The figure formed by joining the
Question: The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only, if (a)ABCD is a rhombus (b)diagonals of ABCD are equal (c)diagonals of ABCD are equal and perpendicular (d)diagonals of ABCD are perpendicular Solution: (c) Given, $A B C D$ is a quadrilateral and $P, Q, R$ and $S$ are the mid-points of sides of $A B, B C$, $C D$ and $D A$, respectively. Then, $P Q R S$ is a square. $\therefore \quad P Q=Q R=R S=P S \quad$ .......(i) and ...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:6x2 13xy+ 2y2 Solution: The given expression is $6 \mathrm{x}^{2}-13 \mathrm{xy}+2 \mathrm{y}^{2}$. (Coefficient of $\mathrm{x}^{2}=6$, coefficient of $\mathrm{x}=-13 \mathrm{y}$ and constant term $=2 y^{2}$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is $-13 \mathrm{y}$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $6 \times\le...
Read More →If A and B are square matrices
Question: If $A$ and $B$ are square matrices of order 3 and that $|A|=-2,|B|=4$, then $|2 A B|=$ Solution: Given:AandBare square matrices of order 3|A| = 2|B| = 4 Now, $|2 A B|=|2 A||B| \quad(\because|A B|=|A||B|$, if they are square matrices of same order $)$ $=|2 A| \times 4 \quad(\because|B|=4)$ $=2^{3}|A| \times 4 \quad(\because$ Order of $A$ is $3 \times 3)$ $=32|A|$ $=32 \times(-2) \quad(\because|A|=-2)$ $=-64$ Hence, $|2 A B|=-64$....
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:6x2 5xy 6y2 Solution: The given expression is $6 \mathrm{x}^{2}-5 \mathrm{xy}-6 \mathrm{y}^{2}$. (Coefficient of $\mathrm{x}^{2}=6$, coefficient of $\mathrm{x}=-5 \mathrm{y}$ and constant term $=-6 \mathrm{y}^{2}$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is $-5 \mathrm{y}$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $6 \tim...
Read More →Let A = [aij] and B = [bij] be a square matrices
Question: Let $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be a square matrices of order 3 such that $b_{i 1}=2 a_{i 1}, b_{i 2}=3 a_{i 2}$ and $b_{i 3}=4 a_{i 3}, i=1,2,3$ If $|A|=5$, then $|B|=$)____________ Solution: Given: $A$ and $B$ are square matrices of order 3 $b_{i 1}=2 a_{i 1}, b_{i 2}=3 a_{i 2}$ and $b_{i 3}=4 a_{i 3}, i=1,2,3$ $|A|=5$ Let $A=\left[\begin{array}{lll}a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33}\end{array}\right]$ Since, $b_{i 1}=2 a_{i...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:12x2 17xy+ 6y2 Solution: The given expression is $12 \mathrm{x}^{2}-17 \mathrm{xy}+6 \mathrm{y}^{2} . \quad$ (Coefficient of $\mathrm{x}^{2}=12$, coefficient of $\mathrm{x}=-17 \mathrm{y}$ and constant term $=6 \mathrm{y}^{2}$ ) We willsplit the coefficient of $x$ into two parts such that their sum is $-17 y$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term i.e., $12 \times 6 ...
Read More →D and E are the mid-points of
Question: D and E are the mid-points of the sides AB and AC of ΔABC and 0 is any point on side BC. 0 is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is (a)a square (b)a rectangle (c)a rhombus (d)a parallelogram Thinking Process Use the mid-point theorem i.e., the line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it. Solution: (d)In ΔABC, D and E are the mid-points of sides AB and AC, respectively. By mid...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:3x2+ 22x+ 35 Solution: The given expression is $3 \mathrm{x}^{2}+22 \mathrm{x}+35$. (Coefficient of $\mathrm{x}^{2}=3$, coefficient of $\mathrm{x}=22$ and constant term $=35$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is 22 and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $3 \times 35=105$. Now, $15+7=22$ and $15 \times 7=105$ Rep...
Read More →The value of the determinant
Question: The value of the determinant $\Delta=\left|\begin{array}{ccc}0 x-y y-z \\ y-x 0 z-x \\ z-y x-z 0\end{array}\right|$ is Solution: Given: $\Delta=\left|\begin{array}{ccc}0 x-y y-z \\ y-x 0 z-x \\ z-y x-z 0\end{array}\right|$ Let $A=\left[\begin{array}{ccc}0 x-y y-z \\ y-x 0 z-x \\ z-y x-z 0\end{array}\right]$ $A^{T}=\left[\begin{array}{ccc}0 x-y y-z \\ y-x 0 z-x \\ z-y x-z 0\end{array}\right]^{T}$ $=\left[\begin{array}{ccc}0 y-x z-y \\ x-y 0 x-z \\ y-z z-x 0\end{array}\right]$ $=(-1)\lef...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:7x 6x2+ 20 Solution: The given expression is $7 \mathrm{x}-6 \mathrm{x}^{2}+20 . \quad$ (Coefficient of $\mathrm{x}^{2}=-6$, coefficient of $\mathrm{x}=7$ and constant term $=20$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is 7 and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i. e., $(-6) \times 20=-120$. Now, $15+(-8)=7$ and $15 \times(...
Read More →The figure obtained by joining the
Question: The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is (a) a rhombus (b) a rectangle (c) a square (d) any parallelogram Solution: (b) Let $A B C D$ be a rhombus in which $P, Q, R$ and $S$ are the mid-points of sides $A B, B C, C D$ and $D A$, respectively. Join $A C, P R$ and $S Q$ In $\triangle A B C, P$ is the mid-point of $A B$ and $Q$ is the mid-point of $B C$. $\Rightarrow \quad P Q \| A C$ and $P Q=\frac{1}{2} A C \quad[$ by using mid-point th...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:11x2 54x+ 63 Solution: The given expression is $11 \mathrm{x}^{2}-54 \mathrm{x}+63 . \quad$ (Coefficient of $\mathrm{x}^{2}=11$, coefficient of $\mathrm{x}=-54$ and constant term $=63$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is $-54$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $11 \times 63=693$. Now, $(-33)+(-21)=-54$ and...
Read More →If APB and CQD are two parallel lines,
Question: If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form (a)a square (b)a rhombus (c)a rectangle (d)any other parallelogram Solution: (c) Given, $A P B$ and $C Q D$ are two parallel lines. Let the bisectors of angles $A P Q$ and $C Q P$ meet at a point $M$ and bisectors of angles $B P Q$ and $P Q D$ meet at a point $N$. Join $P M, M Q, Q N$ and $N P$. Since, $A P B \| C Q D$ Then, $\quad \angle A P Q=\angle P Q D \quad$ [alternate interior angl...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:3 + 23y 8y2 Solution: The given expression is $3+23 y-8 y^{2} . \quad$ (Coefficient of $y^{2}=-8$, coefficient of $y=23$ and constant term $=3)$ We will split the coefficient of y into two parts such that their sum is 23 and their product equals the product of the coefficient of $\mathrm{y}^{2}$ and the constant term, i.e., $(-8) \times 3=-24$. Now, $(-1)+24=23$ and $(-1) \times 24=-24$ Replacing the middle term $23 y$ by $-...
Read More →A race track is in the form of a rig whose inner circumference is 352 m and outer circumference is 396 m.
Question: A race track is in the form of a rig whose inner circumference is 352 m and outer circumference is 396 m. Find the width and the area of the track. Solution: Letrm andRm be the radii of the inner and outer tracks.Now, Circumference of the outer track $=2 \pi R$ $\Rightarrow 396=2 \times \frac{22}{7} \times R$ $\Rightarrow R=\frac{396 \times 7}{44}$ $\Rightarrow R=63$ Circumference of the inner track $=2 \pi r$ $\Rightarrow 352=2 \times \frac{22}{7} \times r$ $\Rightarrow r=\frac{352 \t...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:28 31x 5x2 Solution: The given expression is $28-31 \mathrm{x}-5 \mathrm{x}^{2}$. (Coefficient of $\mathrm{x}^{2}=-5$, coefficient of $\mathrm{x}=-31$ and constant term $=28)$ We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is $-31$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $(-5) \times(28)=-140$. Now, $(-35)+4=-31$ and $(-35) \ti...
Read More →The value of the determinant
Question: The value of the determinant $\Delta=\left|\begin{array}{ccc}\sec ^{2} \theta \tan ^{2} \theta 1 \\ \tan ^{2} \theta \sec ^{2} \theta -1 \\ 22 20 2\end{array}\right|$ is Solution: Given: $\Delta=\left|\begin{array}{ccc}\sec ^{2} \theta \tan ^{2} \theta 1 \\ \tan ^{2} \theta \sec ^{2} \theta -1 \\ 22 20 2\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}\sec ^{2} \theta \tan ^{2} \theta 1 \\ \tan ^{2} \theta \sec ^{2} \theta -1 \\ 22 20 2\end{array}\right|$ Applying $C_{2} \rightarrow...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:7x2 19x 6 Solution: The given expression is $7 \mathrm{x}^{2}-19 \mathrm{x}-6$. (Coefficient of $\mathrm{x}^{2}=7$, coefficient of $\mathrm{x}=-19$ and constant term $=$ $-6)$ We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is $-19$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $7 \times(-6)=-42$. Now, $(-21)+2=-19$ and $(-21) \times ...
Read More →The value of the determinant
Question: The value of the determinant $\Delta=\left|\begin{array}{ccc}1 2 3 \\ 4 5 6 \\ 3 x 6 x 9 x\end{array}\right|$ is__________ Solution: Given: $\Delta=\left|\begin{array}{ccc}1 2 3 \\ 4 5 6 \\ 3 x 6 x 9 x\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}1 2 3 \\ 4 5 6 \\ 3 x 6 x 9 x\end{array}\right|$ Taking $(3 x)$ common from $R_{3}$ $\Rightarrow \Delta=(3 x)\left|\begin{array}{lll}1 2 3 \\ 4 5 6 \\ 1 2 3\end{array}\right|$ $\Rightarrow \Delta=(3 x)(0)$ ( $\because$ The value of deter...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:7x 6 2x2 Solution: The given expression is $7 \mathrm{x}-6-2 \mathrm{x}^{2}$. (Coefficient of $\mathrm{x}^{2}=-2$, coefficient of $\mathrm{x}=7$ and constant term$=-6$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is 7 and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $(-2) \times(-6)=12$. Now, $4+3=7$ and $4 \times 3=12$ Replacing th...
Read More →Prove the following
Question: If bisectors of $\angle A$ and $\angle B$ of a quadrilateral $A B C D$ intersect each other at $P$, of $\angle B$ and $\angle C$ at $Q$, of $\angle C$ and $\angle D$ at $R$ and of $\angle D$ and $\angle A$ at $S$, then $P Q R S$ is a (a) rectangle (b) rhombus (c) parallelogram (d) quadrilateral whose opposite angles are supplementary Solution: (d) Given, $A B C D$ is a quadrilateral and all angles bisectors form a quadrilateral $P Q R S$. We know that, sum of all angles in a quadrilate...
Read More →A path of 8 m width runs around the outside of a circular park whose radius is 17 m.
Question: (i) A path of 8 m width runs around the outside of a circular park whose radius is 17 m. Find the area of the path.(ii) A park is of the shape of a circle of diameter 7 m. It is surrounded by a path of width of 0.7 m. Find the expenditure of cementing the path, if its cost is ₹ 110 per sq m. Solution: (i) The radius (r) of the inner circle is 17 m.The radius (R) of the outer circle is 25 m. [Includes path, i.e., (17 + 8)] Area of the path $=\pi R^{2}-\pi r^{2}$ $=\pi\left(R^{2}-r^{2}\r...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:3x2+ 10x+ 3 Solution: The given expression is $3 \mathrm{x}^{2}+10 \mathrm{x}+3 . \quad$ (Coefficient of $\mathrm{x}^{2}=3$, coefficient of $\mathrm{x}=10$ and constant term $=3$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is 10 and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i. e., $3 \times 3=9$. Now, $9+1=10$ and $9 \times 1=9$ Repla...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:2x2 3x 2 Solution: The given expression is $2 x^{2}-3 x-2$. (Coefficient of $\mathrm{x}^{2}=2$, coefficient of $\mathrm{x}=-3$ and constant term $=-2$ ) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is $-3$ and their product equals the product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $2 \times(-2)=-4$. Now, $(-4)+1=-3$ and $(-4) \times 1=-4$ Replacing the middle t...
Read More →Find the area of a ring whose outer and inner radii are respectively 23 cm and 12 cm.
Question: Find the area of a ring whose outer and inner radii are respectively 23 cm and 12 cm. Solution: Letr1cm andr2cm be the radii of the outer and inner boundaries of the ring, respectively.We have: $r_{1}=23 \mathrm{~cm}$ $r_{2}=12 \mathrm{~cm}$ Now, Area of the outer ring $=\pi r_{1}^{2}$ $=\frac{22}{7} \times 23 \times 23$ $=1662.57 \mathrm{~cm}^{2}$ Area of the inner ring $=\pi r_{2}{ }^{2}$ $=\frac{22}{7} \times 12 \times 12$ $=452.57 \mathrm{~cm}^{2}$ Area of the ring = Area of the ou...
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