If A, B and C are angles of a triangle,
Question: If $A, B$ and $C$ are angles of a triangle, then the determinant $\left|\begin{array}{ccc}-1 \cos C \cos B \\ \cos C -1 \cos A \\ \cos B \cos A -1\end{array}\right|$ is equal to (a) 0 (b) $-1$ (c) 1 (d) none of these Solution: Given: $A, B$ and $C$ are angles of a triangle Therefore, $A+B+C=\pi \ldots$ (1) $\left|\begin{array}{ccc}-1 \cos C \cos B \\ \cos C -1 \cos A \\ \cos B \cos A -1\end{array}\right|$ Expanding through $R_{1}$ $=\left\{-1\left(1-\cos ^{2} A\right)-\cos C(-\cos C-\c...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:49 a2+ 8ab 16b2 Solution: $49-a^{2}+8 a b-16 b^{2}$ $=49-\left(\mathrm{a}^{2}-8 \mathrm{ab}+16 \mathrm{~b}^{2}\right)$ $=49-\left[\mathrm{a}^{2}-2 \times \mathrm{a} \times 4 \mathrm{~b}+(4 \mathrm{~b})^{2}\right]$ $=7^{2}-(\mathrm{a}-4 \mathrm{~b})^{2}$ $=[7-(\mathrm{a}-4 \mathrm{~b})][7+(\mathrm{a}-4 \mathrm{~b})]$ $=(7-\mathrm{a}+4 \mathrm{~b})(7+\mathrm{a}-4 \mathrm{~b})$ $=-(\mathrm{a}-4 \mathrm{~b}-7)(\mathrm{a}-4 \mathrm{~b}+7)...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2+ 9y2 6xy 25a2 Solution: $x^{2}+9 y^{2}-6 x y-25 a^{2}$ $=\left(\mathrm{x}^{2}-6 \mathrm{xy}+9 \mathrm{y}^{2}\right)-25 \mathrm{a}^{2}$ $=\left[\mathrm{x}^{2}-2 \times \mathrm{x} \times 3 \mathrm{y}+(3 \mathrm{y})^{2}\right]-25 \mathrm{a}^{2}$ $=(\mathrm{x}-3 \mathrm{y})^{2}-(5 \mathrm{a})^{2}$ $=[(\mathrm{x}-3 \mathrm{y})-5 \mathrm{a}][(\mathrm{x}-3 \mathrm{y})+5 \mathrm{a}]$ $=(\mathrm{x}-3 \mathrm{y}-5 \mathrm{a})(\mathrm{x}-3 \...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:25 p2q2 2pq Solution: $25-p^{2}-q^{2}-2 p q$ $=25-\left(\mathrm{p}^{2}+2 \mathrm{pq}+\mathrm{q}^{2}\right)$ $=5^{2}-\left(\mathrm{p}^{2}+2 \times \mathrm{p} \times \mathrm{q}+\mathrm{q}^{2}\right)$ $=5^{2}-(\mathrm{p}+\mathrm{q})^{2}$ $=[5-(\mathrm{p}+\mathrm{q})][5+(\mathrm{p}+\mathrm{q})]$ $=(5-\mathrm{p}-\mathrm{q})(5+\mathrm{p}+\mathrm{q})$ $=-(\mathrm{p}+\mathrm{q}-5)(\mathrm{p}+\mathrm{q}+5)$...
Read More →Prove that sum of any two sides of a triangle
Question: Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side. Solution: Given $\ln \triangle A B C, A D$ is a median. Construction Produce $A D$ to a point $E$ such that $A D=D E$ and join $C E$. To prove $A C+A B2 A D$ Proof In $\triangle A B D$ and $\triangle E C D$, $A D=D E$ [by construction] $B D=C D$ [given $A D$ is the median] and $\angle A D B=\angle C D E$ [vertically opposite angle] $\therefore$ $\triangle \mathrm{ABD} \equiv \...
Read More →The number of distinct real root
Question: The number of distinct real root of $\left|\begin{array}{lll}\sin x \cos x \cos x \\ \cos x \sin x \cos x \\ \cos x \cos x \sin x\end{array}\right|=0$ in the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, is (a) 0 (b) 2 (c) 1 (d) 3 Solution: Given: $\left|\begin{array}{lll}\sin x \cos x \cos x \\ \cos x \sin x \cos x \\ \cos x \cos x \sin x\end{array}\right|=0$ $\left|\begin{array}{lll}\sin x \cos x \cos x \\ \cos x \sin x \cos x \\ \cos x \cos x \sin x\end{array}\right|$ Apply...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:(x+ 2)2 6(x+ 2) + 9 Solution: $(x+2)^{2}-6(x+2)+9$ $=(\mathrm{x}+2)^{2}-2 \times(\mathrm{x}+2) \times 3+3^{2}$ $=[(\mathrm{x}+2)-3]^{2}$ $=(\mathrm{x}+2-3)^{2}$ $=(\mathrm{x}-1)^{2}$ $=(\mathrm{x}-1)(\mathrm{x}-1)$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:4x4+y4 Solution: $4 x^{4}+y^{4}$ $=4 \mathrm{x}^{4}+4 \mathrm{x}^{2} \mathrm{y}^{2}+\mathrm{y}^{4}-4 \mathrm{x}^{2} \mathrm{y}^{2}$ $=\left[\left(2 \mathrm{x}^{2}\right)^{2}+2 \times 2 \mathrm{x}^{2} \times \mathrm{y}+\left(\mathrm{y}^{2}\right)^{2}\right]-(2 \mathrm{xy})^{2}$ $=\left(2 \mathrm{x}^{2}+\mathrm{y}^{2}\right)^{2}-(2 \mathrm{xy})^{2}$ $=\left[\left(2 \mathrm{x}^{2}+\mathrm{y}^{2}\right)-2 \mathrm{xy}\right]\left[\left(2 ...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:4x4+ 1 Solution: $4 x^{4}+1$ $=4 \mathrm{x}^{4}+4 \mathrm{x}^{2}+1-4 \mathrm{x}^{2}$ $=\left[\left(2 \mathrm{x}^{2}\right)^{2}+2 \times 2 \mathrm{x}^{2} \times 1+1\right]-4 \mathrm{x}^{2}$ $=\left(2 \mathrm{x}^{2}+1\right)^{2}-(2 \mathrm{x})^{2}$ $=\left[\left(2 \mathrm{x}^{2}+1\right)-2 \mathrm{x}\right]\left[\left(2 \mathrm{x}^{2}+1\right)+2 \mathrm{x}\right]$ $=\left(2 \mathrm{x}^{2}-2 \mathrm{x}+1\right)\left(2 \mathrm{x}^{2}+2 \...
Read More →Find the area of a rhombus each side of which measures 20 cm and one of whose diagonals is 24 cm.
Question: Find the area of a rhombus each side of which measures 20 cm and one of whose diagonals is 24 cm. Solution: Given:Sides are 20 cm each and one diagonal is of 24 cm.The diagonal divides the rhombus into two congruent triangles, as shown in the figure below. We will now use Hero's formula to find the area of triangle ABC.First, we will find the semiperimeter. $s=\frac{1}{2}(a+b+c)=\frac{1}{2}(20+20+24)=\frac{64}{2}=32 \mathrm{~m}$ Area of $\Delta \mathrm{ABC}=\sqrt{s(s-a)(s-b)(s-c)}$ $=\...
Read More →If ABC is an isosceles triangle in
Question: If ABC is an isosceles triangle in which AC = BC, AD and BE are respectively two altitudes to sides BC and AC, then prove that AE = BD. Solution: Given $\triangle A B C$ is an isosceles triangle in which $A C=B C$. Also, $A D$ and $B E$ are two altitudes to sides $B C$ and $A C$, respectively. To prove $A E=B D$. Proof In $\triangle A B C$, $A C=B C$ [given] $\angle A B C=\angle C A B$ [angles opposite to equal sides are equal] i.e., $\angle A B D=\angle E A B$ $\ldots($ i) In $\triang...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a4+ 3a2+4 Solution: $a^{4}+3 a^{2}+4$ $=\mathrm{a}^{4}+4 \mathrm{a}^{2}-\mathrm{a}^{2}+4$ $=\left(\mathrm{a}^{4}+4 \mathrm{a}^{2}+4\right)-\mathrm{a}^{2}$ $=\left[\left(\mathrm{a}^{2}\right)^{2}+2 \times \mathrm{a}^{2} \times 2+2^{2}\right]-\mathrm{a}^{2}$ $=\left(\mathrm{a}^{2}+2\right)^{2}-\mathrm{a}^{2}$ $=\left[\left(\mathrm{a}^{2}+2\right)-\mathrm{a}\right]\left[\left(\mathrm{a}^{2}+2\right)+\mathrm{a}\right]$ $=\left(\mathrm{a}...
Read More →Solve the following equations
Question: The value of $\left|\begin{array}{ccc}1 1 1 \\ { }^{n} C_{1} { }^{n+2} C_{1} { }^{n+4} C_{1} \\ { }^{n} C_{2} { }^{n+2} C_{2} { }^{n+4} C_{2}\end{array}\right|$ is (a) 2 (b) 4 (c) 8 (d) $n^{2}$ Solution: $\left|\begin{array}{ccc}1 1 1 \\ { }^{n} C_{1} { }^{n+2} C_{1} { }^{n+4} C_{1} \\ { }^{n} C_{2} { }^{n+2} C_{2} { }^{n+4} C_{2}\end{array}\right|$ $=\left|\begin{array}{ccc}1 1 1 \\ n n+2 n+4 \\ \frac{n(n-1)}{2} \frac{(n+2)(n+1)}{2} \frac{(n+4)(n+3)}{2}\end{array}\right|$ $=\left|\beg...
Read More →A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3 and its area is 3375 m2.
Question: A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3 and its area is 3375 m2. Find the cost of fencing the lawn at Rs 20 per metre. Solution: Let the length and breadth of the lawn be $5 x \mathrm{~m}$ and $3 x \mathrm{~m}$, respectively. Now, Area of the lawn $=5 x \times 3 x=5 x^{2}$ $\Rightarrow 15 x^{2}=3375$ $\Rightarrow x=\sqrt{\frac{3375}{15}}$ $\Rightarrow x=\sqrt{225}=15$ Length $=5 x=5 \times 15=75 \mathrm{~m}$ Breadth $=3 x=3 \times 15=45 \mathrm{~m}$ $\th...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:96 4xx2 Solution: $96-4 x-x^{2}$ $=100-4-4 \mathrm{x}-\mathrm{x}^{2}$ $=100-\left(\mathrm{x}^{2}+4 \mathrm{x}+4\right)$ $=100-\left(\mathrm{x}^{2}+2 \times \mathrm{x} \times 2+2^{2}\right)$ $=10^{2}-(\mathrm{x}+2)^{2}$ $=[10-(\mathrm{x}+2)][10+(\mathrm{x}+2)]$ $=(10-\mathrm{x}-2)(10+\mathrm{x}+2)$ $=(8-\mathrm{x})(12+\mathrm{x})$ $=(x+12)(-x+8)$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2+ 4ab+ 3b2 Solution: $a^{2}+4 a b+3 b^{2}$ $=\mathrm{a}^{2}+4 \mathrm{ab}+4 \mathrm{~b}^{2}-\mathrm{b}^{2}$ $=\left[\mathrm{a}^{2}+2 \times \mathrm{a} \times 2 \mathrm{~b}+(2 \mathrm{~b})^{2}\right]-\mathrm{b}^{2}$ $=(\mathrm{a}+2 \mathrm{~b})^{2}-\mathrm{b}^{2}$ $=[(\mathrm{a}+2 \mathrm{~b})-\mathrm{b}][(\mathrm{a}+2 \mathrm{~b})+\mathrm{b}]$ $=(\mathrm{a}+2 \mathrm{~b}-\mathrm{b})(\mathrm{a}+2 \mathrm{~b}+\mathrm{b})$ $=(\mathrm{...
Read More →Solve this
Question: If $\left|\begin{array}{lll}a p x \\ b q y \\ c r z\end{array}\right|=16$, then the value of $\left|\begin{array}{ccc}p+x a+x a+p \\ q+y b+y b+q \\ r+z c+z c+r\end{array}\right|$ is (a) 4 (b) 8 (c) 16 (d) 32 Solution: $\left|\begin{array}{ccc}p+x a+x a+p \\ q+y b+y b+q \\ r+z c+z c+r\end{array}\right|=\left|\begin{array}{ccc}p a a \\ q b b \\ r c c\end{array}\right|+\left|\begin{array}{ccc}p a p \\ q b q \\ r c r\end{array}\right|+\left|\begin{array}{ccc}p x a \\ q y b \\ r z c\end{arr...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2+ 2x+ 1 9y2 Solution: $x^{2}+2 x+1-9 y^{2}$ $=\left(x^{2}+2 x+1\right)-9 y^{2}$ $=\left(x^{2}+2 \times x \times 1+1\right)-9 y^{2}$ $=(x+1)^{2}-(3 y)^{2}$ $=[(x+1)-3 y][(x+1)+3 y]$ $=(x+1-3 y)(x+1+3 y)$ $=(x+3 y+1)(x-3 y+1)$...
Read More →Find the area of a triangle whose sides are 42 cm, 34 cm and 20 cm.
Question: Find the area of a triangle whose sides are 42 cm, 34 cm and 20 cm. Solution: To find the area of the triangle, we will first find the semiperimeter of the triangle.Thus, we have: $s=\frac{1}{2}(a+b+c)=\frac{1}{2}(42+34+20)=\frac{1}{2} \times 96=48 \mathrm{~cm}$ Now, Area of the triangle $=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{48(48-42)(48-34)(48-20)}$ $=\sqrt{48 \times 6 \times 14 \times 28}$ $=\sqrt{112896}$ $=336 \mathrm{~cm}^{2}$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2 2ab+b2c2 Solution: $a^{2}-2 a b+b^{2}-c^{2}$ $=\left(a^{2}-2 a b+b^{2}\right)-c^{2}$ $=\left(a^{2}-2 \times a \times b+b^{2}\right)-c^{2}$ $=(a-b)^{2}-c^{2}$ $=[(a-b)-c][(a-b)+c]$ $=(a-b-c)(a-b+c)$...
Read More →There are two values of a which makes the determinant
Question: There are two values of $a$ which makes the determinant $\Delta=\left|\begin{array}{ccc}1 -2 5 \\ 2 a -1 \\ 0 4 2 a\end{array}\right|$ equal to 86 . The sum of these two values is (a) 4 (b) 5 (c) $-4$ (d) 9 Solution: $\Delta=\left|\begin{array}{ccc}1 -2 5 \\ 2 a -1 \\ 0 4 2 a\end{array}\right|=86$ $\Rightarrow 1\left(2 a^{2}+4\right)-2(-4 a-20)=86$ $\Rightarrow 2 a^{2}+4+8 a+40=86$ $\Rightarrow 2 a^{2}+8 a-42=0$ $\Rightarrow a^{2}+4 a-21=0$ $\Rightarrow a^{2}+7 a-3 a-21=0$ $\Rightarrow...
Read More →A B C and D B C are two triangles
Question: $A B C$ and $D B C$ are two triangles on the same base $B C$ such that $A$ and $D$ lie on the opposite sides of $B C, A B=$ $A C$ and $D B=D C$. Show that $A D$ is the perpendicular bisector of $B C$. Solution: Given Two $\triangle A B C$ and $\triangle D B C$ are formed on the same base $B C$ such that $A$ and $D$ lie on the opposite sides of $B C$ such that $A B=A C$ and $D B=D C$. Also $A D$ intersects $B C$ at $O$. To show $A D$ is the perpendicular bisector of $B C$ i.e., $A D \pe...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:16 a6+ 4a3b3 4b6 Solution: $16-a^{6}+4 a^{3} b^{3}-4 b^{6}$ $=16-\left(a^{6}-4 a^{3} b^{3}+4 b^{6}\right)$ $=4^{2}-\left[\left(a^{3}\right)^{2}-2 \times a^{3} \times 2 b^{3}+\left(2 b^{3}\right)^{2}\right]$ $=4^{2}-\left(a^{3}-2 b^{3}\right)^{2}$ $=\left[4-\left(a^{3}-2 b^{3}\right)\right]\left[4+\left(a^{3}-2 b^{3}\right)\right]$ $=\left(4-a^{3}+2 b^{3}\right)\left(4+a^{3}-2 b^{3}\right)$ $=\left(a^{3}-2 b^{3}+4\right)\left(-a^{3}+2...
Read More →Find the area of a rhombus whose diagonals are 48 cm and 20 cm long.
Question: Find the area of a rhombus whose diagonals are 48 cm and 20 cm long. Solution: Area of the rhombus $=\frac{1}{2}$ (Product of diagonals) $=\frac{1}{2}(48 \times 20)$ $=480 \mathrm{~cm}^{2}$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:9a4 24a2b2+ 16b4 256 Solution: $9 a^{4}-24 a^{2} b^{2}+16 b^{4}-256$ $=\left(9 a^{4}-24 a^{2} b^{2}+16 b^{4}\right)-256$ $=\left[\left(3 a^{2}\right)^{2}-2 \times 3 a^{2} \times 4 b^{2}+\left(4 b^{2}\right)^{2}\right]-16^{2}$ $=\left(3 a^{2}-4 b^{2}\right)^{2}-16^{2}$ $=\left[\left(3 a^{2}-4 b^{2}\right)-16\right]\left[\left(3 a^{2}-4 b^{2}\right)+16\right]$ $=\left(3 a^{2}-4 b^{2}-16\right)\left(3 a^{2}-4 b^{2}+16\right)$...
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