⚡ Quick quiz

Test yourself in under a minute: Class 11 Math Practice Test

From counting permutations to evaluating a trigonometric limit, this set targets the pre-calculus skills Class 11 Math is built on.

1. 1. A student says the greatest integer function of 5.1 is 6, since that's the 'closest' integer. What's wrong with this reasoning?

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   A) 3.7 has no greatest integer valueB) The greatest integer function always rounds down, never to the nearest valueC) Nothing is wrong, 4 is correctD) The function only works for whole numbers

2. 2. A student identifies a conic equation with a minus sign between its squared terms. What feature should they expect in its graph?

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   A) The sign only affects the size, not the shape, of the curveB) The minus sign means the equation has no valid graph at allC) Both equations always describe the same shapeD) The plus sign creates a closed, bounded curve (ellipse); the minus sign allows two unbounded branches (hyperbola)

3. 3. A position function s(t) has a derivative ds/dt that equals zero at a certain instant t. What does this tell you about the object's motion at that moment?

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   A) This situation is mathematically impossibleB) The object's instantaneous velocity is zero at that momentC) The object has stopped moving permanentlyD) The object is moving at its maximum possible speed

4. 4. A geometric progression has infinitely many terms and a common ratio of 1.5. Does this series have a finite sum to infinity?

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   A) Yes, every GP has a finite sum to infinityB) No, the sum to infinity formula only applies when |r| < 1C) Yes, because 1.5 is a small numberD) No GP can ever have a finite sum to infinity

5. 5. A quadratic equation with real coefficients has one root equal to 3 + 4i. What must its other root be?

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   A) There isn't enough information to know the other rootB) 3 - 4iC) -3 - 4iD) 3 + 4i (the same root repeated)

6. 6. A student calculates the number of ways to choose a 3-person committee using the permutation formula nPr instead of nCr. What error does this introduce?

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   A) It undercounts, since some valid committees get left outB) It overcounts, since different orderings of the same 3 people get treated as distinct selectionsC) There's no error, nPr and nCr give the same result for committeesD) It only causes an error when n is very large

7. 7. A student uses π as if it meant '180' directly inside a formula that expects a pure number. What is actually true about π radians?

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   A) π only applies to circles, never to angles in formulasB) π radians and 180 degrees are unrelated quantitiesC) π radians equals exactly 180 degrees, a specific angle valueD) π is always just the number 180 in any context

8. 8. Two lines have slopes 2 and -1/2 respectively. What can you conclude about how these two lines are positioned relative to each other?

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   A) They are parallel to each otherB) They must intersect at the originC) No conclusion can be drawn from the slopes aloneD) They are perpendicular to each other

9. 9. A probability calculation gives a final answer of 1.2. What does this tell you about the calculation?

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   A) The answer just needs to be rounded down to 1B) The event is very likely to happenC) There must be an error, since probability values can never exceed 1D) This is a valid, unusually high probability

Sets & Sequences

Covers set union, subset counting, overlapping-group word problems, and arithmetic/geometric progression sums and terms.

1. 1. If Set A contains elements 1, 2, 3 and Set B contains elements 3, 4, 5, find the union of Set A and Set B.

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   ▶Reveal answer

   Set with elements 1, 2, 3, 4, 5🔊

   The union of two sets combines all unique elements present across both sets. Common elements like 3 are listed only once, resulting in the set containing 1, 2, 3, 4, 5.

2. 2. Find the sum of the first 10 terms of the arithmetic progression: 3, 7, 11, 15...

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   ▶Reveal answer

   210🔊

   Here, first term a = 3 and common difference d = 4. The sum formula is S_n = (n/2) × (2a + (n - 1)d). For n = 10: S_10 = 5 × (2(3) + 9(4)) = 5 × (6 + 36) = 5 × 42 = 210.

3. 3. Find the number of proper subsets of a set that has 6 elements.

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   ▶Reveal answer

   63🔊

   A set with n elements has 2^n total subsets; excluding the set itself leaves 2^n - 1 proper subsets, so 2^6 - 1 = 63.

4. 4. In a class, 20 students play cricket, 28 students play football, and 36 students play at least one of the two games. Find the number of students who play both games.

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   ▶Reveal answer

   12🔊

   By the formula n(A∪B) = n(A) + n(B) − n(A∩B), rearranging gives n(A∩B) = 20 + 28 − 36 = 12 students who play both games.

5. 5. Find the sum to infinity of the geometric series 12 + 4 + 4/3 + 4/9 + ...

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   ▶Reveal answer

   18🔊

   This is an infinite GP with first term a = 12 and common ratio r = 1/3 (|r| < 1), so the sum to infinity is a/(1-r) = 12/(1 - 1/3) = 12/(2/3) = 18.

6. 6. The first term of a geometric progression is 3 and the common ratio is 2. Find the 6th term.

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   ▶Reveal answer

   96🔊

   The nth term of a GP is a·r^(n-1); the 6th term is 3 × 2^5 = 3 × 32 = 96.

Complex Numbers, Quadratic Equations & Linear Inequalities

Covers the additive/multiplicative inverse and modulus of a complex number, complex roots of a quadratic equation, and solving linear inequalities.

1. 1. Calculate the additive inverse of the complex number 3 - 4i.

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   ▶Reveal answer

   -3 + 4i🔊

   The additive inverse of a complex number z = a + bi is -z = -a - bi. Thus, the additive inverse of 3 - 4i is found by changing the signs of both parts, yielding -3 + 4i.

2. 2. Find the multiplicative inverse of the complex number 4 + 3i.

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   ▶Reveal answer

   4/25 - 3/25i🔊

   The multiplicative inverse is 1/z. Multiplying the numerator and denominator by the conjugate 4 - 3i yields conjugate/magnitude-squared = (4 - 3i)/(4^2 + 3^2) = (4 - 3i)/25 = 4/25 - 3/25i.

3. 3. Solve the linear inequality for the variable x: 3x - 5 less than 16.

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   ▶Reveal answer

   x less than 7🔊

   Add 5 to both sides to balance the inequality: 3x less than 16 + 5 -> 3x less than 21. Dividing both sides by the positive number 3 gives x less than 7, which in interval form is (negative infinity, 7).

4. 4. Find the modulus of the complex number z = 5 - 12i.

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   ▶Reveal answer

   13🔊

   The modulus of a complex number a + bi is √(a² + b²); here √(5² + (-12)²) = √(25 + 144) = √169 = 13.

5. 5. Solve the quadratic equation x² - 4x + 13 = 0 and express the roots in the form a + bi.

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   ▶Reveal answer

   x = 2 + 3i or x = 2 - 3i🔊

   Using the quadratic formula, x = [4 ± √(16 - 52)]/2 = [4 ± √(-36)]/2 = [4 ± 6i]/2 = 2 ± 3i, since the discriminant is negative the roots are complex conjugates.

6. 6. Solve the inequality -3 < 2x - 1 ≤ 7 and write the solution in interval notation.

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   ▶Reveal answer

   (-1, 4]🔊

   Adding 1 to all three parts gives -2 < 2x ≤ 8, then dividing by 2 gives -1 < x ≤ 4, which in interval notation is (-1, 4].

Permutations, Combinations & Binomial Theorem

Covers permutation and combination word problems and finding a specific coefficient in a binomial expansion.

1. 1. In a race with 8 runners, in how many different ways can the first, second, and third place positions be awarded?

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   ▶Reveal answer

   336🔊

   This is a permutation of 8 runners taken 3 at a time: P(8,3) = 8 × 7 × 6 = 336, since the order of finishing matters.

2. 2. How many distinct arrangements can be made using all the letters of the word SUCCESS?

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   ▶Reveal answer

   420🔊

   SUCCESS has 7 letters where S repeats 3 times and C repeats 2 times, so the number of distinct arrangements is 7!/(3!×2!) = 5040/12 = 420.

3. 3. A school wants to form a committee of 4 students from a group of 10 students. In how many ways can this committee be chosen?

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   ▶Reveal answer

   210🔊

   Since the order of selection does not matter, the number of ways is C(10,4) = 10!/(4!×6!) = 210.

4. 4. From a group of 6 men and 5 women, a committee of 3 men and 2 women is to be formed. In how many ways can this be done?

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   ▶Reveal answer

   200🔊

   Choose 3 men from 6 in C(6,3) = 20 ways and 2 women from 5 in C(5,2) = 10 ways; by the multiplication principle the total number of committees is 20 × 10 = 200.

5. 5. Find the coefficient of x³y² in the binomial expansion of (x + y)^5.

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   ▶Reveal answer

   10🔊

   The general term in the expansion of (x+y)^5 is C(5,r) x^(5-r) y^r; for x³y² we need r = 2, giving coefficient C(5,2) = 10.

6. 6. Using the digits 1, 2, 3, 4, and 5 without repetition, how many 3-digit numbers can be formed?

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   ▶Reveal answer

   60🔊

   This is a permutation problem: choosing and arranging 3 digits out of 5 without repetition gives P(5,3) = 5 × 4 × 3 = 60 distinct 3-digit numbers.

Trigonometric Functions

Covers degree-radian conversion, standard angle values, and the sine addition formula.

1. 1. Convert an angle of 150 degrees into radian measure.

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   ▶Reveal answer

   5π/6 radians🔊

   To convert degrees to radians, multiply by π/180: 150 × π/180 = 150π/180 = 5π/6.

2. 2. Find the value of tan 30°.

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   ▶Reveal answer

   1/√3 (equivalently √3/3)🔊

   tan 30° is a standard trigonometric ratio equal to 1/√3, which can be rationalized to √3/3.

3. 3. If sinθ = 3/5 and θ lies between 0° and 90°, find the value of cosθ.

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   ▶Reveal answer

   4/5🔊

   Using sin²θ + cos²θ = 1, cos²θ = 1 - 9/25 = 16/25, so cosθ = 4/5 (positive since θ is in the first quadrant).

4. 4. Using the sine addition formula, find the value of sin 75°, given that 75° = 45° + 30°.

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   ▶Reveal answer

   (√6 + √2)/4🔊

   sin(A+B) = sinA cosB + cosA sinB, so sin75° = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Coordinate Geometry: Straight Lines & Circles

Covers finding the slope and equation of a line, the distance between two points, and the equation of a circle.

1. 1. Find the slope of the line passing through the points (2, 3) and (6, 11).

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   ▶Reveal answer

   2🔊

   Slope = (y2 - y1)/(x2 - x1) = (11 - 3)/(6 - 2) = 8/4 = 2.

2. 2. Find the equation of the line that passes through the point (3, -2) and has a slope of 4.

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   ▶Reveal answer

   4x - y - 14 = 0 (equivalently y = 4x - 14)🔊

   Using point-slope form y - y1 = m(x - x1): y - (-2) = 4(x - 3), which simplifies to y + 2 = 4x - 12, i.e., 4x - y - 14 = 0.

3. 3. Find the distance between the points (1, 2) and (4, 6).

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   ▶Reveal answer

   5 units🔊

   Using the distance formula, d = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5.

4. 4. Find the equation of a circle with center (2, -3) and radius 5 units.

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   ▶Reveal answer

   (x - 2)² + (y + 3)² = 25🔊

   The standard equation of a circle with center (h,k) and radius r is (x-h)² + (y-k)² = r²; substituting h=2, k=-3, r=5 gives (x-2)² + (y+3)² = 25.

Limits, Probability & Statistics

Covers evaluating a basic limit, probability of a single or combined event, and mean deviation.

1. 1. Evaluate the limit: lim(x→3) of (x² - 9)/(x - 3).

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   ▶Reveal answer

   6🔊

   Factor the numerator as (x-3)(x+3), cancel the common (x-3) factor since x ≠ 3 in the limit, and substitute x=3 into (x+3) to get 6.

2. 2. A die is rolled once. What is the probability of getting a number greater than 4?

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   ▶Reveal answer

   1/3🔊

   Numbers greater than 4 on a die are 5 and 6, so there are 2 favorable outcomes out of 6 total, giving a probability of 2/6 = 1/3.

3. 3. A bag contains 5 red balls and 3 blue balls. If 2 balls are drawn at random without replacement, what is the probability that both balls are red?

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   ▶Reveal answer

   5/14🔊

   The total ways to choose 2 balls from 8 is C(8,2) = 28, and the ways to choose 2 red balls from 5 is C(5,2) = 10, so the probability is 10/28 = 5/14.

4. 4. Find the mean deviation about the mean for the data set: 4, 6, 8, 10, 12.

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   ▶Reveal answer

   2.4🔊

   The mean is (4+6+8+10+12)/5 = 8; the absolute deviations are 4, 2, 0, 2, 4 which sum to 12, and dividing by 5 gives a mean deviation of 2.4.

Relations, Functions & Derivatives

Covers the Cartesian product, domain of a function, the greatest integer function, adding two functions, and finding a derivative using the power rule and standard trig derivatives.

1. 1. If A = {1, 2} and B = {3, 4}, find the Cartesian product A x B.

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   ▶Reveal answer

   {(1,3), (1,4), (2,3), (2,4)}🔊

   The Cartesian product pairs every element of A with every element of B, in that order, giving one ordered pair for each combination. With 2 elements in A and 2 in B, there are 2 x 2 = 4 ordered pairs in total.

2. 2. What is the domain of the real function f(x) = 1/(x - 2)?

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   ▶Reveal answer

   All real numbers except x = 2 (R - {2})🔊

   Division by zero is undefined, so any x that makes the denominator zero must be excluded from the domain. Here, x - 2 = 0 when x = 2, so that single value is removed while every other real number is allowed.

3. 3. What is the greatest integer function ⌊x⌋, and what is the value of ⌊3.7⌋?

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   ▶Reveal answer

   It gives the largest integer less than or equal to x; ⌊3.7⌋ = 3.🔊

   Unlike rounding, the greatest integer function always rounds down toward the nearest integer that doesn't exceed x, never up, regardless of how close x is to the next integer. This is why ⌊3.7⌋ is 3, not 4.

4. 4. If f(x) = x² and g(x) = x + 1 are two functions with domain R, find (f + g)(x).

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   ▶Reveal answer

   x² + x + 1🔊

   Adding two functions means adding their outputs for the same input: (f + g)(x) = f(x) + g(x) = x² + (x + 1) = x² + x + 1.

5. 5. Find the derivative of f(x) = x³ with respect to x, using the power rule.

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   ▶Reveal answer

   f'(x) = 3x²🔊

   The power rule states that the derivative of xⁿ is n·x^(n-1). Applying this with n = 3 gives 3x^(3-1) = 3x².

6. 6. Find the derivative of f(x) = sin x with respect to x.

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   ▶Reveal answer

   f'(x) = cos x🔊

   This is a standard derivative result covered early in differential calculus: the derivative of sin x is cos x, a fact that follows from the definition of the derivative applied to the sine function.

7. 7. A car's position is given by s(t) = t² + 3t (in metres, t in seconds). What does ds/dt represent physically, and what is its value at t = 2 seconds?

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   ▶Reveal answer

   ds/dt represents the car's instantaneous velocity (rate of change of position with respect to time); at t = 2, ds/dt = 7 m/s.🔊

   Differentiating s(t) = t² + 3t gives ds/dt = 2t + 3, using the power rule term by term. Substituting t = 2 gives 2(2) + 3 = 7 m/s, the car's speed at exactly that moment, not its average speed over some interval.

3D Geometry & Additional Conic Sections

Covers the three coordinate axes in 3D space, the 3D distance formula, and the standard equations of a parabola, ellipse, and hyperbola.

1. 1. In three-dimensional geometry, how many mutually perpendicular coordinate axes are used to locate a point in space?

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   ▶Reveal answer

   Three (the x-axis, y-axis, and z-axis)🔊

   Two-dimensional geometry locates a point using two perpendicular axes (x and y); three-dimensional geometry adds a third axis, the z-axis, perpendicular to both of the others, letting a point's position in space be described with three coordinates instead of two.

2. 2. Find the distance between the points (1, 2, 3) and (4, 6, 3) in three-dimensional space.

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   ▶Reveal answer

   5 units🔊

   The 3D distance formula extends the familiar 2D version by adding a third coordinate's difference: sqrt[(4-1)² + (6-2)² + (3-3)²] = sqrt[9 + 16 + 0] = sqrt[25] = 5.

3. 3. What is the standard equation of a parabola that opens to the right with its vertex at the origin?

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   ▶Reveal answer

   y² = 4ax🔊

   In this standard form, a determines how wide or narrow the parabola is, and the parabola opens toward the positive x-direction since y² (always non-negative) forces x to be non-negative too when a is positive.

4. 4. In the standard equation of an ellipse x²/a² + y²/b² = 1 (with a > b), what does the value 'a' represent?

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   ▶Reveal answer

   The length of the semi-major axis (half the ellipse's longest diameter)🔊

   Since a is larger than b in this convention, it corresponds to the longer axis of the ellipse, the semi-major axis, while b corresponds to the shorter semi-minor axis. Swapping which one is larger would mean the ellipse is oriented the other way.

5. 5. What is the key difference between the standard equations of an ellipse and a hyperbola?

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   ▶Reveal answer

   An ellipse's equation has a plus sign between the x² and y² terms (x²/a² + y²/b² = 1); a hyperbola's equation has a minus sign instead (x²/a² - y²/b² = 1).🔊

   That single sign difference is what creates two fundamentally different shapes: the plus sign keeps the curve closed and bounded (an ellipse), while the minus sign allows the curve to open into two separate, unbounded branches (a hyperbola).