JEE Main 2025January, Unspecified (session-level dataset)

Mathematics 146

Mathematics

Q1If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20…Sequence and Series
Q2One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces…Statistics and Probability
Q3Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i + 2 j + k, i + 3 j - 2 k and 2 i + j - k respectively. The altitude from the…Three Dimensional Geometry
Q4If A, B, and (adj(A⁻¹) + adj(B⁻¹)) are non-singular matrices of same order, then the inverse of A (adj(A⁻¹) + adj(B⁻¹))⁻¹ B, is equal toMatrices and Determinants
Q5Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be…Statistics and Probability
Q6Let a curve y = f(x) pass through the points (0, 5) and (log e 2, k). If the curve satisfies the differential equation 2(3 + y) e²ˣ dx - (7 + e²ˣ) dy = 0…Differential Equations
Q7If the function f(x) = 2/x sin (k₁ + 1) x + sin (k₂ - 1) x x < 0 4, x = 0 2/x log e ((2 + k₁ x)/(2 + k₂ x)), x > 0 is continuous at x = 0, then k₁² + k₂² isLimit, Continuity and Differentiability
Q8If the line 3x - 2y + 12 = 0 intersects the parabola 4y = 3x² at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle…Coordinate Geometry
Q10Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that (length of arc AB)/(length of arc…Vector Algebra
Q11Let f(x) = log e x and g(x) = (x⁴ - 2x³ + 3x² - 2x + 2)/(2x² - 2x + 1). Then the domain of f° g isSets, Relations and Functions
Q12If the system of equations (λ - 1)x + (λ - 4)y + λ z = 5, λ x + (λ - 1)y + (λ - 4)z = 7, (λ + 1)x + (λ + 2)y - (λ + 2)z = 9 has infinitely many solutions…Matrices and Determinants
Q13The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, isPermutations and Combinations
Q14Let R = \(1, 2), (2, 3), (3, 3) be a relation defined on the set \1, 2, 3, 4. Then the minimum number of elements, needed to be added in R so that R becomes…Sets, Relations and Functions
Q15Let the area of a PQR with vertices P(5, 4), Q(-2, 4) and R(a, b) be 35 square units. If its orthocenter and centroid are O(2, 14/5) and C(c, d) respectively…Coordinate Geometry
Q18The value of (sin 70°) (cot 10° cot 70° - 1) isTrigonometry
Q26For a 3 × 3 matrix M, let trace(M) denote the sum of all the diagonal elements of M. Let A be a 3 × 3 matrix such that |A|=1/2 and trace(A)=3. If B=adj(adj(2…Matrices and Determinants
Q27In a group of 3 girls and 4 boys, there are two boys B₁ and B₂. The number of ways, in which these girls and boys can stand in a queue such that all the girls…Permutations and Combinations
Q28Let α, β, γ and δ be the coefficients of x⁷, x⁵, x³ and x respectively in the expansion of (x+√(x³-1))⁵+(x-√(x³-1))⁵, x>1. If u and v satisfy the equations α…Binomial Theorem and Its Simple Applications
Q29Let a line pass through two distinct points P(-2,-1,3) and Q, and be parallel to the vector 3i+2j+2k. If the distance of the point Q from the point R(1,3,3)…Three Dimensional Geometry
Q30If A and B are two events such that P(A B)=0.1, and P(A B) and P(B A) are the roots of the equation 12x²-7x+1=0, then the value of (P(A B))/(P(A B)) is:Statistics and Probability
Q32The area of the region enclosed by the curves y=x²-4x+4 and y²=16-8x is:Integral Calculus
Q33Let f(x)=∫₀^(x²) (t²-8t+15)/(e^(t)) dt, x R. Then the numbers of local maximum and local minimum points of f, respectively, are:Limit, Continuity and Differentiability
Q34Let P(4,4√(3)) be a point on the parabola y²=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q…Coordinate Geometry
Q35Let a and b be two unit vectors such that the angle between them is (π)/3. If λ a + 2 b and 3 a - λ b are perpendicular to each other, then the number of…Vector Algebra
Q37Let A=\1,2,3,4 and B=\1,4,9,16. Then the number of many-one functions f: A → B such that 1 f(A) is equal to:Sets, Relations and Functions
Q38Suppose that the number of terms in an A.P. is 2k, k N. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of…Sequence and Series
Q39The perpendicular distance, of the line (x-1)/2=(y+2)/(-1)=(z+3)/2 from the point P(2,-10,1), is:Three Dimensional Geometry
Q40If the system of linear equations: x + y + 2z = 6 2x + 3y + az = a + 1 -x - 3y + bz = 2b where a, b R, has infinitely many solutions, then 7a + 3b is equal to:Matrices and Determinants
Q41If x=f(y) is the solution of the differential equation (1+y²) + (x - 2e^ tan ⁻¹ y) dy/dx = 0, y (-(π)/2, (π)/2) with f(0)=1, then f(1/(√(3))) is equal to:Differential Equations
Q42Let α_θ and β_θ be the distinct roots of 2x² + (cos θ) x - 1 = 0, θ (0, 2π). If m and M are the minimum and the maximum values of α_θ⁴ + β_θ⁴, then 16(M + m)…Complex Numbers and Quadratic Equations
Q43The sum of all values of θ [0, 2π] satisfying 2 sin ² θ = cos 2θ and 2 cos ² θ = 3 sin θ is:Trigonometry
Q44Let the curve z(1+i) + z(1-i) = 4, z C, divide the region |z-3| ≤ 1 into two parts of areas α and β. Then |α - β| equals:Complex Numbers and Quadratic Equations
Q51Let circle C be the image of x² + y² - 2x + 4y - 4 = 0 in the line 2x - 3y + 5 = 0 and A be the point on C such that OA is parallel to x -axis and A lies on…Coordinate Geometry
Q53Let the product of the focal distances of the point (√(3), 1/2) on the ellipse (x²)/(a²) + (y²)/(b²) = 1, (a > b), be 7/4. Then the absolute difference of the…Coordinate Geometry
Q54If the system of equations 2x - y + z = 4, 5x + λ y + 3z = 12, 100x - 47y + μ z = 212 has infinitely many solutions, then μ - 2λ is equal toMatrices and Determinants
Q55For some n ≠ 10, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of (1 + x)ⁿ⁺⁴ be in A.P. Then the largest coefficient in the…Binomial Theorem and Its Simple Applications
Q56The product of all the rational roots of the equation (x² - 9x + 11)² - (x - 4)(x - 5) = 3, is equal toComplex Numbers and Quadratic Equations
Q60For a statistical data x₁, x₂,, x₁₀ of 10 values, a student obtained the mean as 5.5 and Σ i=1 ¹⁰ x i ² = 371. He later found that he had noted two values in…Statistics and Probability
Q61The area of the region \(x, y): x² + 4x + 2 ≤ y ≤ |x + 2| is equal toIntegral Calculus
Q62Let S n = 1/2 + 1/6 + 1/12 + 1/20 + up to n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is √(2026 S₂₀₂₅)…Sequence and Series
Q63Let f: R - \0 → R be a function such that f(x) - 6 f(1/x) = 35/3x - 5/2. If the lim x → 0 (1/(α x) + f(x)) = β; α, β R, then α + 2β is equal toLimit, Continuity and Differentiability
Q64If I(m, n) = ∫₀¹ x^(m-1) (1 - x)ⁿ⁻¹ dx, m, n > 0, then I(9, 14) + I(10, 13) isIntegral Calculus
Q65A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum…Statistics and Probability
Q66Let f(x) = (2ˣ⁺² + 16)/(2²ˣ⁺¹ + 2ˣ⁺⁴ + 32). Then the value of 8(f(1/15) + f(2/15) + + f(59/15)) is equal toSets, Relations and Functions
Q67Let y = y(x) be the solution of the differential equation (xy - 5x² √(1 + x²)) dx + (1 + x²) dy = 0, y(0) = 0. Then y(√(3)) is equal toDifferential Equations
Q68lim x → 0 csc x (√(2 cos ² x + 3 cos x) - √(cos ² x + sin x + 4)) is:Limit, Continuity and Differentiability
Q69Consider the region R = \(x, y): x ≤ y ≤ 9 - 11/3 x², x ≥ 0. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R, is:Limit, Continuity and Differentiability
Q70Let a = i + 2j + 3k, b = 3i + j - k and c be three vectors such that c is coplanar with a and b. If the vector c is perpendicular to b and a · c = 5, then |c|…Vector Algebra
Q76Let A=[ cc1/(√(2)) -2 0 1 ] and P=[ cc cos θ - sin θ sin θ cos θ ], θ>0. If B=PAP^(T), C=P^(T) B¹⁰ P and the sum of the diagonal elements of C is m/n, where…Matrices and Determinants
Q80Let [x] denote the greatest integer less than or equal to x. Then the domain of f(x)= sec ⁻¹(2[x]+1) is:Sets, Relations and Functions
Q81Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The…Permutations and Combinations
Q83Let f be a real valued continuous function defined on the positive real axis such that g(x)=∫₀ˣ t f(t) dt. If g(x³)=x⁶+x⁷, then value of Σ r=1 ¹⁵ f(r³) is:Integral Calculus
Q85Bag B₁ contains 6 white and 4 blue balls, Bag B₂ contains 4 white and 6 blue balls, and Bag B₃ contains 5 white and 5 blue balls. One of the bags is selected…Statistics and Probability
Q87For positive integers n, if 4a n = (n² + 5n + 6) and S n = Σ k=1 ⁿ (1/a k), then the value of 507 S₂₀₂₅ is:Sequence and Series
Q88Let f: R - \0 → (-∞, 1) be a polynomial of degree 2, satisfying f(x) f(1/x) = f(x) + f(1/x). If f(K) = -2K, then the sum of squares of all possible values of…Sets, Relations and Functions
Q89If A and B are the points of intersection of the circle x² + y² - 8x = 0 and the hyperbola (x²)/9 - (y²)/4 = 1 and a point P moves on the line 2x - 3y + 4 =…Coordinate Geometry
Q90If f(x) = ∫ 1/(x^(1/4) (1 + x^(1/4))) dx, f(0) = -6, then f(1) is equal to:Integral Calculus
Q91The area of the region bounded by the curves x(1 + y²) = 1 and y² = 2x is:Integral Calculus
Q94If α + iβ and γ + iδ are the roots of x² - (3 - 2i)x - (2i - 2) = 0, i = √(-1), then αγ + βδ is equal to:Complex Numbers and Quadratic Equations
Q95Two equal sides of an isosceles triangle are along -x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum of all possible distinct…Coordinate Geometry
Q101Let x₁, x₂,, x₁₀ be ten observations such that Σ i=1 ¹⁰(x i -2)=30, Σ i=1 ¹⁰(x i -β)²=98, β>2, and their variance is 4/5. If μ and σ² are respectively the…Statistics and Probability
Q102Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its…Sequence and Series
Q103The number of solutions of the equation (9/x-9/(√(x))+2)(2/x-7/(√(x))+3)=0 is:Complex Numbers and Quadratic Equations
Q105Two parabolas have the same focus (4,3) and their directrices are the x -axis and the y -axis, respectively. If these parabolas intersect at the points A and…Coordinate Geometry
Q106Let P be the set of seven-digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2, and 3 only, then the…Permutations and Combinations
Q107Let a= +2 +k and b=2 +7 +3k. Let L₁: r=(- +2 +k)+λ a, λ R and L₂: r=(+k)+μ b, μ R be two lines. If the line L₃ passes through the point of intersection of L₁…Three Dimensional Geometry
Q108Let a=2 - +3k, b=3 -5 +k and c be a vector such that a × c=c × b and (a+c) · (b+c)=168. Then the maximum value of |c|² is:Vector Algebra
Q110Let the ellipse E₁: (x²)/(a²) + (y²)/(b²) = 1, a > b and E₂: (x²)/(A²) + (y²)/(B²) = 1, A < B have the same eccentricity 1/(√(3)). Let the product of their…Coordinate Geometry
Q111Let A=[a ij ]=[ cc log ₅ 128 log ₄ 5 log ₅ 8 log ₄ 25 ]. If A ij is the cofactor of a ij, C ij =Σ k=1 ² a ik A jk, 1 ≤ i, j ≤ 2, and C=[C ij ], then 8|C| is…Matrices and Determinants
Q112Let |z₁ - 8 - 2i| ≤ 1 and |z₂ - 2 + 6i| ≤ 2, z₁, z₂ C. Then the minimum value of |z₁ - z₂| is:Complex Numbers and Quadratic Equations
Q115Let ABC be a triangle formed by the lines 7x - 6y + 3 = 0, x + 2y - 31 = 0 and 9x - 2y - 19 = 0. Let the point (h, k) be the image of the centroid of ABC in…Coordinate Geometry
Q118Let y = y(x) be the solution of the differential equation cos x (log e (cos x))² dy + (sin x - 3y sin x log e (cos x)) dx = 0, x (0, (π)/2). If y((π)/4) =…Differential Equations
Q119Let the line x + y = 1 meet the circle x² + y² = 4 at the points A and B. If the line perpendicular to AB and passing through the midpoint of the chord AB…Coordinate Geometry
Q120Let the area of the region \(x, y): 2y ≤ x² + 3, y + |x| ≤ 3, y ≥ |x-1| be A. Then 6A is equal to:Integral Calculus
Q126Let f(x)=∫₀ˣ t(t²-9t+20) dt, 1 ≤ x ≤ 5. If the range of f is [α, β], then 4(α+β) equals:Limit, Continuity and Differentiability
Q127Let a be a unit vector perpendicular to the vectors b=i-2j+3k and c=2i+3j-k, and makes an angle of cos ⁻¹(-1/3) with the vector i+j+k. If a makes an angle of…Vector Algebra
Q128If for the solution curve y=f(x) of the differential equation dy/dx+(tan x)y=(2+ sec x)/((1+2 sec x)²), x (-(π)/2, (π)/2), f((π)/3)=(√(3))/10, then f((π)/4)…Differential Equations
Q129Let P be the foot of the perpendicular from the point (1,2,2) on the line L: (x-1)/1=(y+1)/(-1)=(z-2)/2. Let the line r=(-i+j-2k)+λ(i-j+k), λ R, intersect the…Three Dimensional Geometry
Q130Let A=[a ij ] be a matrix of order 3 × 3, with a ij =(√(2))^(i+j). If the sum of all the elements in the third row of A² is α+β√(2), α, β Z, then α+β is equal…Matrices and Determinants
Q132If all the words with or without meaning made using all the letters of the word 'KANPUR' are arranged as in a dictionary, then the word at 440^(th) position…Permutations and Combinations
Q133If the set of all a R, for which the equation 2x²+(a-5)x+15=3a has no real root, is the interval (α, β), and X= Z: α < x < β, then Σ x X x² is equal to:Complex Numbers and Quadratic Equations
Q135Let the function f(x)=(x²+1)|x²-ax+2|+ cos |x| be not differentiable at the two points x=α=2 and x=β. Then the distance of the point (α, β) from the line…Limit, Continuity and Differentiability
Q136Let the area enclosed between the curves |y|=1-x² and x²+y²=1 be α. If 9α=βπ+γ, β, γ are integers, then the value of |β-γ| equals:Integral Calculus
Q137The remainder, when 7¹⁰³ is divided by 23, is equal to:Binomial Theorem and Its Simple Applications
Q138If α x+β y=109 is the equation of the chord of the ellipse (x²)/9+(y²)/4=1, whose midpoint is (5/2, 1/2), then α+β is equal to:Coordinate Geometry
Q140Let a circle C pass through the points (4,2) and (0,2), and its centre lie on 3x+2y+2=0. Then the length of the chord of the circle C, whose midpoint is…Coordinate Geometry
Q142Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to…Statistics and Probability
Q143Let α, β (α ≠ β) be the values of m, for which the equations x+y+z=1, x+2y+4z=m, and x+4y+10z=m² have infinitely many solutions. Then the value of Σ n=1 ¹⁰…Matrices and Determinants
Q145If sin x + sin ² x = 1, x (0, (π)/2), then (cos ¹² x + tan ¹² x) + 3(cos ¹⁰ x + tan ¹⁰ x + cos ⁸ x + tan ⁸ x) + (cos ⁶ x + tan ⁶ x) is equal to:Trigonometry
Q152Let f: R → R be a function defined by f(x) = (2 + 3a) x² + ((a + 2)/(a - 1)) x + b, a ≠ 1. If f(x + y) = f(x) + f(y) + 1 - 2/7 xy, then the value of 28 Σ i=1…Sets, Relations and Functions
Q155Let ⁿ C r-1 = 28, ⁿ C r = 56 and ⁿ C r+1 = 70. Let A(4 cos t, 4 sin t), B(2 sin t, -2 cos t) and C(3r - n, r² - n - 1) be the vertices of a triangle ABC…Coordinate Geometry
Q156Let the equation of the circle, which touches x -axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y -axis be x² + y² - α x + β y + γ =…Coordinate Geometry
Q158Two numbers k₁ and k₂ are randomly chosen from the set of natural numbers. Then, the probability that the value of i^(k₁) + i^(k₂), (i = √(-1)) is non-zero…Statistics and Probability
Q159If the image of the point (4, 4, 3) in the line (x - 1)/2 = (y - 2)/1 = (z - 1)/3 is (α, β, γ), then α + β + γ is equal toThree Dimensional Geometry
Q161Let A(x, y, z) be a point in xy -plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, -1) and C = (2, 0, -2). Then…Three Dimensional Geometry
Q163The sum, of the squares of all the roots of the equation x² + |2x - 3| - 4 = 0, isComplex Numbers and Quadratic Equations
Q164Let T r be the r^(th) term of an A.P. If for some m, T m = 1/25, T₂₅ = 1/20, and 20 Σ r=1 ²⁵ T r = 13, then 5m Σ r=m ^(2m) T r is equal toSequence and Series
Q165Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are…Statistics and Probability
Q168Let a n be a sequence such that a₀ = 0, a₁ = 1/2 and 2 a n+2 = 5 a n+1 - 3 a n, n = 0, 1, 2, 3,. Then Σ k=1 ¹⁰⁰ a k is equal toSequence and Series
Q169The number of different 5-digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and…Permutations and Combinations
Q170The relation R = \(x, y): x, y Z and x + y is even is:Sets, Relations and Functions
Q176Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5…Permutations and Combinations
Q177If the system of equations x + 2y - 3z = 2, 2x + λ y + 5z = 5, 14x + 3y + μ z = 33 has infinitely many solutions, then λ + μ is equal to:Matrices and Determinants
Q178Let A = (0, π) - \(π)/2: log (2 / π) | sin x| + log (2 / π) | cos x| = 2 and B = 0: √(x) (√(x) - 4) - 3 |√(x) - 2| + 6 = 0. Then n(A B) is equal to:Trigonometry
Q179The area of the region enclosed by the curves y = eˣ, y = |eˣ - 1|, and the y -axis is:Integral Calculus
Q180The equation of the chord of the ellipse (x²)/25 + (y²)/16 = 1, whose mid-point is (3, 1), is:Coordinate Geometry
Q182Let f: (0, ∞) → R be a function which is differentiable at all points of its domain and satisfies the condition x² f'(x) = 2x f(x) + 3, with f(1) = 4. Then 2…Differential Equations
Q184Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points where the function f(x) = [x] + |x - 2|, -2 < x < 3…Limit, Continuity and Differentiability
Q185Let A = [a ij ] be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is:Statistics and Probability
Q186Let the position vectors of three vertices of a triangle be 4 p + q - 3 r, -5 p + q + 2 r, and 2 p - q + 2 r. If the position vectors of the orthocenter and…Vector Algebra
Q187Let a = 3 i - j + 2 k, b = a × (i - 2 k), and c = b × k. Then the projection of c - 2 j on a is:Vector Algebra
Q189The function f: (-∞, ∞) → (-∞, 1), defined by f(x) = (2ˣ - 2⁻ˣ)/(2ˣ + 2⁻ˣ), is:Sets, Relations and Functions
Q190In an arithmetic progression, if S₄₀ = 1030 and S₁₂ = 57, then S₃₀ - S₁₀ is equal to:Sequence and Series
Q191Suppose A and B are the coefficients of 30^(th) and 12^(th) terms respectively in the binomial expansion of (1 + x)²ⁿ⁻¹. If 2A = 5B, then n is equal to:Binomial Theorem and Its Simple Applications
Q192Let (2, 3) be the largest open interval in which the function f(x) = 2 log e (x - 2) - x² + a x + 1 is strictly increasing and (b, c) be the largest open…Limit, Continuity and Differentiability
Q193For some a, b, let f(x) = | ccc a + (sin x)/x 1 b a 1 + (sin x)/x b a 1 b + (sin x)/x |, x ≠ 0, lim x → 0 f(x) = λ + μ a + ν b. Then (λ + μ + ν)² is equal to:Matrices and Determinants
Q194If the equation of the parabola with vertex V(3/2, 3) and the directrix x + 2y = 0 is α x² + β y² - γ xy - 30x - 60y + 225 = 0, then α + β + γ is equal to:Coordinate Geometry
Q201Let a₁, a₂, a₃, be a G.P. of increasing positive terms. If a₁ a₅ = 28 and a₂ + a₄ = 29, then a₆ is equal to:Sequence and Series
Q203Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected…Statistics and Probability
Q204The product of all solutions of the equation e^(5(log e x)² + 3) = x⁸, x > 0, is:Complex Numbers and Quadratic Equations
Q205Let the triangle PQR be the image of the triangle with vertices (1,3), (3,1), and (2,4) in the line x + 2y = 2. If the centroid of PQR is the point (α, β)…Coordinate Geometry
Q207Let the parabola y = x² + p x - 3 meet the coordinate axes at the points P, Q, and R. If the circle C with centre at (-1, -1) passes through the points P, Q…Coordinate Geometry
Q209Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y R. Then Σ n=1 ¹⁰⁰ log e f(n) is equal to:Limit, Continuity and Differentiability
Q210From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:Permutations and Combinations
Q211Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 16((sec ⁻¹ x)² + (csc ⁻¹ x)²) is:Trigonometry
Q212Let f: R → R be a twice differentiable function such that f(x + y) = f(x) f(y) for all x, y R. If f'(0) = 4a and f satisfies f''(x) - 3a f'(x) - f(x) = 0, a >…Integral Calculus
Q213The area of the region, inside the circle (x - 2√(3))² + y² = 12 and outside the parabola y² = 2√(3) x is:Integral Calculus
Q214Let the foci of a hyperbola be (1, 14) and (1, -12). If it passes through the point (1, 6), then the length of its latus-rectum is:Coordinate Geometry
Q216A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ² denote the mean and variance of X, then the value of 64(μ +…Statistics and Probability
Q217The number of non-empty equivalence relations on the set \1, 2, 3 is:Sets, Relations and Functions
Q218A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5)…Coordinate Geometry
Q219Let A = \1, 2, 3,, 10 and B = /n: m, n A, m < n and gcd (m, n) = 1. Then n(B) is equal to:Sets, Relations and Functions
Q227Let A=\(x, y) R × R: |x+y| 3 and B=\(x, y) R × R: |x|+|y| ≤ 3. If C=\(x, y) A B: x=0 or y=0, then Σ (x, y) C |x+y| is:Sets, Relations and Functions
Q228Let X=R × R. Define a relation R on X as: (a₁, b₁) R (a₂, b₂) b₁=b₂. Statement I: R is an equivalence relation. Statement II: For some (a, b) X, the set…Sets, Relations and Functions
Q229Let ∫ x³ sin x dx = g(x) + C, where C is the constant of integration. If 8(g((π)/2) + g'((π)/2)) = α π³ + β π² + γ, α, β, γ Z, then α + β - γ equals:Integral Calculus
Q230A rod of length eight units moves such that its ends A and B always lie on the lines x-y+2=0 and y+2=0, respectively. If the locus of the point P, that…Coordinate Geometry
Q233Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r:1 (r>0). If O is the origin and (OQ · OA…Vector Algebra
Q234The length of the chord of the ellipse (x²)/4 + (y²)/2 = 1, whose mid-point is (1, 1/2), is:Coordinate Geometry
Q235The system of equations x + y + z = 6, x + 2y + 5z = 9, x + 5y + λ z = μ has no solution if:Matrices and Determinants
Q236Let the range of the function f(x) = 6 + 16 cos x · cos ((π)/3 - x) · cos ((π)/3 + x) · sin 3x · cos 6x, x R be [α, β]. Then the distance of the point (α, β)…Trigonometry
Q237Let x = x(y) be the solution of the differential equation y = (x - y dx/dy) sin (x/y), y > 0 and x(1) = (π)/2. Then cos (x(2)) is equal to:Differential Equations
Q238A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at…Limit, Continuity and Differentiability
Q240Let A = [a ij ] be a 3 × 3 matrix such that A 0 1 0 = 0 0 1, A 4 1 3 = 0 1 0 and A 2 1 2 = 1 0 0, then a₂₃ equals:Matrices and Determinants
Q242A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:Statistics and Probability
Q243Let the shortest distance from (a, 0), a > 0, to the parabola y² = 4x be 4. Then the equation of the circle passing through the point (a, 0) and the focus of…Coordinate Geometry
Q244If in the expansion of (1 + x)^(p) (1 - x)^(q), the coefficients of x and x² are 1 and -2, respectively, then p² + q² is equal to:Binomial Theorem and Its Simple Applications