CBSE Class 12 Mathematics 2026
305 questions · 654 marks · every one with the board’s own marking-scheme answer.
Marks by chapter
Counted from the 305 questions listed below, as the 2026 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 |
|---|---|---|
| Matrices41 questions | 74 | 11.3% |
| Three Dimensional Geometry29 questions | 73 | 11.2% |
| Application of Derivatives29 questions | 70 | 10.7% |
| Integrals34 questions | 70 | 10.7% |
| Continuity and Differentiability25 questions | 56 | 8.6% |
| Differential Equations29 questions | 53 | 8.1% |
| Linear Programming19 questions | 47 | 7.2% |
| Application of Integrals14 questions | 47 | 7.2% |
| Inverse Trigonometric Functions24 questions | 44 | 6.7% |
| Probability17 questions | 42 | 6.4% |
| Vector Algebra23 questions | 35 | 5.4% |
| Relations and Functions13 questions | 35 | 5.4% |
| Determinants8 questions | 8 | 1.2% |
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 mk
- Q1If matrix A= [-p, q; r, p] 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= [sin θ, cos θ; - cos θ, sin θ] 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 mk
- 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 mk
- Q3The principal value of sec^-1(√2)+2 cosec^-1(-√2) is:1 mk
- Q3If A and B are skew-symmetric matrices of same order, then AB^′+BA^′ is a/an:1 mk
- Q3For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined?1 mk
- 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 mk
- Q3If A= [1, a, b; -1, 2, c; 0, 5, 3] is a symmetric matrix, then the value of 3 a+b+c is1 mk
- 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= [cos x, - sin x; sin x, cos x] and A+A^′=I, then the value of x ∈[0, (π)/2] is1 mk
- Q4Let A= [0, -3, 4; 1, 0, 2] and B= [-3, 0, 1; 2, 4, 0]. If A+B+C=O, then matrix C is:1 mk
- Q4If [4; 1; 3] A= [-4, 8, 4; -1, 2, 1; -3, 6, 3], then order of A must be:1 mk
- Q4If Δ_1= |1, 0, 0; 0, 2, 0; 0, 0, 3| and Δ_2= |0, 2, 0; 1, 0, 0; 0, 0, 6|, then1 mk
- 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 mk
- 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 |cos x, sin x; - cos x, sin x| =1 is1 mk
- 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 mk
- Q5Differential of e^e^x with respect to x is:1 mk
- 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 mk
- Q5If A is a non-singular matrix, then which of the following is not true?1 mk
- 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)= {(x^2-4 x-5)/(x+1), x ≠-1; k, x=-1} is continuous at x=-1, then the value of k is:1 mk
- 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 mk
- Q6The surface area of a sphere when its volume changes at the same rate as its radius is:1 mk
- Q6If |-1, -2, 5; -2, a, -1; 0, 4, 2 a| =-86, then the sum of all possible values of a is1 mk
- Q6If A=[a_ij]_3 × 3 is a scalar matrix then which of the following must be true?1 mk
- Q6If B(adj B)= [1/3, 0, 0; 0, 1/3, 0; 0, 0, 1/3], 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 mk
- 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 mk
- Q7If f(x)= {(sin x)/x+ cos x, x ≠ 0; k, x=0} is continuous at x=0, then the value of k is:1 mk
- 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 mk
- 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 mk
- Q8For f(x)=x+1/x(x ≠ 0)1 mk
- Q8If A= [tan x, cot x; - cot x, tan x] 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 mk
- 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 mk
- Q9The value of ∫_-1^1 (x^3)/(x^2+2|x|+1) d x is1 mk
- Q9∫ dx/(√(25-16 x^2)) is equal to:1 mk
- 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)= [3, 0, 0; 0, 3, 0; 0, 0, 3], 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 mk
- 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 mk
- Q10If ∫_0^1 dx/(e^x+e^-x)= tan^-1 e+k, then the value of k is:1 mk
- Q10∫ dx/(2^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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 d/(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 mk
- 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 mk
- 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 mk
- Q13In the graph, the feasible region representing the Linear Programming Problem for maximising objective function…1 mk
- 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 mk
- Q14For any two vectors a and b, which of the following statements is always true?1 mk
- Q14The region represented by the system of inequations 3 x+y ≥ 3, 2 x-y ≥-5, x, y ≥ 0 is:1 mk
- Q14The general solution for the differential equation (d y)/(d x)=e^3 x-y is:1 mk
- 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 · b|1 mk
- Q15The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40)…1 mk
- Q15If ( a+ b) ·( a- b)=198 and | a|=10| b|, then:1 mk
- Q15If a matrix B is such that B [1, 2, 3] = [1, 2, 3; 0, 1, 1; 2, 0, 1], then the order of matrix B is: matrix B is:1 mk
- Q15If A= [1/2 cos x, - sin x; sin x, 1/2 cos x] 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 mk
- Q16If (3 i-2 j+5 k) ×(4 i+p j+q k)= 0, then the values of p and q are:1 mk
- 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 mk
- Q16In a linear programming problem, the linear function which has to be maximized or minimized is called1 mk
- Q16The order and degree of the differential equation: d/(d x)(y^′)^3+(y^′)^3=1 respectively are where y^′=dy/dx1 mk
- Q17For the feasible region shown below, the non-trivial constraints of the linear programming problem are1 mk
- 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 mk
- Q17If A(adj A)= [2026, 0, 0; 0, 2026, 0; 0, 0, 2026], 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 mk
- 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 mk
- Q18The degree of an objective function of a linear programming problem is1 mk
- 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)= {x^2 sin 1/x, x ≠ 0; k(x+1), x=0} is a continuous function, is:1 mk
- 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 mk
- Q18If 3 P(A)=P(B)=3/5 and P(A ∣ B)=2/3, then P(A ∪ B) is:1 mk
- Q18For two events A and B such that P(A) ≠ 0 and P(B) ≠ 1, P(A^′ / B^′)=1 mk
- 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 mk
- Q19Assertion: The vectors a and (-2 a), where a ≠ 0 are collinear vectors. Reason (R): a ·(-2 a)=0.1 mk
- 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.…1 mk
- 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 mk
- Q19Assertion: In an experiment of throwing an unbiased die, the probability of getting a prime number given that number…1 mk
- 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 mk
- 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…1 mk
- 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 mk
- 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 mk
- Q20Assertion: A line can have direction cosines < 1, 1, 1 > Reason (R): cos θ=1 is possible for θ=0.1 mk
- Q21Check whether function f(x) defined as f(x)={ (|x-3|)/(2(x-3)), x<3; (x-6)/6, x ≥ 3 is continuous at x=3. or not? OR If…2 mk
- 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…2 mk
- Q21Evaluate: tan^-1(-1/(√3))+ cot^-1(1/(√3))+ tan^-1( sin (-(π)/2))+ tan^-1( tan (2 π)/3)2 mk
- 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)= {(cos x)/(-x+(π)/2), x ≠ (π)/2; 1, x=(π)/2} is continuous at x=(π)/2. OR Find whether the…2 mk
- 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 mk
- 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 mk
- 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)}.…2 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- 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 mk
- Q25Find the angle between the following pair of lines: (x-2)/3=(y+5)/2=(1-z)/-6 and (x-7)/1=y/2=(6-z)/-22 mk
- 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 mk
- 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…3 mk
- 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= [1, 2, 2; 2, 1, 2; 2, 2, 1], 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 mk
- Q27If A= [2, 0, 1; 2, 1, 3; 1, -1, 0], then compute A^2-5 A+4 I.3 mk
- Q27If A= [3, 2, 0; 1, 4, 0; 0, 0, 5], 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 mk
- Q27Find: ∫ (x- sin x)/(1- cos x) d x OR Evaluate: ∫_0^2 1/(√(x^2+2 x+3)) d x3 mk
- 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…3 mk
- 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 constraints 3 x+4 y ≤…3 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 mk
- Q28If d/(d x)( F(x))=1/(e^x+1), then find F(x) given that F(0)= log 1/2.3 mk
- Q28Find: ∫ (cos x)/((2+ sin x)(4+ sin x)) d x OR Find: ∫ (x+3)/(x^2+4 x+5) d x3 mk
- Q29The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a…3 mk
- Q29Find the general solution of the following differential equation: x^2 dy/dx=x^2+x y+y^2 OR Find the particular solution…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 mk
- Q29Solve the following Linear Programming Problem graphically: Maximise Z=200 x+120 y subject to the constraints x+y ≤…3 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 mk
- Q29Solve the following Linear Programming Problem graphically: Maximize Z=20 x+10 y subject to constraints x+2 y ≤ 28; 3…3 mk
- Q29Solve the following Linear Programming Problem graphically: Minimize Z=20 x+10 y subject to x+2 y ≤ 40; 3 x+y ≥ 30; 4…3 mk
- Q29Solve the following Linear Programming Problem graphically: Maximize Z=(2 x)/5+(3 y)/10 subject to constraints 2 x+y ≤…3 mk
- Q30Solve the following linear programming problem graphically: Minimize Z=13x-15y Subject to constraints x+y ≤ 7; 2 x-3…3 mk
- 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 mk
- 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 mk
- 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 mk
- 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 constraints x+y ≤ 50;…3 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 ∩…3 mk
- 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 mk
- Q31A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee…3 mk
- Q31Evaluate: ∫_-(π)/2^((π)/2) (cos^2 x)/(2^x+1) d x3 mk
- Q31Solve the following linear programming problem graphically: Maximize Z=4500 x+5000 y Subject to constraints x+y ≤ 250;…3 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 mk
- 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 mk
- Q32If P= [1, -1, 0; 2, 3, 4; 0, 1, 2] and Q= [2, 2, -4; -4, 2, -4; 2, -1, 5], find (QP) and hence solve the following…5 mk
- Q33Solve the following Linear Programming Problem graphically: Maximise Z=600 x+400 y subject to the constraints x+2 y ≤…5 mk
- Q33If A= [0, 2, 1; -2, -1, -2; 1, -1, 0], find A^-1 and use it to solve the following system of equations: -2 y+z=7,2…5 mk
- 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…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 mk
- 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 constraints x-20 ≤ 0; 2…5 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 mk
- Q34On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given…5 mk
- 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 constraints x+y ≤ 1200;…5 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 mk
- 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…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. l_1:…5 mk
- 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 mk
- Q36Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung…4 mk
- Q36At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm…4 mk
- Q36A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the…4 mk
- Q36An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber…4 mk
- 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 mk
- Q37There is a triangular park in the society. The park is divided into two sections as shown in the figure. In the region…4 mk
- Q37A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a…4 mk
- Q37Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks…4 mk
- 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 mk
- 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 mk
- 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 mk
- Q38Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such…4 mk
- Q38An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The…4 mk
- 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 mk
- Q38A school wants the students of class XII to do a project on 'Sustainability' keeping the world environment in mind.…4 mk
Other Mathematics years
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