CBSE Class 12 Mathematics 2023

What 2023 actually asked, chapter by chapter

Marks we counted in the 2023 papers themselves. CBSE allocates marks per unit, not per chapter — this is what was observed, not an official CBSE weightage.

All 303 questions from 2023

• Q1∫ (1+ tan x)/(1- tan x) dx is equal to:1 mk
• Q1If A=[ ] 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 x) f(x)=2 x+3/x and f(1)=1, then f(x) is1 mk
• Q1If A=[ ], 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 [ ][ ]=[ ], 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 mkasked 2×
• 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 mkasked 3×
• Q1If x[ ]+y[ ]=[4/9], then:1 mkasked 3×
• 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 [ ]=P+Q, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to1 mkasked 3×
• 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)=[ ], then the value of A + adj A is equal to:1 mkasked 3×
• Q2sin [(π)/3+ sin^-1(1/2)] is equal to1 mkasked 3×
• 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=[ ] 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 mkasked 2×
• Q3Let A be a 3 × 3 matrix such that adj A =64. Then A is equal to:1 mkasked 3×
• 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 mkasked 2×
• 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 =3/5, then P( / ) is equal to 11 mk
• Q3A and B are skew-symmetric matrices of same order. AB is symmetric, if:1 mkasked 3×
• Q3If [ ] is non-singular matrix and a ∈ A, then the set A is1 mkasked 3×
• Q4If a matrix A=[ ], then the matrix AA^′ (where A^′ is the transpose of A ) is:1 mkasked 3×
• 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=[ ]?1 mkasked 2×
• Q4If A=[a i j] is a square matrix of order 2 such that a ij=., then A^2 is1 mk
• Q4If A=[ ] and 2 A+B is a null matrix, then B is equal to:1 mkasked 2×
• 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=[ ], B=[ ] and A=B^2, then x equals1 mkasked 2×
• Q4If A = kA, where A is a square matrix of order 2, then sum of all possible values of k is1 mkasked 3×
• Q5The value of is1 mk
• Q5The value of the determinant 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 is:1 mk
• Q5If in ABC, BA=2 a and BC=3 b, then AC is1 mk
• Q5If /(d x)[f(x)]=a x+b and f(0)=0, then f(x) is equal to1 mkasked 2×
• Q5X and Y are independent events such that P(X Y)=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 mkasked 2×
• Q5If =0, then the value of α is1 mkasked 2×
• 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 mkasked 3×
• Q6Degree of the differential equation sin x+ cos ((d y)/(d x))=y^2 is1 mkasked 3×
• Q6∫ 2^x+2 dx is equal to:1 mkasked 3×
• Q6The product [ ][ ] is equal to:1 mkasked 2×
• Q6The function f(x)= x is1 mkasked 2×
• Q6∫ 0^(π)/6 sec^2(x-(π)/6) dx is equal to:1 mkasked 3×
• 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 mkasked 2×
• Q7The integrating factor of the differential equation (1-y^2) (d x)/(dy)+y x=ay,(-1<y<1) is1 mkasked 3×
• 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
• 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 mkasked 3×
• Q7The function f(x)=[x], where [x] denotes the greatest integer less than or equal to x, is continuous at1 mkasked 3×
• 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 mkasked 2×
• Q8The solution of the differential equation (d x)/(x)+(d y)/(y)=0 is:1 mkasked 2×
• Q8If x=A cos 4 t+B sin 4 t, then (d^2 x)/(d t^2) is equal to1 mkasked 2×
• Q8Unit vector along PQ, where coordinates of P and Q respectively are (2,1,-1) and (4,4,-7), is1 mkasked 3×
• Q8∫ e^5 log x dx is equal to:1 mkasked 3×
• Q8If A and B are two events such that P(A / B)=2 × P(B / A) and P +P =2/3, then P 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 mkasked 3×
• 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 mkasked 2×
• Q9If ∫ 0^a 3 x^2 dx=8, then the value of ' a ' is:1 mkasked 2×
• 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 mkasked 2×
• 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
• 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 mkasked 3×
• 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 mkasked 2×
• Q11Direction cosines of the line x-1/2=1-y/3=(2 z-1)/12 are:1 mk
• Q11If [ ] 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 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q12If P( / )=0 · 3, P =0 · 4 and P =0 · 8, then P( / ) is equal to:1 mkasked 2×
• 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 mkasked 2×
• Q13If θ is the angle between two vectors a and b, then a · b ≥ 0 only when:1 mkasked 2×
• Q13The value of k for which f(x)=. 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 mkasked 2×
• 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 ^2 is equal to1 mkasked 3×
• Q14If A=[ ] and (3 I+4 A)(3 I-4 A)=x^2 I, then the value (s) x is/are:1 mkasked 3×
• Q14The primitive of 2/(1+ cos 2 x) is1 mk
• Q14The magnitude of the vector 6 i-2 j+3 k is1 mkasked 2×
• Q14Distance of the point (p, q, r) from y -axis is:1 mkasked 3×
• Q14If P(A B)=1/8 and P( A)=3/4, then P( / ) 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)=. is differentiable at x=0 is:1 mkasked 3×
• Q15The general solution of the differential equation x d y-(1+x^2) d x=d x is:1 mkasked 3×
• Q15If a line makes angles of 90^°, 135^° and 45^° with the x, y and z axes respectively, then its direction cosines are1 mkasked 3×
• 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 mkasked 2×
• Q15The function f(x)=x x is1 mkasked 3×
• Q16If y=( cos x- sin x)/( cos x+ sin x), then (d y)/(d x) is:1 mkasked 3×
• Q16Which of the following points satisfies both the inequations 2 x+y ≤ 10 and x+2 y ≥ 8?1 mkasked 3×
• Q16If f(x)=a(x- cos x) is strictly decreasing in R, then ' a ' belongs to1 mkasked 3×
• Q16The angle between the lines 2 x=3 y=-z and 6 x=-y=-4 z is1 mkasked 3×
• Q16For two events A and B, if P =0 · 4, P =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/, 1/, 1/ ), then:1 mkasked 2×
• Q17The number of feasible solutions of the linear programming problem given as Maximize z=15 x+30 y subject to…1 mkasked 3×
• 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 mkasked 3×
• 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 =4/5 and P(A B)=7/10, then P(B / A) is equal to1 mkasked 3×
• 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 mkasked 2×
• Q18If A and B are two independent events such that P =1/3 and P =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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 2×
• Q19Assertion: All trigonometric functions have their inverses over their respective domains. Reason (R): The inverse of…1 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• Q21Consider the statement "There exists at least one value of b ∈ R for which f(x)= /(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 mkasked 3×
• Q21Find the value of k for which the function f given as f(x)= is continuous at x=0.. OR If x=a cos t and y=b sin t, then…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 mkasked 3×
• Q22If (x^2+y^2)^2=x y, then find (d y)/(d x).2 mkasked 2×
• Q22If f(x)=., then show that f is not differentiable at x=1. OR Find the value(s) of ' λ ', if the function f(x)=.2 mkasked 2×
• Q22If y=(x+√x^2-1)^2, then show that (x^2-1)((d y)/(d x))^2=4 y^2.2 mkasked 2×
• 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 mkasked 2×
• 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…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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 2×
• 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. OR…2 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 2×
• 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: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 mkasked 3×
• 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 each other.2 mk
• Q25Find the vector equation of the line passing through the point (2,1,3) and perpendicular to both the lines…2 mkasked 3×
• Q25If y=√a x+b, prove that y((d^2 y)/(d x^2))+((dy)/(dx))^2=0. OR If f(x)=. is a differentiable function in (0, 2), then…2 mkasked 3×
• 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 mkasked 3×
• Q26If A=[ ], 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 mkasked 2×
• 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 mkasked 3×
• 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 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 mkasked 2×
• 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 x^2)=( cos…3 mkasked 3×
• 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^21+5^x d x3 mkasked 3×
• 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^x1+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 constraints3 mk
• Q27Find ∫ e^ cot^-1 x((1-x+x^2)/(1+x^2)) d x. log √33 mkasked 2×
• 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.…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 constraints3 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 constraints3 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. OR If…3 mk
• Q28Evaluate: ∫ 0^2 π 11+e^ sin x d x OR Find: ∫ (x^4)/((x-1)(x^2+1)) d x3 mkasked 2×
• Q29Find: Find: ∫ x^2 log (x^2+1) d x3 mkasked 3×
• Q29Find the particular solution of the differential equation (dy)/(dx)=(x+y)/(x), y(1)=0. OR Find the general solution of…3 mkasked 3×
• Q29Solve graphically the following linear programming problem: Maximise z=6 x+3 y, subject to the constraints3 mk
• Q29Find the area of the following region using integration: (x, y): y^2 ≤ 2 x and y ≥ x-43 mk
• Q29Find the general solution of the differential equation: (d y)/(d x)-(2 y)/(x)= sin 1/(x). OR Find the particular…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 mkasked 3×
• Q29Solve the following linear programming problem graphically: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 mkasked 3×
• Q30Find: ∫ e^x√5-4 e^x-e^2 x dx OR Evaluate: ∫ 0^π / 2 √ sin x cos^5 x d x3 mkasked 2×
• Q30Find the coordinates of the foot of the perpendicular drawn from the point P(0,2,3) to the line…3 mkasked 2×
• 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 mkasked 2×
• Q30Find: ∫ 1^4 1√2 x+1-√2 x-1 d x3 mk
• Q30Determine graphically the minimum value of the following objective function:3 mk
• Q30Solve the following linear programming problem graphically: Minimise: z=-3 x+4 y subject to the constraints3 mk
• Q30Solve the following linear programming problem graphically: Maximize z=600 x+400 y subject to the constraints: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 mkasked 2×
• Q31Evaluate: ∫ 0^(π)/2 e^x sin x d x OR Find: ∫ 1/( cos (x-a) cos (x-b)) d x3 mkasked 3×
• 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 mkasked 3×
• Q31Solve the following linear programming problem graphically: Minimize z=x+2 y subject to the constraints3 mk
• Q31Evaluate: ∫ -π / 2^π / 2 ( sin^100 x)/( sin^100 x+ cos^100 x) d x3 mk
• Q31Find the distance between the lines:3 mkasked 3×
• Q31Solve the following linear programming problem graphically: Maximise z=-3 x-5 y subject to the constraints3 mk
• Q32Find the inverse of the matrix A=[ ]. Using the inverse, A^-1, solve the system of linear equations x-y+2 z=1; 2 y-3…5 mkasked 3×
• Q32If A=[ ], B=[ ], then find AB and use it to solve the following system of equations: OR If f(α)=[ ], prove that f(α) ·…5 mkasked 3×
• 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 mkasked 3×
• Q32Solve the following Linear Programming Problem graphically: Minimise: Z=60 x+80 y subject to constraints: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 mkasked 3×
• Q32Find the image of the point (2,-1,5) in the line gathered (x-11)/10=(y+2)/-4=(z+8)/-11 OR gathered Vertices B and C of…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=[ ] and B^-1=[ ], find (AB)^-1. OR Solve the following system of equations by matrix method:5 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• Q33Evaluate: ∫ 0^(π)/2 sin 2 x tan^-1( sin x) d x5 mkasked 2×
• 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 mkasked 3×
• 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=[ ], then show that A^3-6 A^2+7 A+2 I=O. OR If A=[ ], then find A^-1 and use it to solve the following system of…5 mkasked 3×
• Q34Solve the following Linear Programming Problem graphically: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 mkasked 3×
• 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 mkasked 2×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 mkasked 3×
• 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 the…5 mkasked 2×
• 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 mkasked 3×
• Q36A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as…4 mkasked 3×
• Q36There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as…4 mkasked 3×
• 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 mkasked 3×
• Q36An organization conducted bike race under two different categories - Boys and Girls. There were 28 participants in all.…4 mkasked 3×
• Q36Let f(x) be a real valued function. Then its - Left Hand Derivative (L.H.D.): Lf^′ = lim h → 0 (f(a-h)-f )/-h - Right…4 mkasked 3×
• Q37A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the…4 mkasked 3×
• Q37Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves…4 mkasked 3×
• 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 mkasked 3×
• Q37An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces.…4 mkasked 3×
• Q38Sooraj's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing…4 mkasked 3×
• 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 mkasked 3×
• Q38An equation involving derivatives of the dependent variable with respect to the independent variables is called a…4 mkasked 3×
• 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 mkasked 3×
• Q38The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every…4 mkasked 3×