CBSE Class 12 Mathematics 2024

What 2024 actually asked, chapter by chapter

Marks we counted in the 2024 papers themselves. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage.

All 308 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 is:1 mk
• Q1A function f: R → R defined as f(x)=x^2-4 x+5 is:1 mkasked 3×
• 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 mkasked 3×
• Q1The number of solutions of differential equation (d y)/(d x)-y=1, given that y(0)=1, is:1 mk
• Q1If [ ] is a scalar matrix, then the value of a+2 b+3 c+4 d is:1 mkasked 3×
• Q1If the sum of all the elements of a 3 × 3 scalar matrix is 9, then the product of all its elements is:1 mkasked 3×
• Q1If x=at, y= /(t), then (dy)/(dx) is:1 mk
• Q2If A=[ ] 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 mkasked 3×
• 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 mkasked 2×
• Q2If a matrix has 36 elements, the number of possible orders it can have, is:1 mkasked 3×
• 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=[ ] 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[ ], matrix A is:1 mkasked 3×
• Q3If A=[ ], then the value of I-A+A^2-A^3+… is:1 mkasked 3×
• Q3If =kabc, then the value of k is:1 mkasked 3×
• Q3Let A=[ ] be a square matrix such that adj A=A. Then, (a+b+c+d) is equal to:1 mkasked 3×
• Q3Which of the following statements is true for the function f(x)=.?1 mkasked 2×
• Q4If inverse of matrix [ ] is the matrix [ ], then value of λ is:1 mkasked 3×
• Q4A function f(x)= 1-x+ x is:1 mkasked 3×
• Q4The number of points of discontinuity of f(x)=. is:1 mkasked 3×
• Q4If A=[a ij]=[ ] and c ij is the cofactor of element a ij, then the value of a 21 · c 11+a 22 · c 12+a 23 · c 13 is:1 mk
• Q4If A=[ ] and A^2+7 I=kA, then the value of k is:1 mk
• Q4If A=[ ], then the value of A (adj. A) is:1 mk
• Q5If A=[ ] and B=[ ], then value of x for which A^2=B is:1 mk
• Q5If [ ][ ]=[ ][ ], then value of x is:1 mk
• Q5If A=[ ] and A^2-kA-5 I=O, then the value of k is:1 mk
• Q5If [ ]=[ ], then the value of (24/(x)+24/(y)) is:1 mkasked 3×
• Q5Let A=[ ] and B=1/3[ ]. If AB=I, then the value of λ is:1 mk
• Q5Given that [ ][ ]=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 mkasked 3×
• Q5If for the matrix A=[ ], 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 [ ] respectively, then the value of a 11 A 21+a 12 A…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 mkasked 3×
• Q7The derivative of sin (x^2) w.r.t. x, at x=√π is:1 mkasked 2×
• Q7x log x (d y)/(d x)+y=2 log x is an example of a:1 mkasked 3×
• Q7The value of constant c that makes the function f defined by f(x)= casesx^2-c^2, & if x<4 c x+20, & if x ≥ 4 cases…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 mkasked 3×
• 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 mkasked 2×
• 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)=.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)=, 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 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• Q9The number of arbitrary constants in the particular solution of the differential equation log ((dy)/(dx))=3 x+4 y;…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 mkasked 3×
• Q9The function f(x)=x/2+2/x has a local minima at x equal to:1 mkasked 3×
• Q9The coordinates of the foot of the perpendicular drawn from the point (0,1,2) on the x -axis are given by:1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q10A linear programming problem deals with the optimization of a/an:1 mkasked 3×
• Q10The common region determined by all the constraints of a linear programming problem is called:1 mkasked 3×
• Q10The restrictions imposed on decision variables involved in an objective function of a linear programming problem are…1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 2×
• Q12The value of ∫ 1^e log x d x is:1 mk
• Q12∫ -a^a f(x) d x=0, if:1 mk
• Q12is equal to:1 mkasked 2×
• Q12The integral ∫ d x√9-4 x^2 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 mkasked 2×
• 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 mkasked 3×
• Q12If f(x) is an odd function, then ∫ -π / 2^π / 2 f(x) cos^3 x d x equals:1 mk
• Q13If = ± 6, then the value of k is:1 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 2×
• Q14The degree of the differential equation (y^′ ′)^2+(y^′)^3=x sin (y^′) is:1 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• Q15If a =2 and -3 ≤ k ≤ 2, then k a ∈:1 mkasked 3×
• Q15The angle which the line x/1=(y)/-1=(z)/0 makes with the positive direction of Y -axis is:1 mkasked 3×
• Q15The unit vector perpendicular to both vectors i+ k and i- k is:1 mkasked 3×
• 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 mkasked 3×
• Q16Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly…1 mkasked 3×
• 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 mkasked 3×
• Q16Direction ratios of a vector parallel to line x-1/2=-y=(2 z+1)/6 are:1 mkasked 3×
• Q16The maximum value of Z=4 x+y for a L.P.P. whose feasible region is given below is:1 mkasked 3×
• Q16The distance of point P(a, b, c) from y -axis is:1 mkasked 3×
• Q17The number of corner points of the feasible region determined by constraints x ≥ 0, y ≥ 0, x+y ≥ 4 is:1 mkasked 3×
• Q17The probability distribution of a random variable X is: where k is some unknown constant. The probability that the…1 mkasked 3×
• Q17If F(x)=[ ] and [F(x)]^2=F(k x), then the value of k is:1 mkasked 2×
• 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 mkasked 3×
• Q17Of the following, which group of constraints represents the feasible region given below?1 mkasked 3×
• Q18If ∫ 0^2 2 e^2 x dx=∫ 0^a e^x dx, the value of ' a ' is:1 mk
• Q18If A and B are events such that P(A / B)=P(B / A) ≠ 0, then:1 mkasked 3×
• Q18The function f(x)=k x- sin x is strictly increasing for1 mkasked 3×
• Q18If a line makes an angle of 30^° with the positive direction of x -axis, 120^° with the positive direction of y -axis,…1 mkasked 3×
• Q18If A and B are two non-zero square matrices of same order such that (A+B)^2=A^2+B^2, then:1 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• Q19Assertion: Every scalar matrix is a diagonal matrix. Reason (R): In a diagonal matrix, all the diagonal elements are 0.1 mkasked 3×
• Q19Assertion: For matrix A=[ ], where θ ∈[0,2 π], A ∈[2,4]. Reason (R): cos θ ∈[-1,1], θ ∈[0,2 π].1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 2×
• Q20Assertion: The vectors represent the sides of a right angled triangle. Reason (R): Three non-zero vectors of which none…1 mkasked 3×
• 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 mkasked 2×
• Q20Assertion: A line in space cannot be drawn perpendicular to x, y and z axes simultaneously. Reason ( R ): For any line…1 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• 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^22]; 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 mkasked 3×
• 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 mkasked 2×
• 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)=. at x=1.2 mkasked 3×
• Q22Verify whether the function f defined by f(x)=. is continuous at x=0 or not. OR Check for differentiability of the…2 mkasked 2×
• 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 mkasked 3×
• 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: gathered ∫ x √1+2 x dx OR gathered Evaluate: ∫ 0^(π)/4^2 sin √x√x d x2 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q24Find: ∫ (e^4 x-1)/(e^4 x+1) d x2 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 the…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, i.e., a ×…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 mkasked 3×
• Q25Show that f(x)=e^x-e^-x+x- tan^-1 x is strictly increasing in its domain.2 mkasked 2×
• Q25Find the vector equation of the line passing through the point (2,3,-5) and making equal angles with the co-ordinate…2 mkasked 3×
• 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: /(dx)( x )=(x)/( x ), x ≠ 03 mk
• Q26Solve the following linear programming problem graphically: Maximize z=x+y subject to constraints3 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 mkasked 3×
• 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 mkasked 3×
• Q26Solve the following linear programming problem graphically: Minimise z=5 x-2 y subject to the constraints3 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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)= casesa x+b, & x…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: gathered ∫ -2^2 √2-x/(2+x) dx OR gathered Find: ∫ 1/(x[( log x)^2-3 log x-4]) d x3 mkasked 3×
• Q28Solve the following linear programming problem graphically: Maximise z=5 x+4 y subject to the constraints3 mk
• Q28Find: ∫ (2 x+3)/(x^2(x+3)) d x3 mk
• Q28Evaluate: ∫ 0^π e^ cos xe^ cos x+e^- cos x d x OR Find: ∫ (2 x+1)/((x+1)^2(x-1)) d x3 mkasked 3×
• 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 mkasked 3×
• Q28Solve the following linear programming problem graphically: Minimize z=600 x+400 y, subject to the constraints3 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• 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 constraints3 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 mkasked 3×
• 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 constraints3 mk
• Q30Solve the following linear programming problem graphically: Maximise Z=2 x+3 y subject to the constraints: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: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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• Q31E and F are two independent events such that P( E)=0.6 and P(E F)=0 · 6. Find P(F) and P( E F).3 mkasked 3×
• 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 mkasked 3×
• Q32Find the value of p for which the lines are perpendicular to each other and also intersect. Also, find the point of…5 mk
• Q32If A=[ ], then find A^-1 and hence solve the following system of equations: OR Find the product of the matrices [ ][ ]…5 mkasked 3×
• 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:…5 mk
• Q32If A=[ ], find A^-1 and use it to solve the following system of equations: x-2 y=10,2 x-y-z=8,-2 y+z=7 OR If A=[ ] and…5 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• 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. OR…5 mkasked 3×
• 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=[ ] and B^-1=[ ], find (AB)^-1. Also, find (AB)^-1. OR Given A=[ ], find A^-1. Use it to solve the following…5 mk
• Q33Evaluate: gathered ∫ 0^(π)/4 ( sin x+ cos x)/(9+16 sin 2 x) dx OR gathered Evaluate: ∫ 0^(π)/2 sin 2 x tan^-1( sin x) d…5 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. Also,…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,…5 mkasked 2×
• Q34Find A^-1, if A=[ ]. Hence, solve the following system of equations: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 mkasked 3×
• Q34If A=[ ], find A^-1 and hence solve the following system of equations: 2 x+y-3 z=13 3 x+2 y+z=4 x+2 y-z=85 mk
• Q34Use the product of matrices ( )( ) to solve the following system of equations: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 mkasked 3×
• 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 constraints5 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 mkasked 3×
• 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 ≤ 125 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 mkasked 3×
• Q35Solve the following L.P.P. graphically: Maximise Z=60 x+40 y Subject to x+2 y ≤ 125 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 mkasked 3×
• Q36Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against…4 mkasked 3×
• 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 mkasked 3×
• Q36Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys…4 mkasked 3×
• Q36Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the…4 mkasked 3×
• Q37Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air…4 mkasked 3×
• Q37A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using…4 mkasked 3×
• Q37An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume…4 mkasked 3×
• Q37The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and…4 mkasked 3×
• Q37According to recent research, air turbulence has increased in various regions around the world due to climate change.…4 mkasked 3×
• Q37A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card…4 mkasked 3×
• 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 mkasked 3×
• Q38A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of…4 mkasked 3×
• Q38A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted…4 mkasked 3×
• Q38A departmental store sends bills to charge its customers once a month. Past experience shows that 70 % of its customers…4 mkasked 3×