Class 12 Mathematics - Chapter Integrals NCERT Solutions | \begin{align} \int\left(1-x\right).\sqrt
\begin{align} \int\left(1-x\right).\sqrt {x}.dx\end{align}
\begin{align} \int\left(1-x\right).\sqrt {x}.dx\end{align}
\begin{align} =\int \left(\sqrt {x} - x^\frac32\right).dx\end{align}
\begin{align} =\frac{x^{\displaystyle\frac32}}{\displaystyle\frac32} - \frac{x^{\displaystyle\frac52}}{\displaystyle\frac52}+C \end{align}
\begin{align} =\frac23\;x^\frac32\; - \frac25\;x^\frac52\;+C \end{align}
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Popular Questions of Class 12 Mathematics
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Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
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Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}
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Represent graphically a displacement of 40 km, 30° east of north.
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If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
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Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
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Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
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\begin{vmatrix} \mathbf{2} & \mathbf{4} \\ \mathbf{-5} & \mathbf{-1} \end{vmatrix} - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A = {1, 2, 3,13, 14} defined as
R = {(x, y): 3x − y = 0}
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
R = {(x, y): x − y is as integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
(e) R = {(x, y): x is father of y} - Q:- Find the rate of change of the area of a circle with respect to its radius r when
(a) r = 3 cm
(b) r = 4 cm - Q:-
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).
Recently Viewed Questions of Class 12 Mathematics
- Q:-
Check the injectivity and surjectivity of the following functions:
(i) f : N → N given by f(x) = x2
(ii) f : Z → Z given by f(x) = x2
(iii) f : R → R given by f(x) = x2
(iv) f : N → N given by f(x) = x3
(v) f : Z → Z given by f(x) = x3
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Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
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y = Ax : xy' = y (x ≠ 0)
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(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive. - Q:- Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
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(A) (2, 4) ∈ R
(B) (3, 8) ∈R
(C) (6, 8) ∈R
(D) (8, 7) ∈ R - Q:-
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
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