Science (English Medium) Class 12 - CBSE Question Bank Solutions

If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?

Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.

Let A = {3, 5}. Then number of reflexive relations on A is ______.

The sum of the order and the degree of the differential equation `d/dx[(dy/dx)^3]` is ______.

If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.

The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.

Read the following passage:

Based on the above information, answer the following questions:

1. How many relations are possible from B to G? (1)
2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
3. Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
   OR
   A function f : B `rightarrow` G be defined by f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}. Check if f is bijective. Justify your answer. (2)

The order and the degree of the differential equation `(1 + 3 dy/dx)^2 = 4 (d^3y)/(dx^3)` respectively are ______.

Find 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`.

Find the value of `sin^-1(cos((33π)/5))`.

Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].

Find the coordinates of the image of the point (1, 6, 3) with respect to the line `vecr = (hatj + 2hatk) + λ(hati + 2hatj + 3hatk)`; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.

If `(a + bx)e^(y/x)` = x then prove that `x(d^2y)/(dx^2) = (a/(a + bx))^2`.

The degree of the differential equation `[1 + (dy/dx)^2]^3 = ((d^2y)/(dx^2))^2` is ______.

Discuss the continuity of the following function:

f(x) = sin x + cos x

Discuss the continuity of the following function:

f(x) = sin x – cos x

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  `hati + 2hatj - hatk` and `-hati + hatj + hatk`  respectively, internally the ratio 2:1.

If `sin (sin^(−1)(1/5)+cos^(−1) x)=1`, then find the value of x.

Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `

Prove that `2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)=pi/4`