CBSE 12 Maths Question Paper-2016 Set-1

Find the maximum value of 1 + sin θ / 1 + cos θ

If A is a square matrix such that A² = I, then find the simplified value of (A - I)³ + (A + I)³ - 7A.

Matrix A = [ [0, 2b, -2], [3, 1, 3], [3a, 3, -1] ] is given to be symmetric. Find values of a and b.

Find the position vector of a point which divides the join of points with position vectors →a – 2 →b and 2 →a + →b externally in the ratio 2 : 1.

The two vectors ^j + ^k and 3^i – ^j + 4^k represent the two sides AB and AC, respectively, of a triangle ABC. Find the length of the median through A.

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is 2^i – 3^j + 6^k.

Prove that tan–1(1/5) + tan–1(1/7) + tan–1(1/3) + tan–1(1/8) = π/4

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹15,000 per month, find their monthly incomes using matrix method.

If x = a sin 2t(1 + cos 2t) and y = b cos 2t(1 – cos 2t), find the values of dy/dx at t = π/4 and t = π/3.

Find the values of p and q, for which f(x) is continuous at x = π/2, where f(x) is given by the piecewise function:

The vectors x = 3 cos t – cos3t, y = 3 sin t – sin3t satisfy the equation 4(y cos3t – x sin3t) = 3 sin 4t. Show that this equation holds for all values of t.

Find the integral ∫(3 sin θ - 2) cos θ / (5 - cos²θ - 4 sin θ) dθ.

Evaluate the integral ∫x / (a³ - x³) dx.

Evaluate the integral ∫|x³ - x| dx from -1 to 2.

Find the particular solution of the differential equation (1 - y²)(1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.

Find the general solution of the differential equation (1 + y²) + (x - e^(tan⁻¹y)) dy/dx = 0.

Show that the vectors →a, →b, and →c are coplanar if →a + →b, →b + →c, and →c + →a are coplanar.

Find the vector and Cartesian equations of the line through the point (1, 2, -4) and perpendicular to the two lines r = (8^i - 19^j + 10^k) + λ(3^i - 16^j + 7^k) and r = (15^i + 29^j + 5^k) + µ(3^i + 8^j - 5^k).

Find the probability that the job of Manager in a private company will be assigned to C, given the chances of selection of A, B, and C are in the ratio 1:2:4, and the probabilities of A, B, and C introducing changes are 0.8, 0.5, and 0.3 respectively.

Let f(x) = 9x² + 6x – 5. Show that f(x) is invertible and find its inverse. Then find f⁻¹(43) and f⁻¹(163).

Using elementary transformations, find the inverse of the matrix A = [[8, 4, 3], [2, 1, 1], [1, 2, 2]] and use it to solve the system of equations 8x + 4y + 3z = 19, 2x + y + z = 5, x + 2y + 2z = 7.

Find the area of the region {(x, y) : x² + y² ≤ 2ax, y² ≥ ax, x, y ≥ 0} using integration.

Find the vector equation of a line passing through the point (1, 2, -4) and perpendicular to the two lines r = (8^i - 19^j + 10^k) + λ(3^i - 16^j + 7^k) and r = (15^i + 29^j + 5^k) + µ(3^i + 8^j - 5^k).