetry of y² = 18x is:

The parabola y² = 4ax is symmetric about the x-axis.

Q37 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.

Q38 Logical Reasoning

The locus of a point for which y = 0 and z = 0 is:

When both y and z coordinates are zero, the x-coordinate can be any real number. This is the exact definition of the x-axis.

Q39 Mathematical Reasoning

The value of 20C19 is:

20C19 = 20C1 = 20.

Q40 Logical Reasoning

What is lim(x→0) [ 1 / x ] ?

As x approaches 0 from the right, 1/x approaches +∞. As x approaches 0 from the left, 1/x approaches −∞. Since LHL ≠ RHL, the limit does not exist.

Q41 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.

Q42 Achievers Section

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

5x−2x=3x.

Q43 Logical Reasoning

Which parabola opens towards the left?

Negative coefficient in y² = 4ax means opening left.

Q44 Mathematical Reasoning

If A = [−2, 5] and B = (2, 7], then A ∩ B is the interval:

The intersection looks for overlapping real numbers. The overlap between −2 ≤ x ≤ 5 and 2 < x ≤ 7 is 2 < x ≤ 5, which is written as (2, 5].

Q45 Achievers Section

The symmetric difference of two sets A and B, denoted by A Δ B, is mathematically equivalent to:

The symmetric difference is the set of elements in A or in B, but not in both. This can be expressed as (A ∪ B) − (A ∩ B) or (A − B) ∪ (B − A).

Q46 Everyday Mathematics

A stationery shop owner in Mumbai has sets of pens. Set A has 10 Cello pens, Set B has 10 Reynolds pens. Sets A and B are:

Since both sets have exactly 10 items, their cardinality is the same [n(A) = n(B) = 10]. However, the items (elements) are different. Thus, they are equivalent but not equal.

Q47 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.

Q48 Everyday Mathematics

Distance between (0,0,0) and (6,8,0) is:

Distance = √(6²+8²+0²)=√100=10.

Q49 Everyday Mathematics

The correct radius of the park is:

Radius²=25, therefore radius=5.

Q50 Mathematical Reasoning

The line through origin making inclination 135° is:

Slope = tan135° = −1.