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

Mathematics 147

Mathematics

Q1If the image of the point P(1, 0, 3) in the line joining the points A(4, 7, 1) and B(3, 5, 3) is Q(α, β, γ), then α + β + γ is equal to:Three Dimensional Geometry
Q3The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by 21/2…Sequence and Series
Q5If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:Coordinate Geometry
Q6The line L₁ is parallel to the vector a = -3i + 2j + 4k and passes through the point (7, 6, 2) and the line L₂ is parallel to the vector b = 2i + j + 3k and…Three Dimensional Geometry
Q7Let (a, b) be the point of intersection of the curve x² = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral I = ∫ a ^(b)…Integral Calculus
Q8If the system of equations 2x + λ y + 3z = 5 3x + 2y - z = 7 4x + 5y + μ z = 9 has infinitely many solutions, then (λ² + μ²) is equal to:Matrices and Determinants
Q9If θ [-(7π)/6, (4π)/3], then the number of solutions of √(3) csc ²θ - 2(√(3) - 1) csc θ - 4 = 0, is equal to:Trigonometry
Q10Given three identical bags each containing 10 balls, whose colours are as follows: Bag I: Red = 3, Blue = 2, Green = 5 Bag II: Red = 4, Blue = 3, Green = 3…Statistics and Probability
Q11If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a + b + ab is equal to:Statistics and Probability
Q12If the domain of the function f(x) = 1/(√(10 + 3x - x²)) + 1/(√(x + |x|)) is (a, b), then (1 + a)² + b² is equal to:Sets, Relations and Functions
Q134∫₀¹ (1/(√(3 + x²) + √(1 + x²))) dx - 3 log e (√(3)) is equal to:Integral Calculus
Q14If lim x → 0 (cos (2x) + a cos (4x) - b)/(x⁴) is finite, then (a + b) is equal to:Limit, Continuity and Differentiability
Q15If Σ r=0 ¹⁰ ((10^(r+1) - 1)/(10^(r))) · ¹¹C r+1 = (α¹¹ - 11¹¹)/(10¹⁰), then α is equal to:Binomial Theorem and Its Simple Applications
Q17Let the point P of the focal chord PQ of the parabola y² = 16x be (1, -4). If the focus of the parabola divides the chord PQ in the ratio m: n, gcd (m, n) =…Coordinate Geometry
Q18Let a = 2i - 3j + k, b = 3i + 2j + 5k and a vector c be such that (a - c) × b = -18i - 3j + 12k and a · c = 3. If b × c = d, then |a · d| is equal to:Vector Algebra
Q19Let the area of the triangle formed by a straight line L: x + by + c = 0 with coordinate axes be 48 square units. If the perpendicular drawn from the origin…Coordinate Geometry
Q20Let A be a 3 × 3 real matrix such that A²(A - 2I) - 4(A - I) = O, where I and O are the identity and null matrices, respectively. If A⁵ = α A² + β A + γ I…Matrices and Determinants
Q26The largest n such that 3ⁿ divides 50! is:Permutations and Combinations
Q27The number of sequences of ten terms, whose terms are either 0, 1, or 2, that contain exactly five 1 ’s and exactly three 2 ’s is equal to:Permutations and Combinations
Q28Let one focus of the hyperbola H:(x²)/(a²)-(y²)/(b²)=1 be at (√(10),0) and the corresponding directrix be x=9/(√(10)). If e and l respectively are the…Coordinate Geometry
Q29Let f:R be a twice-differentiable function such that (sin x cos y) [f(2x+2y)-f(2x-2y) ]=(cos x sin y) [f(2x+2y)+f(2x-2y) ] for all x,y. If f'(0)=1/2, then the…Limit, Continuity and Differentiability
Q32If θ [-2π, 2π], then the number of solutions of 2√(2) cos ²θ + (2-√(6)) cos θ - √(3) = 0 is equal to:Trigonometry
Q33Let a₁,a₂,a₃, be in an A.P. such that Σ k=1 ¹²a 2k-1 =-72/5a₁, a₁≠0. If Σ k=1 ⁿ a k =0, then n is:Sequence and Series
Q34If the function f(x)=2x³ - 9a x² + 12a² x + 1, where a>0, attains its local maximum and local minimum at p and q respectively, such that p² = q, then f(3) is…Limit, Continuity and Differentiability
Q35Let z be a complex number such that |z|=1. If (2 + k² z)/(k + z) = k z, k, then the maximum distance of k + i k² from the circle |z - (1 + 2i)| = 1 is:Complex Numbers and Quadratic Equations
Q36If a is a nonzero vector such that its projections on the vectors 2i-j+2k, i+2j-2k, and k are equal, then a unit vector along a is:Vector Algebra
Q37Let A be the set of all functions f: Z → Z and R be a relation on A such that R = \(f,g): f(0)=g(1) and f(1)=g(0). Then R is:Sets, Relations and Functions
Q38For α,β,γ, if lim x→0 (x² sin (α x)+(γ-1)e^(x²))/(sin (2x)-β x)=3, then β+γ-α is equal to:Limit, Continuity and Differentiability
Q39Let P n =αⁿ+βⁿ, n. If P₁₀=123, P₉=76, P₈=47 and P₁=1, then the quadratic equation having roots 1/(α) and 1/(β) is:Complex Numbers and Quadratic Equations
Q40If the system of linear equations 3x + y + β z = 3, 2x + α y - z = -3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β - 9α is:Matrices and Determinants
Q41If S and S' are the foci of the ellipse (x²)/18 + (y²)/9 = 1 and P is a point on the ellipse, then min (SP· S'P) + max (SP· S'P) is equal to:Coordinate Geometry
Q42Let the vertices Q and R of the triangle PQR lie on the line (x+3)/5=(y-1)/2=(z+4)/3, QR = 5, and the coordinates of the point P be (0,2,3). If the area of…Three Dimensional Geometry
Q43Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square…Vector Algebra
Q44Let a and A be a matrix of order 3×3 such that det (A)=-4 and A+I= 1 a 1 2 1 0 a 1 2, where I is the 3×3 identity. If det ((a+1) ((a-1)A))=2^(m)3ⁿ, m,n…Matrices and Determinants
Q45Let the focal chord PQ of the parabola y² = 4x make an angle of 60° with the positive x-axis, where P lies in the first quadrant. If the circle, whose one…Coordinate Geometry
Q52Each of the angles β and γ that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the…Three Dimensional Geometry
Q53If the four distinct points (4,6), (-1,5), (0,0) and (k,3k) lie on a circle of radius r, then 10k + r² is equal to:Coordinate Geometry
Q54Let the mean and variance of five observations x₁=1, x₂=3, x₃=a, x₄=7 and x₅=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n +…Statistics and Probability
Q55Let A=\-2,-1,0,1,2,3. Define a relation R on A by xRy if and only if y= max (x,1). If l is the number of elements in R, and m and n are the minimum numbers of…Sets, Relations and Functions
Q56Consider the lines x(3λ+1)+y(7λ+2)=17λ+5, λ being a parameter, all passing through a point P. One of these lines (say L) is farthest from the origin. If the…Coordinate Geometry
Q57Let the equation x(x+2)(12-k)=2 have equal roots. Then the distance of the point (k,k/2) from the line 3x+4y+5=0 is:Coordinate Geometry
Q58Line L₁ of slope 2 and line L₂ of slope 1/2 intersect at the origin O. In the first quadrant, P₁,P₂,,P₁₂ are 12 points on L₁ and Q₁,Q₂,,Q₉ are 9 points on L₂…Permutations and Combinations
Q59Let f be a function such that f(x) + 3 f (24/x) = 4x, x≠0. Then f(3) + f(8) is equal to:Sets, Relations and Functions
Q60The integral ∫₀^(π)8x/(4 cos ²x + sin ²x) dx is equal to:Integral Calculus
Q61The area of the region \(x,y): x-y ≤ y ≤ 4√(x) is:Integral Calculus
Q62If the probability that the random variable X takes the value x is given by P(X=x)=k (x+1)3⁻ˣ, x=0,1,2,, where k is a constant, then P(X≥3) is equal to:Statistics and Probability
Q63Let y=y(x) be the solution of the differential equation dy/dx + 3(tan ²x) y + 3y = sec ²x, with y(0)=1/3+e³. Then y ((π)/4) is equal to:Differential Equations
Q64If z₁, z₂, z₃ are the vertices of an equilateral triangle whose centroid is z₀, then Σ k=1 ³ (z k - z₀)² is equal to:Complex Numbers and Quadratic Equations
Q65The number of solutions of the equation (4-√(3)) sin x - 2√(3) cos ²x = -4/(1+√(3)), x [-2π,(5π)/2] is equal to:Trigonometry
Q66The shortest distance between the curves y² = 8x and x² + y² + 12y + 35 = 0 is:Coordinate Geometry
Q68The distance of the point (7,10,11) from the line (x-4)/1=(y-4)/0=(z-2)/3 along the line (x-9)/2=(y-13)/(-3)=(z-17)/6 is:Three Dimensional Geometry
Q69The sum 1 + (1+3)/(2!) + (1+3+5)/(3!) + (1+3+5+7)/(4!) + up to infinitely many terms is equal to:Sequence and Series
Q70If the domain of the function f(x)= log ₇ (1- log ₄(x²-9x+18)) is (α,β) (γ,δ), then α+β+γ+δ is equal to:Sets, Relations and Functions
Q76Let A be a matrix of order 3×3 with A =5. If 2 adj (3A adj(2A)) =2^(α)3^(β)5^(γ), where α,β,γ, then α+β+γ is equal to:Matrices and Determinants
Q78Let α and β be the roots of x² + √(3) x - 16 = 0, and γ and δ be the roots of x² + 3x - 1 = 0. If P n = αⁿ + βⁿ and Q n = γⁿ + δⁿ, then (P₂₅ + √(3)…Complex Numbers and Quadratic Equations
Q79The sum of all rational terms in the expansion of (2+√(3))⁸ is:Binomial Theorem and Its Simple Applications
Q80Let A = \-3, -2, -1, 0, 1, 2, 3. Let R be a relation on A defined by xRy if and only if 0 ≤ x² + 2y ≤ 4. Let be the number of elements in R and m be the…Sets, Relations and Functions
Q81A line passing through the point P(√(5),√(5)) intersects the ellipse (x²)/36+(y²)/25=1 at points A and B such that (PA)·(PB) is maximum. Then 5 (PA²+PB²) is…Coordinate Geometry
Q82The sum 1 + 3 + 11 + 25 + 45 + 71 + up to 20 terms is equal to:Sequence and Series
Q83If the domain of the function f(x)= log e ((2x-3)/(5+4x))+ sin ⁻¹ ((4+3x)/(2-x)) is [α,β], then α²+4β is equal to:Sets, Relations and Functions
Q85If y(x)= sin x cos x sin x+ cos x+1 27 28 27 1 1 1, x, then (d²y)/(dx²)+y is equal to:Matrices and Determinants
Q86The number of solutions of the equation 2x + 3 tan x = π, x [-2π,2π] \±(π)/2,±(3π)/2 is:Trigonometry
Q88A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines L₁: 2x+y+6=0 and L₂: 4x+2y-p=0, p>0, at the…Coordinate Geometry
Q89Let z be such that (z² + 3i)/(z - 2 + i) = 2 + 3i. Then the sum of all possible values of z² is:Complex Numbers and Quadratic Equations
Q90Let f(x)=∫ x³√(3 - x²) dx. If 5f(√(2)) = -4, then f(1) is equal to:Integral Calculus
Q91Let a₁, a₂, a₃, be a G.P. of increasing positive numbers. If a₃ a₅ = 729 and a₂ + a₄ = 111/4, then 24(a₁ + a₂ + a₃) is equal to:Sequence and Series
Q93The radius of the smallest circle which touches the parabolas y = x² + 2 and x = y² + 2 is:Coordinate Geometry
Q94Let f(x)= (1+ax)^(1/x), x<0, 1+b, x=0, (√(x+4)-2)/(3√(x+c)-2), x>0 be continuous at x=0. Then e^(a) b c is equal to:Limit, Continuity and Differentiability
Q95Line L1 passes through the point (1, 2, 3) and is parallel to the z-axis. Line L2 passes through the point (λ, 5, 6) and is parallel to the y-axis. If for λ =…Three Dimensional Geometry
Q101Let a>0. If the function f(x)=6x³-45a x²+108a²x+1 attains its local maximum and minimum values at the points x₁ and x₂ respectively such that x₁x₂=54, then…Limit, Continuity and Differentiability
Q102The sum of the infinite series cot ⁻¹(7/4)+ cot ⁻¹(19/4)+ cot ⁻¹(39/4)+ cot ⁻¹(67/4)+ is:Trigonometry
Q103Let f be a differentiable function on R such that f(2)=1 and f'(2)=4. If lim x→0 (f(2+x))^(3/x)=e^(α), then the number of times the curve y=4x³-4x²-4(α-7)x-α…Limit, Continuity and Differentiability
Q104Let A = \-3,-2,-1,0,1,2,3 and R be a relation on A defined by xRy iff 2x - y \0,1. Let be the number of elements in R. Let m and n be the minimum numbers of…Sets, Relations and Functions
Q105Let the product of ω₁ = (8+i) sin θ + (7+4i) cos θ and ω₂ = (1+8i) sin θ + (4+7i) cos θ be α + iβ, where i=√(-1). Let p and q be the maximum and minimum…Complex Numbers and Quadratic Equations
Q106Let the values of p, for which the shortest distance between the lines (x+1)/3=y/4=z/5 and r=(pi+2j+k)+λ(2i+3j+4k) is 1/(6), be a,b with a<b. Then the length…Three Dimensional Geometry
Q107The axis of a parabola is the line y = x and its vertex and focus are in the first quadrant at distances √(2) and 2√(2) from the origin, respectively. If the…Coordinate Geometry
Q109A line passing through the point A(-2,0) touches the parabola P: y² = x - 2 at the point B in the first quadrant. The area of the region bounded by the line…Integral Calculus
Q111Let the matrix A= 1 0 0 1 0 1 0 1 0 satisfy Aⁿ = Aⁿ⁻² + A² - I for n≥3. Then the sum of all the elements of A⁵⁰ is:Matrices and Determinants
Q112If the sum of the first 20 terms of the series (4·1)/(4+3·1²+1⁴)+(4·2)/(4+3·2²+2⁴)+(4·3)/(4+3·3²+3⁴)+ is m/n, where m and n are coprime, then m+n is equal to:Sequence and Series
Q115Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively, and the sum and…Sequence and Series
Q116If a curve y=y(x) passes through the point (1,(π)/2) and satisfies the differential equation (7x⁴ cot y - eˣ csc y)dx/dy = x⁵, x≥1, then at x=2, the value of…Differential Equations
Q117The centre of a circle C is at the centre of the ellipse E: (x²)/(a²)+(y²)/(b²)=1, a>b. Let C pass through the foci F₁ and F₂ of E such that the circle C and…Coordinate Geometry
Q119Let A be the point of intersection of the lines L₁: (x-7)/1=(y-5)/0=(z-3)/(-1), and L₂: (x-1)/3=(y+3)/4=(z+7)/5. Let B and C be points on L₁ and L₂…Three Dimensional Geometry
Q120Let the mean and the standard deviation of the observations 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these…Statistics and Probability
Q126Let f, g: (1, ∞) → R be defined as f(x) = (2x+3)/(5x+2) and g(x) = (2-3x)/(1-x). If the range of the function f° g: [2,4] → R is [α, β], then 1/(β - α) is…Sets, Relations and Functions
Q128Let A = \1, 6, 11, 16, and B = \9, 16, 23, 30, be the sets consisting of the first 2025 terms of two arithmetic progressions. Then n(A B) is:Sequence and Series
Q129The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:Statistics and Probability
Q130For an integer n ≥ 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)²ⁿ⁻³ is 16, then the distance of the point P(2n-1, n²-4n)…Binomial Theorem and Its Simple Applications
Q131Let the shortest distance between the lines (x-3)/3 = (y-α)/(-1) = (z-3)/1 and (x+3)/(-3) = (y+7)/2 = (z-β)/4 be 3√(30). Then the positive value of 5α + β is:Three Dimensional Geometry
Q132If lim x→1 ((x-1)(6 + λ cos (x-1)) + μ sin (1-x))/((x-1)³) = -1, where λ, μ R, then λ + μ is equal to:Limit, Continuity and Differentiability
Q134Let A and B be two distinct points on the line L: (x-6)/3 = (y-7)/2 = (z-7)/-2. Both A and B are at a distance 2√(17) from the foot of the perpendicular drawn…Three Dimensional Geometry
Q135Let f: R → R be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x R. If lim n→∞ [f(x) - f(x/2ⁿ)] = G(x), then Σ r=1 ¹⁰ G(r²) is equal to:Limit, Continuity and Differentiability
Q1361 + 3 + 5² + 7 + 9² + up to 40 terms is equal to:Sequence and Series
Q138Considering the principal values of the inverse trigonometric functions, sin ⁻¹ ((√(3))/2x + 1/2√(1-x²)), -1/2 < x < 1/(√(2)), is equal to:Trigonometry
Q139Consider two vectors u=3i-j and v=2i+j-, λ > 0. The angle between them is given by cos ⁻¹ ((√(5))/(2√(7))). Let v=v₁+v₂, where v₁ is parallel to u and v₂ is…Vector Algebra
Q140Let the three sides of a triangle be on the lines 4x - 7y + 10 = 0, x + y = 5, and 7x + 4y = 15. Then the distance of its orthocentre from the orthocentre of…Coordinate Geometry
Q142The length of the latus-rectum of the ellipse whose foci are (2,5) and (2,-3) and eccentricity is 4/5 is:Coordinate Geometry
Q143Consider the equation x² + 4x - n = 0, where n [20,100] is a natural number. Then the number of all distinct values of n, for which the given equation has…Complex Numbers and Quadratic Equations
Q144A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is:Statistics and Probability
Q145If 10 sin ⁴θ + 15 cos ⁴θ = 6, then the value of (27 csc ⁶θ + 8 sec ⁶θ)/(16 sec ⁸θ) is:Trigonometry
Q151If the orthocentre of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = m x + c is at (3, -1), then m - c is:Coordinate Geometry
Q152Let the vectors a and b be of the same magnitude such that (a + b + a - b)/(a + b - a - b) = √(2) + 1. Then (a + b ²)/(a ²) is:Vector Algebra
Q153Let A = \(α, β) R × R: |α - 1| ≤ 4 and |β - 5| ≤ 6 and B = \(α, β) R × R: 16(α - 2)² + 9(β - 6)² ≤ 144. Then:Sets, Relations and Functions
Q154If the range of the function f(x) = (5 - x)/(x² - 3x + 2), x ≠ 1,2, is (-∞, α) [β, ∞), then α² + β² is equal to:Sets, Relations and Functions
Q155A bag contains 19 unbiased coins and one coin with heads on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the…Statistics and Probability
Q156Let a random variable X take values 0, 1, 2 and 3 with P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) = (1 - 2p)/2. If E(X²) = 2E(X), then the value of 8p…Statistics and Probability
Q157If the area of the region \(x,y): 1 + x² ≤ y ≤ min + 7, 11 - 3x is A, then 3A is equal to:Integral Calculus
Q158Let f: R → R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If lim x → 0 (f(x))/(x²) = 5, then f(2) is equal to:Limit, Continuity and Differentiability
Q159The number of solutions of the equation cos 2θ cos (θ)/2 + cos (5θ)/2 = 2 cos ³(5θ)/2 in [-(π)/2,(π)/2] is:Trigonometry
Q160Let a n be the n th term of an A.P. If S n = a₁ + a₂ + + a n = 700 for some n, a₆ = 7 and S₇ = 7, then a n is equal to:Sequence and Series
Q161If the locus of z C, such that ((z-1)/(2z + i)) + ((z-1)/(2z - i)) = 2, is a circle of radius r and center (a,b), then 15ab/(r²) is equal to:Complex Numbers and Quadratic Equations
Q162Let the length of a latus rectum of an ellipse (x²)/(a²) + (y²)/(b²) = 1 be 10. If its eccentricity is the minimum value of the function f(t) = t² + t +…Coordinate Geometry
Q163Let y = y(x) be the solution of the differential equation (x² + 1)y' - 2xy = (x⁴ + 2x² + 1) cos x, with y(0) = 1. Then ∫₋₃³ y(x) dx is:Differential Equations
Q164If the equation of the line passing through the point (0, -1/2, 0) and perpendicular to the lines r = λ(i + aj + bk) and r = (i - j - 6k) + μ(-bi + aj + 5k)…Three Dimensional Geometry
Q165Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals…Permutations and Combinations
Q167The number of real roots of the equation x x - 2 + 3 x - 3 + 1 = 0 is:Complex Numbers and Quadratic Equations
Q169Let the system of equations x + 5y - z = 1, 4x + 3y - 3z = 7, and 24x + y + λ z = μ, with λ,μ R, have infinitely many solutions. Then the number of solutions…Matrices and Determinants
Q170If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum…Sequence and Series
Q177If the shortest distance between the lines (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and x/1 = y/(α) = (z - 5)/1 is 5/(√(6)), then the sum of all possible values of α…Three Dimensional Geometry
Q179The remainder when (64⁶⁴)⁶⁴ is divided by 7 is equal to:Binomial Theorem and Its Simple Applications
Q180Let P be the parabola, whose focus is (-2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:Coordinate Geometry
Q181Let y = y(x) be the solution curve of the differential equation x(x² + eˣ)dy + (eˣ(x - 2)y - x³)dx = 0, x > 0, passing through the point (1, 0). Then y(2) is…Differential Equations
Q182From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman…Permutations and Combinations
Q183If for θ [ -(π)/3, 0 ], the points (x, y) = (3 tan (θ + (π)/3), 2 tan (θ + (π)/6)) lie on xy + α x + β y + γ = 0, then α² + β² + γ² is equal to:Trigonometry
Q184Let C₁ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C₂ be the circle with centre (1, 3) that touches C₁ externally…Coordinate Geometry
Q186Among the statements: (S1): The set z C \-i: |z| = 1 and (z - i)/(z + i) is purely real contains exactly two elements, and (S2): The set z C \-1: |z| = 1 and…Complex Numbers and Quadratic Equations
Q187The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake, one observation is taken as 50 instead of 40. If the correct…Statistics and Probability
Q188Let x₁, x₂, x₃, x₄ be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x₁, x₂, x₃, x₄, then the resulting numbers are in an…Sequence and Series
Q189Let the set of all values of p R, for which both the roots of the equation x² - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the interval (α, β]…Complex Numbers and Quadratic Equations
Q190Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 81. If S = n Z: |adj(adj A)|^(((n-1)²)/2) = |A|^(3n² - 5n - 4) then Σ n S | A^(n² + n) | is equal to:Matrices and Determinants
Q191If the area of the region bounded by the curves y = 4 - (x²)/4 and y = (x - 4)/2 is equal to α, then 6α equals:Integral Calculus
Q192Let the system of equations: 2x + 3y + 5z = 9, 7x + 3y - 2z = 8, 12x + 3y - (4 + λ)z = 16 - μ have infinitely many solutions. Then the radius of the circle…Matrices and Determinants
Q193Let the line L pass through (1, 1, 1) and intersect the lines (x - 1)/2 = (y + 1)/3 = (z - 1)/4 and (x - 3)/1 = (y - 4)/2 = z/1. Then, which of the following…Three Dimensional Geometry
Q194Let the angle θ, 0 < θ < (π)/2, be the angle between two unit vectors a and b, and θ = sin ⁻¹((√(65))/9). If the vector c = 3a + 6b + 9(a × b), then the value…Vector Algebra
Q195Let ABC be the triangle such that the equations of lines AB and AC be 3y - x = 2 and x + y = 2, respectively, and the points B and C lie on the x-axis. If P…Coordinate Geometry
Q201Let the values of λ for which the shortest distance between the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x - λ)/3 = (y - 4)/4 = (z - 5)/5 is 1/(√(6)) be λ₁ and…Three Dimensional Geometry
Q204If 1/(1⁴) + 1/(2⁴) + 1/(3⁴) + = (π⁴)/90, 1/(1⁴) + 1/(3⁴) + 1/(5⁴) + = α, 1/(2⁴) + 1/(4⁴) + 1/(6⁴) + = β, then (α)/(β) is equal toSequence and Series
Q205If A and B are two events such that P(A) = 0.7, P(B) = 0.4 and P(A B) = 0.5, where B denotes the complement of B, then P(B (A B)) is equal to:Statistics and Probability
Q207Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle α with the positive x-axis and the equations of its…Coordinate Geometry
Q209Let a = i + 2j + k and b = 2i + j - k. Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a. Then such a vector c is:Vector Algebra
Q210Let the ellipse 3x² + py² = 4 pass through the centre C of the circle x² + y² - 2x - 4y - 11 = 0 of radius r. Let f₁, f₂ be the focal distances of the point C…Coordinate Geometry
Q211The integral ∫₋₁^(3/2) | π² x sin (π x) | dx is equal to:Integral Calculus
Q213There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that…Permutations and Combinations
Q214Let A = θ [0, 2π]: 1 + 10 Re ((2 cos θ + i sin θ)/(cos θ - 3i sin θ)) = 0 \. Then Σ θ A θ² is equal to:Complex Numbers and Quadratic Equations
Q216Let A = \0, 1, 2, 3, 4, 5. Let R be a relation on A defined by (x, y) R if and only if max, y \3, 4. Then among the statements (S₁): The number of elements in…Sets, Relations and Functions
Q218The value of cot ⁻¹ ((√(1 + tan ²(2)) - 1)/(tan (2))) - cot ⁻¹ ((√(1 + tan ²(1/2)) + 1)/(tan (1/2))) is equal to:Trigonometry
Q219Let A = 2 2 + p 2 + p + q 4 6 + 2p 8 + 3p + 2q 6 12 + 3p 20 + 6p + 3q. If det (adj(adj(3A))) = 2^(m) · 3ⁿ, m, n N, then m + n is equal to:Matrices and Determinants