CBSE Class 12 Mathematics 2026

What 2026 actually asked, chapter by chapter

Marks we counted in the 2026 papers themselves. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage.

All 305 questions from 2026

• Q1If ∫ (3 a x)/( b^2+c^2 x^2) d x=A log b^2+c^2 x^2 +K, then the value of A is1 mkasked 3×
• Q1If matrix A=[ ] is such that A^2=I, then:1 mk
• Q1The following graph represents:1 mk
• Q1The following graph represents:1 mk
• Q1If tan^-1 x=y, then (dy)/(dx) is equal to:1 mk
• Q1If 2 cos^-1 x=y, then1 mk
• Q1The following graph represents:1 mk
• Q1If A=[ ] and A+A^′=I, then θ is equal to:1 mk
• Q1∫ (d x)/(1+ cos x) is equal to1 mk
• Q1For any square matrix A with real entries, if A + A' is a symmetric matrix then:1 mk
• Q1The domain of f(x)= cos^-1(2 x-5) is:1 mk
• Q1A relation R on set A= 1,2,3 defined as R= (1,1),(2,2),(1,2) is1 mk
• Q1Direction cosines of line 1-x/0=y=z are1 mk
• Q2If A=[a ij] is a 2 × 2 matrix whose elements are given by a ij=( i-3 j )/2, then A' is:1 mk
• Q2If A and B are square matrices of same order, then which of the following statements is/are always true? (i)…1 mkasked 3×
• Q2Let A=[a ij] be a 2 × 2 matrix whose elements are given by a ij=((i+2 j)^2)/3. Then A^′ is given by:1 mk
• Q2The length of perpendicular drawn from the point (3, 4, 2) on the line x/0=(y)/0=(z)/1 is1 mk
• Q2If A is a square matrix such that A^2=A, then (A-I)^3-A is equal to:1 mk
• Q2Let A=[a ij] be a 2 × 2 matrix whose elements are given by a ij=((2 i-j)^2)/3. Then A^′ is:1 mk
• Q2Which of the following cannot be the order of a row-matrix?1 mk
• Q2If A is a symmetric matrix, then for any matrix B of order same as A, BAB' is a/an:1 mk
• Q2A matrix B=[b ij] m × m is said to be a diagonal matrix, if:1 mk
• Q2If A^2=4 A+3 I and A^-1=xA+yI, then the value of (x+y) is:1 mkasked 3×
• Q3The principal value of sec^-1(√2)+2 cosec^-1(-√2) is:1 mkasked 3×
• Q3If A and B are skew-symmetric matrices of same order, then AB^′+BA^′ is a/an:1 mkasked 3×
• Q3For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined?1 mkasked 3×
• Q3∫ dx√e^-2 x-1 is equal to:1 mk
• Q3Which of the following properties is/are true for two matrices of suitable orders? (i) (A+B)^′=A^′+B^′ (ii)…1 mkasked 3×
• Q3If A=[ ] is a symmetric matrix, then the value of 3 a+b+c is1 mkasked 3×
• Q3Direction ratios of lines l 1 and l 2 respectively are <1,0,0 and <0,-1,0. The direction ratios of the line…1 mk
• Q4If A=[ ] and A+A^′=I, then the value of x ∈[0, (π)/2] is1 mk
• Q4Let A=[ ] and B=[ ]. If A+B+C=O, then matrix C is:1 mkasked 3×
• Q4If [ ] A=[ ], then order of A must be:1 mk
• Q4If Δ 1= and Δ 2=, then1 mkasked 3×
• Q4The general solution of the differential equation x dy-y d x=0 is1 mk
• Q4If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is:1 mkasked 3×
• Q5A relation R on set A= 1,2,3 is defined as R= (1,3),(3,3),(1,1),(2,2), (3,1) is1 mk
• Q5One of the values of x for which =1 is1 mkasked 3×
• Q5The order and degree of the differential equation: d/(d x)( sin y)=y^2 respectively are1 mk
• Q5For a square matrix A, (3 A)^-1=1 mkasked 3×
• Q5Differential of e^e^x with respect to x is:1 mkasked 3×
• Q5The domain of cos^-1(4 x+1) is:1 mk
• Q5If sin^-1 x+π=y, then1 mk
• Q5If a square matrix A is such that A^2=A and (I-A)^3=xA+I, then value of x must be:1 mkasked 3×
• Q5If A is a non-singular matrix, then which of the following is not true?1 mkasked 3×
• Q6A cylindrical tank is being filled with sand at a rate of 314 m^3 / h. If the radius of the tank is 10 m, then the…1 mk
• Q6If f(x)=. is continuous at x=-1, then the value of k is:1 mkasked 3×
• Q6If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric?1 mk
• Q6The value of p for which vectors i+2 j+3 k and 2 i-p j+ k are perpendicular to each other is1 mkasked 3×
• Q6The surface area of a sphere when its volume changes at the same rate as its radius is:1 mk
• Q6If =-86, then the sum of all possible values of a is1 mkasked 3×
• Q6If A=[a ij] 3 × 3 is a scalar matrix then which of the following must be true?1 mk
• Q6If B(adj B)=[ ], then the value of det(B^-1)=1 mk
• Q6If the distance travelled by a particle in t seconds is given by S=72 t+3 t^2-t^3, then time taken by the particle to…1 mk
• Q7The least value of f(x)=x^3-12 x, x ∈[0,3] is1 mk
• Q7If the area of Δ ABC with vertices A(3, 1), B(-2,1) and C(0, k) is 5 sq. units, then values of k are:1 mkasked 3×
• Q7The value of m for which the points with position vectors - i- j+2 k, 2 i+m j+5 k and 3 i+11 j+6 k are collinear, is1 mkasked 3×
• Q7If f(x)=. is continuous at x=0, then the value of k is:1 mkasked 3×
• Q7If e^-x+e^-y=2, then (dy)/(dx) is1 mk
• Q8If a matrix X is such that [2 1] X = [3 4 5], then the order of matrix X is:1 mk
• Q8Derivative of cos^-1(( sin x+ cos x)/(√2)),-(π)/4<x<(π)/4 with respect to x is:1 mkasked 3×
• Q8Direction cosines of line x=y=1-z are1 mk
• Q8The greatest integer function, f(x)=[x], 0<x<3 is not differentiable at how many points?1 mkasked 3×
• Q8For f(x)=x+1/x(x ≠ 0)1 mkasked 3×
• Q8If A=[ ] and A+A^′=2 I, then value of x ∈[0, (π)/2] is1 mk
• Q8If sin^-1 x=y, then (dy)/(dx) is:1 mk
• Q9The length of perpendicular drawn from the point (1,2,3) on line x/0=y/1=z/0 is1 mk
• Q9Absolute minimum value of f(x)=(x-2)^2+5 in the interval [-3,2] is:1 mkasked 3×
• Q9If ∫ 0^2 a 1/(1+4 x^2) d x=(π)/6, then the value of a is1 mk
• Q9The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is:1 mkasked 3×
• Q9The value of ∫ -1^1 (x^3)/(x^2+2 x +1) d x is1 mkasked 3×
• Q9∫ dx√25-16 x^2 is equal to:1 mkasked 3×
• Q10Direction ratios of lines l 1 and l 2 respectively are <1,-2,3 and <-2, p,-6. The value of p for which l 1 l 2, is:1 mk
• Q10If A(adj A)=[ ], then the value of 2 A is:1 mk
• Q10∫ 1√1+ cos 2 x dx is equal to:1 mk
• Q10If A and B are symmetric matrics of same order, then (AB-BA) is a1 mk
• Q10Which of the following expressions will give the area of region bounded by the curve y=x^2 and line y=16?1 mkasked 3×
• Q10∫ √(1- cos 2 x)/(1+ cos 2 x) dx is equal to:1 mk
• Q10The area bounded by the curve y=x x, x -axis and the ordinates x=-1 and x=1 is given by1 mkasked 3×
• Q10If ∫ 0^1 dxe^x+e^-x= tan^-1 e+k, then the value of k is:1 mkasked 3×
• Q10∫ dx2^x+2^-x is equal to:1 mk
• Q10∫ √(1+ cos x)/(1- cos x) dx is equal to:1 mk
• Q11The integrating factor of differential equation R (d x)/(dy)+P x=Q where P, Q, R are functions of y is1 mkasked 3×
• Q11The value of ∫ 0^1 1/(3 x-4) dx is:1 mk
• Q11The least value of f(x)=e^-x in [0, 3] is1 mk
• Q11∫ -1^1(1- x ) dx is equal to:1 mkasked 3×
• Q11The general solution of the differential equation (dy)/(dx)= √y√x is1 mk
• Q11If x+y=x y, then (dy)/(dx) is1 mk
• Q11The value of ∫ -5^-1 1/(x) dx is equal to:1 mk
• Q11The area of the region bounded by the curve y=x and x -axis, between x=0 and x=2 is:1 mkasked 3×
• Q11If ∫ 0^1(6 x^2-4 x+k) dx=0, then the value of k is:1 mk
• Q12Product of the order and degree of differential equation 1+((dy)/(dx))^3=λ((d^3 y)/(dx^3))^2 is:1 mk
• Q12The integrating factor of the differential equation 2 x (dy)/(dx)-y=3 is1 mkasked 3×
• Q12An ant is observed crawling on a sheet of paper along a straight line given by equation y=2 x-4. Area of the surface…1 mkasked 3×
• Q12∫ (d x)/( sec x+ tan x) is equal to1 mk
• Q12The sum of the order and the degree of the differential equation y=x (dy)/(dx)+3[2-((dy)/(dx))^2] is:1 mk
• Q12The area of the shaded region of the circle given below is equal to:1 mkasked 3×
• Q12If tan^-1 x=3 y, then1 mk
• Q12If 2 cos^-1 x = y, then dy/dx is:1 mk
• Q12The sum of the order and the degree of the differential equation 4[( d^2 y)/(dx^2)]^2+3[1+((dy)/(dx))^2-y]=0 is:1 mk
• Q12A relation R on set A= 1,2,3 defined as R= (1,2),(2,1),(2,2) is1 mk
• Q12The domain of sin^-1(1-2 x) is:1 mk
• Q12The order and degree of the differential equation /(dx)(e^y)=0 respectively are1 mk
• Q13The order and degree of the differential equation 1+((d^3 y)/(dx^3))^3=λ (d^2 y)/(dx^2) is:1 mk
• Q13If a =5 and -2 ≤ λ ≤ 1, then the sum of greatest and the smallest value of λ a is1 mkasked 3×
• Q13The order and degree of differential equation y=((d^2 y)/(dx^2))^2-λ (dy)/(dx) is:1 mk
• Q13Which of the following is not a Linear Differential Equation?1 mkasked 3×
• Q13(d y)/(d x)=F(x, y) will be a homogeneous differential equation for which of the following functions? (i) F(x, y)=3 x+2…1 mkasked 3×
• Q13In the graph, the feasible region representing the Linear Programming Problem for maximising objective function…1 mkasked 3×
• Q13Which of the following cannot be an order of a column-matrix?1 mk
• Q13The order and degree of differential equation (d^2 y)/(dx^2)=1-((d^3 y)/(dx^3))^2 is:1 mk
• Q14Vector of magnitude 3 making equal angles with x and y axes and perpendicular to z axis is1 mkasked 3×
• Q14For any two vectors a and b, which of the following statements is always true?1 mkasked 3×
• Q14The region represented by the system of inequations 3 x+y ≥ 3, 2 x-y ≥-5, x, y ≥ 0 is:1 mkasked 3×
• Q14The general solution for the differential equation (d y)/(d x)=e^3 x-y is:1 mkasked 3×
• Q14∫ cos x√ sin^2 x+1 dx is equal to:1 mk
• Q15The general solution of the differential equation: x^2 dy+y^2 d x=0 is1 mk
• Q15If a =8, b =3 and a × b =12, then the value of a · b1 mkasked 3×
• Q15The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40)…1 mkasked 3×
• Q15If ( a+ b) ·( a- b)=198 and a =10 b, then:1 mkasked 3×
• Q15If a matrix B is such that B[ ]=[ ], then the order of matrix B is: matrix B is:1 mk
• Q15If A=[ ] and A+A^′=I, then value of x ∈[(-π)/2, (π)/2] is1 mk
• Q15Direction cosines of the line given by equations: (2 x-1)/4=(1-y)/3=(-z)/6 are1 mk
• Q16If l 1, m 1, n 1 and l 2, m 2, n 2 are direction cosines of lines L 1 and L 2 respectively and θ is the acute angle…1 mkasked 3×
• Q16If (3 i-2 j+5 k) ×(4 i+p j+q k)= 0, then the values of p and q are:1 mkasked 3×
• Q16The length of perpendicular drawn from point (2,5,7) on line x/1=y/0=z/0 is1 mk
• Q16If position vector p of a point (24, n) is such that p =25, then the value of n is:1 mkasked 3×
• Q16In a linear programming problem, the linear function which has to be maximized or minimized is called1 mkasked 3×
• Q16The order and degree of the differential equation: /(d x)(y^′)^3+(y^′)^3=1 respectively are where y^′=(dy)/(dx)1 mk
• Q17For the feasible region shown below, the non-trivial constraints of the linear programming problem are1 mkasked 3×
• Q17If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix?1 mk
• Q17Three points A(0, 1, 1), B(2, 0, -1) and C(1, 0, 3) form Δ ABC. The ar( ABC) is:1 mkasked 3×
• Q17If A(adj A)=[ ], then the value of adj A is equal to:1 mk
• Q17Direction ratios of lines l 1 and l 2 are <12, -3, 9 and <4, q, -p respectively. The values of p and q for which l 1…1 mk
• Q17The feasible region of a linear programming problem with objective function Z=5 x+7 y is shown below: The maximum value…1 mkasked 3×
• Q17If vectors a=3 i+2 j+λ k and b=2 i-4 j+5 k, represent the two strips of the Red Cross sign placed outside a doctor's…1 mkasked 3×
• Q18The degree of an objective function of a linear programming problem is1 mkasked 3×
• Q18The probability that a particular item is available in three shops A, B and C is 4/5, 3/4 and 2/3 respectively. If a…1 mk
• Q18The value of k for which the function f(x)= casesx^2 sin 1/x, & x ≠ 0 k(x+1), & x=0 cases is a continuous function, is:1 mkasked 3×
• Q18If e^x+y=3 x, then (dy)/(dx) is1 mk
• Q18The probability that it will rain tomorrow in cities A, B and C is 60 %, 70 % and 80 % respectively. The probability…1 mk
• Q18The absolute maximum value of f(x)=x^2+1 in [-5,2] is1 mk
• Q18A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the…1 mk
• Q18If E and F are two independent events such that P(E)=3/10, P(E F)=1/2, then P(E F)-P(F E) is equal to:1 mkasked 3×
• Q18If 3 P =P =3/5 and P(A B)=2/3, then P(A B) is:1 mkasked 3×
• Q18For two events A and B such that P ≠ 0 and P ≠ 1, P(A^′ / B^′)=1 mkasked 3×
• Q19Assertion: One of the particular solutions of the differential equation (d y)/(d x)=e^x+y can be e^x+e^-y=-2. Reason…1 mkasked 3×
• Q19Assertion: The vectors a and (-2 a), where a ≠ 0 are collinear vectors. Reason (R): a ·(-2 a)=0.1 mkasked 3×
• Q19Assertion: A relation R on the set 1,2,3 defined as R= (1,1),(1,2),(2,1),(2,2),(3,3) is an equivalence relation. Reason…1 mkasked 3×
• Q19Assertion: If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB=BA.…1 mkasked 3×
• Q19Assertion: In an experiment of throwing an unbiased die, the probability of getting a prime number given that number…1 mkasked 3×
• Q19For two vectors a and b Assertion: a × b ^2+( a · b)^2= a ^2 b ^2 Reason (R): a × b =( a · b) tan θ,(θ ≠ (π)/2)1 mkasked 3×
• Q20Assertion: A function f: N → N given by f(x)=x^3+2, x ∈ N is one-one but not onto. Reason (R): Since y ∈ N (Codomain),…1 mkasked 3×
• Q20Assertion: Lines given by x=py+q, z=ry+s and x=p^′ y+q^′, z=r^′ y+s^′ are perpendicular to each other when pp^′+rr^′=1.…1 mkasked 3×
• Q20Assertion: Consider a Linear Programming Problem with minimise Z=x+2 y subject to constraints 2 x+y ≥ 3, x+2 y ≥ 6, x,…1 mkasked 3×
• Q20Assertion: A line can have direction cosines < 1, 1, 1 Reason (R): cos θ=1 is possible for θ=0.1 mkasked 3×
• Q21Check whether function f(x) defined as f(x)= is continuous at x=3. or not? OR If √3(x^2+y^2)=4 x y, then find (dy)/(dx)…2 mkasked 3×
• Q21Evaluate tan [ cos^-1( tan (3 π)/4)].2 mk
• Q21Check whether f: R- 3 → R defined as f(x)=x-2/x-3 is onto or not. OR Check whether f: Z × Z → Z × Z (where Z is the set…2 mkasked 3×
• Q21Evaluate: tan^-1(-1/(√3))+ cot^-1(1/(√3))+ tan^-1( sin (-(π)/2))+ tan^-1( tan (2 π)/3)2 mkasked 3×
• Q21Evaluate sin [ cos^-1 cos ((7 π)/6)].2 mk
• Q21If x=t+1/(t) and y=t-1/(t), find (dy)/(dx) at t=2.2 mk
• Q21Evaluate sin [ tan^-1 tan ((3 π)/4)].2 mk
• Q21A vector a of magnitude 14 has direction ratios <2,3,-6. Find the projection of the vector a on i.2 mk
• Q21Find the co-ordinates of the point on line x=(y-1)/2=(z-2)/3 whose y co- ordinate is 3 times the x co-ordinate.2 mk
• Q22Find the sub-interval(s) of (0, (π)/2) in which f(x)= tan x-4 x is increasing.2 mk
• Q22Show that the function f(x)=. is continuous at x=(π)/2. OR Find whether the function f(x)=. at x=2 is differentiable or…2 mkasked 3×
• Q22A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the…2 mk
• Q22Differentiate x^x with respect to x log x. OR If y=P cos ux+Q sin ux, show that (d^2 y)/(dx^2)+u^2 y=0.2 mkasked 3×
• Q22If x=a^2 sin^3 t, y=b cos^3 t, then find (dy)/(d x) at t=(π)/4.2 mk
• Q23Find the interval(s) for which the function f(x)=(x)/3+3/(x), x ≠ 0 is increasing.2 mk
• Q23Find the interval(s) in which the function f(x)=(x)/( log x), where x ∈(0,1) (1, ∞), is increasing.2 mk
• Q23Find the sub-interval of (0, ∞) in which f(x)=x log x is increasing.2 mk
• Q23If x=e^ sin^-1 t, y=e^ cos^-1 t, find (dy)/(dx) at t=1/(√2)2 mkasked 2×
• Q23Find the value of sin [ cot^-1 √2( cos ( tan^-1 1))]. OR A relation R on A= 1,2,3 is defined as R= (1,1)(3,3),(1,2). Is…2 mkasked 3×
• Q23Find the absolute maximum value of f(x)= cos x+ sin^2 x, x ∈[0, π] OR If the volume of a solid hemisphere increases at…2 mkasked 3×
• Q23Find the vector of magnitude 14 in the direction of QP, where P and Q are the points (1,3,2) and (-1,0,8) respectively.2 mk
• Q23Simplify: sin^-1 √(1+ cos 2 x)/2, 0<x<(π)/2. OR Evaluate: cos [ sin^-1(-1)- tan^-1(-√3)].2 mk
• Q23Find the values of x for which f(x)=x^x, x 0 is increasing.2 mk
• Q23Determine the values of x for which f(x)=(x-3)/(x+1), x ≠-1 is an increasing function.2 mk
• Q23Find the co-ordinates of foot of perpendicular drawn from (0,0,0) to line x/1=(y+1)/-1=(z-3)/-2.2 mk
• Q23Determine the interval(s) in which f(x)=(x)/3+3/(x), x ≠ 0 is increasing.2 mk
• Q23Determine the interval(s) in which f(x)=5 x^3 / 2-3 x^5 / 2, x 0 is increasing.2 mk
• Q24Using vectors, find the area of ABC with vertices A(1,2,3), B(2, -1,4) and C(4, 5, -1).2 mk
• Q24Vectors a=3 i-2 j+2 k and b= i+2 k represent the two adjacent sides of a parallelogram. Find the vectors representing…2 mkasked 3×
• Q24Three honey bees were found flying along the vectors a=2 i-3 j+ k, b=4 j-2 k and c=3 i+2 k respectively. Find the value…2 mkasked 3×
• Q24If AB= j+ k and AC=3 i- j+4 k represent the two vectors along the sides AB and AC of ABC, prove that the median AD= AB+…2 mkasked 3×
• Q24If x=e^t+1/t and y=e^t-1/t, then find (dy)/(dx) at t=-2.2 mk
• Q24If the position vectors of three points A, B and C are 3 i+ j, 5 i+6 j-3 k and 4 j respectively, then show that they…2 mk
• Q24If ( 2 i+3 j+ k ) and ( 2 i+ j- k ) represent the sides A B and A C respectively of Δ ABC, find the vector representing…2 mk
• Q24If for two unit vectors a and b, a+2 b = 2 a- b, then find the angle between a and b.2 mkasked 3×
• Q24If in a parallelogram PQRS, PQ=2 i+3 j+ k and QR= i-2 j+ k, then find the unit vectors parallel to the diagonals PR and…2 mk
• Q25If the lines (x-3)/1=(1-y)/1=(z+2)/(p) and (2-x)/3=(y+1)/5=(z+56)/(2 p) are perpendicular to each other, then find the…2 mkasked 3×
• Q25Find the sub-interval of (0, ∞) in which f(x)=x^2 e^-x is increasing.2 mk
• Q25If x= sin t- cos t, y= sin t cos t, find (dy)/(d x) at t=(π)/4.2 mk
• Q25Find the co-ordinates of the point on the line r=- j+3 k+λ(2 i-2 j+ k) such that the sum of co-ordinates is 3.2 mk
• Q25Simplify: cot^-1 √(1+ cos 2 x)/(1- cos 2 x), x ∈(0, (π)/2). OR Evaluate: sin ( tan^-1(-√3)- sec^-1 2)2 mk
• Q25Let two rods placed on the ground be represented by vectors 4 i- j+3 k and -2 i+ j-2 k. Find a vector representing a…2 mkasked 3×
• Q25Simplify: tan^-1(( cos 2 x- sin 2 x)/( cos 2 x+ sin 2 x)), 0<x<(π)/4. OR Evaluate: tan ( sin^-1 1- cos^-1(-1/2))2 mkasked 3×
• Q25Find the angle between the following pair of lines:2 mkasked 3×
• Q26Find: ∫ x+2√9 x-x^2 d x3 mk
• Q26A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated.…3 mk
• Q26Evaluate: ∫ 0^1 log (1+x^2) d x3 mk
• Q26Find: ∫ (d x)/(x^1 / 2+x^1 / 3) OR Find: ∫ tan^-1((1-x)/(1+x)) d x3 mkasked 3×
• Q26Let A=R- 3 and B=R- 1. A function f: A → B is defined by f(x)=((x-2)/(x-3)). Find whether f is one-one and onto. OR Let…3 mkasked 3×
• Q26A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant…3 mk
• Q26Evaluate: ∫ 0^1 x tan^-1 x d x.3 mk
• Q26The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the…3 mk
• Q27Evaluate: ∫ 0^π ( sin^2026 x)/( sin^2026 x+ cos^2026 x) d x3 mk
• Q27If A=[ ], then compute A^2-4 A-5 I.3 mk
• Q27Find the general solution of the differential equation: y^2 d x+(x^2-x y+y^2) dy=0 OR Find the particular solution of…3 mk
• Q27Find ∫ √(x+2)/x-2 d x OR Find: ∫ (x^2)/((x^2+9)(x^2+16)) d x3 mkasked 3×
• Q27If A=[ ], then compute A^2-5 A+4 I.3 mk
• Q27If A=[ ], then compute A^2-7 A+10 I.3 mk
• Q27Evaluate: ∫ (π)/12^(5 π)/12 ( d x)/(1+√ cot x) OR Evaluate: ∫ (-π)/6^(π)/2( sin x + cos x ) d x3 mkasked 3×
• Q27Find: ∫ (x- sin x)/(1- cos x) d x OR Evaluate: ∫ 0^2 1√x^2+2 x+3 d x3 mkasked 3×
• Q27Find the general solution of the differential equation (x^2+y^2) d y=x y d x. OR Find the particular solution of the…3 mk
• Q27Find: ∫ 3 x-1√x^2-4 x d x3 mk
• Q28Evaluate: ∫ 0^(π)/2 ( sin^2 x)/(1+ sin 2 x) d x3 mk
• Q28Solve the differential equation (x- sin y) d y+ tan y d x=0.3 mk
• Q28If xy=e^x-y, then find (dy)/(dx). OR Differentiate tan^-1( √1+x^2-√1-x^2√1+x^2+√1-x^2) with respect to cos^-1 x^2.3 mkasked 3×
• Q28Solve the differential equation (x+2 y^3) d y=y d x.3 mk
• Q28Solve the following linear programming problem graphically: Maximize Z=8000 x+12000 y Subject to constraints3 mk
• Q28Solve the differential equation y dx+(x-y^3) dy=0.3 mk
• Q28If I 1=∫ -π / 4^π / 4 ( d x)/(1+ cos 2 x) and I 2=∫ -1 / 2^1 / 2 x d x, then show that I 1-4 I 2=0.3 mkasked 3×
• Q28If /(d x)( F(x))=1/(e^x+1), then find F(x) given that F(0)= log 1/2.3 mkasked 3×
• Q28Find: ∫ ( cos x)/((2+ sin x)(4+ sin x)) d x OR Find: ∫ (x+3)/(x^2+4 x+5) d x3 mkasked 3×
• Q29The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a…3 mkasked 3×
• Q29Find the general solution of the following differential equation: x^2 (dy)/(dx)=x^2+x y+y^2 OR Find the particular…3 mk
• Q29Find the general solution of the differential equation 2 x^2 (dy)/(dx)=y^2+2 xy. OR Find a particular solution of the…3 mk
• Q29Solve the following differential equation: x (dy)/(d x)=y-x sin^2((y)/x), given that y(1)=(π)/6 OR Find the general…3 mkasked 3×
• Q29Solve the following Linear Programming Problem graphically: Maximise Z=200 x+120 y subject to the constraints3 mk
• Q29Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown.…3 mkasked 3×
• Q29Solve the following Linear Programming Problem graphically: Maximize Z=20 x+10 y subject to constraints3 mk
• Q29Solve the following Linear Programming Problem graphically: Minimize Z=20 x+10 y subject to3 mk
• Q29Solve the following Linear Programming Problem graphically: Maximize Z=(2 x)/5+(3 y)/10 subject to constraints3 mkasked 3×
• Q30Solve the following linear programming problem graphically: Minimize Z=13x-15y Subject to constraints3 mkasked 3×
• Q30Find the general solution of the differential equation (y^2-x^2) d x=2 x y dy OR Find the particular solution of the…3 mk
• Q30If ( sin x)^y=y^ cos x, then find (dy)/(dx).3 mkasked 3×
• Q30Evaluate: ∫ 0^1 x sin^-1 x d x3 mk
• Q30Find a point on the line (x-2)/3=(1-y)/2=(z-3)/2 at a distance of √2 units from the point (1, 2, 3). OR Find the…3 mkasked 3×
• Q30Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are 55 i-2 j, 5…3 mkasked 3×
• Q30Find the general solution of the differential equation (x^2-y^2) d x+2 x y d y=0 OR Solve the differential equation sin…3 mk
• Q30Find: ∫ 2 x+1√6 x+x^2 d x3 mk
• Q30Solve the following linear programming problem graphically: Maximize Z=10500 x+9000 y Subject to constraints3 mk
• Q31A die is rolled. Consider events: A= 1,2,5, B= 3,5, C= 2,3,4,5 and hence find: (i) P(A C) and P(C A) (ii) P(A B C) and…3 mkasked 3×
• Q31In a school, the probability of holding a debate competition is 1/3 and that of a quiz competition is 2/3. In the two…3 mkasked 3×
• Q31A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee…3 mkasked 3×
• Q31Evaluate: ∫ -(π)/2^(π)/2 cos^2 x2^x+1 d x3 mk
• Q31Solve the following linear programming problem graphically: Maximize Z=4500 x+5000 y Subject to constraints3 mk
• Q32A relation R is defined on Z, the set of integers, as R= (x, y): x-y is divisible by a prime number ' p ', x, y ∈ Z…5 mkasked 3×
• Q32Find cost (per kg) of each fertilizer A, B and C that the farmer needs to buy, such that 1 kg each of fertilizer A and…5 mk
• Q32A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60;…5 mk
• Q32Show that line AB passing through points A(0,4,1), B(2,3,-1) and the line CD passing through points C(4,5,0), D(2,6,2)…5 mk
• Q32Show that f: R → R defined as f(x)= x√1+x^2 is one-one but not onto.5 mk
• Q32If x=3 sin t- sin 3 t, y=3 cos t- cos 3 t find (dy)/(d x) and prove that (d^2 y)/(d x^2)=(-cosec^3 2 t · cosec t)/35 mk
• Q32Find the sub-interval of (0, (π)/2) in which f(x)= log ( sin x+ cos x) is increasing and decreasing. OR A rectangle of…5 mk
• Q32Find the domain of p(x)= sin^-1(1-2 x^2). Hence, find the value of x for which p(x)=(π)/6. Also, write the range of 2…5 mk
• Q32Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each…5 mk
• Q32Find: ∫ (x)/((x-1)(x^2+4)) dx OR Evaluate: ∫ 0^1 (x tan^-1 x)/((1+x^2)^3 / 2) d x5 mkasked 3×
• Q32If P=[ ] and Q=[ ], find (QP) and hence solve the following system of equations using matrices: x-y=3,2 x+3 y+4 z=17,…5 mkasked 3×
• Q33Solve the following Linear Programming Problem graphically: Maximise Z=600 x+400 y subject to the constraints5 mk
• Q33If A=[ ], find A^-1 and use it to solve the following system of equations: -2 y+z=7,2 x-y-z=8, x-2 y=10 OR If [ ] is a…5 mkasked 3×
• Q33Prove that the line through points A(0,-1,-1) and B(4,5,1) intersects the line through points C(3, 9, 4) and D(-4, 4,…5 mk
• Q33Using integration, find the area of the region bounded by y=5 x+4, y=0, x=-1 and x=1.5 mk
• Q33Find the sub intervals in which f(x)= cot^-1( sin x+ cos x), x ∈(0, π) is increasing and decreasing. OR A rectangle of…5 mk
• Q33Using integration, find the area of the region enclosed by the curve y= x-6, the x -axis, and between x=4 and x=8.5 mk
• Q33Show that a function f: R + → A N, defined as f(x)=4 x^2+12 x+15 is one-one. Find set A so that f is onto where R +=[0,…5 mk
• Q33Find the domain of q(x)= cos^-1(4 x^2-3). Hence, find the value of x for which q(x)=0. Also, write the range of 3…5 mk
• Q33If y √x^2+1= log √x^2+1-x, show that (x^2+1) (dy)/(dx)+xy+1=0. OR Find the differential of x^ cot x+(2 x^2-3)/(2…5 mkasked 3×
• Q33Using integration, find the area of the region bounded by the curve y=x x, x -axis, x=-2 and x=2.5 mk
• Q34Find the domain of g(x)= cos^-1(x^2-1). Hence, find the value of x for which g(x)=(π)/3. Also, write the range of…5 mk
• Q34Sketch the curve described by (x, y): 9 x^2+16 y^2=144 and find the area of the region enclosed by it, using…5 mk
• Q34Sketch the curve (x, y): 100 x^2+25 y^2=2500 and find the area of the region enclosed by it, using integration.5 mk
• Q34Sketch the graph defined by (x, y): (x^2)/25+(y^2)/25=1. Find the area of the region of minor segment cut off by the…5 mk
• Q34Show that f: R + →[-5, ∞) given by f(x)=4 x^2+4 x-5 is both one-one and onto where R +=[0, ∞). Also, find p ∈ R + such…5 mk
• Q34Solve the following Linear Programming Problem graphically: Maximize Z=8 x+8 y subject to the constraints5 mk
• Q34Solve the differential equation y e^y dx=(y^3+2 x e^y) dy, when y(0)=1. OR Find the general solution of the…5 mkasked 3×
• Q34On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given…5 mkasked 3×
• Q34If x= cos t, y= cos mt, prove that (1-x^2) (d^2 y)/(d x^2)-x (dy)/(d x)+m^2 y=0.5 mk
• Q35If x=a( sin t-t cos t) and y=b( cos t+t sin t), then find (d y)/(d x) and (d^2 y)/(d x^2).5 mk
• Q35Solve the following Linear Programming Problem graphically: Maximise Z=12 x+18 y subject to the constraints5 mk
• Q35Find the foot of the perpendicular from the point (0,2,3) on the line (-x-3)/-5=(1-y)/-2=(3 z+12)/9 and hence find the…5 mkasked 3×
• Q35Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines…5 mk
• Q35Find the vector and cartesian equations of the line passing through the point of intersection of the lines r=( i+ j-…5 mk
• Q35Check whether the lines given by x-1/2=y-2/3=z-3/4 and x-4/5=y-1/2=z are parallel or not. If parallel, find the…5 mk
• Q35Find the length of the perpendicular drawn from the point P(1, 2, 3) to the line (x-6)/3=(y-7)/2=(z-7)/-2. Also, find…5 mk
• Q35Represent the equations of lines l 1 and l 2 in vector form and check whether they are intersecting or not. OR Opposite…5 mkasked 3×
• Q35Find the sub-interval of (0, π) in which f(x)= tan^-1( sin x- cos x) is increasing and decreasing. OR A rectangle of…5 mk
• Q35A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line r=4 i+ j+λ(5 i+2 j+ k). Find the…5 mkasked 3×
• Q36Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung…4 mkasked 3×
• Q36At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm…4 mkasked 3×
• Q36A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the…4 mkasked 3×
• Q36An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber…4 mkasked 3×
• Q37In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third…4 mkasked 3×
• Q37There is a triangular park in the society. The park is divided into two sections as shown in the figure. In the region…4 mkasked 3×
• Q37A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a…4 mkasked 3×
• Q37Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks…4 mkasked 3×
• Q37There are three types of vaccines A 1, A 2, A 3, available in the market to protect the population of the country from…4 mkasked 3×
• Q37A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm…4 mkasked 3×
• Q37A racing track is build around an elliptical ground whose equation is given by 9 x^2+16 y^2=144. The width of the track…4 mkasked 3×
• Q38Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such…4 mkasked 3×
• Q38An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The…4 mkasked 3×
• Q38Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the…4 mkasked 3×
• Q38A school wants the students of class XII to do a project on 'Sustainability' keeping the world environment in mind.…4 mkasked 3×