CBSE Class 12 Mathematics 2023

Marks by chapter

Counted from the 311 questions listed below, as the 2023 papers print them. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage. Every year compared →

311 questions from 2023

• Q1∫ (1+ tan x)/(1- tan x) dx is equal to:1 mk
• Q1If A= [1, 4, x; z, 2, y; -3, -1, 3] is a symmetric matrix, then the value of x+y+z is:1 mk
• Q1If the angle between the vectors a and b is (π)/4 and | a × b|=1, then a · b is equal to1 mk
• Q1If d/(d x) f(x)=2 x+3/x and f(1)=1, then f(x) is1 mk
• Q1If A= [0, 1; 0, 0], then A^2023 is equal to1 mk
• Q1If A is a 3 × 4 matrix and B is a matrix such that A^′ B and AB^′ are both defined, then the order of the matrix B is:1 mk
• Q1If [1, 1, 1; 0, 1, 1; 0, 0, 1] [x; y; z] = [6; 3; 2], then the value of (2 x+y-z) is:1 mk
• Q1Let R be a relation in the set N given by R={(a, b): a=b-2, b>6}. Then1 mk
• Q1If the angle between a and b is (π)/3 and | a × b|=3 √3, then the value of a · b is1 mk
• Q1Let A={3,5}. Then number of reflexive relations on A is1 mk
• Q1If for a square matrix A, A^2-3 A+I=O and A^-1=xA+yI, then the value of x+y is:1 mk
• Q1If x [1; 2] +y [2; 5] =[4/9], then:1 mk
• Q2The position vectors of three consecutive vertices of a parallelogram ABCD are A(4 i+2 j-6 k), B(5 i-3 j+ k) and C(12…1 mk
• Q2If [2, 0; 5, 4] =P+Q, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to1 mk
• Q2If |A|=2, where A is a 2 × 2 matrix, then |4 A^-1| equals:1 mk
• Q2The direction ratios of a line parallel to z -axis are:1 mk
• Q2Let A be a skew-symmetric matrix of order 3. If |A|=x, then (2023)^x is equal to:1 mk
• Q2If A ·(adj A)= [3, 0, 0; 0, 3, 0; 0, 0, 3], then the value of |A|+|adj A| is equal to:1 mk
• Q2sin [(π)/3+ sin^-1(1/2)] is equal to1 mk
• Q2If the area of a triangle with vertices ( 2,-6 ), ( 5,4 ) and ( k, 4 ) is 35 sq units, then k is1 mk
• Q2If A= [5, x; y, 0] and A=A^T, where A^T is the transpose of the matrix A, then1 mk
• Q3If A is a square matrix and A^2=A, then (I+A)^2-3 A is equal to:1 mk
• Q3Let A be a 3 × 3 matrix such that |adj A|=64. Then |A| is equal to:1 mk
• Q3Ashima can hit a target 2 out of 3 times. She tried to hit the target twice. The probability that she missed the target…1 mk
• Q3If for a square matrix A, A^2-A+I=O, then A^-1 equals1 mk
• Q3If f(x)=2|x|+3| sin x|+6, then the right hand derivative of f(x) at x=0 is:1 mk
• Q3If for two events A and B, P(A-B)=1/5 and P(A)=3/5, then P(B/A) is equal to 11 mk
• Q3A and B are skew-symmetric matrices of same order. AB is symmetric, if:1 mk
• Q3∫_0^2 √(4-x^2) dx equals:1 mk
• Q3If [1, 2, 1; 2, 3, 1; 3, a, 1] is non-singular matrix and a ∈ A, then the set A is1 mk
• Q4If a matrix A= [1, 2, 3], then the matrix AA^′ (where A^′ is the transpose of A ) is:1 mk
• Q4If ∫_0^2 π cos^2 x d x=k ∫_0^π / 2 cos^2 x d x, then the value of k is1 mk
• Q4The function f(x)=|x|-x is:1 mk
• Q4For what value of x ∈[0, (π)/2], is A+A^′=√3 I, where A= [cos x, sin x; - sin x, cos x]?1 mk
• Q4If A=[a_i j] is a square matrix of order 2 such that a_ij= {1, when i ≠ j; 0, when i=j}, then A^2 is1 mk
• Q4If A= [3, 4; 5, 2] and 2 A+B is a null matrix, then B is equal to:1 mk
• Q4The equation of a line passing through point (2,-1,0) and parallel to the line x/1=(y-1)/2=(2-z)/2 is:1 mk
• Q4If A= [1, 0; 2, 1], B= [x, 0; 1, 1] and A=B^2, then x equals1 mk
• Q4If |A|=|kA|, where A is a square matrix of order 2, then sum of all possible values of k is1 mk
• Q5The value of |x+y, y+z, z+x; z, x, y; 1, 1, 1| is1 mk
• Q5The value of the determinant |6, 0, -1; 2, 1, 4; 1, 1, 3| is1 mk
• Q5If d/(d x)(f(x))= log x, then f(x) equals:1 mk
• Q5If the vector i-b j+ k is equally inclined to the coordinate axes, then the value of b is:1 mk
• Q5The value of the determinant |2, 7, 1; 1, 1, 1; 10, 8, 1| is:1 mk
• Q5If in ABC, BA=2 a and BC=3 b, then AC is1 mk
• Q5If d/(d x)[f(x)]=a x+b and f(0)=0, then f(x) is equal to1 mk
• Q5X and Y are independent events such that P(X ∩ Ȳ)=2/5 and P(X)=3/5. Then P(Y) is equal to:1 mk
• Q5Let A be the area of a triangle having vertices (x_1, y_1),(x_2, y_2) and (x_3, y_3). Which of the following is correct?1 mk
• Q5If |α, 3, 4; 1, 2, 1; 1, 4, 1| =0, then the value of α is1 mk
• Q6If | a × b|=√3 and a · b=-3, then angle between a and b is1 mk
• Q6If a+ b= i and a=2 i-2 j+2 k, then | b| equals:1 mk
• Q6Degree of the differential equation sin x+ cos ((d y)/(d x))=y^2 is1 mk
• Q6∫ 2^x+2 dx is equal to:1 mk
• Q6The product [a, b; -b, a] [a, -b; b, a] is equal to:1 mk
• Q6The function f(x)=|x| is1 mk
• Q6∫_0^((π)/6) sec^2(x-(π)/6) dx is equal to:1 mk
• Q6The direction cosines of vector BA, where coordinates of A and B are (1,2,-1) and ( 3,4,0 ) respectively, are:1 mk
• Q6If f(x)=| cos x|, then f((3 π)/4) is1 mk
• Q6The derivative of x^2 x w.r.t. x is1 mk
• Q7The integrating factor of the differential equation (1-y^2) (d x)/dy+y x=ay,(-1<y<1) is1 mk
• Q7∫ (2 cos 2 x-1)/(1+2 sin x) dx is equal to:1 mk
• Q7Equation of line passing through origin and making 30^°, 60^° and 90^° with x, y, z axes respectively is1 mk
• Q7Direction cosines of a line perpendicular to both x -axis and z -axis are:1 mk
• Q7The sum of the order and the degree of the differential equation (d^2 y)/(d x^2)+((d y)/(d x))^3= sin y is:1 mk
• Q7The function f(x)=[x], where [x] denotes the greatest integer less than or equal to x, is continuous at1 mk
• Q7If y= log ( sin e^x), then (d y)/(d x) is:1 mk
• Q7If y= sin^2(x^3), then (d y)/(d x) is equal to:1 mk
• Q8The solution of the differential equation (d x)/x+(d y)/y=0 is:1 mk
• Q8If x=A cos 4 t+B sin 4 t, then (d^2 x)/(d t^2) is equal to1 mk
• Q8Unit vector along PQ, where coordinates of P and Q respectively are (2,1,-1) and (4,4,-7), is1 mk
• Q8∫ e^5 log x dx is equal to:1 mk
• Q8If A and B are two events such that P(A / B)=2 × P(B / A) and P(A)+P(B)=2/3, then P(B) is equal to1 mk
• Q8The value of p for which the vectors 2 i+p j+ k and -4 i-6 j+26 k are perpendicular to each other, is:1 mk
• Q9For what value of k may the function f(x)= {k(3 x^2-5 x), x ≤ 0; cos x, x>0} become continuous?1 mk
• Q9If the point P(a, b, 0) lies on the line (x+1)/2=(y+2)/3=(z+3)/4, then (a, b) is:1 mk
• Q9The interval in which the function f(x)=2 x^3+9 x^2+12 x-1 is decreasing, is1 mk
• Q9If ∫_0^a 3 x^2 dx=8, then the value of ' a ' is:1 mk
• Q9∫_0^4(e^2 x+x) dx is equal to:1 mk
• Q9What is the product of the order and degree of the differential equation (d^2 y)/(d x^2) sin y+((d y)/(d x))^3 cos y=√y?1 mk
• Q9The value of ( i × j) · j+( j × i) · k is:1 mk
• Q9The function f(x)=x|x|, x ∈ R is differentiable1 mk
• Q9The function f(x)=x^3+3 x is increasing in interval1 mk
• Q9Position vector of the mid-point of line segment AB is 3 i+2 j-3 k. If position vector of the point A is 2 i+3 j-4 k,…1 mk
• Q10If a vector makes an angle of (π)/4 with the positive directions of both x -axis and y -axis, then the angle which it…1 mk
• Q10For any two events A and B, if P(Ā)=1/2, P(B̄)=2/3 and P(A ∩ B)=1/4, then P((Ā)/(B̄)) equals:1 mk
• Q10Number of symmetric matrices of order 3 × 3 with each entry 1 or -1 is1 mk
• Q10A and B are square matrices of same order. If (A+B)^2=A^2+B^2, then:1 mk
• Q10The sine of the angle between the vectors a=3 i+ j+2 k and b= i+ j+2 k is:1 mk
• Q10∫ (sec x)/(sec x- tan x) d x equals1 mk
• Q10Projection of vector 2 i+3 j on the vector 3 i-2 j is1 mk
• Q10The integrating factor for solving the differential equation x (d y)/(d x)-y=2 x^2 is:1 mk
• Q11Direction cosines of the line (x-1)/2=(1-y)/3=(2 z-1)/12 are:1 mk
• Q11If [x, 2; 3, x-1] is a singular matrix, then the product of all possible values of x is:1 mk
• Q11The value of ∫_0^((π)/4)( sin 2 x) d x is1 mk
• Q11Equation of a line passing through point (1,1,1) and parallel to z -axis is1 mk
• Q11∫_-1^1 (|x-2|)/(x-2) d x, x ≠ 2 is equal to1 mk
• Q11a and b are two non-zero vectors such that the projection of a on b is 0. The angle between a and b is:1 mk
• Q11The order and degree (if defined) of the differential equation, ((d^2 y)/(d x^2))^2+((d y)/(d x))^3=x sin ((d y)/(d x))…1 mk
• Q12The order and the degree of the differential equation (1+3 (d y)/(d x))^2=4 (d^3 y)/(d x^3) respectively are:1 mk
• Q12If tan ((x+y)/(x-y))=k, then dy/dx is equal to1 mk
• Q12In Δ ABC, AB= i+ j+2 k and AC=3 i- j+4 k. If D is mid-point of BC, then vector AD is equal to:1 mk
• Q12The sum of the order and the degree of the differential equation d/(d x)((dy/(d x))^3) is1 mk
• Q12If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the…1 mk
• Q12A unit vector along the vector 4 i-3 k is:1 mk
• Q12If P(A/B)=0 · 3, P(A)=0 · 4 and P(B)=0 · 8, then P(B/A) is equal to:1 mk
• Q13The value of λ for which the angle between the lines r= i+ j+ k+p(2 i+ j+2 k) and r=(1+q) i+(1+q λ) j+(1+q) k is (π)/2…1 mk
• Q13Anti-derivative of (tan x-1)/(tan x+1) with respect to x is1 mk
• Q13If θ is the angle between two vectors a and b, then a · b ≥ 0 only when:1 mk
• Q13The value of k for which f(x)= {3 x+5, x ≥ 2; kx^2, x<2} is a continuous function, is:1 mk
• Q13If a · i= a ·( i+ j)= a ·( i+ j+ k)=1, then a is1 mk
• Q13Two vectors a=a_1 i+a_2 j+a_3 k and b=b_1 i+b_2 j+b_3 k are collinear if1 mk
• Q13The number of solutions of the differential equation (d y)/(d x)=(y+1)/(x-1), when y(1)=2, is:1 mk
• Q14If (a, b), (c, d) and (e, f) are the vertices of ABC and Δ denotes the area of ABC, then |a, c, e; b, d, f; 1, 1, 1|^2…1 mk
• Q14If A= [0, 1; -1, 0] and (3 I+4 A)(3 I-4 A)=x^2 I, then the value (s) x is/are:1 mk
• Q14The primitive of 2/(1+ cos 2 x) is1 mk
• Q14The magnitude of the vector 6 i-2 j+3 k is1 mk
• Q14Distance of the point (p, q, r) from y -axis is:1 mk
• Q14If P(A ∩ B)=1/8 and P(Ā)=3/4, then P(B/A) is equal to:1 mk
• Q14A unit vector a makes equal but acute angles on the co-ordinate axes. The projection of the vector a on the vector b=5…1 mk
• Q15The value of k for which function f(x)= {x^2, x ≥ 0; kx, x<0} is differentiable at x=0 is:1 mk
• Q15If A= [1, 0; 0, 0] and B= [1, 1; 0, 0], then B^′ A^′ is equal to:1 mk
• Q15The general solution of the differential equation x d y-(1+x^2) d x=d x is:1 mk
• Q15If a line makes angles of 90^°, 135^° and 45^° with the x, y and z axes respectively, then its direction cosines are1 mk
• Q15If A is a 2 × 3 matrix such that AB and AB^′ both are defined, then order of the matrix B is1 mk
• Q15The solution set of the inequation 3 x+5 y<7 is:1 mk
• Q15The function f(x)=x|x| is1 mk
• Q16If y=(cos x- sin x)/(cos x+ sin x), then (d y)/(d x) is:1 mk
• Q16Which of the following points satisfies both the inequations 2 x+y ≤ 10 and x+2 y ≥ 8?1 mk
• Q16If f(x)=a(x- cos x) is strictly decreasing in R, then ' a ' belongs to1 mk
• Q16The angle between the lines 2 x=3 y=-z and 6 x=-y=-4 z is1 mk
• Q16∫ e^-x((x+1)/(x^2)) dx is equal to:1 mk
• Q16For two events A and B, if P(A)=0 · 4, P(B)=0 · 8 and P(B / A)=0 · 6, then P(A ∪ B) is:1 mk
• Q16The point (x, y, 0) on the x y -plane divides the line segment joining the points (1,2,3) and (3,2,1) in the ratio:1 mk
• Q16If |(A^-1)/2|=1/(k|A|), where A is a 3 × 3 matrix, then the value of k is:1 mk
• Q17If the direction cosines of a line are (1/a, 1/a, 1/a), then:1 mk
• Q17The number of feasible solutions of the linear programming problem given as Maximize z=15 x+30 y subject to…1 mk
• Q17The value of ∫_0^π / 2 log tan x dx is:1 mk
• Q17The objective function Z=a x+ by of an LPP has maximum value 42 at (4,6) and minimum value 19 at (3,2). Which of the…1 mk
• Q17The events E and F are independent. If P(E)=0 · 3 and P(E ∪ F)=0 · 5, then P(E / F)-P(F / E) equals:1 mk
• Q17If for any two events A and B, P(A)=4/5 and P(A ∩ B)=7/10, then P(B / A) is equal to1 mk
• Q18The integrating factor for solving the differential equation x dy/dx-y=2 x^2 is:1 mk
• Q18Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is1 mk
• Q18The corner points of the feasible region of a linear programming problem are (0,4),(8,0) and (20/3, 4/3). If Z=30 x+24…1 mk
• Q18If A and B are two independent events such that P(A)=1/3 and P(B)=1/4, then P(B^′ / A) is1 mk
• Q18The number of corner points of the feasible region determined by the constraints x-y ≥ 0,2 y ≤ x+2, x ≥ 0, y ≥ 0 is:1 mk
• Q18For what value of λ, the projection of vector i+λ j on vector i- j is √2?1 mk
• Q18The feasible region of a linear programming problem is shown in the figure below: Which of the following are the…1 mk
• Q18Equation of a line passing through point ( 1,2,3 ) and equally inclined to the coordinate axis, is1 mk
• Q18The probability that A speaks the truth is 4/5 and that of B speaking the truth is 3/4. The probability that they…1 mk
• Q19Assertion: Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one…1 mk
• Q19Assertion: Maximum value of ( cos^-1 x)^2 is π^2. Reason (R): Range of the principal value branch of cos^-1 x is…1 mk
• Q19Assertion: Range of [ sin^-1 x+2 cos^-1 x] is [0, π]. Reason ( R ): Principal value branch of sin^-1 x has range…1 mk
• Q19Assertion: The range of the function f(x)=2 sin^-1 x+(3 π)/2, where x ∈[-1,1], is [(π)/2, (5 π)/2]. Reason ( R ): The…1 mk
• Q19Assertion: All trigonometric functions have their inverses over their respective domains. Reason (R): The inverse of…1 mk
• Q20Assertion: If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin^2 α+ sin^2 β+ sin^2…1 mk
• Q20Assertion ( A ): The lines r= a_1+λ b_1 and r= a_2+μ b_2 are perpendicular, when b_1 · b_2=0. Reason (R): The angle θ…1 mk
• Q20Assertion: ∫_2^8 (√(10-x))/(√x+√(10-x)) d x=3 Reason (R): ∫_a^b f(x) d x=∫_a^b f(a+b-x) d x1 mk
• Q20Assertion: A line through the points (4, 7, 8) and (2, 3, 4) is parallel to a line through the points (-1,-2,1) and…1 mk
• Q20Assertion: Equation of a line passing through the points ( 1,2,3 ) and (3,-1,3) is (x-3)/2=(y+1)/3=(z-3)/0 Reason (R):…1 mk
• Q20Assertion: The number of onto functions from a set P containing 5 elements to a set Q containing 2 elements is 30.…1 mk
• Q21If the product of two positive numbers is 9, find the numbers so that the sum of their squares is minimum.2 mk
• Q21Evaluate sin^-1( sin (3 π)/4)+ cos^-1( cos π)+ tan^-1(1) OR Draw the graph of cos^-1 x, where x ∈[-1,0]. Also, write…2 mk
• Q21Write the domain and range (principle value branch) of the following functions: f(x)= tan^-1 x2 mk
• Q21Draw the graph of the principal branch of the function f(x)= cos^-1 x.2 mk
• Q21If xy=e^x-y, then show that dy/dx=(y(x-1))/(x(y+1)).2 mk
• Q21A function f: A → B defined as f(x)=2 x is both one-one and onto. If A={1,2,3,4}, then find the set B. OR Evaluate:…2 mk
• Q21Consider the statement "There exists at least one value of b ∈ R for which f(x)=b/x, b ≠ 0 is strictly increasing in…2 mk
• Q21Find the domain of y= sin^-1(x^2-4). OR Evaluate: cos^-1[ cos (-(7 π)/3)]2 mk
• Q21Find the value of k for which the function f given as f(x)={ (1- cos x)/(2 x^2), if x ≠ 0; k, if x=0 is continuous at…2 mk
• Q21Find the interval in which the function f(x)=2 x^3-3 x is strictly increasing.2 mk
• Q21If points A, B and C have position vectors 2 i, j and 2 k respectively, then show that ABC is an isosceles triangle.2 mk
• Q21If r=3 i-2 j+6 k, find the value of ( r × j) ·( r × k)-12.2 mk
• Q22If (x^2+y^2)^2=x y, then find (d y)/(d x).2 mk
• Q22If f(x)= {x^2, if x ≥ 1; x, if x<1}, then show that f is not differentiable at x=1. OR Find the value(s) of ' λ ', if…2 mk
• Q22If y=(x+√(x^2-1))^2, then show that (x^2-1)((d y)/(d x))^2=4 y^2.2 mk
• Q22Find the value of tan^-1[2 cos (2 sin^-1 1/2)]+ tan^-1 1.2 mk
• Q22Find all the vectors of magnitude 3 √3 which are collinear to vector i+ j+ k.2 mk
• Q22A particle moves along the curve 3 y=a x^3+1 such that at a point with x -coordinate 1, y -coordinate is changing twice…2 mk
• Q22If the angle between the lines (x-5)/(α)=(y+2)/-5=(z+24/5)/(β) and x/1=y/0=z/1 is (π)/4, find the relation between α…2 mk
• Q23Find the maximum and minimum values of the function given by f(x)=5+ sin 2 x.2 mk
• Q23Find the sub-intervals in which f(x)= log (2+x)-x/(2+x), x>-2 is increasing or decreasing.2 mk
• Q23Find the direction cosines of the line whose Cartesian equations are 5 x-3=15 y+7=3-10 z.2 mk
• Q23If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm / sec, how fast is the…2 mk
• Q23If the equation of a line is x=ay+b, z=cy+d, then find the direction ratios of the line and a point on the line.3 mk
• Q23If f(x)=a( tan x- cot x), where a>0, then find whether f(x) is increasing or decreasing function in its domain.2 mk
• Q23Sketch the region bounded by the lines 2 x+y=8, y=2, y=4 and the y -axis. Hence, obtain its area using integration.2 mk
• Q23If a, b, c are three non-zero unequal vectors such that a · b= a · c, then find the angle between a and b- c.2 mk
• Q23Position vectors of the points A, B and C as shown in the figure below are a, b and c respectively. If AC=5/4 AB,…2 mk
• Q24If the projection of the vector i+ j+ k on the vector p i+ j-2 k is 1/3, then find the value(s) of p.2 mk
• Q24Find the points on the curve 6 y=x^3+2 at which ordinate is changing 8 times as fast as abscissa.2 mk
• Q24Find the point on the curve y^2=8 x for which the abscissa and ordinate change at the same rate.2 mk
• Q24If the vectors a and b are such that | a|=3,| b|=2/3 and a × b is a unit vector, then find the angle between a and b.…2 mk
• Q24Evaluate: 3 sin^-1(1/(√2))+2 cos^-1((√3)/2)+ cos^-1(0) OR Draw the graph of f(x)= sin^-1 x, x ∈[-1/(√2), 1/(√2)]. Also,…2 mk
• Q24Find the coordinates of points on line x/1=(y-1)/2=(z+1)/2 which are at a distance of √11 units from origin.2 mk
• Q25Find the vector and the cartesian equations of a line that passes through the point A(1,2,-1) and parallel to the line…2 mk
• Q25Show that the function f(x)=(16 sin x)/(4+ cos x)-x, is strictly decreasing in ((π)/2, π).2 mk
• Q25Find the angle between the following two lines: r=2 i-5 j+ k+λ(3 i+2 j+6 k); r=7 i-6 k+μ( i+2 j+2 k)2 mk
• Q25If y=x^(1/x), then find dy/dx at x=1. OR If x=a sin 2 t, y=a( cos 2 t+ log tan t), then find (d y)/(d x).2 mk
• Q25Find the value of p, so that lines (x-1)/-2=(y-4)/(3 p)=(z-3)/4 and (x-2)/(4 p)=(y-5)/2=(1-z)/7 are perpendicular to…2 mk
• Q25Find the vector equation of the line passing through the point (2,1,3) and perpendicular to both the lines…2 mk
• Q25If y=√(a x+b), prove that y((d^2 y)/(d x^2))+(dy/dx)^2=0. OR If f(x)= {ax+b; 0<x ≤ 1; 2 x^2-x; 1<x<2} is a…2 mk
• Q25For two non-zero vectors a and b, if | a- b|=| a+ b|, then find the angle between a and b.2 mk
• Q25If x=√(a^tan^-1 t), y=√(a^cot^-1 t), then show that x dy/dx+y=0.2 mk
• Q25If a=4 i- j+ k and b=2 i-2 j+ k, then find a unit vector along the vector a × b.2 mk
• Q25If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is…3 mk
• Q26Evaluate ∫ 1/((e^x+e^-x)(e^x-e^-x)) d x log √23 mk
• Q26If A= [1, 2, 3; 3, -2, 1; 4, 2, 1], then show that A^3-23 A-40 I=O.3 mk
• Q26Evaluate: ∫_0^((π)/2)[ log ( sin x)- log (2 cos x)] d x3 mk
• Q26Find: ∫ 2/((1-x)(1+x^2)) d x3 mk
• Q26Find: ∫ (e^x)/(√(e^2 x-4 e^x-5)) d x3 mk
• Q26Evaluate ∫_0^π / 4 log (1+ tan x) d x. OR Find ∫ (d x)/(√(sin^3 x cos (x-α))).3 mk
• Q26Find: ∫ (x^2+x+1)/((x+1)^2(x+2)) d x3 mk
• Q26Find the general solution of the differential equation: d/(d x)(x y^2)=2 y(1+x^2) OR Solve the following differential…3 mk
• Q26Using determinants, find the area of PQR with vertices P(3,1), Q(9,3) and R(5,7). Also, find the equation of line PQ…3 mk
• Q26Show that the determinant |x, sin θ, cos θ; - sin θ, -x, 1; cos θ, 1, x| is independent of θ.3 mk
• Q26Solve the following Linear Programming problem graphically: Maximize: Z=3 x+3.5 y subject to constraints: x+2 y ≥ 240,…3 mk
• Q26Find the general solution of the differential equation: (d x)/(d y)=(e^x / y(x/y-1))/(1+e^x / y). OR Find the…3 mk
• Q26Find: ∫ (x^2)/(x^2+6 x+12) d x3 mk
• Q27Find: ∫ 1/(√x(√x+1)(√x+2)) d x3 mk
• Q27Differentiate sec^-1(1/(√(1-x^2))) w.r.t. sin^-1(2 x √(1-x^2)). OR If y= tan x+ sec x, then prove that (d^2 y)/(d…3 mk
• Q27Using integration, find the area of the region bounded by y=m x( m>0), x=1, x=2 and the x -axis.3 mk
• Q27Two fair dice are thrown simultaneously. If X denotes the number of sixes, find the mean of X.3 mk
• Q27Evaluate: ∫_π / 4^π / 2 e^2 x((1- sin 2 x)/(1- cos 2 x)) d x OR Evaluate: ∫_-2^2 (x^2)/(1+5^x) d x3 mk
• Q27Evaluate: ∫_1 / 3^1 ((x-x^3)^1 / 3)/(x^4) d x OR Evaluate: ∫_1^3{|(x-1)|+|(x-2)|} dx3 mk
• Q27Find ∫ (x+2)/(√(x^2-4 x-5)) d x. OR Evaluate ∫_-a^a f(x) d x, where f(x)=(9^x)/(1+9^x).3 mk
• Q27Evaluate: ∫_1^3 (√(4-x))/(√x+√(4-x)) d x3 mk
• Q27Solve the following linear programming problem graphically: Maximize P=100 x+5 y subject to the constraints x+y ≤ 300;…3 mk
• Q27Find ∫ e^cot^-1 x((1-x+x^2)/(1+x^2)) d x. log √33 mk
• Q28Find: ∫ (x^2)/((x^2+4)(x^2+9)) d x3 mk
• Q28Find the particular solution of the differential equation dy/dx+ sec^2 x · y= tan x · sec^2 x, given that y(0)=0. OR…3 mk
• Q28Evaluate: ∫_-(π)/4^((π)/4) (cos 2 x)/(1+ cos 2 x) d x OR Find: ∫ e^x^2(x^5+2 x^3) d x3 mk
• Q28Evaluate: ∫_1^e 1/(√(4 x^2-(x log x)^2)) d x3 mk
• Q28Solve the following linear programming problem graphically: Maximise z=5 x+3 y subject to the constraints 3 x+5 y ≤ 15;…3 mk
• Q28Evaluate: ∫_0^π / 4 log (1+ tan x) d x3 mk
• Q28Solve the following linear programming problem graphically: Maximize z=3 x+9 y subject to the constraints x+y ≥ 10; x+3…3 mk
• Q28Find the coordinates of the foot of the perpendicular drawn from point (5,7,3) to the line (x-15)/3=(y-29)/8=(z-5)/-5.…3 mk
• Q28Evaluate: ∫_0^2 π 1/(1+e^sin x) d x OR Find: ∫ (x^4)/((x-1)(x^2+1)) d x3 mk
• Q29Find: ∫ (cos x)/(sin 3 x) d x; OR Find: ∫ x^2 log (x^2+1) d x3 mk
• Q29Find the particular solution of the differential equation dy/dx=(x+y)/x, y(1)=0. OR Find the general solution of the…3 mk
• Q29Solve graphically the following linear programming problem: Maximise z=6 x+3 y, subject to the constraints 4 x+y ≥ 80;…3 mk
• Q29Find the area of the following region using integration: {(x, y): y^2 ≤ 2 x and y ≥ x-4}3 mk
• Q29Find the general solution of the differential equation: (d y)/(d x)-(2 y)/x= sin 1/x. OR Find the particular solution…3 mk
• Q29Find the area of the minor segment of the circle x^2+y^2=4 cut off by the line x=1, using integration.3 mk
• Q29Find the general solution of the differential equation: (x y-x^2) d y=y^2 d x OR Find the general solution of the…3 mk
• Q29Solve the following linear programming problem graphically: Maximize: Z=x+2 y subject to constraints: x+2 y ≥ 100, 2…3 mk
• Q30Evaluate ∫_-1^1|x^4-x| d x. OR Find ∫ (sin^-1 x)/((1-x^2)^3 / 2) d x.3 mk
• Q30The probability distribution of a random variable X is given below: (i) Find the value of k. (ii) Find P(1 ≤ X<3).…3 mk
• Q30Find: ∫ (e^x)/(√(5-4 e^x-e^2 x)) dx OR Evaluate: ∫_0^π / 2 √(sin x) cos^5 x d x3 mk
• Q30Find the coordinates of the foot of the perpendicular drawn from the point P(0,2,3) to the line…3 mk
• Q30Evaluate: ∫_0^((π)/2)| sin x- cos x| d x3 mk
• Q30Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green…3 mk
• Q30Find: ∫_1^4 1/(√(2 x+1)-√(2 x-1)) d x3 mk
• Q30Determine graphically the minimum value of the following objective function: z=500 x+400 y subject to constraints; x+y…3 mk
• Q30Solve the following linear programming problem graphically: Minimise: z=-3 x+4 y subject to the constraints x+2 y ≤ 8;…3 mk
• Q30Solve the following linear programming problem graphically: Maximize z=600 x+400 y subject to the constraints: x+2 y ≤…3 mk
• Q31From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with…3 mk
• Q31Evaluate: ∫_0^((π)/2) e^x sin x d x OR Find: ∫ 1/(cos (x-a) cos (x-b)) d x3 mk
• Q31Solve the following linear programming problem graphically: Minimize: Z=5 x+10 y subject to constraints: x+2 y ≤ 120,…3 mk
• Q31Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of…3 mk
• Q31Find ∫ e^x((1- sin x)/(1- cos x)) d x3 mk
• Q31A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice,…3 mk
• Q31Solve the following linear programming problem graphically: Minimize z=x+2 y subject to the constraints 2 x+y ≥ 3; x+2…3 mk
• Q31Evaluate: ∫_-π / 2^π / 2 (sin^100 x)/(sin^100 x+ cos^100 x) d x3 mk
• Q31Find the distance between the lines: r=( i+2 j-4 k)+λ(2 i+3 j+6 k); r=(3 i+3 j-5 k)+μ(4 i+6 j+12 k)3 mk
• Q31Solve the following linear programming problem graphically: Maximise z=-3 x-5 y subject to the constraints -2 x+y ≤ 4;…3 mk
• Q32Find the inverse of the matrix A= [1, -1, 2; 0, 2, -3; 3, -2, 4]. Using the inverse, A^-1, solve the system of linear…5 mk
• Q32If A= [-3, -2, -4; 2, 1, 2; 2, 1, 3], B= [1, 2, 0; -2, -1, -2; 0, -1, 1], then find AB and use it to solve the…5 mk
• Q32A relation R is defined on a set of real numbers R as R={(x, y): x · y is an irrational number }. Check whether R is…5 mk
• Q32Solve the following Linear Programming Problem graphically: Minimise: Z=60 x+80 y subject to constraints: 3 x+4 y ≥ 8;…5 mk
• Q32Using integration, find the area of the region bounded by the circle x^2+y^2=16, line y=x and y -axis, but lying in the…5 mk
• Q32The median of an equilateral triangle is increasing at the rate of 2 √3 cm / s. Find the rate at which its side is…5 mk
• Q32Find the image of the point (2,-1,5) in the line (x-11)/10=(y+2)/-4=(z+8)/-11; OR Vertices B and C of ABC lie on the…5 mk
• Q32Show that a function f: R → R defined as f(x)=(5 x-3)/4 is both one-one and onto.5 mk
• Q33Check whether a function f: R →[-1/2, 1/2] defined as f(x)=x/(1+x^2) is one-one and onto or not.5 mk
• Q33If A= [1, 2, -2; -1, 3, 0; 0, -2, 1] and B^-1= [3, -1, 1; -15, 6, -5; 5, -2, 2], find (AB)^-1. OR Solve the following…5 mk
• Q33Find the area of the region {(x, y): x^2+y^2 ≤ 1 ≤ x+y}, using integration.5 mk
• Q33Using integration, find the area of the region bounded by the parabola y^2=4 a x and its latus rectum.5 mk
• Q33The area of the region bounded by the line y=mx(m>0), the curve x^2+y^2=4 and the x -axis in the first quadrant is…5 mk
• Q33Evaluate: ∫_0^((π)/2) sin 2 x tan^-1( sin x) d x5 mk
• Q33Using Integration, find the area of triangle whose vertices are ( -1,1 ), (0,5) and (3,2).5 mk
• Q33Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4,2,-6), Q(5,-3,1), R(12,4,5) and…5 mk
• Q34Find the area of the region bounded by the lines y=4 x+5, x+y=5 and 4 y=x+5, using integration.5 mk
• Q34Prove that a function f:[0, ∞) →[-5, ∞) defined as f(x)=4 x^2+4 x-5 is both one-one and onto.5 mk
• Q34If A= [1, 0, 2; 0, 2, 1; 2, 0, 3], then show that A^3-6 A^2+7 A+2 I=O. OR If A= [3, 2; 5, -7], then find A^-1 and use…5 mk
• Q34Solve the following Linear Programming Problem graphically: Maximize: P=70 x+40 y subject to: 3 x+2 y ≤ 9; 3 x+y ≤ 9; x…5 mk
• Q34If N denotes the set of all natural numbers and R is the relation on N × N defined by (a, b) R(c, d), if…5 mk
• Q34A function f:[-4,4] →[0,4] is given by f(x)=√(16-x^2). Show that f is an onto function but not a one-one function.…5 mk
• Q34Using integration, find the area of region bounded by line y=√3 x, the curve y=√(4-x^2) and y -axis in first quadrant.5 mk
• Q34Find the vector and the Cartesian equations of a line passing through the point ( 1,2,-4 ) and parallel to the line…5 mk
• Q35A function f:[-4,4] →[0,4] is given by f(x)=√(16-x^2). Show that f is an onto function but not a one-one function.…5 mk
• Q35Solve the following Linear Programming Problem graphically:5 mk
• Q35In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the…5 mk
• Q35Find the area of the smaller region bounded by the curves (x^2)/25+(y^2)/16=1 and x/5+y/4=1, using integration.5 mk
• Q35Find the value of b so that the lines (x-1)/2=(y-b)/3=(z-3)/4 and (x-4)/5=(y-1)/2=z are intersecting lines. Also, find…5 mk
• Q35Show that the following lines do not intersect each other: (x-1)/3=(y+1)/2=(z-1)/5; (x+2)/4=(y-1)/3=(z+1)/-2 OR Find…5 mk
• Q35Find the area of the region bounded by the curves x^2=y, y=x+2 and x-axis, using integration.5 mk
• Q36Recent studies suggest that roughly 12 % of the world population is left handed. Depending upon the parents, the…4 mk
• Q36A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as…4 mk
• Q36There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as…4 mk
• Q36In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a…4 mk
• Q36An organization conducted bike race under two different categories - Boys and Girls. There were 28 participants in all.…4 mk
• Q36Let f(x) be a real valued function. Then its - Left Hand Derivative (L.H.D.): Lf^′(a)= lim_h → 0 (f(a-h)-f(a))/-h -…4 mk
• Q37A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the…4 mk
• Q37Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves…4 mk
• Q37Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag…4 mk
• Q37An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces.…4 mk
• Q38Sooraj's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing…4 mk
• Q38The equation of the path traced by a roller-coaster is given by the polynomial f(x)=a(x+9)(x+1)(x-3). If the…4 mk
• Q38An equation involving derivatives of the dependent variable with respect to the independent variables is called a…4 mk
• Q38A volleyball player serves the ball which takes a parabolic path given by the equation h(t)=-7/2 t^2+13/2 t+1, where…4 mk
• Q38The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every…4 mk