XII MATHS 2marks, 3 marks ,5 marks expected questions for CBSE examination



EXPECTED QUESTIONS 

XII MATHEMATICS 





2 Marks 

: Q.no. 21 --  Inverse trigonometric functions (OR) Relations and functions

                 Q.no. 22 – Rate of change / Increasing and decreasing functions/Maxima and minima

                 Q.no. 23 –Unit vector / Scalar product / Vector product / Projection / Direction ratios and  

                 direction cosines (OR) Vector and cartesianequations of a line / angle between two lines

                 Q.no. 24 – Derivative / second order derivative / parametric forms / logarithmic  

                 differentiation / continuity / Integration

                 Q.no. 25 –Anyone question from vector algebra / Three dimensional geometry






3 Marks:



 Q.no. 26 –Integration by parts / Partial fractions / Using formulas to find integration

                 Q.no. 27 –probability distribution / Total theorem on probability / Conditional probability /  

                 Independent events 

                 Q.no. 28 – Definite integrals (OR) definite integrals

                 Q.no. 29 – Variable separable method / Homogeneous DE / Linear DE

                 Q.no. 30 – Linear programming problem

                 Q.no. 31 – Integrals / Derivative / second order derivative / parametric forms / logarithmic  

                 differentiation / continuity 




5 Marks: 




Q.no. 32 – Applicartions of integrals

                 Q.no. 33 – Relations (OR) Function

                 Q.no. 34 – Solving linear equations of three variables / Matrices

                 Q.no. 35 – Foot of the perpendicular / Image of a point / Shortest distance 

CSA (4 Marks) Q.no. 36 – Maxima or minima / increasing and decreasing functions

  Q.no. 37 – Three dimensional geometry / Vector algebra / Determinants / any other chap.

                 Q.no. 38 –  Bayes’ theorem / probability distribution / Conditional probability






5 Marks questions :

Q. no. 32:

TYPE 1: 

  1. Using integration, find the area of  the region in the first quadrant enclosed by the line x +y =2, the parabola y2 = x and the x – axis

  2. Find the area enclosed by the curve y = - x2 and the line x+y+2=0

  3. Find the area of the region bounded by y2 = 4ax and the line y = 2a and y – axis

  4. Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12 

  5. Find the area of the region lying in the first quadrant and bounded by  y = 4x2 , x = 0, 

y = 1 and y = 4

  1. Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y – axis in the first quadrant

  2. Find the area of the region bounded by x2 = 4y and the line x = 4y-2

TYPE II: 

  1. Find the area of the smaller region bounded by the ellipse x2a2 + y2b2 = 1 and the line xa+yb = 1

  2. Find the area of the smaller region bounded by the ellipse x29 + y24 = 1 and the line x3+y2 = 1

  3. Find the area of the smaller region bounded by the ellipse x29 + y24 = 1 and the line 2x+3y = 6

TYPE III: 

  1. Find the area of the region {(x ,y) : 0 ≤ y ≤ x2, 0 ≤ y ≤ x+1, 0 ≤ x ≤ 2}

  2. Find the area bounded by curves {(x ,y) : y≥x2 and y = x

  3. Using integration, find the area of region {(x,y) : x2≤y≤x}

  4. Find the area of following region using integration  {(x,y): y ≤ x +2, y ≥ x2}

  5. Find the area of following region using integration {(x,y):  x2+y2≤ 1≤ x+y}

  6. Find the area of following region using integration {(x,y):  x2+y2≤ 4 , y ≤x3}

TYPE IV:

  1. Find the area of the region bounded by the line y = 3x+2, the x-axis and the ordinates x = -1 and x = 1.

  1. Find the area of the region bounded by the line y -1 = x, x - axis and the ordinates x = -2, x = 3

TYPE V: 

  1. Find the area enclosed by the curves x = 3 cos and y = sin

  2. Using integration, find the area of triangle ABC whose vertices have   coordinates A(2,5), B(4,7) and C(6,2)

  1. Find the area bounded by the curve y = x | x | , x-axis and the ordinates x = – 1 and x = 1.

TYPE VI:

  1. Using the method of integration find the area bounded by the curve x + y = 1

  2. Sketch the graph of y = x+3 and evaluate -60x+3 dx

  3. Sketch the graph of y = x+1 and evaluate -31x+1 dx

TYPE VII: 

  1. Find the area enclosed by the curve  y = sin x between x = 0 and x = 2π

  2. Find the area enclosed by the curve  y = cos x between x = 0 and x = 2π

TYPE VIII:

  1. Draw a rough sketch of graph of function y = 21-x2 , x [0,1] and  evaluate the area between curve and axes

  1. Find the area under the curve y = 2x included between the lines x = 0 and x = 1

Q. no. 33

        



 























        18. Let N be the set of natural numbers and R be the relation on N x N defined by 

        (a, b) R (c, d) if only if ad=bc for all a, b, c, d∈ N. Show that R is an equivalence relation.    

        19. Let A= {1,2, 3, …,9} and R be the relation on A×A defined as (a, b) R (c, d) if and only 

         if a+ d=b+ c. Prove that R is an equivalence relation also obtain the equivalence class [(2,5)] 

         20. Let R be the relation on N×N defined by (a, b) R (c, d) if and only if ad (b+ c) =bc (a+ d),      

         Prove that R is an equivalence relation

         21. Show that the relation R defined on the set N×N defined as (a, b) R (c, d) if and only if  𝑎2+𝑑2=𝑏2+𝑐2 is an equivalence relation

(OR)

      









     



 

    10. Show that the function f: R R given by f(x)= xx2+1 is neither one – one nor onto.

Q. no. 34

  1. Given A = 1 -1 0 2 3 4 0 1 2 and B = 2 2 -4 -4 2 -4 2 -1 5 verify that BA = 6I. How we can use the result to find the values of x,y,z from the given equations.  x - y = 3 , 2x +3y +4z = 17, y +2z = 7.

  2. If A = 2 3 5 3 2 -4 1 1 -2 , find A-1. Use A-1 to solve the following system of equations: x+3y+5z = 16,  3x+2y-4z = -4; x+y-2z = -3

  1. Use the product 1 -1 2 0 2 -3 3 -2 4 -2 0 1 9 2 -3 6 1 -2 to solve the system of equations

x – y + 2z = 1 , 2y – 3z = 1 , 3x – 2 y + 4 z = 2

  1. Given that A= 1 -1 0 2 3 4 0 1 2 and B = 2 2 -4 -4 2 -4 2 -1 5 find AB. Use this product to solve the following system of equations: x – y = 3 ; 2 x + 3 y + 4 z = 17 ; y + 2 z = 7.

  2. If A-1 = 3 -1 1 -15 6 -5 5 -2 2   B = 1 2 -2 -1 3 0 0 -2 1 find (AB)-1

  3. If A = 3 1 7 5 find x and y such that A2+ x I = y A . Hence find A-1 

  4. If A = 12 -2 3 0 -1 4 -2 2 1 , find (AT)-1

  5. If A = 2 3 1 -4   B = 1 -2 -1 3 , then verify that (AB )-1 = B-1 A-1

  6. For the matrix A = 3 2 1 1 , find the numbers a and b such that A2 + aA + bI = O.

  7. If A =1 1 1 1 2 -3 2 -1 3 Show that A3– 6A2 + 5A + 11 I = O. Hence, find A–1

  8. Find the equation of the line joining A ( 1 , 3) and B ( 0, 0) using determinants and find k 

if D ( k , 0) is a point such that area of triangle ABD is 3 sq. units.

  1. If A = 2 -1 1 -1 2 -1 1 -1  2 Verify that A3 – 6A2 + 9A – 4I = O and hence find A–1

  2. Find the equation of line joining P (11, 7) and Q (5, 5) using determinants. Also, find the value of k, if R (–1, k) is the point such that area of DPQR is 9 sq m

  3. If A = 1 1 1 1 0 2 3 1 1 , find A–1. Hence, solve the system of equations: x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

  1. If A = 2 3 10 4 -6 5 6 9 -20 , find A–1. Using A–1 solve the system of equations: 2x+ 3y+10z ;  4x- 6y+5z  ; 6x+9y- 20z

  2. Determine the product -4 4 4 -7 1 3 5 -3 -1 1 -1 1 1 -2 -2 2 1 3 and use it to solve the system of equation : x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1

Q. no. 35

 
























         

  1. Show that the linesx+13 = y+35 = z+57 and x-21 = y-43 = z-65 intersect. Find their point of intersection 

  2. Find the shortest distance between the following lines :x-31 = y-5-2 = z-71 and x+17 = y+1-6 = z+11

  3. Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines: x-83 = y+19-16 = z-107 and x-153 = y-298 = z-5-5

  4. Find the shortest distance between the following lines  r= 6i + 2j+ 2k +(i - 2j+2k)  r = -4i -k +(3i -2j-2k) 

  1. Find the co-ordinates of the point where the line through (-1, 1, -8) and (5, -2, 10) crosses the ZX - plane.

  2. A line passing through the point A with position vector a= 4i + 2j+ 2k is parallel to the vector a= 2i + 3j+ 6k , Find the foot of perpendicular and length of the perpendicular drawn on this line from a point P with position vector  r= i + 2j+ 3k









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