Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions

Prove that `int_0^"a" "f(x)" "dx" = int_0^"a" "f"("a"-"x")"dx"` ,and hence evaluate `int_0^1 "x"^2(1 - "x")^"n""dx"`.

Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f⁻¹. 

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ₹ 35 per package of nuts and ₹ 14 per package of bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates each machine for almost 12 hours a day? convert it into an LPP and solve graphically.

If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.

If f(x) = x + 1, find `d/dx (fof) (x)`

if  `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`

Choose the correct option from the given alternatives :

Let f(x) = x³ – 6x² + 9x + 18, then f(x) is strictly decreasing in ______.

Choose the correct alternative :

Area of the region bounded by the curve x² = y, the X-axis and the lines x = 1 and x = 3 is _______.

Find the values of x such that f(x) = 2x³ – 15x² + 36x + 1 is increasing function

Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f^–1 

If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f^–1

Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto

Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f

Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f^–1.

Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.

Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.

Let f: R → R be defined by f(x) = 3x – 4. Then f^–1(x) is given by ______.

Let f: R → R be defined by f(x) = x² + 1. Then, pre-images of 17 and – 3, respectively, are ______.