if the
Question: $(\log x)^{\log x}, x1$ Solution: Let $y=(\log x)^{\log x}$ Taking logarithm on both the sides, we obtain $\log y=\log x \cdot \log (\log x)$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \frac{d y}{d x}=\frac{d}{d x}[\log x \cdot \log (\log x)]$ $\Rightarrow \frac{1}{y} \frac{d y}{d x}=\log (\log x) \cdot \frac{d}{d x}(\log x)+\log x \cdot \frac{d}{d x}[\log (\log x)]$ $\Rightarrow \frac{d y}{d x}=y\left[\log (\log x) \cdot \frac{1}{x}+\log x \cdot \frac{1}{\log ...
Read More →Identify the following as rational or irrational numbers.
Question: Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers: (i) $(\sqrt{4})$ (ii) $3 \sqrt{18}$ (iii) $\sqrt{1.44}$ (iv) $\sqrt{\frac{9}{27}}$ (v) $-\sqrt{64}$ (vi) $\sqrt{100}$ Solution: (i) Given number is $x=\sqrt{4}$ $x=2$, which is a rational number (ii) Given number is $3 \sqrt{18}$ $\Rightarrow 3 \sqrt{18}=3 \sqrt{3 \times 3 \times 2}$ $\Rightarrow 3 \sqrt{18}=3 \times 3 \sqrt{2}$ $\Rightarrow 3 \sqrt{18}=3 \times 3 \sqrt{2}$ $\...
Read More →Find the equation of the set of points which are equidistant from the points (1, 2, 3)
Question: Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, 1). Solution: Let P (x,y,z) be the point that is equidistant from points A(1, 2, 3) and B(3, 2, 1). Accordingly, PA = PB $\Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2}$ $\Rightarrow(x-1)^{2}+(y-2)^{2}+(z-3)^{2}=(x-3)^{2}+(y-2)^{2}+(z+1)^{2}$ $\Rightarrow x^{2}-2 x+1+y^{2}-4 y+4+z^{2}-6 z+9=x^{2}-6 x+9+y^{2}-4 y+4+z^{2}+2 z+1$ $\Rightarrow-2 x-4 y-6 z+14=-6 x-4 y+2 z+14$ $\Rightarrow-2 x-6 z+...
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Question: $\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0x\frac{-}{2}$ Solution: Let $y=\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$ ...(1) Then, $\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}$ $=\frac{(\sqrt{1+\sin x}+\sqrt{1-\sin x})^{2}}{(\sqrt{1+\sin x}-\sqrt{1-\sin x})(\sqrt{1+\sin x}+\sqrt{1-\sin x})}$ $=\frac{(1+\sin x)+(1-\sin x)+2 \sqrt{(1-\sin x)(1+\sin x)}}{(...
Read More →Verify the following:
Question: Verify the following: (i) (0, 7, 10), (1, 6, 6) and (4, 9, 6) are the vertices of an isosceles triangle. (ii) (0, 7, 10), (1, 6, 6) and (4, 9, 6) are the vertices of a right angled triangle. (iii) (1, 2, 1), (1, 2, 5), (4, 7, 8) and (2, 3, 4) are the vertices of a parallelogram. Solution: (i) Let points (0, 7, 10), (1, 6, 6), and (4, 9, 6) be denoted by A, B, and C respectively. $\mathrm{AB}=\sqrt{(1-0)^{2}+(6-7)^{2}+(-6+10)^{2}}$ $=\sqrt{(1)^{2}+(-1)^{2}+(4)^{2}}$ $=\sqrt{1+1+16}$ $=\...
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Question: $\frac{\cos ^{-1} \frac{x}{2}}{\sqrt{2 x+7}},-2x2$ Solution: Let $y=\frac{\cos ^{-1} \frac{x}{2}}{\sqrt{2 x+7}}$ By quotient rule, we obtain $\frac{d y}{d x}=\frac{\sqrt{2 x+7} \frac{d}{d x}\left(\cos ^{-1} \frac{x}{2}\right)-\left(\cos ^{-1} \frac{x}{2}\right) \frac{d}{d x}(\sqrt{2 x+7})}{(\sqrt{2 x+7})^{2}}$ $=\frac{\sqrt{2 x+7}\left[\frac{-1}{\sqrt{1-\left(\frac{x}{2}\right)^{2}}} \cdot \frac{d}{d x}\left(\frac{x}{2}\right)\right]-\left(\cos ^{-1} \frac{x}{2}\right) \frac{1}{2 \sqrt...
Read More →Examine, whether the following numbers are rational or irrational:
Question: Examine, whether the following numbers are rational or irrational: (i) $\sqrt{7}$ (ii) $\sqrt{4}$ (iii) $2+\sqrt{3}$ (iv) $\sqrt{3}+\sqrt{2}$ (v) $\sqrt{3}+\sqrt{5}$ (vi) $(\sqrt{2}-2)^{2}$ (vii) $(2-\sqrt{2})(2+\sqrt{2})$ (viii) $(\sqrt{2}+\sqrt{3})^{2}$ (ix) $\sqrt{5}-2$ (x) $\sqrt{23}$ (xi) $\sqrt{225}$ (xii) $0.3796$ (xiii) $7.478478$ (xiv) $1.101001000100001$ Solution: (i) Let $x=\sqrt{7}$ Therefore, $x=2.645751311064 \ldots$ It is non-terminating and non-repeating Hence $\sqrt{7}...
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Question: $\sin ^{-1}(x \sqrt{x}), 0 \leq x \leq 1$ Solution: Let $y=\sin ^{-1}(x \sqrt{x})$ Using chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x} \sin ^{-1}(x \sqrt{x})$ $=\frac{1}{\sqrt{1-(x \sqrt{x})^{2}}} \times \frac{d}{d x}(x \sqrt{x})$ $=\frac{1}{\sqrt{1-x^{3}}} \cdot \frac{d}{d x}\left(x^{\frac{3}{2}}\right)$ $=\frac{1}{\sqrt{1-x^{3}}} \times \frac{3}{2} \cdot x^{\frac{1}{2}}$ $=\frac{3 \sqrt{x}}{2 \sqrt{1-x^{3}}}$ $=\frac{3}{2} \sqrt{\frac{x}{1-x^{3}}}$...
Read More →Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
Question: Show that the points (2, 3, 5), (1, 2, 3) and (7, 0, 1) are collinear. Solution: Let points (2, 3, 5), (1, 2, 3), and (7, 0, 1) be denoted by P, Q, and R respectively. Points P, Q, and R are collinear if they lie on a line. $\mathrm{PQ}=\sqrt{(1+2)^{2}+(2-3)^{2}+(3-5)^{2}}$ $=\sqrt{(3)^{2}+(-1)^{2}+(-2)^{2}}$ $=\sqrt{9+I+4}$ $=\sqrt{14}$ $\mathrm{QR}=\sqrt{(7-1)^{2}+(0-2)^{2}+(-1-3)^{2}}$ $=\sqrt{(6)^{2}+(-2)^{2}+(-4)^{2}}$ $=\sqrt{36+4+16}$ $=\sqrt{56}$ $=2 \sqrt{14}$ $\mathrm{PR}=\sq...
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Question: $(5 x)^{3 \cos 2 x}$ Solution: Let $y=(5 x)^{3 \cos 2 x}$ Taking logarithmon both the sides, we obtain $\log y=3 \cos 2 x \log 5 x$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \frac{d y}{d x}=3\left[\log 5 x \cdot \frac{d}{d x}(\cos 2 x)+\cos 2 x \cdot \frac{d}{d x}(\log 5 x)\right]$ $\Rightarrow \frac{d y}{d x}=3 y\left[\log 5 x(-\sin 2 x) \cdot \frac{d}{d x}(2 x)+\cos 2 x \cdot \frac{1}{5 x} \cdot \frac{d}{d x}(5 x)\right]$ $\Rightarrow \frac{d y}{d x}=3 y\lef...
Read More →Find the distance between the following pairs of points:
Question: Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1) (ii) (3, 7, 2) and (2, 4, 1) (iii) (1, 3, 4) and (1, 3, 4) (iv) (2, 1, 3) and (2, 1, 3) Solution: The distance between points $\mathrm{P}\left(x_{1}, y_{1}, z_{1}\right)$ and $\mathrm{P}\left(x_{2}, y_{2}, z_{2}\right)$ is given by $\mathrm{PQ}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$ (i) Distance between points (2, 3, 5) and (4, 3, 1) $=\sqrt{(4-2)...
Read More →find the
Question: $\sin ^{3} x+\cos ^{6} x$ Solution: Let $y=\sin ^{3} x+\cos ^{6} x$ $\therefore \frac{d y}{d x}=\frac{d}{d x}\left(\sin ^{3} x\right)+\frac{d}{d x}\left(\cos ^{6} x\right)$ $=3 \sin ^{2} x \cdot \frac{d}{d x}(\sin x)+6 \cos ^{5} x \cdot \frac{d}{d x}(\cos x)$ $=3 \sin ^{2} x \cdot \cos x+6 \cos ^{5} x \cdot(-\sin x)$ $=3 \sin x \cos x\left(\sin x-2 \cos ^{4} x\right)$...
Read More →Fill in the blanks:
Question: Fill in the blanks: Solution: (i) The $x$-axis and $y$-axis taken together determine a plane known as $\underline{X Y \text {-plane }}$. (ii) The coordinates of points in the $X Y$-plane are of the form $(x, y, 0)$. (iii) Coordinate planes divide the space intoeight octants....
Read More →Name the octants in which the following points lie:
Question: Name the octants in which the following points lie: (1, 2, 3), (4, 2, 3), (4, 2, 5), (4, 2, 5), (4, 2, 5), (4, 2, 5), (3, 1, 6), (2, 4, 7) Solution: Thex-coordinate,y-coordinate, andz-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octantI. Thex-coordinate,y-coordinate, andz-coordinate of point (4, 2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octantIV. Thex-coordinate,y-coordinate, andz-coordinate of point (4, 2, 5...
Read More →A point is in the XZ-plane. What can you say about its y-coordinate?
Question: A point is in the XZ-plane. What can you say about itsy-coordinate? Solution: If a point is in the XZ plane, then itsy-coordinate is zero....
Read More →A point is on the x-axis. What are its y-coordinates and z-coordinates?
Question: A point is on thex-axis. What are itsy-coordinates andz-coordinates? Solution: If a point is on thex-axis, then itsy-coordinates andz-coordinates are zero....
Read More →An equilateral triangle is inscribed in the parabola y2 = 4 ax,
Question: An equilateral triangle is inscribed in the parabola $y^{2}=4 \mathrm{ax}$, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle. Solution: Let $O A B$ be the equilateral triangle inscribed in parabola $y^{2}=4 a x$. Let $\mathrm{AB}$ intersect the $x$-axis at point $\mathrm{C}$. Let $\mathrm{OC}=k$ From the equation of the given parabola, we have $y^{2}=4 a k \Rightarrow y=\pm 2 \sqrt{a k}$ $\therefore$ The respective coordinates of points A a...
Read More →Explain, how irrational numbers differ from rational numbers?
Question: Explain, how irrational numbers differ from rational numbers? Solution: Every rational number must have either terminating or non-terminating but irrational number must have non- terminating and non-repeating decimal representation. A rational number is a number that can be written as simple fraction (ratio) and denominator is not equal to zero while an irrational is a number that cannot be written as a ratio....
Read More →A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m.
Question: A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m. find the equation of the posts traced by the man. Solution: Let A and B be the positions of the two flag posts and P(x,y) be the position of the man. Accordingly, PA + PB = 10. We know that if a point moves in a plane in such a way that the sum of its distances from two fixed points is constant, then the path is an ellipse and th...
Read More →Define an irrational number.
Question: Define an irrational number. Solution: An irrational number is a real number that cannot be reduced to any ratio between an integerpand a natural numberq. If the decimal representation of an irrational number is non-terminating and non-repeating, then it is called irrational number. For example $\sqrt{3}=1.732 \ldots \ldots$...
Read More →Express each of the following decimals in the form
Question: Express each of the following decimals in the form $\frac{p}{q}$ : (i) $0 . \overline{4}$ (ii) $0 . \overline{37}$ (iii) $0 . \overline{54}$ (iv) $0 . \overline{621}$ (v) $125 . \overline{3}$ (vi) $4 . \overline{7}$ (vii) $0 . \overline{47}$ Solution: (i) Let $x=0 . \overline{4}$ $\Rightarrow x=0.44444 \ldots$ $10 x=4.444 \ldots$ $\Rightarrow 10 x=4+x$ $\Rightarrow 9 x=4$ $\Rightarrow x=\frac{4}{9}$ (ii) Let $x=0 . \overline{37}$ $\Rightarrow x=0.373737 \ldots$ $\Rightarrow 100 x=37.37...
Read More →find
Question: $\left(3 x^{2}-9 x+5\right)^{9}$ Solution: Let $y=\left(3 x^{2}-9 x+5\right)^{9}$ Using chain rule, weobtain $\frac{d y}{d x}=\frac{d}{d x}\left(3 x^{2}-9 x+5\right)^{9}$ $=9\left(3 x^{2}-9 x+5\right)^{8} \cdot \frac{d}{d x}\left(3 x^{2}-9 x+5\right)$ $=9\left(3 x^{2}-9 x+5\right)^{8} \cdot(6 x-9)$ $=9\left(3 x^{2}-9 x+5\right)^{8} \cdot 3(2 x-3)$ $=27\left(3 x^{2}-9 x+5\right)^{8}(2 x-3)$...
Read More →Find the area of the triangle formed by the lines joining the vertex of the parabola
Question: Find the area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12 y$ to the ends of its latus rectum. Solution: The given parabola is $x^{2}=12 y$. On comparing this equation with $x^{2}=4 a y$, we obtain $4 a=12 \Rightarrow a=3$ $\therefore$ The coordinates of foci are $S(0, a)=S(0,3)$ Let AB be the latus rectum of the given parabola. The given parabola can be roughly drawn as At $y=3, x^{2}=12(3) \Rightarrow x^{2}=36 \Rightarrow x=\pm 6$ $\therefore$ The ...
Read More →A rod of length 12 cm moves with its ends always touching the coordinate axes.
Question: A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with thex-axis. Solution: Let AB be the rod making an anglewith OX and P (x,y) be the point on it such that AP = 3 cm. Then, PB = AB AP = (12 3) cm = 9 cm [AB = 12 cm] From $\mathrm{P}$, draw $\mathrm{PQ} \perp \mathrm{OY}$ and $\mathrm{PR} \perp \mathrm{OX}$. In $\triangle \mathrm{PBQ}, \cos \theta=\frac{\math...
Read More →An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
Question: An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end. Solution: Since the height and width of the arc from the centre is 2 m and 8 m respectively, it is clear that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m. The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is taken along thex-axis. Hence, the semi-ellipse can be...
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