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CBSE Class 12 Sample / Model Paper 2026 : Mathematics

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CBSE Class 12 Mathematics Advanced Practice Test (Board + JEE Level) Time: 3 Hours Maximum Marks: 80 Section A Very Short Answer (1 20 = 20 marks) Choose the correct answer. 1. If AAA is a 3 33 \times 33 3 matrix and |A| = 5, then |3A| equals a) 15 b) 45 c) 135 d) 125 2. If lim (x 0) [sin(ax) / sin(bx)] = 2 then the value of a/b is a) 1 b) 2 c) 3 d) 1/2 3. If f(x) = |x 4|, the number of non-differentiable points is a) 1 b) 2 c) 3 d) 4 4. If (x + ax + bx + 1) dx = 2 then a) a + b = 2 b) a + b = 3 c) a + b = 4 d) a + b = 5 5. If a = 2i + j k b = i 2j + 3k Then a b equals a) 1 b) 3 c) 5 d) 5 6. Eigenvalues of the matrix |2 1| |1 2| are a) 1, 3 b) 2, 2 c) 3, 3 d) 1, 1 7. The differential equation dy/dx = y tan x has solution a) y = C sin x b) y = C cos x c) y = C sec x d) y = C tan x 8. If vectors satisfy a + 2b + 3c = 0 then scalar triple product [a b c ] equals a) 0 b) 1 c) 1 d) 3 9. If y = e^(x ), then d y/dx at x = 0 equals a) 0 b) 1 c) 2 d) e 10. Evaluate ^ sin x dx a) 2 b) 4/3 c) 3/2 d) 1 Section B Short Answer (2 6 = 12 marks) 11. Show that ln(1 + x) dx = 2 ln2 1 12. Find the equation of the tangent to the curve y = x 3x + 1 at the point where slope is zero. 13. If a = i + 2j + 3k Find a unit vector perpendicular to both a and (i + j + k). 14. Evaluate the limit lim (x 0) (e^(2x) 1 2x) / x 15. Solve the differential equation dy/dx + y = e^x 16. If A = |1 2| |3 4| Find A . Section C Long Answer (4 8 = 32 marks) 17. If y = x^x Find dy/dx. 18. Using integration, find the area enclosed between the curves y = x y = 2x 19. Using vector method prove that diagonals of a parallelogram bisect each other. 20. Evaluate x / (x + 1) dx 21. Find maximum and minimum values of f(x) = x 6x + 9x + 1 22. Find equation of plane passing through point (1, 2, 3) and perpendicular to line (x 1)/2 = (y 2)/( 1) = (z 3)/3 Section D Case Study / JEE Level Problems (6 4 = 24 marks) 23. Matrices If A =|1 1 1| |1 2 3| |1 3 6| 1. Find determinant of A 2. Find inverse of A using adjoint method 3. Solve the system of equations x+y+z=3 x + 2y + 3z = 6 x + 3y + 6z = 10 24. Vector Geometry Find the shortest distance between the lines r = (i + 2j + k) + (2i j + 2k) r = (3i j + 4k) + (i + 2j k) 25. Integration (Advanced) Evaluate ^( /2) ln(sin x) dx 26. Differential Equation Solve dy/dx = (x + y) / x

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