CBSE Class 12 Mathematics 2024
311 questions · 654 marks · 308 with the board’s own marking-scheme answer, 3 with no official answer on the site.
Marks by chapter
Counted from the 311 questions listed below, as the 2024 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 |
|---|---|---|
| Integrals42 questions | 88 | 13.5% |
| Continuity and Differentiability40 questions | 73 | 11.2% |
| Three Dimensional Geometry23 questions | 60 | 9.2% |
| Application of Derivatives27 questions | 58 | 8.9% |
| Matrices35 questions | 54 | 8.3% |
| Linear Programming20 questions | 53 | 8.1% |
| Vector Algebra29 questions | 48 | 7.3% |
| Relations and Functions19 questions | 48 | 7.3% |
| Probability17 questions | 46 | 7.0% |
| Differential Equations23 questions | 38 | 5.8% |
| Application of Integrals10 questions | 38 | 5.8% |
| Determinants14 questions | 26 | 4.0% |
| Inverse Trigonometric Functions12 questions | 24 | 3.7% |
311 questions from 2024
- Q1A function f: R_+ → R (where R_+ is the set of all non-negative real numbers) defined by f(x)=4 x+3 is:1 mk
- Q1The value of |8, 2, 7; 12, 3, 5; 16, 4, 3| is:1 mk
- Q1A function f: R → R defined as f(x)=x^2-4 x+5 is:1 mk
- Q1If y= cos^-1(e^x), then (d y)/(d x) is:1 mk
- Q1If A=[a_ij] is an identity matrix, then which of the following is true?1 mk
- Q1The number of solutions of differential equation (d y)/(d x)-y=1, given that y(0)=1, is:1 mk
- Q1If [a, c, 0; b, d, 0; 0, 0, 5] is a scalar matrix, then the value of a+2 b+3 c+4 d is:1 mk
- Q1If the sum of all the elements of a 3 × 3 scalar matrix is 9, then the product of all its elements is:1 mk
- Q1If x=at, y=a/t, then dy/dx is:1 mk
- Q2If A= [a, c, -1; b, 0, 5; 1, -5, 0] is a skew-symmetric matrix, then the value of 2 a-(b+c) is:1 mk
- Q2For any two vectors a and b, which of the following statements is always true?1 mk
- Q2If A is a square matrix of order 2 and |A|=-2, then value of |5 A^′| is:1 mk
- Q2The derivative of 5^x w.r.t. e^x is:1 mk
- Q2Let f: R_+ →[-5, ∞) be defined as f(x)=9 x^2+6 x-5, where R_+ is the set of all non-negative real numbers. Then, f is:1 mk
- Q2If a matrix has 36 elements, the number of possible orders it can have, is:1 mk
- Q2The solution of the differential equation (d y)/(d x)=1/(log y) is:1 mk
- Q2The degree and order of differential equation y^′ ′ 2+ log (y^′)=x^5 respectively are:1 mk
- Q2Let R_+ denote the set of all non-negative real numbers. Then the function f: R_+ → R_+ defined as f(x)=x^2+1 is:1 mk
- Q2The integrating factor of the differential equation x (d y)/(d x)-y=x^4-3 x is:1 mk
- Q2For the matrix A= [2, -1, 1; λ, 2, 0; 1, -2, 3] to be invertible, the value of λ is:1 mk
- Q2If y= sin^-1 x, then (d^2 y)/(d x^2) is:1 mk
- Q2Given that A^-1=1/7 [2, 1; -3, 2], matrix A is:1 mk
- Q3If A= [2, 1; -4, -2], then the value of I-A+A^2-A^3+… is:1 mk
- Q3If |-a, b, c; a, -b, c; a, b, -c| =kabc, then the value of k is:1 mk
- Q3Let A= [a, b; c, d] be a square matrix such that adj A=A. Then, (a+b+c+d) is equal to:1 mk
- Q3Which of the following statements is true for the function f(x)= {x^2+3, x ≠ 0; 1, x=0}?1 mk
- Q4If inverse of matrix [7, -3, -3; -1, 1, 0; -1, 0, 1] is the matrix [1, 3, 3; 1, λ, 3; 1, 3, 4], then value of λ is:1 mk
- Q4A function f(x)=|1-x+|x|| is:1 mk
- Q4The number of points of discontinuity of f(x)= {|x|+3, if x ≤-3; -2 x, if -3<x<3; 6 x+2, if x ≥ 3} is:1 mk
- Q4If A=[a_ij]= [2, -1, 5; 1, 3, 2; 5, 0, 4] and c_ij is the cofactor of element a_ij, then the value of a_21 · c_11+a_22…1 mk
- Q4If A= [3, 1; -1, 2] and A^2+7 I=kA, then the value of k is:1 mk
- Q4If A= [-2, 0, 0; 1, 2, 3; 5, 1, -1], then the value of ∣ A (adj. A) is:1 mk
- Q5If A= [x, 0; 1, 1] and B= [4, 0; -1, 1], then value of x for which A^2=B is:1 mk
- Q5If [x, 2, 0] [5; -1; x] = [3, 1] [-2; x], then value of x is:1 mk
- Q5If A= [1, 3; 3, 4] and A^2-kA-5 I=O, then the value of k is:1 mk
- Q5If [x+y, 2; 5, xy] = [6, 2; 5, 8], then the value of (24/x+24/y) is:1 mk
- Q5Let A= [1, -1, 2; 0, 2, -3; 3, -2, 4] and B=1/3 [-2, 0, 1; 9, 2, -3; 6, 1, λ]. If AB=I, then the value of λ is:1 mk
- Q5Given that [1, x] [4, 0; -2, 0] =0, the value of x is:1 mk
- Q5The product of matrix P and Q is equal to a diagonal matrix. If the order of matrix Q is 3 × 2, then order of matrix P…1 mk
- Q5If the sides of a square are decreasing at the rate of 1.5 cm / s, the rate of decrease of its perimeter is:1 mk
- Q5The function f(x)=x^3-3 x^2+12 x-18 is:1 mk
- Q5If for the matrix A= [tan x, 1; -1, tan x], A+A^′=2 √3 I, then the value of x ∈[0, (π)/2] is:1 mk
- Q6If a_ij and A_ij represent the (ij)th element and its cofactor of [2, -3, 5; 6, 0, 4; 1, 5, -7] respectively, then the…1 mk
- Q6∫_a^b f(x) d x is equal to:1 mk
- Q6If e^x^2 y=c, then dy/dx is:1 mk
- Q6Derivative of e^2 x with respect to e^x, is:1 mk
- Q6∫_0^π / 2 (sin x- cos x)/(1+ sin x cos x) d x is equal to:1 mk
- Q6If matrices A and B are of order 1 × 3 and 3 × 1 respectively, then the order of A^′ B^′ is:1 mk
- Q6Derivative of x^2 with respect to x^3, is:1 mk
- Q6The number of all scalar matrices of order 3, with each entry -1,0 or 1, is:1 mk
- Q6∫_-a^a f(x) d x=0, if:1 mk
- Q6Find the matrix A^2, where A=[a_ij] is a 2 × 2 matrix whose elements are given by a_ij= maximum ( i, j ) - minimum ( i,…1 mk
- Q7The derivative of sin (x^2) w.r.t. x, at x=√π is:1 mk
- Q7x log x (d y)/(d x)+y=2 log x is an example of a:1 mk
- Q7The value of constant c that makes the function f defined by f(x)= {x^2-c^2, if x<4; c x+20, if x ≥ 4} continuous for…1 mk
- Q7If x e^y=1, then the value of dy/dx at x=1 is:1 mk
- Q7Let θ be the angle between two unit vectors a and b such that sin θ=3/5. Then, a · b is equal to:1 mk
- Q7The function f(x)=|x|+|x-2| is1 mk
- Q7For any square matrix A,(A-A^′) is always1 mk
- Q7The differential equation (d y)/(d x)=F(x, y) will not be a homogeneous differential equation, if F(x, y) is:1 mk
- Q7A relation R defined on a set of human beings as R={(x, y): x is 5 cm shorter than y } is:1 mk
- Q7For what value of k, the function given below is continuous at x=0? f(x)= {(√(4+x)-2)/x, x ≠ 0; k, x=0}1 mk
- Q7If sin (x y)=1, then dy/(d x) is equal to:1 mk
- Q7d/(d x)[ cos ( log x+e^x)] at x=1 is:1 mk
- Q7Let f(x)= |x^2, sin x; p, -1|, where p is a constant. The value of p for which f^′(0)=1 is:1 mk
- Q8The integrating factor of the differential equation (1-x^2) (d y)/(d x)+x y=a x, -1<x<1, is:1 mk
- Q8A function f: R → A defined as f(x)=x^2+1 is onto, if A is:1 mk
- Q8Let Z denote the set of integers, then function f: Z → Z defined as f(x)=x^3-1 is:1 mk
- Q8The value of ∫_0^π tan^2((θ)/3) d θ is:1 mk
- Q8Derivative of e^sin^2 x with respect to cos x is:1 mk
- Q8The value of ∫_-1^1|x| d x is:1 mk
- Q8If a=2 i- j+ k and b= i+ j- k, then a and b are:1 mk
- Q8The value of ∫_0^3 (d x)/(√(9-x^2)) is:1 mk
- Q9If the direction cosines of a line are √3 k, √3 k, √3 k, then the value of k is:1 mk
- Q9The number of arbitrary constants in the particular solution of the differential equation log (dy/dx)=3 x+4 y; y(0)=0…1 mk
- Q9If α, β and γ are the angles which a line makes with positive directions of x, y and z axes respectively, then which of…1 mk
- Q9The function f(x)=x/2+2/x has a local minima at x equal to:1 mk
- Q9The coordinates of the foot of the perpendicular drawn from the point (0,1,2) on the x -axis are given by:1 mk
- Q9The integrating factor of the differential equation dy/(d x)+2/x y=0, x ≠ 0 is:1 mk
- Q9The general solution of the differential equation x dy+ydx=0 is:1 mk
- Q10The integrating factor of the differential equation (x+2 y^2) (d y)/(d x)=y(y>0) is:1 mk
- Q10If A is a square matrix of order 3 such that the value of |adj · A|=8, then the value of |A^T| is:1 mk
- Q10Given a curve y=7 x-x^3 and x increases at the rate of 2 units per second. The rate at which the slope of the curve is…1 mk
- Q10A linear programming problem deals with the optimization of a/an:1 mk
- Q10The common region determined by all the constraints of a linear programming problem is called:1 mk
- Q10The restrictions imposed on decision variables involved in an objective function of a linear programming problem are…1 mk
- Q11The rate of change of surface area of a sphere with respect to its radius ' r ', when r=4 cm, is:1 mk
- Q11Let E be an event of a sample space S of an experiment, then P(S ∣ E)=1 mk
- Q11Let E and F be two events such that P(E)=0 · 1, P(F)=0 · 3, P(E ∪ F)=0 · 4, then P(F ∣ E) is:1 mk
- Q11If a and b are two vectors such that | a|=1,| b|=2 and a · b=√3, then the angle between 2 a and - b is:1 mk
- Q11The value of ∫_π / 4^π / 2 cot θ cosec^2 θ d θ is:1 mk
- Q11If ∫_-2^3 x^2 d x=k ∫_0^2 x^2 d x+∫_2^3 x^2 d x, then the value of k is:1 mk
- Q11The point of inflexion of a function f(x) is the point where1 mk
- Q11∫ 1/(x( log x)^2) d x is equal to:1 mk
- Q11If P(A ∣ B)=P(A^′ ∣ B), then which of the following statements is true?1 mk
- Q12Anti-derivative of √(1+ sin 2 x), x ∈[0, (π)/4] is:1 mk
- Q12The value of ∫_-1^1 x|x| d x is:1 mk
- Q12If A and B are two skew symmetric matrices, then (AB+BA) is:1 mk
- Q12The value of ∫_1^e log x d x is:1 mk
- Q12∫_-a^a f(x) d x=0, if:1 mk
- Q12|x+1, x-1; x^2+x+1, x^2-x+1| is equal to:1 mk
- Q12The integral ∫ (d x)/(√(9-4 x^2)) is equal to:1 mk
- Q12If g(x) is a continuous function satisfying g(-x)=-g(x), then ∫_0^2 a g(x) dx is equal to:1 mk
- Q12If A=[a_ij] be a 3 × 3 matrix, where a_ij=i-3 j, then which of the following is false?1 mk
- Q12The vectors a=2 i- j+ k, b= i-3 j-5 k and c=-3 i+4 j+4 k represents the sides of1 mk
- Q12If f(x) is an odd function, then ∫_-π / 2^π / 2 f(x) cos^3 x d x equals:1 mk
- Q13If |1, 3, 1; k, 0, 1; 0, 0, 1| = ± 6, then the value of k is:1 mk
- Q13The derivative of tan^-1(x^2) w.r.t. x is:1 mk
- Q13Let a be any vector such that | a|=a. The value of | a × i|^2+| a × j|^2+| a × k|^2 is:1 mk
- Q13The area bounded by the curve y=√x, Y -axis and between the lines y=0 and y=3 is:1 mk
- Q13A relation R defined on set A={x: x ∈ Z and 0 ≤ x ≤ 10} as R={(x, y): x=y} is given to be an equivalence relation. The…1 mk
- Q13The area of the region bounded by the curve y^2=4 x and x=1 is:1 mk
- Q13Area of the region bounded by curve y^2=4 x and the X -axis between x=0 and x=1 is:1 mk
- Q14Which of the following statements is not true about equivalence classes A_i(i=1,2, …. n) formed by an equivalence…1 mk
- Q14The derivative of 2^x w.r.t. 3^x is:1 mk
- Q14The general solution of the differential equation dy/dx=e^x+y is:1 mk
- Q14A vector perpendicular to the line r= i+ j- k+λ(3 i- j) is:1 mk
- Q14The order of the differential equation (d^4 y)/(d x^4)- sin ((d^2 y)/(d x^2))=5 is:1 mk
- Q14If a=2 i- j+ k and b= i+ j- k, then a and b are:1 mk
- Q14The vector equation of a line passing through the point ( 1,-1,0 ) and parallel to Y -axis is:1 mk
- Q14The Cartesian equation of a line passing through the point with position vector a= i- j and parallel to the line r= i+…1 mk
- Q14The order and degree of the differential equation [1+((d y)/(d x))^2]^3=(d^2 y)/(d x^2) respectively are:1 mk
- Q14The degree of the differential equation (y^′ ′)^2+(y^′)^3=x sin (y^′) is:1 mk
- Q14The order of the following differential equation (d^3 y)/(d x^3)+x(dy/(d x))^5=4 log ((d^4 y)/(d x^4)) is:1 mk
- Q15The position vectors of points P and Q are p and q respectively. The point R divides line segment PQ in the ratio 3: 1…1 mk
- Q15The number of points, where f(x)=[x], 0<x<3 ([•] denotes greatest integer function) is not differentiable is:1 mk
- Q15The vector with terminal point A(2,-3,5) and initial point B(3,-4,7) is:1 mk
- Q15If | a|=2 and -3 ≤ k ≤ 2, then |k a| ∈:1 mk
- Q15The angle which the line x/1=y/-1=z/0 makes with the positive direction of Y -axis is:1 mk
- Q15The unit vector perpendicular to both vectors i+ k and i- k is:1 mk
- Q15The lines (1-x)/2=(y-1)/3=z/1 and (2 x-3)/(2 p)=y/-1=(z-4)/7 are perpendicular to each other for p equal to:1 mk
- Q16Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly…1 mk
- Q16If a line makes an angle of (π)/4 with the positive directions of both x -axis and z -axis, then the angle which it…1 mk
- Q16Direction ratios of a vector parallel to line (x-1)/2=-y=(2 z+1)/6 are:1 mk
- Q16The maximum value of Z=4 x+y for a L.P.P. whose feasible region is given below is:1 mk
- Q16The distance of point P(a, b, c) from y -axis is:1 mk
- Q17The number of corner points of the feasible region determined by constraints x ≥ 0, y ≥ 0, x+y ≥ 4 is:1 mk
- Q17The probability distribution of a random variable X is: where k is some unknown constant. The probability that the…1 mk
- Q17If F(x)= [cos x, - sin x, 0; sin x, cos x, 0; 0, 0, 1] and [F(x)]^2=F(k x), then the value of k is:1 mk
- Q17The Cartesian equation of the line passing through the point (1,-3,2) and parallel to the line: r=(2+λ) i+λ j+(2 λ-1) k…1 mk
- Q17Of the following, which group of constraints represents the feasible region given below?1 mk
- Q18If ∫_0^2 2 e^2 x dx=∫_0^a e^x dx, the value of ' a ' is:1 mk
- Q18If A= [2, 0, 0; 0, 3, 0; 0, 0, 5], then A^-1 is:1 mk
- Q18If A and B are events such that P(A / B)=P(B / A) ≠ 0, then:1 mk
- Q18The function f(x)=k x- sin x is strictly increasing for1 mk
- Q18If a line makes an angle of 30^° with the positive direction of x -axis, 120^° with the positive direction of y -axis,…1 mk
- Q18If A and B are two non-zero square matrices of same order such that (A+B)^2=A^2+B^2, then:1 mk
- Q18If ∫_0^a 1/(4+x^2) dx=(π)/6, then the value of ' a ' is:1 mk
- Q19Assertion: For any symmetric matrix A, B^′ AB is a skew-symmetric matrix. Reason (R): A square matrix P is…1 mk
- Q19Assertion: Domain of y= cos^-1(x) is [-1,1]. Reason (R): The range of the principal value branch of y= cos^-1(x) is [0,…1 mk
- Q19Assertion: Every scalar matrix is a diagonal matrix. Reason (R): In a diagonal matrix, all the diagonal elements are 0.1 mk
- Q19Assertion: For matrix A= [1, cos θ, 1; - cos θ, 1, cos θ; -1, - cos θ, 1], where θ ∈[0,2 π], |A| ∈[2,4]. Reason (R):…1 mk
- Q19Assertion: For any non-zero unit vector a, a ·(- a)=(- a) · a=-1. Reason (R): Angle between a and (- a) is (π)/2.1 mk
- Q19Assertion: The relation R={(x, y):(x+y) is a prime number and x, y ∈ N} is not a reflexive relation. Reason (R): The…1 mk
- Q20Assertion: The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of Z=x+2 y…1 mk
- Q20Assertion: Projection of a on b is same as projection of b on a. Reason (R): Angle between a and b is same as angle…1 mk
- Q20Assertion: The vectors a=6 i+2 j-8 k; b=10 i-2 j-6 k; c=4 i-4 j+2 k represent the sides of a right angled triangle.…1 mk
- Q20Assertion: ( b · c) a is a scalar quantity. Reason (R): Dot product of two vectors is a scalar quantity.1 mk
- Q20Assertion: For two non-zero vectors a and b, a · b= b · a. Reason (R): For two non-zero vectors a and b, a × b= b × a.1 mk
- Q20Assertion: A line in space cannot be drawn perpendicular to x, y and z axes simultaneously. Reason ( R ): For any line…1 mk
- Q21Check whether the function f(x)=x^2|x| is differentiable at x=0 or not. OR If y=√(tan √x), prove that √x (d y)/(d…2 mk
- Q21Find value of k if sin^-1[k tan (2 cos^-1 (√3)/2)]=(π)/32 mk
- Q21Find the value of tan^-1(-1/(√3))+ cot^-1(1/(√3))+ tan^-1[ sin (-(π)/2)]. OR Find the domain of the function f(x)=…2 mk
- Q21Determine whether the function f(x)=x^2-6 x+3 is increasing or decreasing in [4,6].2 mk
- Q21a, b and c are three mutually perpendicular unit vectors. If θ is the angle between a and ( 2 a+3 b+6 c ), find the…2 mk
- Q21Express tan^-1((cos x)/(1- sin x)), where (-π)/2<x<(π)/2 in the simplest form. OR Find the principal value of…2 mk
- Q21If a= sin^-1((√2)/2)+ cos^-1(-1/2) and b= tan^-1(√3)- cot^-1(-1/(√3)) then find the value of a+b.2 mk
- Q21Evaluate: sec^2( tan^-1 1/2)+cosec^2( cot^-1 1/3)2 mk
- Q21Simplify: cos^-1 x+ cos^-1[x/2+(√(3-3 x^2))/2]; 1/2 ≤ x ≤ 12 mk
- Q22If a and b are two non-zero vectors such that ( a+ b) ⊥ a and (2 a+ b) ⊥ b, then prove that | b|=√2| a|.2 mk
- Q22Check the differentiability of function f(x)=[x] at x=-3, where [·] denotes greatest integer function. OR If x^1 /…2 mk
- Q22If f(x)=| tan 2 x|, then find the value of f^′(x) at x=(π)/3. OR If y=cosec( cot^-1 x), then prove that √(1+x^2)…2 mk
- Q22Evaluate: cot^2{cosec^-1 3}+ sin^2{ cos^-1(1/3)}2 mk
- Q22Show that the function f(x)=4 x^3-18 x^2+27 x-7 has neither maxima nor minima.2 mk
- Q22Evaluate: ∫_0^a^3 (x^2)/(x^6+a^6) d x2 mk
- Q22Find the value of sin^-1(-1/2)+ cos^-1(-(√3)/2)+ cot^-1( tan (4 π)/3). OR Find the domain of f(x)= cos^-1(1-x^2). Also,…2 mk
- Q22If x=e^x / y, prove that dy/dx=(log x-1)/(( log x)^2) OR Check the differentiability of f(x)= {x^2+1, 0 ≤ x<1; 3-x, 1 ≤…2 mk
- Q22Verify whether the function f defined by f(x)= {x sin (1/x), x ≠ 0; 0, x=0} is continuous at x=0 or not. OR Check for…2 mk
- Q23Show that the function f given by f(x)= sin x+ cos x, is strictly decreasing in the interval ((π)/4, (5 π)/4).2 mk
- Q23Find the interval in which the function f(x)=x^4-4 x^3+10 is strictly decreasing.2 mk
- Q23Show that f(x)=(4 sin x)/(2+ cos x)-x is an increasing function of x in [0, (π)/2].2 mk
- Q23Evaluate: ∫_0^π / 2 sin 2 x cos 3 x d x OR Given d/(d x) F(x)=1/(√(2 x-x^2)) and F(1)=0, find F(x).2 mk
- Q23Sand is pouring from a pipe at the rate of 15 cm^3 / minute. The falling sand forms a cone on the ground such that the…2 mk
- Q23The area of the circle is increasing at a uniform rate of 2 cm^2 / sec. How fast is the circumference of the circle…2 mk
- Q23Find: ∫ x √(1+2 x) dx; OR Evaluate: ∫_0^(π)/4^2 (sin √x)/(√x) d x2 mk
- Q23If a, b and c are three unit vectors such that a+ b- c= 0, find the angle between vectors a and c.2 mk
- Q23If M and m denote the local maximum and local minimum values of the function f(x)=x+1/x(x ≠ 0) respectively, find the…2 mk
- Q23Find local maximum value and local minimum value (whichever exists) for the function f(x)=4 x^2+1/(x)(x ≠ 0).2 mk
- Q23The surface area of a cube increases at the rate of 72 cm^2 / sec. Find the rate of change of its volume, when the edge…2 mk
- Q24Find the position vector of point C which divides the line segment joining points A and B having position vectors i+2…2 mk
- Q24Find the value of [ sin^2{ cos^-1(3/5)}+ tan^2{ sec^-1(3)}].2 mk
- Q24The volume of a cube is increasing at the rate of 6 cm^3 / s. How fast is the surface area of cube increasing, when the…2 mk
- Q24Find: ∫ (e^4 x-1)/(e^4 x+1) d x2 mk
- Q24If y= cos^3( sec^2 2 t), find dy/dt. OR If x^y=e^x-y, prove that dy/dx=(log x)/((1+ log x)^2).2 mk
- Q24If y=√(cos x+y), prove that (d y)/(d x)=(sin x)/(1-2 y). OR Show that the function f(x)=|x|^3 is differentiable at all…2 mk
- Q24Find: ∫ cos^3 x e^log sin x d x OR Find: ∫ 1/(5+4 x-x^2) d x2 mk
- Q25Verify whether the function f defined by is continuous at x=0 or not. OR Check for differentiability of the function f…2 mk
- Q25Evaluate: ∫_(-1/2)^(1/2) cos x · log ((1+x)/(1-x)) d x2 mk
- Q25Find: ∫ (2 x)/((x^2+1)(x^2-4)) d x.2 mk
- Q25Check the differentiability of f(x)=| cos x| at x=(π)/2. OR If y=A sin 2 x+B cos 2 x and (d^2 y)/(d x^2)-k y=0, find…2 mk
- Q25Find the absolute maximum and minimum values of the function f(x)=12 x^4 / 3-6 x^1 / 3, x ∈[0,1].2 mk
- Q25Find: ∫ 1/(x(x^2-1)) d x.2 mk
- Q25Let a and b be two non-zero vectors. Prove that | a × b| ≤| a|| b|. State the condition under which equality holds,…2 mk
- Q25In the given figure, ABCD is a parallelogram. If AB=2 i-4 j+5 k and D B=3 i-6 j+2 k, then find A D and hence find the…2 mk
- Q25Show that f(x)=e^x-e^-x+x- tan^-1 x is strictly increasing in its domain.2 mk
- Q25Find the vector equation of the line passing through the point (2,3,-5) and making equal angles with the co-ordinate…2 mk
- Q26If x=e^cos 3 t and y=e^sin 3 t, prove that dy/dx=-(y log x)/(x log y). OR Show that: d/(dx)(|x|)=x/(|x|), x ≠ 03 mk
- Q26Solve the following linear programming problem graphically: Maximize z=x+y subject to constraints 2 x+5 y ≤ 100; 8 x+5…3 mk
- Q26Find dy/dx, if ( cos x)^y=( cos y)^x. OR If √(1-x^2)+√(1-y^2)=a(x-y), prove that (d y)/(d x)=√((1-y^2)/(1-x^2)).3 mk
- Q26Given that y=( sin x)^x · x^sin x+a^x, find dy/(d x).3 mk
- Q26Given that x^y+y^x=a^b, where a and b are positive constants, find (d y)/(d x).3 mk
- Q26A relation R on set A={1,2,3,4,5} is defined as R={(x, y):|x^2-y^2|<8}. Check whether the relation R is reflexive,…3 mk
- Q26Solve the following linear programming problem graphically: Minimise z=5 x-2 y subject to the constraints x+2 y ≤ 120;…3 mk
- Q26Find: ∫ (x^-1)/(( log x)^2-5 log x+4) d x3 mk
- Q26Find dy/dx, if y=( cos x)^x+ cos^-1 √x is given.3 mk
- Q26If x cos (p+y)+ cos p sin (p+y)=0, prove that cos p (d y)/(d x)=- cos^2(p+y), where p is a constant. OR Find the value…3 mk
- Q27Find the general solution of the differential equation (d y)/(d x)=(x^2+y^2)/(2 x y).3 mk
- Q27If y=( tan^-1 x)^2, show that (x^2+1)^2 (d^2 y)/(d x^2)+2 x(x^2+1) dy/dx=2.3 mk
- Q27If √(1-x^2)+√(1-y^2)=a(x-y), prove that dy/dx=√((1-y^2)/(1-x^2)). OR If y=( tan x)^x, then find dy/dx.3 mk
- Q27Evaluate: ∫_0^((π)/4) (x d x)/(1+ cos 2 x+ sin 2 x) OR Find: ∫ e^x[1/((1+x^2)^(3/2))+x/(√(1+x^2))] d x3 mk
- Q27Find: ∫ x^2 log (x^2-1) d x3 mk
- Q27Find the intervals in which the function f(x)=(log x)/x is strictly increasing or strictly decreasing. OR Find the…3 mk
- Q27If x=a sin^3 θ, y=b cos^3 θ, then find (d^2 y)/(d x^2) at θ=(π)/4.3 mk
- Q27Find the values of a and b so that the following function is differentiable for all values of x: f(x)= {a x+b, x>-1; b…3 mk
- Q27If vectors a, b and 2 a+3 b are unit vectors, then find the angle between a and b.3 mk
- Q27Evaluate: ∫_-2^2 √((2-x)/(2+x)) dx; OR Find: ∫ 1/(x[( log x)^2-3 log x-4]) d x3 mk
- Q28Solve the following linear programming problem graphically: Maximise z=5 x+4 y subject to the constraints x+2 y ≥ 4; 3…3 mk
- Q28Find: ∫ (2 x+3)/(x^2(x+3)) d x3 mk
- Q28Evaluate: ∫_0^π (e^cos x)/(e^cos x+e^- cos x) d x OR Find: ∫ (2 x+1)/((x+1)^2(x-1)) d x3 mk
- Q28If y=( log x)^2, prove that x^2 y^′ ′+xy^′=2. OR If y= sin ( tan^-1 e^x), then find dy/dx at x=0.3 mk
- Q28Find: ∫ (x^2)/((x^2+4)(x^2+9)) d x OR Evaluate: ∫_1^3(|x-1|+|x-2|+|x-3|) dx3 mk
- Q28Solve the following linear programming problem graphically: Minimize z=600 x+400 y, subject to the constraints x+y ≥ 8;…3 mk
- Q28Find the particular solution of the differential equation given by 2 x y+y^2-2 x^2 (d y)/(d x)=0; y=2, when x=1. OR…3 mk
- Q28Find: ∫ (3 x+5)/(√(x^2+2 x+4)) d x3 mk
- Q28Find: ∫ sec^3 θ d θ3 mk
- Q28Find: ∫ (x^2+1)/((x^2+2)(x^2+4)) d x3 mk
- Q29Find the particular solution of the differential equation given by x^2 (d y)/(d x)-x y=x^2 cos^2(y/(2 x)), given that…3 mk
- Q29Find the particular solution of the differential equation dy/dx=y cot 2 x, given that y((π)/4)=2. OR Find the…3 mk
- Q29Find: ∫ (2+ sin 2 x)/(1+ cos 2 x) e^x d x OR Evaluate: ∫_0^π / 4 1/(sin x+ cos x) d x3 mk
- Q29The position vectors of vertices of Δ ABC are A(2 i- j+ k), B( i-3 j-5 k) and C(3 i-4 j-4 k). Find all the angles of Δ…3 mk
- Q29Find the particular solution of the differential equation dy/dx-2 x y=3 x^2 e^x^2; y(0)=5 OR Solve the following…3 mk
- Q30Find a vector of magnitude 4 units perpendicular to each of the vectors 2 i- j+ k and i+ j- k and hence verify your…3 mk
- Q30Given a=2 i- j+ k, b=3 i- k and c=2 i+ j-2 k. Find a vector d which is perpendicular to both a and b and c · d=3.3 mk
- Q30Solve the following linear programming problem graphically: Maximise z=500 x+300 y, subject to constraints x+2 y ≤ 12;…3 mk
- Q30The corner points of the feasible region determined by the system of linear constraints are as shown in the following…3 mk
- Q30If x^30 y^20=(x+y)^50, prove that dy/dx=y/x. OR Find (d y)/(d x), if 5^x+5^y=5^x+y.3 mk
- Q30A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the…3 mk
- Q30Find the projection of vector ( b+ c ) on vector a, where a=2 i+2 j+ k, b= i+3 j+ k and c= i+ k.3 mk
- Q30Solve the following linear programming problem graphically: Maximise z=4 x+3 y, subject to the constraints x+y ≤ 800; 2…3 mk
- Q30Solve the following linear programming problem graphically: Maximise Z=2 x+3 y subject to the constraints: x+y ≤ 6; x ≥…3 mk
- Q30Find: ∫ (√x)/((x+1)(x-1)) d x3 mk
- Q30Solve the following L.P.P. graphically: Maximise Z=x+3 y subject to the constraints: x+2 y ≤ 200; x+y ≤ 150; y ≤ 75; x,…3 mk
- Q31Solve the following differential equation: ( tan^-1 y-x) d y=(1+y^2) d x3 mk
- Q31The chances of P, Q and R getting selected as CEO of a company are in the ratio 4: 1: 2 respectively. The probabilities…3 mk
- Q31Bag I contains 3 red and 4 black balls, Bag II contains 5 red and 2 black balls. Two balls are transferred at random…3 mk
- Q31Find: ∫ x^2 · sin^-1(x^3 / 2) d x3 mk
- Q31An urn contains 3 red and 2 white marbles. Two marbles are drawn one by one with replacement from the urn. Find the…3 mk
- Q31A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at…3 mk
- Q31E and F are two independent events such that P(Ē)=0.6 and P(E ∪ F)=0 · 6. Find P(F) and P(Ē ∪ F̄).3 mk
- Q31The random variable X has the following probability distribution where a and b are some constants: If the mean E(X)=3,…3 mk
- Q32Sketch the graph of y=x|x| and hence find the area bounded by this curve, X -axis and the ordinates x=-2 and x=2, using…5 mk
- Q32A relation R on set A={-4,-3,-2,-1,0,1,2,3,4} be defined as R={(x, y): x+y is an integer divisible by 2 }. Show that R…5 mk
- Q32Find the value of p for which the lines r=λ i+(2 λ+1) j+(3 λ+2) k and r= i-3 μ j+(p μ+7) k are perpendicular to each…5 mk
- Q32If A= [1, 2, -3; 2, 0, -3; 1, 2, 0], then find A^-1 and hence solve the following system of equations: x+2 y-3 z=1; 2…5 mk
- Q32Evaluate: ∫_0^π / 2 e^x((1+ sin x)/(1+ cos x)) d x OR Find: ∫_π / 6^π / 3 (sin x+ cos x)/(√(sin 2 x)) d x5 mk
- Q32Find the equation of a line l_2 which is the mirror image of the line l_1 with respect to line l: x/1=(y-1)/2=(z-2)/3,…5 mk
- Q32If A= [1, -2, 0; 2, -1, -1; 0, -2, 1], find A^-1 and use it to solve the following system of equations: x-2 y=10,2…5 mk
- Q32Show that a function f: R → R defined by f(x)=(2 x)/(1+x^2) is neither one-one nor onto. Further, find set A so that…5 mk
- Q33Find the area of the region bounded by the lines x-2 y=4, x=-1, x=6 and x -axis, using integration.5 mk
- Q33Find the area of the region bounded by the curve 4 x^2+y^2=36 using integration.5 mk
- Q33It is given that function f(x)=x^4-62 x^2+a x+9 attains local maximum value at x=1. Find the value of ' a ', hence…5 mk
- Q33Let A=R-{5} and B=R-{1}. Consider the function f: A → B, defined by f(x)=(x-3)/(x-5). Show that f is one-one and onto.…5 mk
- Q33Show that a function f: R → R defined as f(x)=x^2+x+1 is neither one-one nor onto. Also, find all the values of x for…5 mk
- Q33If A= [5, 0, 4; 2, 3, 2; 1, 2, 1] and B^-1= [1, 3, 3; 1, 4, 3; 1, 3, 4], find (AB)^-1. Also, find |(AB)^-1|. OR Given…5 mk
- Q33Evaluate: ∫_0^((π)/4) (sin x+ cos x)/(9+16 sin 2 x) dx; OR Evaluate: ∫_0^((π)/2) sin 2 x tan^-1( sin x) d x5 mk
- Q33Let A=R-{3} and B=R-{a}. Find the value of 'a' such that the function f: A → B defined by f(x)=(x-2)/(x-3) is onto.…5 mk
- Q33Find the equation of the line which bisects the line segment joining points A(2,3,4) and B(4,5,8) and is perpendicular…5 mk
- Q34Find: ∫ ((3 cos x-2) sin x)/(5- sin^2 x-4 cos x) d x OR Evaluate: ∫_-2^2 (x^3+|x|+1)/(x^2+4|x|+4) d x5 mk
- Q34Solve the following system of equations, using matrices: 2/x+3/y+10/z=4, 4/x-6/y+5/z=1, 6/x+9/y-20/z=2 where x, y, z ≠…5 mk
- Q34Find A^-1, if A= [1, 2, 1; 2, 3, -1; 1, 0, 1]. Hence, solve the following system of equations: x+2 y+z=5; 2 x+3 y=1;…5 mk
- Q34Using integration, find the area of the region enclosed between the circle x^2+y^2=16 and the lines x=-2 and x=2.5 mk
- Q34Using integration, find the area of the ellipse (x^2)/16+(y^2)/4=1, included between the lines x=-2 and x=2.5 mk
- Q34If A= [2, 1, -3; 3, 2, 1; 1, 2, -1], find A^-1 and hence solve the following system of equations: 2 x+y-3 z=13 3 x+2…5 mk
- Q34Use the product of matrices (1, 2, -3; 3, 2, -2; 2, -1, 1) (0, 1, 2; -7, 7, -7; -7, 5, -4) to solve the following…5 mk
- Q34Equations of sides of a parallelogram ABCD are as follows: AB: (x+1)/1=(y-2)/-2=(z-1)/2 BC: (x-1)/3=(y+2)/-5=(z-5)/3…5 mk
- Q34Find the co-ordinates of the foot of the perpendicular drawn from the point (2,3,-8) to the line (4-x)/2=y/6=(1-z)/3.…5 mk
- Q34The vertices of Δ ABC are A(1,1,0), B(1,2,1) and C(-2,2,-1). Find the equations of the medians through A and B. Use the…5 mk
- Q35Solve the following L.P.P. graphically: Minimise Z=6 x+3 y Subject to constraints 4 x+y ≥ 80; x+5 y ≥ 115; 3 x+2 y ≤…5 mk
- Q35Find the distance between the line x/2=(2 y-6)/4=(1-z)/-1 and another line parallel to it passing through the point…5 mk
- Q35A relation R on set A={x:-10 ≤ x ≤ 10, x ∈ Z} is defined as R={(x, y):(x-y) is divisible by 5 }. Show that R is an…5 mk
- Q35Solve the following Linear Programming problem graphically: Maximise Z=300 x+600 y Subject to x+2 y ≤ 12 2 x+y ≤ 12;…5 mk
- Q35Using integration, find the area of the region enclosed between the curve y=√(4-x^2) and the lines x=-1, x=1 and the x…5 mk
- Q35The image of point P(x, y, z) with respect to line x/1=(y-1)/2=(z-2)/3 is P^′(1,0,7). Find the coordinates of point P.5 mk
- Q35Find the equation of the line passing through the point of intersection of the lines x/1=(y-1)/2=(z-2)/3 and…5 mk
- Q35Solve the following L.P.P. graphically: Maximise Z=60 x+40 y Subject to x+2 y ≤ 12 2 x+y ≤ 12; 4 x+5 y ≥ 20; x, y ≥ 05 mk
- Q35If A_1 denotes the area of region bounded by y^2=4 x, x=1 and x -axis in the first quadrant and A_2 denotes the area of…5 mk
- Q36Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against…4 mk
- Q36If a function f: X → Y defined as f(x)=y is one-one and onto, then we can define a unique function g: Y → X such that…4 mk
- Q36Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys…4 mk
- Q36Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the…4 mk
- Q37Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air…4 mk
- Q37A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using…4 mk
- Q37An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume…4 mk
- Q37The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and…4 mk
- Q37According to recent research, air turbulence has increased in various regions around the world due to climate change.…4 mk
- Q37A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card…4 mk
- Q38The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These…4 mk
- Q38Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's…4 mk
- Q38A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of…4 mk
- Q38A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted…4 mk
- Q38A departmental store sends bills to charge its customers once a month. Past experience shows that 70 % of its customers…4 mk
Other Mathematics years
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