CBSE Class 12 Mathematics 2025
367 questions · 782 marks · 365 with the board’s own marking-scheme answer, 2 with no official answer on the site.
Marks by chapter
Counted from the 367 questions listed below, as the 2025 papers print them. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage. Every year compared →
| Chapter | Marks | Share |
|---|---|---|
| Integrals45 questions | 112 | 14.3% |
| Matrices50 questions | 89 | 11.4% |
| Application of Derivatives36 questions | 87 | 11.1% |
| Probability32 questions | 72 | 9.2% |
| Three Dimensional Geometry21 questions | 62 | 7.9% |
| Continuity and Differentiability30 questions | 61 | 7.8% |
| Linear Programming31 questions | 58 | 7.4% |
| Differential Equations25 questions | 55 | 7.0% |
| Vector Algebra30 questions | 52 | 6.6% |
| Application of Integrals16 questions | 51 | 6.5% |
| Relations and Functions24 questions | 46 | 5.9% |
| Inverse Trigonometric Functions19 questions | 27 | 3.5% |
| Determinants8 questions | 10 | 1.3% |
367 questions from 2025
- Q1Let both AB^′ and B^′ A be defined for matrices A and B. If order of A is n × m, then the order of B is:1 mk
- Q1The values of x for which the angle between the vectors a=2 x^2 i+4 x j+ k and b=7 i-2 j+x k is obtuse, is:1 mk
- Q1If p and q are unit vectors, then which of the following values of p · q is not possible?1 mk
- Q1If A= [5, 0, 0; 0, 5, 0; 0, 0, 5], then A^3 is:1 mk
- Q1The given graph illustrates:1 mk
- Q1Domain of sin^-1(2 x^2-3) is:1 mk
- Q1The given graph illustrates:1 mk
- Q1The principal value of sin^-1( sin (-(10 π)/3)) is:1 mk
- Q1The principal branch of cos^-1 x is:1 mk
- Q1Study the given graph. It illustrates:1 mk
- Q1The projection vector of vector a on vector b is1 mk
- Q2If A= [0, -3, 8; 3, 0, 5; -8, -5, 0], then A is a:1 mk
- Q2If vector a=3 i+2 j- k and vector b= i- j+ k, then which of the following is correct?1 mk
- Q2Domain of f(x)= cos^-1 x+ sin x is:1 mk
- Q2The unit vector perpendicular to the vectors i- j and i+ j is1 mk
- Q2If A and B are square matrices of same order such that AB=A and BA=B, then A^2+B^2 is equal to:1 mk
- Q2If P(A ∪ B)=0 · 9 and P(A ∩ B)=0 · 4, then P(Ā)+P(B̄) is:1 mk
- Q2If A= [-1, 0, 0; 0, 3, 0; 0, 0, 5], then A is a/an:1 mk
- Q2If α= i-4 j+9 k and β=2 i-8 j+λ k are two mutually parallel vectors, then λ is equal to:1 mk
- Q2If A= [0, 0, -5; 0, 3, 0; 4 · 3, 0, 0], then A is a:1 mk
- Q2The function f(x)=x^2-4 x+6 is increasing in the interval1 mk
- Q3The graph shown below depicts:1 mk
- Q3The graph shown below depicts:1 mk
- Q3If ∫_0^1 (e^x)/(1+x) d x=α, then ∫_0^1 (e^x)/((1+x)^2) d x is equal to1 mk
- Q3If f(2 a-x)=f(x), then ∫_0^2 a f(x) d x is1 mk
- Q3What is the total number of possible matrices of order 3 × 3 with each entry as √2 or √3?1 mk
- Q3For real x, let f(x)=x^3+5 x+1. Then:1 mk
- Q3∫_-1^1 (|x|)/x d x, x ≠ 0 is equal to1 mk
- Q3∫ (1-2 sin x)/(cos^2 x) d x is equal to:1 mk
- Q3∫ (d x)/(sin^2 x cos^2 x) is equal to1 mk
- Q3If A and B are two square matrices of the same order, then (A+B)(A-B) is equal to:1 mk
- Q3If ∫_0^a x d x ≤ a/2+6, then1 mk
- Q3If A=kB, where A and B are two square matrices of order n and k is a scalar, then:1 mk
- Q3The following graph is a combination of:1 mk
- Q3If A= [1, 2, 3; -4, 3, 7] and B= [4, 3; -1, 2; 0, 5], then the correct statement is:1 mk
- Q4Let P be a skew-symmetric matrix of order 3. If det(P)=α, then (2025)^α is1 mk
- Q4Which of the following is not a homogeneous function of x and y?1 mk
- Q4If f(x)= {(1- sin^3 x)/(3 cos^2 x), for x ≠ (π)/2; k, for x=(π)/2} is continuous at x=(π)/2, then the value of k is:1 mk
- Q4The matrix A= [√3, 0, 0; 0, √2, 0; 0, 0, √5] is a/an:1 mk
- Q4If A= [1, 12, 4 y; 6 x, 5, 2 x; 8 x, 4, 6] is a symmetric matrix, then (2 x+y) is1 mk
- Q4If y= sin^-1 x, then (1-x^2) (d^2 y)/(d x^2) is equal to:1 mk
- Q4If y= log_2 x(√(2 x)), then (d y)/(d x) is equal to:1 mk
- Q4If A and B are square matrices of same order, then (AB^T-BA^T) is a1 mk
- Q4If |2 x, 5; 12, x| = |6, -5; 4, 3|, then the value of x is:1 mk
- Q4Sum of two skew-symmetric matrices of same order is always a/an:1 mk
- Q5If f(x)= {(sin^2 a x)/(x^2), x ≠ 0; 1, x=0} is continuous at x=0, then the value of a is:1 mk
- Q5If f(x)=|x|+|x-1|, then which of the following is correct?1 mk
- Q5If A and B are two square matrices each of order 3 with |A|=3 and |B|=5, then |2 AB| is:1 mk
- Q5The value of cos ((π)/6+ cot^-1(-√3)) is1 mk
- Q5[ sec^-1(-√2)- tan^-1(1/(√3))] is equal to:1 mk
- Q5The values of λ so that f(x)= sin x- cos x-λ x+C decreases for all real values of x are:1 mk
- Q5The principal value of sin^-1( cos (43 π)/5) is1 mk
- Q5If y= sin^-1 x,-1 ≤ x ≤ 0, then the range of y is1 mk
- Q6If f(x)= {(log (1+a x)+ log (1-b x))/x, for x ≠ 0; k, for x=0} is continuous at x=0, then the value of k is:1 mk
- Q6If P is a point on the line segment joining ( 3,6,-1 ) and ( 6,2,-2 ) and y -coordinate of P is 4, then its z…1 mk
- Q6The coordinates of the foot of the perpendicular drawn from the point A(-2,3,5) on the y -axis is:1 mk
- Q6If the direction cosines of a line are λ, λ, λ, then λ is equal to:1 mk
- Q6If A=[a_ij] is a 3 × 3 diagonal matrix such that a_11=1, a_22=5 and a_33=-2, then |A| is:1 mk
- Q6If A is a square matrix of order 2 such that det(A)=4, then det(4 adj A) is equal to:1 mk
- Q6If A is a square matrix of order 3 such that det(A)=9, then det(9 A^-1) is equal to1 mk
- Q6If a line makes angles of (3 π)/4, (π)/3 and θ with the positive directions of x, y and z -axis respectively, then θ is1 mk
- Q6Let A be a square matrix of order 3. If |A|=5, then |adj A| is:1 mk
- Q6If [x+y, 3 y; 3 x, x+3] = [9, 4 x+y; x+6, y], then (x-y)=?1 mk
- Q7Let M and N be two events such that P(M)=0.6, P(N)=0.2 and P(M ∩ N)=0.5, then P(M^′ / N^′) is1 mk
- Q7If M and N are square matrices of order 3 such that det(M)=m and MN=mI, then det(N) is equal to:1 mk
- Q7The principal value of cot^-1(-1/(√3)) is:1 mk
- Q7If |-1, 2, 4; 1, x, 1; 0, 3, 3 x| =-57, the product of the possible values of x is:1 mk
- Q7If [2 x-1, 3 x; 0, y^2-1] = [x+3, 12; 0, 35], then the value of (x-y) is:1 mk
- Q7If P(A ∪ B)=0.9 and P(A ∩ B)=0.4, then P(Ā)+P(B̄) is:1 mk
- Q7If E and F are two events such that P(E)>0 and P(F) ≠ 1, then P(Ē / F̄) is1 mk
- Q7If tan^-1(x^2-y^2)=a, where ' a ' is a constant, then (d y)/(d x) is:1 mk
- Q7If A and B are invertible matrices of order 3 × 3 such that det(A)=4 and det[(AB)^-1]=1/20, then det(B) is equal to:1 mk
- Q7If E and F are two independent events such that P(E)=2/3, P(F)=3/7, then P(E / F̄) is equal to:1 mk
- Q8If a matrix A is both symmetric and skew-symmetric, then A is a:1 mk
- Q8Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify 4 AB+3(AB+BA)-4 BA, where A and B are both matrices…1 mk
- Q8Let A and B be two matrices of suitable orders. Then, which of the following is not correct?1 mk
- Q8If f(x)= {1, if x ≤ 3; ax+b, if 3<x<5; 7, if 5 ≤ x} is continuous in R, then the values of a and b are:1 mk
- Q8If [4+x, x-1; -2, 3] is a singular matrix, then the value of x is:1 mk
- Q8The absolute maximum value of function f(x)=x^3-3 x+2 in [0,2] is:1 mk
- Q8Which of the following can be both a symmetric and skew-symmetric matrix?1 mk
- Q8If f(x)= {3 x-2, 0<x ≤ 1; 2 x^2+ax, 1<x<2} is continuous for x ∈(0,2), then a is equal to:1 mk
- Q8If y=a cos ( log x)+b sin ( log x), then x^2 y_2+xy_1 is:1 mk
- Q8If f(x)= {3 a x-b, x>1; 11, x=1; -5 a x-2 b, x<1} is continuous at x=1, then the values of a and b are:1 mk
- Q9If f: N → W is defined as f(n)= {n/2, if n is even; 0, if n is odd}, then f is:1 mk
- Q9If f(x)=-2 x^8, then the correct statement is:1 mk
- Q9f(x)=x^x has a critical point at:1 mk
- Q9If f(x)={[x], x ∈ R} is the greatest integer function, then the correct statement is:1 mk
- Q9Let f(x)=|x|, x ∈ R. Then, which of the following statements is incorrect?1 mk
- Q9Let A= [1, -2, -1; 0, 4, -1; -3, 2, 1], B= [-2; -5; -7], C= [9, 8, 7], which of the following is defined?1 mk
- Q9Let A be a matrix of order m × n and B is a matrix such that A^T B and B A^T are defined. Then, the order of B is:1 mk
- Q9Let f(x)=x^2, x ∈ R. Then, which of the following statements is incorrect?1 mk
- Q9The equation of a line parallel to the vector 3 i+ j+2 k and passing through the point (4,-3,7) is:1 mk
- Q10If ∫ (2^(1/x))/(x^2) d x=k · 2^(1/x)+C, then k is equal to1 mk
- Q10The slope of the curve y=-x^3+3 x^2+8 x-20 is maximum at:1 mk
- Q10Edge of a variable cube increases at the rate of 5 cm / s. The rate at which the surface area of the cube increases…1 mk
- Q10Let f^′(x)=3(x^2+2 x)-4/(x^3)+5, f(1)=0. Then, f(x) is:1 mk
- Q10If R be a relation defined as a R b iff |a-b|>0 a, b ∈ R then R is:1 mk
- Q10A spherical ball has a variable diameter 5/2(3 x+1). The rate of change of its volume w.r.t. x, when x=1, is:1 mk
- Q10The matrix [0, 1, -2; -1, 0, -7; 2, 7, 0] is a:1 mk
- Q11A system of linear equations is represented as AX=B, where A is coefficient matrix, X is variable matrix and B is the…1 mk
- Q11In the following probability distribution, the value of p is:1 mk
- Q11The probability distribution of a random variable X is given by: Then E(X) of distribution is1 mk
- Q11If f: R → R is defined as f(x)=2 x- sin x, then f is:1 mk
- Q11If the sides A B and A C of A B C are represented by vectors j+ k and 3 i- j+4 k respectively, then the length of the…1 mk
- Q11A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100 π cm^3 / s. The rate, at which the…1 mk
- Q11If A= [-1, 0, 0; 0, 1, 0; 0, 0, 1], then A^-1 is1 mk
- Q11If a+ b+ c= 0,| a|=√37,| b|=3 and | c|=4, then angle between b and c is1 mk
- Q11∫ √(1+ sin x) dx is equal to:1 mk
- Q11∫ (cos 2 x)/(sin^2 x cos^2 x) dx is equal to:1 mk
- Q12Let p and q be two unit vectors and α be the angle between them. Then ( p+ q) will be a unit vector for what value of α?1 mk
- Q12If PQ × PR=4 i+8 j-8 k, then the area ( PQR ) is1 mk
- Q12The integrating factor of differential equation (x+2 y^3) (d y)/(d x)=2 y is1 mk
- Q12The order and degree of the differential equation (d^2 y)/(d x^2)+4((d y)/(d x))=x log ((d^2 y)/(d x^2)) are…1 mk
- Q12The order and degree of the differential equation [(d^2 y)/(d x^2)-1]^2=(d y)/(d x) are, respectively:1 mk
- Q12∫ (e^9 log x-e^8 log x)/(e^6 log x-e^5 log x) dx is equal to:1 mk
- Q12The function f defined by f(x)= {x, if x ≤ 1; 5, if x>1} is not continuous at:1 mk
- Q12If ∫_0^a 1/(1+4 x^2) dx=(π)/8, then the value of 'a' is:1 mk
- Q12The order and degree of the following differential equation are, respectively: -(d^4 y)/(d x^4)+2 e^d y / d x+y^2=01 mk
- Q12∫_0^π / 2 cos x · e^sin x dx is equal to:1 mk
- Q12∫_0^1 (2 x)/(5 x^2+1) dx is equal to:1 mk
- Q12The integrating factor of the differential equation dy/dx+y=(1+y)/x is1 mk
- Q12If projection of a=α i+ j+4 k on b=2 i+6 j+3 k is 4 units, then α is1 mk
- Q13The area of the region enclosed between the curve y=x|x|, x -axis, x=-2 and x=2 is:1 mk
- Q13The solution for the differential equation log ((d y)/(d x))=3 x+4 y is:1 mk
- Q13The line x=1+5 μ, y=-5+μ, z=-6-3 μ passes through which of the following point?1 mk
- Q13If A= [7, 0, x; 0, 7, 0; 0, 0, y] is a scalar matrix, then y^x is equal to1 mk
- Q13For a function f(x), which of the following holds true?1 mk
- Q13If f(x)=2 x+ cos x, then f(x):1 mk
- Q13The order and degree of differential function [1+(dy/dx)^2]^5=(d^2 y)/(d x^2) are1 mk
- Q13Let A=[a_ij] be a square matrix of order 3 such that a_ij= j-2 i. Then which of the following is true?1 mk
- Q14∫ (e^-x)/(16+9 e^-2 x) dx is equal to:1 mk
- Q14For a Linear Programming Problem (LPP), the given objective function is Z=x+2 y. The feasible region PQRS determined by…1 mk
- Q14If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following…1 mk
- Q14∫ (cos 2 x- cos 2 α)/(cos x- cos α) d x is equal to:1 mk
- Q14∫ (e^x)/(√(4-e^2 x)) dx is equal to:1 mk
- Q14∫ (a^x)/(√(1-a^2 x)) dx is equal to:1 mk
- Q14The integrating factor of the differential equation (d x)/(d y)=(x log x)/(2/x log x-y) is:1 mk
- Q14The integrating factor of the differential equation (d x)/(d y)=(-(1+ sin x))/(x+y cos x) is:1 mk
- Q14∫ e^x( cos x- sin x) dx is equal to:1 mk
- Q14The integrating factor of the differential equation ((e^-2 √x)/(√x)-y/(√x)) (d x)/(d y)=1 is:1 mk
- Q14If ∫ e^-3 log x dx=f(x)+C, then f(x) is:1 mk
- Q15The area of the shaded region (figure) represented by the curves y=x^2, 0 ≤ x ≤ 2 and y -axis is given by1 mk
- Q15The value of ∫_0^1 (d x)/(e^x+e^-x) is:1 mk
- Q15If A and B are invertible matrices, then which of the following is not correct?1 mk
- Q15A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the…1 mk
- Q15The sum of the order and degree of the differential equation [1+((d y)/(d x))^2]^3=(d^2 y)/(d x^2) is:1 mk
- Q16∫ (x+5)/((x+6)^2) e^x dx is equal to:1 mk
- Q16For a Linear Programming Problem (LPP), the given objective function Z=3 x+2 y is subject to constraints: x+2 y ≤ 10; 3…1 mk
- Q16A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function…1 mk
- Q16Let | a|=5 and -2 ≤ λ ≤ 1. Then, the range of |λ a| is:1 mk
- Q16The solution of the differential equation (d y)/(d x)=-x/y represents family of:1 mk
- Q16The order and degree of the differential equation ((d^2 y)/(d x^2))^2+((d y)/(d x))^2=x sin ((d y)/(d x)) are:1 mk
- Q16If the feasible region of a linear programming problem with objective function Z=a x+by, is bounded, then which of the…1 mk
- Q16The integrating factor of the differential equation dy/dx+y tan x- sec x=0 is:1 mk
- Q16If | a+ b|=| a- b| for any two vectors, then vectors a and b are:1 mk
- Q17If A and B are two events such that P(B)=1/5, P(A ∣ B)=2/3 and P(A ∪ B)=3/5, then P(A) is:1 mk
- Q17If P(A)=1/5, P(B)=3/5 and P(A/B)=2/5, then P(A^′ ∩ B^′) is:1 mk
- Q17The corner points of the feasible region in graphical representation of a L. P. P. are (2,72), (15,20) and (40,15). If…1 mkasked 2×
- Q17The area of the region bounded by the curve y^2=x between x=0 and x=1 is:1 mk
- Q17If A and B are square matrices of order m such that A^2-B^2=(A-B)(A+B), then which of the following is always correct?1 mk
- Q17In a Linear Programming Problem (LPP), the objective function Z=2 x+5 y is to be maximised under the following…1 mk
- Q17If P(A)=1/7, P(B)=5/7 and P(A ∩ B)=4/7, then P(Ā ∣ B) is:1 mk
- Q17The area of the region enclosed by the curve y=√x and the lines x=0 and x=4 and x -axis is:1 mk
- Q17Let a be a position vector whose tip is the point (2, -3). If AB= a, where coordinates of A are (-4,5), then the…1 mk
- Q18A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the…1 mk
- Q18The corner points of the feasible region of a Linear Programming Problem are (0,2),(3,0),(6,0),(6,8) and (0,5). If…1 mk
- Q18If p and q are respectively the order and degree of the differential equation d/(d x)(dy/(d x))^3=0, then (p-q) is1 mk
- Q18The respective values of | a| and | b|, if given ( a- b) ·( a+ b)=512 and | a|=3| b|, are:1 mk
- Q18The area of the shaded region bounded by the curves y^2=x, x=4 and the x -axis is given by1 mk
- Q18Chances that three persons A, B, and C go to the market are 30 %, 60 % and 50 % respectively. The probability that at…1 mk
- Q18The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse?1 mk
- Q18A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of…1 mk
- Q18A meeting will be held only if all three members A, B and C are present. The probability that member A does not turn up…1 mk
- Q19Assertion: The shaded portion of the graph represents the feasible region for the given Linear Programming Problem…1 mk
- Q19Assertion: If | a × b|^2+| a · b|^2=256 and | b|=8, then | a|=2 Reason (R): sin^2 θ+ cos^2 θ=1 and | a × b|=| a|| b|…1 mk
- Q19Assertion ( A ): If A and B are two events such that P(A ∩ B)=0, then A and B are independent events. Reason (R): Two…1 mk
- Q19Assertion: Let Z be the set of integers. A function f: Z → Z defined as f(x)=3 x-5, ∀ x ∈ Z is a bijective. Reason (R):…1 mk
- Q19Assertion: f(x)= {x sin 1/x, x ≠ 0; 0, x=0} is continuous at x=0. Reason (R): When x → 0, sin 1/x is a finite value…1 mk
- Q20Assertion: A =diag [3, 5, 2] is a scalar matrix of order 3 × 3. Reason (R): If a diagonal matrix has all non-zero…1 mk
- Q20Assertion: Let f(x)=e^x and g(x)= log x. Then (f+g) x=e^x+ log x where domain of ( f+g ) is R. Reason (R):…1 mk
- Q20Assertion: f(x)= {3 x-8, x ≤ 5; 2 k, x>5} is continuous at x=5 for k=5/2. Reason (R): For a function f to be continuous…1 mk
- Q20Assertion: Set of values of sec^-1((√3)/2) is a null set. Reason (R): sec^-1 x is defined for x ∈ R-(-1,1).1 mk
- Q20Assertion: Every point of the feasible region of a Linear Programming Problem is an optimal solution. Reason (R): The…1 mk
- Q20Assertion: Let A={x ∈ R:-1 ≤ x ≤ 1}. If f: A → A be defined as f(x)=x^2, then f is not an onto function. Reason (R): If…1 mk
- Q20Assertion: In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no…1 mk
- Q21Let f: A → B be defined by f(x)=(x-2)/(x-3), where A=R-{3} and B=R-{1}. Discuss the bijectivity of the function.2 mk
- Q21A vector a makes equal angles with all the three axes. If the magnitude of the vector is 5 √3 units, then find a. OR If…2 mk
- Q21Find the domain of f(x)= sin^-1(-x^2).2 mk
- Q21Find the domain of sec^-1(2 x+1).2 mk
- Q21Using matrices and determinants, find the value(s) of k for which the pair of equations 5 x-ky=2; 7 x-5 y=3 has a…2 mk
- Q21Find the domain of the function f(x)= cos^-1(x^2-4).2 mk
- Q21Let A and B be two square matrices of order 3 such that det(A)=3 and det(B)=-4. Find the value of det(-6 A B).2 mk
- Q21If A= [2, -2; -2, 2] and A^2=kA, then find the value of k.2 mk
- Q21Differentiate 2^cos^2 x w. r. t cos^2 x. OR If tan^-1(x^2+y^2)=a^2, then find dy/dx.2 mk
- Q21Find the domain of sin^-1(x^2-3).2 mk
- Q22Find the values of 'a' for which f(x)=x^2-2 a x+b is an increasing function for x>0.2 mk
- Q22Find the least value of ' a ' so that f(x)=2 x^2-a x+3 is an increasing function on [2,4]. OR If f(x)=x+1/x, x ≥ 1,…2 mk
- Q22Evaluate: ∫_0^π (sin 2 p x)/(sin x) d x, p ∈ N.2 mk
- Q22If [3, -1; 0, 1; 2, -3] A= [2; -5; -17], then find matrix A.2 mk
- Q22Evaluate: ∫_0^((π)/4) √(1+ sin 2 x) d x2 mk
- Q22If A= [1, 0; -1, 5], then find the value of K if A^2=6 A+K I_2, where I_2 is an identity matrix.2 mk
- Q22Find the values of ' a ' for which f(x)=√3 sin x- cos x-2 a x+b is decreasing on R.2 mk
- Q22Let the volume of a metallic hollow sphere be constant. If the inner radius increases at the rate of 2 cm / s, find the…2 mk
- Q22The radius of a cylinder is decreasing at a rate of 2 cm / s and the altitude is increasing at the rate of 3 cm / s.…2 mk
- Q22Differentiate √(e^√(2 x)) with respect to e^√(2 x) for x>0. OR If (x)^y=(y)^x, then find (d y)/(d x).2 mk
- Q22If A= [2, 3; -1, 2], then show that A^2-4 A+7 I=0.2 mk
- Q22Evaluate: tan^-1[2 sin (2 cos^-1 (√3)/2)]2 mk
- Q22Find: ∫ 2 x^3 e^x^2 d x.2 mk
- Q22Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm^2 / s. When the radius of…2 mk
- Q23The diagonals of a parallelogram are given by a=2 i- j+ k and b= i+3 j- k. Find the area of the parallelogram.2 mk
- Q23Simplify sin^-1(x/(√(1+x^2))). OR Find domain of sin^-1 √(x-1).2 mk
- Q23Differentiate (sin x)/(√(cos x)) with respect to x. OR If y=5 cos x-3 sin x, prove that (d^2 y)/(d x^2)+y=0.2 mk
- Q23Differentiate ((5^x)/(x^5)) with respect to x. OR If -2 x^2-5 x y+y^3=76, then find (d y)/(d x).2 mk
- Q23Determine the values of x for which f(x)=(x-4)/(x+1), x ≠-1 is an increasing or a decreasing function.2 mk
- Q23Find the interval in which f(x)=x+1/x is always increasing, x ≠ 0.2 mk
- Q23Find the values of' a 'for which f(x)= sin x-a x+b is increasing on R.2 mk
- Q23Determine those values of x for which f(x)=2/x-5, x ≠ 0 is increasing or decreasing.2 mk
- Q24If a and b are two non-collinear vectors, then find x, such that α=(x-2) a+ b and β=(3+2 x) a-2 b are collinear.2 mk
- Q24If | a|=2,| b|=3 and a · b=4, then evaluate | a+2 b|.2 mk
- Q24Find a vector of magnitude 5 which is perpendicular to both the vectors 3 i-2 j+ k and 4 i+3 j-2 k. OR Let a, b and c…2 mk
- Q24In a Linear Programming Problem, the objective function Z=5 x+4 y needs to be maximised under constraints 3 x+y ≤ 6, x…2 mk
- Q24In a Linear Programming Program (LPP) for objective function Z=14 x-10 y subject to constraints x+y ≤ 8; 3 x-2 y ≥-6;…2 mk
- Q24If a and b are position vectors of point A and point B respectively, find the position vector of point C on BA produced…2 mk
- Q24Let p=2 i-3 j- k, q=-3 i+4 j+ k and r= i+ j+2 k. Express r in the form of r=λ p+μ q and hence find the values of λ and…2 mk
- Q24Calculate the area of the region bounded by the curve (x^2)/9+(y^2)/4=1 and the x -axis using integration.2 mk
- Q24For a Linear Programming Problem, find min Z=5 x+3 y (where Z is the objective function) for the feasible region shaded…2 mk
- Q24Evaluate: sin^-1( sin (3 π)/5)2 mk
- Q24Solve for x, 2 tan^-1 x+ sin^-1((2 x)/(1+x^2))=4 √32 mk
- Q24Find the intervals in which function f(x)=5 x^(3/2)-3 x^(5/2) is (i) increasing (ii) decreasing.2 mk
- Q25Find the local maxima and local minima of the function f(x)=8/3 x^3-12 x^2+18 x+5.2 mk
- Q25Find the value of λ if the following lines are perpendicular to each other: l_1: (1-x)/-3=(3 y-2)/(2 λ)=(z-3)/3; l_2:…2 mk
- Q25If x=e^(x/y), then prove that dy/(d x)=(x-y)/(x log x). OR If f(x)= {2 x-3,-3 ≤ x ≤-2; x+1,-2<x ≤ 0} Check the…2 mk
- Q2510 identical blocks are marked with ' 0 ' on two of them, ' 1 ' on three of them, ' 2 ' on four of them and ' 3 ' on…2 mk
- Q25A man needs to hang two lanterns on a straight wire whose end points have coordinates A(4,1,-2) and B(6,2,-3). Find the…2 mk
- Q25Determine if the lines r=( i+ j- k)+λ(3 i- j) and r=(4 i- k)+μ(2 i+3 k) intersect with each other.2 mk
- Q25Find the angle at which the given lines are inclined to each other: l_1: (x-5)/2=(y+3)/1=(z-1)/-3; l_2:…2 mk
- Q25Two friends while flying kites from different locations, find the strings of their kites crossing each other. The…2 mk
- Q25For the curve y=5 x-2 x^3, if x increases at the rate of 2 units / s, then how fast is the slope of the curve changing…2 mk
- Q25A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm^3 / s from…2 mk
- Q26The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side…3 mk
- Q26Let A= [1; 4; -2] and C= [3, 4, 2; 12, 16, 8; -6, -8, -4] be two matrices. Then, find the matrix B if AB=C.3 mk
- Q26Find the value of x, if [1, x, 1] [1, 3, 2; 2, 5, 1; 15, 3, 2] [1; 2; x] =O.3 mk
- Q26If f: R^+ → R is defined as f(x)= log_a x(a>0 and a ≠ 1), prove that f is a bijection. ( R^+ is a set of all positive…3 mk
- Q26Solve the following linear programming problem graphically: Maximize Z=8 x+9 y Subject to the constraints: 2 x+3 y ≤ 6;…3 mk
- Q26Show that the function f: R → R defined by f(x)=4 x^3-5, ∀ x ∈ R is one-one and onto. OR Let R be a relation defined on…3 mk
- Q26Find: ∫ (√x)/(1+√(x^3 / 2)) d x3 mk
- Q26Show that f(x)= tan^-1( sin x+ cos x) is an increasing function in [0, (π)/4].3 mk
- Q26If A= [1, -1, 0], B= [2, 0, 1; -1, 3, 4; 0, 5, 1] and C= [2; 3; 4], are three matrices, then find ABC.3 mk
- Q26Find the interval/intervals in which the function f(x)= sin 3 x- cos 3 x, 0<x<(π)/2 is strictly increasing.3 mk
- Q26Solve the differential equation 2(y+3)-x y (d y)/(d x)=0; given y(1)=-2. OR Solve the following differential equation:…3 mk
- Q27Find: ∫ (2 x)/((x^2+3)(x^2-5)) d x OR Evaluate: ∫_1^4(|x-2|+|x-4|) d x3 mk
- Q27Let R be a relation on set of real numbers R defined as {(x, y): x-y+√3 is an irrational number, x, y ∈ R} Verify R for…3 mk
- Q27Let 2 x+5 y-1=0 and 3 x+2 y-7=0 represent the equations of two lines on which the ants are moving on the ground. Using…3 mk
- Q27Differentiate y= sin^-1(3 x-4 x^3) w.r.t. x, if x ∈[-1/2, 1/2]. OR Differentiate y= cos^-1((1-x^2)/(1+x^2)) with…3 mk
- Q27Solve the following linear programming problem graphically: Maximise Z=x+2 y Subject to the constraints: x-y ≥ 0; x-2 y…3 mk
- Q27Find: ∫ (2 x-1)/((x-1)(x+2)(x-3)) d x OR Evaluate: ∫_0^5(|x-1|+|x-2|+|x-5|) d x3 mk
- Q27Let R be a relation defined over N, where N is set of natural numbers, defined as" m R n if and only if m is a multiple…3 mk
- Q27Find k so that f(x)= {(x^2-2 x-3)/(x+1), x ≠-1; k, x=-1} is continuous at x=-1. OR Check the differentiability of…3 mk
- Q27Solve the following linear programming problem graphically: Maximise Z=20 x+30 y Subject to the constraints: x+y ≤ 80;…3 mk
- Q27Prove that f: N → N defined as f(x)=a x+b(a, b ∈ N) is one-one but not onto.3 mk
- Q28Differentiate y=√(log { sin ((x^3)/3-1)}) with respect to x.3 mk
- Q28A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any…3 mk
- Q28Find the particular solution of the differential equation dy/dx-y/x+cosec(y/x)=0; given that y=0, when x=1.3 mk
- Q28Find: ∫ (cos x d x)/(1+ cos x+ sin x)3 mk
- Q28Find the distance of the point P(2,4,-1) from the line (x+5)/1=(y+3)/4=(z-6)/-9. OR Let the position vectors of the…3 mk
- Q28Solve the differential equation x cos (y/x) (d y)/(d x)=y cos (y/x)+x.3 mk
- Q28Differentiate log (x^x+cosec^2 x) with respect to x.3 mk
- Q28A student wants to pair up natural numbers in such a way that they satisfy the equation 2 x+y=41, x, y ∈ N. Find the…3 mk
- Q28The area of an expanding rectangle is increasing at the rate of 48 cm^2 / s. The length of the rectangle is always…3 mk
- Q28Find: ∫ (x+ sin x)/(1+ cos x) d x OR Evaluate: ∫_0^((π)/4) (d x)/(cos^3 x √(2 sin 2 x))3 mk
- Q28Find the particular solution of the differential equation [x sin^2(y/x)-y] d x+x d y=0 given that y=(π)/4, when x=1.3 mk
- Q28Solve the following linear programming problem graphically: Minimise Z=20 subject to the cc; 3 x+y ≥ 9; x+y ≥ 7; x+2 y…3 mk
- Q28Evaluate: ∫_π / 2^π e^x((1- sin x)/(1- cos x)) d x3 mk
- Q28The feasible region along with corner points for a linear programming problem are shown in the graph. Write all the…3 mk
- Q28Solve the following linear programming problem graphically: Minimise Z=x-5 y subject to the constraints: x-y ≥ 0; -x+2…3 mk
- Q28Show that the derivative of tan^-1( sec x+ tan x),[-(π)/2<x<(π)/2] with respect to x is equal to 1/2.3 mk
- Q29Show that of all the rectangles with a fixed perimeter, the square has the greatest area.3 mk
- Q29In the Linear Programming Problem (LPP), find the point/points giving maximum value for Z=5 x+10 y subject to…3 mk
- Q29Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.3 mk
- Q29If y= log (√x+1/(√x))^2, then show that x(x+1)^2 y_2+(x+1)^2 y_1=2. OR If x √(1+y)+y √(1+x)=0,-1<x<1, x ≠ y, then prove…3 mk
- Q29Find the probability distribution of the number of boys in families having three children, assuming equal probability…3 mk
- Q29Verify that lines given by r=(1-λ) i+(λ-2) j+(3-2 λ) k and r=(μ+1) i+(2 μ-1) j-(2 μ+1) k are skew lines. Hence, find…3 mk
- Q29Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a…3 mk
- Q29Consider the Linear Programming Problem, where the objective function Z=(x+4 y) needs to be minimized subject to…3 mk
- Q29Find dimensions of a rectangle of perimeter 12 cm which will generate maximum volume when swept along a circular…3 mk
- Q29The probability distribution of a random variable X is given by: (i) Determine the value of p. (ii) Calculate P(X ≥ 1).…3 mk
- Q30If a+ b+ c= 0 such that | a|=3,| b|=5,| c|=7, then find the angle between a and b. OR If a and b are unit vectors…3 mk
- Q30In the Linear Programming Problem for objective function Z=18 x+10 y subject to constraints 4 x+y ≥ 20; 2 x+3 y ≥ 30;…3 mk
- Q30The probability distribution for the number of students being absent in a class on a Saturday is as follows: Where X is…3 mk
- Q30Find: ∫ (cos 2 x)/(( sin x+ cos x)^2) d x OR Evaluate: ∫_0^((π)/2) (5 sin x+3 cos x)/(sin x+ cos x) d x3 mk
- Q30Find the distance of the point (-1,-5,-10) from the point of intersection of the lines (x-1)/2=(y-2)/3=(z-3)/4 and…3 mk
- Q30A die with number 1 to 6 is biased such that P(2)=3/10 and probability of other numbers is equal. Find the mean of the…3 mk
- Q31For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.3 mk
- Q31A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in…3 mk
- Q31Find: ∫ 1/x √((x+a)/(x-a)) d x.3 mk
- Q31f and g are continuous functions on interval [a, b]. Given that f(a-x)=f(x) and g(x)+g(a-x)=a, show that ∫_0^a f(x)…3 mk
- Q31Solve the following Linear Programming Problem using graphical method: Maximise Z=100 x+50 y subject to the constraints…3 mk
- Q31If ∫_a^b x^3 d x=0 and ∫_a^b x^2 d x=2/3, then find the values of a and b.3 mk
- Q31The scalar product of the vector a= i- j+2 k with a unit vector along sum of vectors b=2 i-4 j+5 k and c=λ i-2 j-3 k is…3 mk
- Q31Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly by from bag…3 mk
- Q31A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, find the…3 mk
- Q31The probability that a student buys a colouring book is 0 · 7 and that she buys a box of colours is 0 · 2. The…3 mk
- Q31Sketch the graph of y=|x+3| and find the area of the region enclosed by the curve, x -axis, between x=-6 and x=0, using…3 mk
- Q31Solve the following Linear Programming Problem graphically: Minimise Z=3 x+5 y subject to the constraints x+2 y ≥ 10;…3 mk
- Q32Find: ∫ (x^2+1)/((x-1)^2(x+3)) d x OR Evaluate: ∫_0^π / 2 x/(sin x+ cos x) d x5 mk
- Q32Find: ∫ x/((x-1)(x^2+4)) d x OR Evaluate: ∫_0^π (x sin x)/(1+ cos^2 x) d x5 mk
- Q32Using integration, find the area of the region bounded by the line y=5 x+2, the x -axis and the ordinates x=-2 and x=2.5 mk
- Q32In a rough sketch, mark the region bounded by y=1+|x+1|, x=-2, x=2 and y=0. Using integration, find the area of the…5 mk
- Q32If A is a 3 × 3 invertible matrix, show that for any scalar k ≠ 0, (kA)^-1=1/k A^-1. Hence calculate (3 A)^-1, where A=…5 mk
- Q32Differentiate tan^-1 (√(1-x^2))/x w. r. t. cos^-1(2 x √(1-x^2)), x ∈(1/(√2), 1) OR Find dy/dx, if y=x^tan…5 mk
- Q32Sketch a graph of y=x^2. Using integration, find the area of the region bounded by y=9, x=0 and y=x^2.5 mk
- Q32Find: ∫ (x^2+1)/((x^2+2)(2 x^2+1)) d x OR Evaluate: ∫_0^π (x tan x)/(sec x+ tan x) d x5 mk
- Q32Draw a rough sketch of the curve y=√x. Using integration, find the area of the region bounded by the curve y=√x, x=4…5 mk
- Q32If √(1-x^2)+√(1-y^2)=a(x-y), then prove that dy/dx=√((1-y^2)/(1-x^2)). OR If x=a( cos θ+ log tan (θ)/2) and y= sin θ,…5 mk
- Q32Find: ∫ (3 x+1)/((x-2)^2(x+2)) d x OR Evaluate: ∫_0^π / 2 x/(cos x+ sin x) d x5 mk
- Q33Find the absolute maximum and absolute minimum of function f(x)=2 x^3-15 x^2+36 x+1 on [1,5].5 mk
- Q33Find: ∫ (x^2+x+1)/((x+2)(x^2+1)) d x.5 mk
- Q33A furniture workshop produces three types of furniture - chairs, tables and beds each day. On a particular day the…5 mk
- Q33Use integration to find the area of the region enclosed by curve y=-x^2 and the straight lines x=-3, x=2 and y=0.…5 mk
- Q33Find: ∫ (5 x)/((x+1)(x^2+9)) d x.5 mk
- Q33Find: ∫ sin^-1 √(x/(a+x)) d x.5 mk
- Q33Find: ∫(√(tan x)+√(cot x)) d x.5 mk
- Q33Three students run on a racing track such that their speeds add up to 6 km / h. However, double the speed of the third…5 mk
- Q33An amount of ₹ 10,000 is put into three investments at the rate of 10 %, 12 % and 15 % per annum. The combined annual…5 mk
- Q33The relation between the height of the plant ( ycm ) with respect to exposure to sunlight is governed by the equation…5 mk
- Q33Using integration, find the area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 bounded between the lines x=-a/2 to x=a/2.5 mk
- Q33Draw a rough sketch for the curve y=2+|x+1|. Using integration, find the area of the region bounded by the curve…5 mk
- Q33A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top…5 mk
- Q34Find the image A^′ of the point A(1,6,3) in the line x/1=(y-1)/2=(z-2)/3. Also, find the equation of the line joining A…5 mk
- Q34Solve the differential equation (d y)/(d x)= cos x-2 y.5 mk
- Q34Evaluate: ∫_0^π / 4 (sin x cos x)/(cos^4 x+ sin^4 x) d x OR Find: ∫ (√(x^2+1)[ log (x^2+1)-2 log x])/(x^2) d x5 mk
- Q34Find: ∫ (cos x)/((4+ sin^2 x)(5-4 cos^2 x)) d x OR Evaluate: ∫_0^π (d x)/(a^2 cos^2 x+b^2 sin^2 x)5 mk
- Q34Solve the differential equation: x^2 y d x-(x^3+y^3) d y=0. OR Solve the differential equation (1+x^2) (d y)/(d x)+2 x…5 mk
- Q34Solve the differential equation (x^2+y^2) d x+x y d y=0, y(1)=1.5 mk
- Q34Find the shortest distance between the lines: (x+1)/2=(y-1)/1=(z-9)/-3 and (x-3)/2=(y+15)/-7=(z-9)/5; OR Find the image…5 mk
- Q34Solve the differential equation (x- sin y) d y+( tan y) d x=0, given y(0)=0.5 mk
- Q34For a positive constant 'a', differentiate a^t+1/t with respect to (t+1/t)^a, where t is a non-zero real number. OR…5 mk
- Q34Evaluate: ∫_0^3 / 2|x cos π x| d x Find: ∫ (d x)/(sin x+ sin 2 x)5 mk
- Q35Given A= [-4, 4, 4; -7, 1, 3; 5, -3, -1] and B= [1, -1, 1; 1, -2, -2; 2, 1, 3], find AB. Hence, solve the system of…5 mk
- Q35Find the foot of the perpendicular drawn from point (2,-1,5) to the line (x-11)/10=(y+2)/-4=(z+8)/-11. Also, find the…5 mk
- Q35Find the foot of the perpendicular drawn from the point ( 1,1,4 ) on the line (x+2)/5=(y+1)/2=(-z+4)/-3. OR Find the…5 mk
- Q35Show that the line passing through the points A(0,-1,-1) and B(4,5,1) intersects the line joining points C(3,9,4) and…5 mk
- Q35Let the polished side of the mirror be along the line x/1=(1-y)/-2=(2 z-4)/6. A point P(1,6,3), some distance away from…5 mk
- Q35Find the point Q on the line (2 x+4)/6=(y+1)/2=(-2 z+6)/-4 at a distance of 3 √2 from the point P(1,2,3). OR Find the…5 mk
- Q35A school wants to allocate students into three clubs: Sports, Music and Drama, under following conditions: - The number…5 mk
- Q35Show that the area of a parallelogram whose diagonals are represented by a and b is given by 1/2| a × b|. Also find the…5 mk
- Q36Three friends A, B and C move out from the same location O at the same time in three different directions to reach…4 mk
- Q36Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2…4 mk
- Q36A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since…4 mk
- Q36A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage…4 mk
- Q36A school is organizing a debate competition with participants as speakers S={S_1, S_2, S_3, S_4} and these are judged…4 mk
- Q36Some students are having a misconception while comparing decimals. For example, a student may mention that 78 · 56>78 ·…4 mk
- Q36A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary…4 mk
- Q36Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their…4 mk
- Q37A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She…4 mk
- Q37Three friends A, B and C move out from the same location O at the same time in three different directions to reach…4 mk
- Q37Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their…4 mk
- Q37An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each…4 mk
- Q37A ladder of fixed length ' h ' is to be placed along the wall such that it is free to move along the height of the…4 mk
- Q37Let A be the set of 30 students of class XII in a school. Let f: A → N, N is a set of natural numbers such that…4 mk
- Q38A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their…4 mk
- Q38Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in…4 mk
- Q38A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary…4 mk
- Q38A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the…4 mk
- Q38A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate.…4 mk
- Q38Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very…4 mk
- Q38During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the…4 mk
Other Mathematics years
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