JEE Advanced Maths questions with solutions

Q1. In a chemistry class, there are 20 students, while a physics class has 30 students. If 10 students are enrolled in both classes, and the two classes are held at separate times, determine the value of k/8, where k represents the total number of students attending either class.

1. 5
2. 6
3. 7
4. 8

Answer: 5

Students in either class = 20 + 30 - 10 = 40 by inclusion-exclusion, so k = 40 and k/8 = 5. The stored answer (6) is wrong; correct value is 5.

Q2. Given that A represents the divisors of 15, B contains prime numbers less than 10, and C includes even numbers less than 9, how many elements are there in the intersection of (A ∪ C) and B?

1. 1
2. 2
3. 3
4. 4

Answer: 3

A (divisors of 15) = {1,3,5,15}, B (primes <10) = {2,3,5,7}, C (even <9) = {2,4,6,8}. A U C = {1,2,3,4,5,6,8,15}; intersecting with B gives {2,3,5}, which has 3 elements (index 2). The stored answer (1) is wrong.

Q3. Identify the periodic function among the following:

1. Sign function of e^x
2. Sine of x added to the absolute value of sine of x
3. Minimum of sine of x and absolute value of x
4. Floor of x + 1/2 added to floor of x - 1/2 and twice the floor of -x

Answer: Floor of x + 1/2 added to floor of x - 1/2 and twice the floor of -x

The periodic function among the given options is identified by its repeating values over intervals, and in this case, the correct answer is the floor of x + 1/2 added to floor of x - 1/2 and twice the floor of -x.

Q4. The function f(x) = √|x² - 5| x + 6 + √8 + 2|x| - |x|² is defined as a real number for values of x within which range?

1. [-4, -3]
2. [-3, -2]
3. [-2, 2]
4. [3, 4]

Answer: [-2, 2]

The domain of the function f(x) = √|x² - 5| x + 6 + √8 + 2|x| - |x|² is determined by ensuring the square root terms are non-negative, yielding a domain of [-2, 2].

Q5. Let f(x) = 4x / (4x² + 2). If the sum of the integrals ∫(1/1997) + ∫(2/1997) +... + ∫(1196/1997) equals 499q, what is the value of q?

1. 1
2. 2
3. 3
4. 4

Answer: 2

The value of q is found by evaluating the sum of the integrals ∫(1/1997) + ∫(2/1997) +... + ∫(1196/1997) and using properties of definite integrals and symmetry to arrive at the correct value of 2.

Q6. Given that α, β, γ, and θ are the smallest positive angles in increasing order for which their sine values equal a positive constant k, what is the result of 4 sin(α/2) + 3 sin(β/2) + 2 sin(γ/2) + sin(θ/2)?

1. 2√(1−k)
2. 2√(1+k)
3. 2√k
4. None of the above

Answer: 2√(1+k)

The result of the expression 4 sin(α/2) + 3 sin(β/2) + 2 sin(γ/2) + sin(θ/2) is 2√(1+k), which can be derived by applying trigonometric identities and analyzing the relationships between the angles.

Q7. Consider the function fn(θ) = tan(θ/2)(1 + sec²θ)(1 + sec⁴θ)... (1 + sec²ⁿθ). Which of the following is true?

1. f₂(π/16) equals 1
2. f₃(π/32) equals 1
3. f₄(π/64) equals 1
4. f₅(π/128) equals 1

Answer: f₄(π/64) equals 1

The correct statement is f₄(π/64) equals 1, which can be determined by analyzing the function fn(θ) and evaluating it for the given values.

Q8. Given that (a−b)sin(θ+ϕ) = (a+b)sin(θ−ϕ) and a tan(θ/2) − b tan(ϕ/2) = c, which of the following holds true?

1. b tan(ϕ) = a tan(θ)
2. a tan(ϕ) = b tan(θ)
3. sin(ϕ) = −2bc/(a²−b²−c²)
4. sin(θ) = −2ac/(a²−b²+c²)

Answer: sin(ϕ) = −2bc/(a²−b²−c²)

The correct relationship is sin(ϕ) = −2bc/(a²−b²−c²), which can be determined by analyzing the given equations and applying trigonometric identities.

Q9. If n is an odd number, how many ways can three terms in an arithmetic progression be chosen from the sequence 1, 2, 3,..., n?

1. (n - 1)² / 2
2. (n + 1)² / 2
3. (n² - 1) / 4
4. (n - 1)² / 4

Answer: (n - 1)² / 4

A 3-term AP (a, a+d, a+2d) from 1..n with common difference d>=1 needs a+2d<=n, giving n-2d choices, summed over d=1..(n-1)/2. The total is sum(n-2d) = ((n-1)/2)^2 = (n-1)^2/4. Check n=5: d=1 gives 3, d=2 gives 1, total 4 = (5-1)^2/4 = 4. So the answer is (n-1)^2/4 (index 3), not (n+1)^2/2; stored index 1 is wrong.

Q10. How many triangles can be created by connecting the vertices of a regular polygon with n (> 5) sides, ensuring that no triangle shares a side with the polygon?

1. n / n-3C3
2. nC3 - n - n(n-4)
3. n-4C2 + n-3C3
4. n+2C3

Answer: nC3 - n - n(n-4)

The number of triangles that can be created by connecting the vertices of a regular polygon with n sides is given by nC3 - n - n(n-4), which subtracts the triangles that share a side with the polygon.

Q11. For n > 1, consider the product E = (2n + 1)(2n + 3)(2n + 5)...(4n - 3)(4n - 1). Which of the following statements is true?

1. 2^nE is divisible by the binomial coefficient 4^nC2n
2. 2^nE is divisible by the factorial of n
3. The value of 2^nE divided by n! is a positive integer
4. The value of 2^nE divided by (4n)! is not an integer

Answer: The value of 2^nE divided by n! is a positive integer

The value of 2^nE divided by n! is a positive integer because the product E contains a sequence of consecutive odd numbers, ensuring divisibility by n!.

Q12. An individual has 6 friends and during a holiday period, he attended several dinners with them. He observed that he dined with all 6 friends on one occasion, with any group of 5 friends on 3 occasions, with any group of 4 friends on 3 occasions, with any group of 3 friends on 4 occasions, and with any pair of friends on 5 occasions. Additionally, each friend joined him for 7 dinners and missed 7 dinners. How many dinners did he attend alone?

1. 1
2. 2
3. 3
4. 4

Answer: 1

The individual attended 1 dinner alone, which can be determined by analyzing the number of dinners attended with each group of friends and applying the principle of inclusion-exclusion.

Q13. How many natural numbers n exist such that the factorial of n concludes with exactly 26 trailing zeros?

1. 4
2. 5
3. 6
4. 7

Answer: 5

Trailing zeros Z(n)=floor(n/5)+floor(n/25)+... Z(110)=22+4=26 and this holds for n=110,111,112,113,114; Z(115)=23+4=27. So exactly 5 natural numbers give 26 trailing zeros, which is option index 1, not the stored index 2.

Q14. In the expansion of (1 + x)ⁿ, let the binomial coefficients be represented as C₀, C₁, C₂,..., Cₙ. If p and q are such that p + q = 1, what is the value of Σᵣ₌₀ nCr pʳqⁿ⁻ʳ?

1. n multiplied by p
2. n times the product of p and q
3. n squared times p squared plus n times p times q
4. None of the above

Answer: n multiplied by p

The summation Σᵣ₌₀ nCr pʳqⁿ⁻ʳ simplifies to n times p using binomial expansion properties and the given condition p + q = 1.

Q15. What is the result of the summation Σ₀≤k≤n ∑ᵢ i * Cᵢ?

1. n(n + 1)2ⁿ⁻³
2. n²2ⁿ⁻³
3. n(n − 1)2ⁿ⁻³
4. None of the above

Answer: n²2ⁿ⁻³

The value of Σ₀≤k≤n ∑ᵢ i * Cᵢ is calculated using summation formulas and binomial coefficients, resulting in the value n²2ⁿ⁻³.

Q16. If a, b, and c are in harmonic progression, then e raised to the power of -a, e raised to the power of -b, and e raised to the power of -c will be in which progression?

1. Arithmetic progression
2. Geometric progression
3. Harmonic progression
4. None of the above

Answer: Arithmetic progression

If a, b, and c are in harmonic progression, then e raised to the power of -a, e raised to the power of -b, and e raised to the power of -c will be in arithmetic progression because the exponential function is the inverse of the logarithmic function.

Q17. If x, y, and z represent the pᵗʰ, qᵗʰ, and rᵗʰ terms of both an arithmetic progression and a geometric progression, what is the value of (xʳ)(yᵖ)(zᵠ)?

1. 1
2. -1
3. 0
4. 2

Answer: 1

Since x, y, and z represent the pᵗʰ, qᵗʰ, and rᵗʰ terms of both an arithmetic progression and a geometric progression, the value of (xʳ)(yᵖ)(zᵠ) simplifies to 1 due to the properties of these progressions.

Q18. Determine the range of β for which the point (0, β) is located on or within the triangle formed by the equations y + 3x + 2 = 0, 3y − 2x − 5 = 0, and 4y − x − 14 = 0.

1. 5 ≤ β ≤ 7
2. 1/2 ≤ β ≤ 1
3. 5/3 ≤ β ≤ 7/2
4. None of these

Answer: 5/3 ≤ β ≤ 7/2

The triangle has vertices (-1,1),(-1.69,3.08),(4.4,4.6). On the line x=0, points inside the triangle lie between y=5/3 (from 3y-2x-5=0) and y=7/2 (from 4y-x-14=0). So 5/3 <= beta <= 7/2 (idx 2); stored '5<=beta<=7' is wrong.

Q19. For the equation a² + b² − c² − 2ab = 0, the common intersection point of the set of lines represented by ax + by + c = 0 is located on which line?

1. y = x
2. y = x + 1
3. y = −x
4. y + x = 1

Answer: y = −x

The given equation can be rearranged to represent a set of lines, and by finding the common intersection point, it is evident that this point lies on the line y = −x, which signifies a specific relationship between the coefficients of the equation.

Q20. If the points (a³/(a − 3), (a² − 3)/(a − 1)), (b³/(b − 3), (b² − 3)/(b − 1)), and (c³/(c − 3), (c² − 3)/(c − 1)), where a, b, and c are not equal to 1, are situated on the line l₁x + m₁y + n₁ = 0, then which of the following is true?

1. The sum a + b + c equals −m/l.
2. The product ab + bc + ca equals n/l.
3. The product abc equals (m + n)/l.
4. The expression abc − (bc + ca + ab) + 3(a + b + c) equals 0.

Answer: The expression abc − (bc + ca + ab) + 3(a + b + c) equals 0.

The points given are situated on a line, and by using the equation of the line and the coordinates of the points, it can be derived that a specific expression involving the coefficients a, b, and c equals zero, which is a necessary condition for the points to lie on the line.

Q21. For the circle represented by (x + c)² + y² = a² and the ellipse given as (x - h)² / b² + y² / a² = 1 (where a, b, c, and h are all positive), if they share a tangent that is horizontal, which condition must be satisfied?

1. c > b + a - h
2. c < b + a - h
3. c > b + a
4. None of the above

Answer: c < b + a - h

The condition for the circle and the ellipse to share a horizontal tangent can be derived by examining the equations of the circle and the ellipse, and it can be found that the correct condition is c < b + a - h, which ensures that the tangent is horizontal.

Q22. The expression √(x² + (y - 1)²) - √(x² + (y + 1)²) = K describes a hyperbola when:

1. K lies in the interval (0, 2)
2. K lies in the interval (0, 1)
3. K lies in the interval (1, ∞)
4. K lies in the interval (0, ∞)

Answer: K lies in the interval (0, 2)

The expression describes a hyperbola when K lies in the interval (0, 2), which means the difference between the distances of any point on the hyperbola from the two foci is constant and equal to 2a.

Q23. When the circle defined by x² + y² = 1 intersects the rectangular hyperbola xy = 1 at four points (x_i, y_i) for i = 1, 2, 3, 4, which of the following is true?

1. The product of x-coordinates, x₁x₂x₃x₄, equals -1.
2. The product of y-coordinates, y₁y₂y₃y₄, equals 1.
3. The sum of x-coordinates, x₁ + x₂ + x₃ + x₄, equals 0.
4. The sum of y-coordinates, y₁ + y₂ + y₃ + y₄, equals 0.

Answer: The sum of x-coordinates, x₁ + x₂ + x₃ + x₄, equals 0.

The sum of x-coordinates of the intersection points of the circle and the rectangular hyperbola is zero because the circle is symmetric about the origin and the hyperbola is symmetric about the line y = x.

Q24. The line 3x + 4y = 24 meets the x-axis and y-axis at points A and B, respectively, while the line 4x + 3y = 24 intersects at points C and D. The four points A, B, C, and D are located on which type of curve?

1. a circle
2. a parabola
3. an ellipse
4. a hyperbola

Answer: a circle

3x+4y=24 cuts the axes at A(8,0), B(0,6); 4x+3y=24 cuts them at C(6,0), D(0,8). The x-intercepts multiply to 8*6=48 and the y-intercepts to 6*8=48 (equal), the condition for the four points to lie on a circle. So the curve is a circle, option index 0, not the stored ellipse (index 2).

Q25. Let ϕ(x) represent a quadratic polynomial. Given that ϕ(1) equals ϕ(−1) and the terms a₁, a₂, a₃ form an arithmetic progression, then the values ϕ(a₁), ϕ(a₂), ϕ(a₃) will be in which sequence?

1. Arithmetic progression
2. Geometric progression
3. Harmonic progression
4. None of the above

Answer: None of the above

phi(1)=phi(-1) forces no linear term: phi(x)=Ax^2+C. For an AP a-d,a,a+d, phi values are A(a-d)^2+C, Aa^2+C, A(a+d)^2+C; check 2*mid=A(a-d)^2+A(a+d)^2 requires 2a^2=(a-d)^2+(a+d)^2=2a^2+2d^2, false for d!=0. So they are not in AP (nor GP/HP generally): None of the above (idx 3).

Q26. Evaluate f(x) = lim x→∞ (x²n − 1) / (x²n + 1). Which of the following is true?

1. f(x) equals 1 when |x| is greater than 1
2. f(x) equals −1 when |x| is less than 1
3. f(x) is undefined for all values of x
4. f(x) equals 1 when |x| equals 1

Answer: f(x) equals 1 when |x| is greater than 1

The function f(x) equals 1 when |x| is greater than 1 because the limit of (x²n - 1) / (x²n + 1) as x approaches infinity is 1 for |x| > 1.

Q27. Determine the value of L if L = lim x→0 [(sin x + ae^x + be^−x + c ln(1 + x)) / x³], given that L is finite and not infinite.

1. 1/2
2. −1/3
3. −1/6
4. 3

Answer: −1/3

Expanding gives a+b=0, 1+a-b+c=0, and a+b-c=0 so c=0, a=-1/2, b=1/2. The x^3 coefficient is -1/6 + (a-b)/6 = -1/6 - 1/6 = -1/3, so L = -1/3 (idx 1). The stored idx 2 (-1/6) is wrong.

Q28. Given that p is a proposition, t represents a tautology, and c denotes a contradiction, which of the following statements is accurate?

1. p OR (NOT p) equals c
2. p OR t results in t
3. p AND t is equivalent to p
4. p AND c is equal to c

Answer: p OR t results in t

The statement p OR t results in t because when combined with a tautology, the result is always true, regardless of the value of p.

Q29. Which of the following pairs of statements correctly represent a logical dual?

1. (p OR q) AND (r OR s), (p AND q) OR (r AND s)
2. [p OR (NOT q) AND (NOT p)] AND (NOT q) OR (NOT p)
3. (p AND q) OR r, (p OR q) AND r
4. (p OR q) OR s, AND (p AND q) OR s

Answer: (p AND q) OR r, (p OR q) AND r

The pair of statements (p AND q) OR r and (p OR q) AND r are logical duals because they have the same structure but with the operators AND and OR interchanged.

Q30. Which of the following statements is accurate?

1. p → q is equivalent to ∼p ∨ q in logic.
2. If p, q, and r have truth values T, F, and T respectively, then (p ∨ q) ∧ (q ∨ r) evaluates to T.
3. The expression ∼(p ∨ q ∨ r) is equal to ∼p ∧ ∼q ∧ ∼r.
4. The logical value of p ∧ ∼(p ∨ q) is always T.

Answer: p → q is equivalent to ∼p ∨ q in logic.

The statement p → q is equivalent to ∼p ∨ q in logic because the implication p → q is true if p is false or q is true, which is the same as ∼p ∨ q.