CBSE Class 12 Mathematics 2025

What 2025 actually asked, chapter by chapter

Marks we counted in the 2025 papers themselves. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage.

All 365 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 mkasked 3×
• 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=[ ], then A^3 is:1 mkasked 3×
• 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=[ ], 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 mkasked 3×
• 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 mkasked 3×
• Q2If P(A B)=0 · 9 and P(A B)=0 · 4, then P( A)+P( B) is:1 mk
• Q2If A=[ ], 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=[ ], then A is a:1 mk
• Q2The function f(x)=x^2-4 x+6 is increasing in the interval1 mkasked 3×
• 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 mkasked 3×
• Q3For real x, let f(x)=x^3+5 x+1. Then:1 mkasked 3×
• 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 ≤ /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=[ ] and B=[ ], 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 mkasked 3×
• Q4If f(x)=. is continuous at x=(π)/2, then the value of k is:1 mk
• Q4The matrix A=[ ] is a/an:1 mkasked 3×
• Q4If A=[ ] 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 =, then the value of x is:1 mkasked 3×
• Q4Sum of two skew-symmetric matrices of same order is always a/an:1 mkasked 3×
• Q5If f(x)=. is continuous at x=0, then the value of a is:1 mkasked 4×
• Q5If f(x)= x + x-1, then which of the following is correct?1 mkasked 3×
• Q5If A and B are two square matrices each of order 3 with A =3 and B =5, then 2 AB is:1 mkasked 3×
• Q5The value of cos ((π)/6+ cot^-1(-√3)) is1 mk
• Q5[ sec^-1(-√2)- tan^-1(1/(√3))] is equal to:1 mkasked 3×
• Q5The values of λ so that f(x)= sin x- cos x-λ x+C decreases for all real values of x are:1 mkasked 3×
• 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)=. is continuous at x=0, then the value of k is:1 mkasked 3×
• 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 mkasked 3×
• Q6If A is a square matrix of order 2 such that det =4, then det(4 adj A) is equal to:1 mk
• Q6If A is a square matrix of order 3 such that det =9, then det(9 A^-1) is equal to1 mk
• Q6Let A be a square matrix of order 3. If A =5, then adj A is:1 mkasked 3×
• Q6If [ ]=[ ], 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 mkasked 3×
• Q7If =-57, the product of the possible values of x is:1 mk
• Q7If [ ]=[ ], then the value of (x-y) is:1 mkasked 3×
• Q7If P(A B)=0.9 and P(A B)=0.4, then P( A)+P( B) is:1 mkasked 2×
• Q7If E and F are two events such that P(E) 0 and P(F) ≠ 1, then P( E / F) is1 mkasked 3×
• Q7If tan^-1(x^2-y^2)=a, where ' a ' is a constant, then (d y)/(d x) is:1 mkasked 3×
• Q7If A and B are invertible matrices of order 3 × 3 such that det =4 and det[(AB)^-1]=1/20, then det 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 / F) is equal to:1 mkasked 2×
• 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 mkasked 3×
• Q8Let A and B be two matrices of suitable orders. Then, which of the following is not correct?1 mk
• Q8If f(x)=. is continuous in R, then the values of a and b are:1 mk
• Q8If [ ] 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 mkasked 3×
• Q8Which of the following can be both a symmetric and skew-symmetric matrix?1 mkasked 3×
• Q8If f(x)=. is continuous for x ∈(0,2), then a is equal to:1 mkasked 3×
• Q8If y=a cos ( log x)+b sin ( log x), then x^2 y 2+xy 1 is:1 mkasked 3×
• Q8If f(x)=. is continuous at x=1, then the values of a and b are:1 mk
• Q9If f: N → W is defined as f(n)=,. then f is:1 mkasked 3×
• Q9If f(x)=-2 x^8, then the correct statement is:1 mkasked 3×
• 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 mkasked 3×
• Q9Let f(x)= x, x ∈ R. Then, which of the following statements is incorrect?1 mk
• Q9Let A=[ ], B=[ ], C=[ ], which of the following is defined?1 mkasked 2×
• 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 mkasked 3×
• Q10If ∫ (2^1/x)/(x^2) d x=k · 2^1/x+C, then k is equal to1 mkasked 3×
• Q10The slope of the curve y=-x^3+3 x^2+8 x-20 is maximum at:1 mkasked 3×
• 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 mkasked 3×
• 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 [ ] is a:1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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=[ ], then A^-1 is1 mkasked 3×
• Q11If a+ b+ c= 0, a =√37, b =3 and c =4, then angle between b and c is1 mkasked 3×
• 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 xe^6 log x-e^5 log x dx is equal to:1 mkasked 3×
• Q12The function f defined by f(x)= casesx, & if x ≤ 1 5, & if x 1 cases is not continuous at:1 mkasked 3×
• 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 mkasked 3×
• Q13The solution for the differential equation log ((d y)/(d x))=3 x+4 y is:1 mkasked 3×
• Q13The line x=1+5 μ, y=-5+μ, z=-6-3 μ passes through which of the following point?1 mkasked 3×
• Q13If A=[ ] is a scalar matrix, then y^x is equal to1 mk
• Q13For a function f(x), which of the following holds true?1 mkasked 3×
• Q13If f(x)=2 x+ cos x, then f(x):1 mkasked 2×
• 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^-x16+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 mkasked 3×
• Q14If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following…1 mkasked 3×
• Q14∫ ( cos 2 x- cos 2 α)/( cos x- cos α) d x is equal to:1 mkasked 2×
• 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 mkasked 3×
• Q15The value of ∫ 0^1 (d x)/(e^x+e^-x) is:1 mkasked 3×
• Q15If A and B are invertible matrices, then which of the following is not correct?1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q16∫ (x+5)/((x+6)^2) e^x dx is equal to:1 mkasked 3×
• Q16For a Linear Programming Problem (LPP), the given objective function Z=3 x+2 y is subject to constraints: The correct…1 mkasked 3×
• Q16A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function…1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q17If A and B are two events such that P =1/5, P(A B)=2/3 and P(A B)=3/5, then P is:1 mk
• Q17If P =1/5, P =3/5 and P( / )=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 6×
• Q17The area of the region bounded by the curve y^2=x between x=0 and x=1 is:1 mkasked 3×
• 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 mkasked 3×
• Q17In a Linear Programming Problem (LPP), the objective function Z=2 x+5 y is to be maximised under the following…1 mkasked 3×
• Q17If P =1/7, P =5/7 and P(A B)=4/7, then P( A 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q18If p and q are respectively the order and degree of the differential equation /(d x)((dy)/(d x))^3=0, then (p-q) is1 mkasked 3×
• Q18The respective values of a and b, if given ( a- b) ·( a+ b)=512 and a =3 b, are:1 mkasked 3×
• Q18The area of the shaded region bounded by the curves y^2=x, x=4 and the x -axis is given by1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 sin θ and a ·…1 mkasked 3×
• 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 mkasked 3×
• 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): A…1 mkasked 3×
• Q19Assertion: f(x)=. is continuous at x=0. Reason (R): When x → 0, sin 1/(x) is a finite value between -1 and 1.1 mkasked 3×
• Q20Assertion: A =diag[ ] is a scalar matrix of order 3 × 3. Reason (R): If a diagonal matrix has all non-zero elements…1 mkasked 3×
• 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 mkasked 3×
• Q20Assertion: f(x)=. is continuous at x=5 for k=5/2. Reason (R): For a function f to be continuous at x=a, lim x → a^-…1 mkasked 3×
• 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 mkasked 3×
• Q20Assertion: Every point of the feasible region of a Linear Programming Problem is an optimal solution. Reason (R): The…1 mkasked 3×
• 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 mkasked 3×
• Q20Assertion: In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no…1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q21Find the domain of f(x)= sin^-1(-x^2).2 mkasked 3×
• 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 =3 and det =-4. Find the value of det(-6 A B).2 mk
• Q21If A=[ ] 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 mkasked 3×
• 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 mkasked 3×
• Q22Evaluate: ∫ 0^π ( sin 2 p x)/( sin x) d x, p ∈ N.2 mk
• Q22If [ ] A=[ ], then find matrix A.2 mk
• Q22Evaluate: ∫ 0^(π)/4 √1+ sin 2 x d x2 mk
• Q22If A=[ ], 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 mkasked 2×
• 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 mkasked 3×
• Q22If A=[ ], 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 mkasked 3×
• Q23Simplify sin^-1( x√1+x^2). OR Find domain of sin^-1 √x-1.2 mkasked 3×
• 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 mkasked 3×
• Q23Differentiate ( 5^xx^5) with respect to x. OR If -2 x^2-5 x y+y^3=76, then find (d y)/(d x).2 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 shade the feasible…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 mkasked 3×
• 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 mkasked 3×
• 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:2 mk
• Q25If x=e^x/(y), then prove that (dy)/(d x)=(x-y)/(x log x). OR If f(x)=. Check the differentiability of f(x) at x=-2.2 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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:2 mk
• Q25Two friends while flying kites from different locations, find the strings of their kites crossing each other. The…2 mkasked 3×
• 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=[ ] and C=[ ] be two matrices. Then, find the matrix B if AB=C.3 mk
• Q26Find the value of x, if [ ][ ][ ]=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 mkasked 3×
• Q26Solve the following linear programming problem graphically: Maximize Z=8 x+9 y Subject to the constraints: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 a…3 mkasked 3×
• Q26Find: ∫ √x1+√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=[ ], B=[ ] and C=[ ], 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 mkasked 3×
• Q27Find: ∫ (2 x)/((x^2+3)(x^2-5)) d x OR Evaluate: ∫ 1^4( x-2 + x-4 ) d x3 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q27Solve the following linear programming problem graphically: Maximise Z=x+2 y Subject to the constraints: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)=. is continuous at x=-1. OR Check the differentiability of function f(x)=x x at x=0.3 mkasked 2×
• Q27Solve the following linear programming problem graphically: Maximise Z=20 x+30 y Subject to the constraints: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 mkasked 3×
• 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 mkasked 3×
• 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: gathered Minimise Z=20 subject to the cc 3 x+y ≥ 9 x+y ≥ 7…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: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 mkasked 3×
• 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 mkasked 3×
• Q29Find the probability distribution of the number of boys in families having three children, assuming equal probability…3 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 inclined…3 mkasked 3×
• Q30In the Linear Programming Problem for objective function Z=18 x+10 y subject to constraints find the minimum value of Z.3 mkasked 3×
• Q30The probability distribution for the number of students being absent in a class on a Saturday is as follows: Where X is…3 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 constraints3 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q31Solve the following Linear Programming Problem graphically: Minimise Z=3 x+5 y subject to the constraints3 mk
• Q32Find: ∫ (x^2+1)/((x-1)^2(x+3)) d x OR Evaluate: ∫ 0^π / 2 (x)/( sin x+ cos x) d x5 mkasked 3×
• 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 mkasked 3×
• 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…5 mkasked 3×
• Q32Differentiate tan^-1 √1-x^2x w. r. t. cos^-1(2 x √1-x^2), x ∈(1/(√2), 1) OR Find (dy)/(dx), if y=x^ tan x+ √x^2+12.5 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 mkasked 2×
• 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 mkasked 2×
• 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 mkasked 3×
• Q33Using integration, find the area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 bounded between the lines x=- /2 to x= /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 y=2+…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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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: Find the image A^′ of the point A(2,1,2) in the line l: r=4 i+2 j+2 k+λ(…5 mkasked 3×
• 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 mkasked 3×
• Q34Evaluate: ∫ 0^3 / 2 x cos π x d x Find: ∫ (d x)/( sin x+ sin 2 x)5 mk
• Q35Given A=[ ] and B=[ ], find AB. Hence, solve the system of linear equations: OR If A=[ ], then find A^-1. Hence, solve…5 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q35A school wants to allocate students into three clubs: Sports, Music and Drama, under following conditions: - The number…5 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q36A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since…4 mkasked 3×
• Q36A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage…4 mkasked 3×
• Q36A school is organizing a debate competition with participants as speakers S= S 1, S 2, S 3, S 4 and these are judged by…4 mkasked 3×
• Q36Some students are having a misconception while comparing decimals. For example, a student may mention that 78 · 56 78 ·…4 mkasked 3×
• Q36A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary…4 mkasked 2×
• Q36Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their…4 mkasked 2×
• Q37A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She…4 mkasked 3×
• Q37Three friends A, B and C move out from the same location O at the same time in three different directions to reach…4 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q38A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their…4 mkasked 3×
• Q38Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in…4 mkasked 3×
• 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 mkasked 3×
• Q38A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate.…4 mkasked 3×
• Q38Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very…4 mkasked 3×
• Q38During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the…4 mkasked 3×