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In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm then the length of AP is

In the given figure, three circles with centres A, B, C respectively touch each other externally. If AB = 5 cm, BC = 7 cm and CA = 6 cm then the radius of the circle with centre A is

In the given figure, DE and DF are tangents from an external point D to a circle with centre A. If DE = 5 cm and DE  DF then the radius of the circle is (a) 3 cm (b) 4 cm (c) 5 cm (d) 6 cm

In the given figure, PA and PB are tangents to the given circle such that PA = 5 cm and APB  60 . The length of chord AB is

Quadrilateral ABCD is circumscribed to a circle. If AB= 6 cm, BC = 7cm and CD = 4cm then the length of AD is (a) 3 cm (b) 4 cm (c) 6 cm (d) 7 cm

In the given figure, ABC is right-angled at B such that BC = 6 cm and AB = 8 cm. A circle with centre O has been inscribed the triangle. OP  AB,OQ  BCand OR  AC. If OP = OQ = OR = x cm then x = ?

In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and AD at P, Q, R and S respectively. If the radius of the circle is 10 cm, BC = 38 cm, PB = 27 cm and AD  CD then the length of CD is

In a right triangle ABC, right angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle is (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm

In the given figure, O is the centre of a circle, AB is a chord and AT is the tangent at A. If AOB 100 then BAT is equal to

In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 6 cm, BP = 5 cm, CQ = 3 cm and DR = 4 cm then perimeter of quad. ABCD is

In the given figure, quad. ABCD is circumscribed touching the circle at P, Q, R and S. If AP = 5 cm, BC= 7 c m and CS = 3 cm. Then, the length of AB = ?

In the given figure, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT= 3.8 cm then the length of QR is

In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of PQR  189 cm2 then the length of side of PQ is (a) 17.5 cm (b) 20 cm (c) 22.5 cm (d) 25 cm

In the given figure, O is the centre of a circle; PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that SQL  50 and DE  DF OQ  BC and OR  AC.

To draw a pair of tangents to a circle, which are inclined to each other at an angle of 45, we have to draw tangents at the end points of those two radii, the angle between which is (a) 105 (b) 135 (c) 140 (d) 145

In the given figure, a circle touches the side DF of EDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm then the perimeter of EDF is

In the given figure, O is the centre of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T. If PBO  30 then PTA  ?

The length of the tangent form an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is (a) 8 cm (b)

In the given figure, PQR is a tangent to the circle at Q, whose centre is O and AB is a chord parallel to PR such that BQR  70 . Then, AQB  ? (a) 20 (b) 35 (c) 40 (d) 45

O is the centre of a circle of radius 5 cm. At a distance of 13 cm form O, a point P is taken. From this point, two tangents PQ and PR are drawn to the circle. Then, the area of quad. PQOR is

In the given figure, two circles touch each other at C and AB is a tangent to both the circles. The measure of ACB is

In the given figure, PQ is a tangent to a circle with centre O, A is the point of contact. If PAB  67 , then the measure of

In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If PAO  30 then CPB ACP is equal to

In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If APQ  58 then the measure of PQB is

If PA and PB are two tangents to a circle with centre O such that APB  80 . Then, AOP  ?

In the given figure, PQ and PR are tangents to a circle with centre A. If QPA  27 then QAR equals

In the given figure, PQ and PR are tangents to a circle with centre A. If QPA  27 then QAR equals

If two tangents inclined at an angle of 60 are drawn to a circle of a radius 3 cm then the length of each tangent is

In the given figure, PA and PB are two tangents to th4e circle with centre O. If APB  60 then OAB is

In the given figure, O is the centre of a circle and PT is the tangent to the circle. If PQ is a chord such that QPT  50then POQ  ?

In the given figure, If AOD  135 then BOC equal to

In the given figure, the length of BC is

If PA and PB are two tangents to a circle with centre O such that AOB  110 then APB is equal to

In the given figure, AT is a tangent to the circle with center O such that OT = 4 cm and OTA  30, Then, AT  ?

In the given figure, O is the center of a circle, PQ is a chord and Pt is the tangent at P. If POQ  70 , then TPQ is equal to

In the given figure, 0 is the centre of a circle, AOC is its diameter such that ACB  50. If AT is the tangent to the circle at the point A, then BAT  ?

In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA=17 cm, then the length of AC (in cm) is

In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of chord AB is

If a chord AB subtends an angle of 60 at the center of a circle, then he angle between the tangents to the circle drawn form A and B is (a) 30 (b) 60 (c) 90 (d) 120

In the given figure, AB and AC are tangents to the circle with center O such that BAC  40 . Then , BOC = 40.

PQ is a tangent to a circle with centre O at the point P. If OQP is equal to

In the given figure, point P is 26 cm away from the center O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then, the radius of the circle is

In the given figure, PT is tangent to the circle with centre O. If OT = 6 cm and OP = 10 cm then the length of tangent PT is

The chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is

Which of the following pairs of lines in a circle cannot be parallel?
(a) two chords (b) a chord and tangent (c) two tangents (d) two diameters

Answer: (d) two diameters Sol: Two diameters cannot be parallel as they perpendicularly bisect each other.

In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT =24 cm. If O is the centre of the circle, then length OP = ?

(a) 30 cm (b) 28 cm (c) 25 cm (d) 18 cm Answer: (c) 25 cm Sol: The tangent at any point of a circle is perpendicular to the radius at the...

In the given figure, RQ is a tangent to the circle with centre O, If SQ = 6 cm and QR = 4 cm. then OR is equal to

The number of tangents that can be drawn form an external point to a circle is (a) 1 (b) 2 (3) (d) 4

In the given figure, PA and PB are two tangents to the circle with centre O. If APB  60 , then find the measure of OAB.

13. In the given figure, PQ is chord of a circle with centre O an PT is a tangent. If QPT  60, find the PRQ.

In two concentric circles, a chord of length 8cm of the large circle touches he smaller circle. If the radius of the larger circle is 5cm then find the radius of the smaller circle.

In the given figure, O is the centre of the circle. PA and PB are tangents. Show that AOBP is cyclic quadrilateral.

In the given figure, a cradle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14cm, BC = 8cm and CA=12 cm. Find the length AD, BE and CF.

BD = 5cm = BE Solving (3) and (4), we get and AD = 9cm

In the given figure, two tangents RQ, and RP and RP are drawn from an external point R to the circle with centre O. If PRQ 120 , then prove that OR = PR + RQ.

Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre. Sol:

Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle (in cm) which touches the smaller circle.

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of sides AB and AC.

If PT is a tangent to a circle with center O and PQ is a chord of the circle such that QPT  70, then find the measure of POQ.

In the given figure common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that AB=CD.

In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If TPQ  70 , find the TRQ.

In the given figure, PA and PB are two tangents to the circle with centre O. If APB  50 then what is the measure of OAB.

In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.

Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. \ AB + CD = AD + BC Þ 6 + 8 = AD = 9 Þ AD = 5 cm...

In the given figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If PBT  30 , prove that BA : AT = 2 : 1.

In the given figure, a circle with center O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29cm, AD = 23cm, B  90 and DS=5cm then find the radius of the circle.

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

PQ is a chord of length 4.8 cm of a circle of radius 3cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BC and DC into which BC is divided by the point of contact D, are of lengths 6cm and 9cm respectively. If the area of sides AB and AC. ABC  54cm2 then find the lengths of

In the given figure, O is the centre of the two concentric circles of radii 4 cm and 6cm respectively. AP and PB are tangents to the outer and inner circle respectively. If PA = 10cm, find the length of PB up to one place of the decimal.

In the given figure, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that point of contact P bisects the base BC.

In the given figure, PA and PB are the tangent segemtns to a circle with centre O. Show that he points A, O, B and P are concyclic.

A circle is inscribed in a ABC touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR=7cm and CR=5cm, find the length of BC.

From an external point P, tangents PA and PB are drawn to a circle with center O. If CD is the tangent to the circle at a point E and PA = 14cm, find the perimeter of PCD

In the given figure, the chord AB of the larger of the two concentric circles, with center O, touches the smaller circle at C. Prove that AC = CB.

In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6cm, BC=7cm and CD=4 cm. Find AD.

In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length of AD, BE and CF.

Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the larger circle which touches the smaller circle.

A point P is 25 cm away from the center of a circle and the length of tangent drawn from P to the circle is 24 cm. Find the radius of the circle.

Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from the center of the circle

CBSE, Circles, Class 10, Maths, RS Aggarwal

Using properties of determinants prove that: $\left|\begin{array}{ccc} (b+c)^{2} & a b & c a \\ a b & (a+c)^{2} & b c \\ a c & b c & (a+b)^{2} \end{array}\right|=2 a b c(a+b+c)^{3}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $=\left|\begin{array}{ccc} b^{2}+c^{2}+2 b c & a b & a c \\ a b & a^{2}+c^{2}+2 a c & b c \\ a c & b c & a^{2}+b^{2}+2 a b \end{array}\right|$ Operating $R_{1}...

Using properties of determinants prove that: $\left|\begin{array}{ccc} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{array}\right|=(a+b+c)\left(a^{2}+b^{2}+c^{2}\right)$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: Operating $\mathrm{C}_{1} \rightarrow \mathrm{aCl}_{1}$ $\frac{1}{a}\left|\begin{array}{ccc} a^{2} & b-c & c+b \\ a^{2}+a c & b & c-a \\ a^{2}-a b & a+b & c...

Using properties of determinants prove that: $\left|\begin{array}{ccc} 1+a^{2}-b^{2} & 2 a b & -2 b \\ 2 a b & 1-a^{2}+b^{2} & 2 a \\ 2 b & -2 a & 1-a^{2}-b^{2} \end{array}\right|=\left(1+a^{2}+b^{2}\right)^{3}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: Operating $R_{1} \rightarrow R_{1}+b R_{3}, R_{2} \rightarrow R_{2^{-}} a R_{3}$ $\begin{array}{l} \left|\begin{array}{ccc} 1+a^{2}-b^{2}+2 b^{2} & 2 a b-2 a b & -2 b+b-a^{2}...

Using properties of determinants prove that: $\left|\begin{array}{ccc} b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right|=4 a^{2} b^{2} c^{2}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right| \\...

Using properties of determinants prove that: $\left|\begin{array}{lll} (b+c)^{2} & a^{2} & b c \\ (c+a)^{2} & b^{2} & c a \\ (a+b)^{2} & c^{2} & a b \end{array}\right|=\left(a^{2}+b^{2}+c^{2}\right)(a-b)(b-c)(c-a)(a+b+c)$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{lll} (b+c)^{2} & a^{2} & b c \\ (c+a)^{2} & b^{2} & c a \\ (a+b)^{2} & c^{2} & a b \end{array}\right| \\...

Using properties of determinants prove that: $\begin{array}{l} \left|\begin{array}{ccc} (\mathrm{m}+\mathrm{n})^{2} & 1^{2} & \mathrm{mn} \\ (\mathrm{n}+1)^{2} & \mathrm{~m}^{2} & \ln \\ (1+\mathrm{m})^{2} & \mathrm{n}^{2} & \operatorname{lm} \end{array}\right|=\left(1^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}\right)(1-\mathrm{m}) \\ (\mathrm{m}-\mathrm{n})(\mathrm{n}-1) \end{array}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\left|\begin{array}{ccc}(\mathrm{m}+\mathrm{n})^{2} & \mathrm{l}^{2} & \mathrm{mn} \\ (\mathrm{n}+\mathrm{l})^{2} & \mathrm{~m}^{2} & \mathrm{ln} \\ (1+\mathrm{m})^{2}...

Using properties of determinants prove that: $\left|\begin{array}{ccc} (x-2)^{2} & (x-1)^{2} & x^{2} \\ (x-1)^{2} & x^{2} & (x+1)^{2} \\ x^{2} & (x+1)^{2} & (x+2)^{2} \end{array}\right|=-8$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} (\mathrm{x}-2)^{2} & (\mathrm{x}-1)^{2} & \mathrm{x}^{2} \\ (\mathrm{x}-1)^{2} & \mathrm{x}^{2} & (\mathrm{x}+1)^{2} \\...

Using properties of determinants prove that: $\left|\begin{array}{ccc} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{array}\right|=2(a+b)(b+c)(c+a)$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{array}\right| \\ =\left|\begin{array}{ccc} a+b & a+b &...

Using properties of determinants prove that: $\left|\begin{array}{ccc} a & a+2 b & a+2 b+3 c \\ 3 a & 4 a+6 b & 5 a+7 b+9 c \\ 6 a & 9 a+12 b & 11 a+15 b+18 c \end{array}\right|=-a^{3}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\left|\begin{array}{ccc}a & a+2 b & a+2 b+3 c \\ 3 a & 4 a+6 b & 5 a+7 b+9 c \\ 6 a & 9 a+12 b & 11 a+15 b+18 c\end{array}\right|$...

Using properties of determinants prove that: $\left|\begin{array}{lll} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{array}\right|=3 a b c-a^{3}-b^{3}-c^{3}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{lll} \mathrm{b}+\mathrm{c} & \mathrm{a}-\mathrm{b} & \mathrm{a} \\ \mathrm{c}+\mathrm{a} & \mathrm{b}-\mathrm{c} & \mathrm{b} \\...

Using properties of determinants prove that: $\left|\begin{array}{ccc} \mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x}^{2} & \mathrm{y}^{2} & \mathrm{z}^{2} \\ \mathrm{x}^{3} & \mathrm{y}^{3} & \mathrm{z}^{3} \end{array}\right|=\mathrm{xyz}(\mathrm{x}-\mathrm{y})(\mathrm{y}-\mathrm{z})(\mathrm{z}-\mathrm{x})$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x}^{2} & \mathrm{y}^{2} & \mathrm{z}^{2} \\ \mathrm{x}^{3} & \mathrm{y}^{3}...

Using properties of determinants prove that: $\left|\begin{array}{ccc} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z \end{array}\right|=3(x+y+z)(x y+y z+z x)$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z \end{array}\right| \\ =\left|\begin{array}{ccc} x+y+z & -x+y...

Using properties of determinants prove that: $\left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right|=9 y^{2}(x+y)$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{x} & \mathrm{x}+\mathrm{y} & \mathrm{x}+2 \mathrm{y} \\ \mathrm{x}+2 \mathrm{y} & \mathrm{x} & \mathrm{x}+\mathrm{y} \\...

Using properties of determinants prove that: $\left|\begin{array}{ccc} \mathrm{x}+\lambda & 2 \mathrm{x} & 2 \mathrm{x} \\ 2 \mathrm{x} & \mathrm{x}+\lambda & 2 \mathrm{x} \\ 2 \mathrm{x} & 2 \mathrm{x} & \mathrm{x}+\lambda \end{array}\right|=(5 \mathrm{x}+\lambda)(\lambda-\mathrm{x})^{2}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{x}+\lambda & 2 \mathrm{x} & 2 \mathrm{x} \\ 2 \mathrm{x} & \mathrm{x}+\lambda & 2 \mathrm{x} \\ 2 \mathrm{x} & 2...

Using properties of determinants prove that: $\left|\begin{array}{lll} \mathrm{x} & \mathrm{a} & \mathrm{a} \\ \mathrm{a} & \mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{a} & \mathrm{x} \end{array}\right|=(\mathrm{x}+2 \mathrm{a})(\mathrm{x}-\mathrm{a})^{2}$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{lll} \mathrm{x} & \mathrm{a} & \mathrm{a} \\ \mathrm{a} & \mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{a} & \mathrm{x}...

Using properties of determinants prove that: $\left|\begin{array}{ccc} \mathrm{a}+\mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{a}+\mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{y} & \mathrm{a}+\mathrm{z} \end{array}\right|=\mathrm{a}^{2}(\mathrm{a}+\mathrm{x}+\mathrm{y}+\mathrm{z})$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{a}+\mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{a}+\mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{y}...

Using properties of determinants prove that: $\left|\begin{array}{ccc} 1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q \end{array}\right|=1$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} 1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q \end{array}\right|\\ =\left|\begin{array}{ccc} -1 &...

Using properties of determinants prove that: $\left|\begin{array}{lll} 1 & \mathrm{~b}+\mathrm{c} & \mathrm{b}^{2}+\mathrm{c}^{2} \\ 1 & \mathrm{c}+\mathrm{a} & \mathrm{c}^{2}+\mathrm{a}^{2} \\ 1 & \mathrm{a}+\mathrm{b} & \mathrm{a}^{2}+\mathrm{b}^{2} \end{array}\right|=(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{llll} 1 & \mathrm{~b}+\mathrm{c} & \mathrm{b}^{2}+\mathrm{c}^{2} \\ 1 & \mathrm{c}+\mathrm{a} & \mathrm{c}^{2}+\mathrm{a}^{2} \\ 1...

Using properties of determinants prove that: $\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ b c & c a & a b \end{array}\right|=(a-b)(b-c)(c-a)$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\begin{array}{l} \left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ b c & c a & a b \end{array}\right| \\ =\left|\begin{array}{ccc} 0 & 0 & 1 \\ a-b &...

Evaluate : $\left|\begin{array}{lll} 1^{2} & 2^{2} & 3^{2} \\ 2^{2} & 2^{2} & 4^{2} \\ 3^{2} & 4^{2} & 5^{2} \end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\left|\begin{array}{lll} 1^{2} & 2^{2} & 3^{2} \\ 2^{2} & 3^{2} & 4^{2} \\ 3^{2} & 4^{2} & 5^{2} \end{array}\right|=\left|\begin{array}{ccc} 1 & 4 & 9 \\ 4...

Evaluate : $\left|\begin{array}{ccc} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\left|\begin{array}{ccc} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{array}\right|=6 \times\left|\begin{array}{ccc} 17 & 18 & 6 \\ 1 & 6 & 4 \\...

Evaluate: $\left|\begin{array}{lll} 29 & 26 & 22 \\ 25 & 31 & 27 \\ 63 & 54 & 46 \end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6B, Maths, RS Aggarwal

Solution: $\left|\begin{array}{lll}29 & 26 & 22 \\ 25 & 31 & 27 \\ 63 & 54 & 46\end{array}\right|$ $=\left|\begin{array}{ccc}4 & -5 & -5 \\ 25 & 31 & 27 \\ 63...

Evaluate $\left|\begin{array}{cc}\sqrt{3} & \sqrt{5} \\ -\sqrt{5} & 3 \sqrt{3}\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: $\left|\begin{array}{cc}\sqrt{3} & \sqrt{5} \\ -\sqrt{5} & 3 \sqrt{3}\end{array}\right| \cdot=3 \sqrt{3} \times \sqrt{3}-(-\sqrt{5} \times \sqrt{5})$ $=14$

For what value of $x$, the given matrix $A=\left[\begin{array}{cc}3-2 x & x+1 \\ 2 & 4\end{array}\right]$ is a singular matrix?

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: For $A$ to be singular matrix its determinant should be equal to 0 . $\begin{array}{l} 0=(3-2 x) \times 4-(x+1) \times 2 \\ 0=12-8 x-2 x-2 \\ 0=10-10 x \\ x=1 \end{array}$

Without expanding the determinant, prove that $\left|\begin{array}{ccc}41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3\end{array}\right|=0$. SINGULAR MATRIX A square matrix $A$ is said to be singular if $|A|=0$. Also, $A$ is called non singular if $|A| \neq 0$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: We know that $C_{1} \Rightarrow C_{1}-C_{2}$, would not change anything for the determinant. Applying the same in above determinant, we get $\left[\begin{array}{lll}40 & 1 & 5 \\...

Evaluate $\left|\begin{array}{lll}0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: We know that expansion of determinant with respect to first row is $a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13}$. $0(3 \times 6-5 \times 4)-2(2 \times 6-4 \times 4)+0(2 \times 5-4 \times 3)$...

Evaluate $\left|\begin{array}{ll}\cos 15^{\circ} & \sin 15^{\circ} \\ \sin 75^{\circ} & \cos 75^{\circ}\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: $\begin{array}{l} \cos 15^{\circ} \cos 75^{\circ}-\sin 75^{\circ} \sin 15^{\circ} \\ =\cos \left(15^{\circ}+75^{\circ}\right) \because \cos A \cos B-\sin A \sin B=\cos (A+B) \\ =\cos...

Evaluate $\left|\begin{array}{cc}\cos 65^{\circ} & \sin 65^{\circ} \\ \sin 25^{\circ} & \cos 25^{\circ}\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: By directly opening this determinant $\begin{array}{l} \cos 65^{\circ} \times \cos 25^{\circ}-\sin 25^{\circ} \times \sin 65^{\circ} \\ =\cos \left(65^{\circ}+25^{\circ}\right) \because...

Evaluate $\left|\begin{array}{ll}\sin 60^{\circ} & \cos 60^{\circ} \\ -\sin 30^{\circ} & \cos 30^{\circ}\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: After finding determinant we will get, $\begin{array}{l} \operatorname{Sin} 60^{\circ}=\frac{\sqrt{3}}{2}=\cos 30^{\circ} \\ \operatorname{Cos} 60^{\circ}=\frac{1}{2}=\sin 30^{\circ} \\...

Evaluate $\left|\begin{array}{cc}2 \cos \theta & -2 \sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: After finding determinant we will get a trigonometric identity. $\begin{array}{l} 2 \cos ^{2} \theta+2 \sin ^{2} \theta \\ =2 \\ \because \sin ^{2} \theta+\cos ^{2} \theta=1...

Evaluate $\left|\begin{array}{ll}\sqrt{6} & \sqrt{5} \\ \sqrt{20} & \sqrt{24}\end{array}\right|$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: Find determinant $\begin{array}{l} \sqrt{6} \times \sqrt{24-\sqrt{2}} 20 \times \sqrt{5} \\ \sqrt{1} 144-\sqrt{1} 100 \\ =12-10 \\ =2 \end{array}$

If $\mathrm{A}=\left[\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right]$, find the value of $3|\mathrm{~A}|$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: Find the determinant of $A$ and then multiply it by 3 $\begin{array}{l} |A|=2 \\ 3|A|=3 \times 2 \\ =6 \end{array}$

If $\left|\begin{array}{cc}2 x & x+3 \\ 2(x+1) & x+1\end{array}\right|=\left|\begin{array}{ll}1 & 5 \\ 3 & 3\end{array}\right|$, write the value of $x$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: Simply by equating both sides we can get the value of $x$. $\begin{array}{l} 2 x^{2}+2 x-2\left(x^{2}+4 x+3\right)=-12 \\ \Rightarrow-6 x-6=-12 \\ \Rightarrow-6 x=-6 \\ \Rightarrow x=1...

If $\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}6 & -2 \\ 7 & 3\end{array}\right|$, write the value of $x$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: This question is having the same logic as above. $\begin{array}{l} 2 x^{2}-40=18+14 \\ \Rightarrow 2 x^{2}=72 \\ \Rightarrow x^{2}=36 \\ \Rightarrow x=\pm 6 \end{array}$

If $\left|\begin{array}{cc}3 x & 7 \\ -2 & 4\end{array}\right|=\left|\begin{array}{ll}8 & 7 \\ 6 & 4\end{array}\right|$, write the value of $x$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: Here the determinant is compared so we need to take determinant both sides then find $\mathrm{x}$. $\begin{array}{l} 12 x+14=32-42 \\ \Rightarrow 12 x=-10-14 \\ \Rightarrow 12 x=-24 \\...

Evaluate $\left|\begin{array}{cc}\mathrm{a}+\mathrm{ib} & \mathrm{c}+\mathrm{id} \\ -\mathrm{c}+\mathrm{id} & \mathrm{a}-\mathrm{ib}\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: This we can very simply go through directly. $\begin{array}{l} ((a+i b)(a-i b))-((-c+i d)(c+i d)) \\ \Rightarrow\left(a^{2}+b^{2}\right)-\left(-c^{2}-d^{2}\right) \\ \Rightarrow...

Evaluate $\left|\begin{array}{cc}\mathrm{x}^{2}-\mathrm{x}+1 & \mathrm{x}-1 \\ \mathrm{x}+1 & \mathrm{x}+1\end{array}\right|$

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: Theorem: This evaluation can be done in two different ways either by taking out the common things anc then calculating the determinants or simply take determinant. I will prefer first...

Let $A$ be a square matrix of order 3, write the value of $|2 A|$, where $|A|=4$.

CBSE, Class 12, Determinants, Maths, RS Aggarwal

Solution: Theorem: If $A$ be $k \times k$ matrix then $|p A|=p^{k}|A|$. Given: $p=2, k=3$ and $|A|=4$ $\begin{array}{l} |2 A|=2^{3} \times|A| \\ =8 \times 4 \\ =32 \end{array}$

If $A$ is a $2 \times 2$ matrix such that $|A| \neq 0$ and $|A|=5$, write the value of $|4 A|$.

CBSE, Class 12, Determinants, Exercise 6A, Maths, RS Aggarwal

Solution: Theorem: If $A$ be $k \times k$ matrix then $|p A|=p^{k}|A|$. Given, $\mathrm{p}=4, \mathrm{k}=2$ and $|\mathrm{A}|=5$. $\begin{array}{l} |4 \mathrm{~A}|=4^{2} \times 5 \\ =16 \times 5 \\...

In a four-sided field, the length of the longer diagonal is 128 m. The lengths of perpendiculars from the opposite vertices upon this diagonal are 22.7 m and 17.3 m. Find the area of the field.

The adjacent sides of a parallelogram are 36 cm and 27 cm in length. If the distance between the shorter sides is 12 cm, find the distance between the longer sides.

The diagonals of a rhombus are 48 cm and 20 cm long. Find the perimeter of the rhombus.

A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120 m and 44 m. If one of the sides of the || gm is 66 m long, find its corresponding altitude.

Find the area of quadrilateral ABCD in which AB = 42cm, BC = 21 cm,CD = 29 cm, DA = 34 cm and diagram BD = 20cm.

The cost of fencing a square lawn at 14 per metre is 2800. Find the cost of mowing the lawn at ₹ 54 per 100 m2.

The adjacent sides of a ||gm ABCD measure 34 cm and 20 cm and the diagonal AC is 42 cm long. Find the area of the ||gm.

Find the area of a trapezium whose parallel sides are 11 cm and 25 cm long and non- parallel sides are 15 cm and 13 cm.

Find the area of a rhombus each side of which measures 20 cm and one of whose diagonals is 24 cm.

A lawn is in the form of a rectangle whose sides are in the ratio 5:3 and its area is Find the cost of fencing the lawn at ₹ 20 per metre.

The longer side of a rectangular hall is 24 m and the length of its diagonal is 26 m. Find the area of the hall.

Find the area of an isosceles triangle each of whose equal sides is 13 cm and whose base is 24 cm.

Find the area of an equilateral triangle having each side of length 10 cm. (Take

The parallel sides of a trapezium are 9.7cm and 6.3 cm, and the distance between them is 6.5 cm. The area of the trapezium is (a) 104 cm2 (b) 78 cm2 (c) 52 cm2 (d) 65 cm2

The sides of a triangle are in the ratio 12: 14 : 25 and its perimeter is 25.5 cm. The largest side of the triangle is (a) 7 cm (b) 14 cm (c) 12.5 cm (d) 18 cm

In the given figure ABCD is a trapezium in which AB =40 m, BC=15m,CD = 28m, AD= 9 m and CE = AB. Area of trapezium ABCD is

In the given figure ABCD is a quadrilateral in which

Find the area of trapezium whose parallel sides are 11 m and 25 m long, and the nonparallel sides are 15 m and 13 m long.

The shape of the cross section of a canal is a trapezium. If the canal is 10 m wide at the top, 6 m wide at the bottom and the area of its cross section is 640 m2 , find the depth of the canal.

The parallel sides of trapezium are 12 cm and 9cm and the distance between them is 8 cm. Find the area of the trapezium.

The area of rhombus is 480 c m2 , and one of its diagonal measures 48 cm. Find
(i) the length of the other diagonal,
(ii) the length of each of the sides
(iii) its perimeter

The perimeter of a rhombus is 60 cm. If one of its diagonal us 18 cm long, find (i) the length of the other diagonal, and (ii) the area of the rhombus.

Find the area of the rhombus, the length of whose diagonals are 30 cm and 16 cm. Also, find the perimeter of the rhombus.

The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of the parallelogram.

The area of a parallelogram is 392 m2 . If its altitude is twice the corresponding base, determined the base and the altitude.

The adjacent sides of a parallelogram are 32 cm and 24 cm. If the distance between the longer sides is 17.4 cm, find the distance between the shorter sides.

Find the area of a parallelogram with base equal to 25 cm and the corresponding height measuring 16.8 cm.

Sol: Given: Base = 25 cm Height = 16.8 cm \Area of the parallelogram = Base ´ Height = 25cm ´16.8 cm = 420 cm2

Find the area of the quadrilateral ABCD in which in AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diagonal BD = 20 cm.

Find the perimeter and area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, ACB  90 and AC = 15 cm.

Find the area of the quadrilateral ABCD in which AD = 24 cm, BAD  90 and BCD is an equilateral triangle having each side equal to 26 cm. Also, find the perimeter of the quadrilateral. Sol:

In the given figure ABCD is quadrilateral in which diagonal BD = 24 cm, AL  BD and CM  BD such that AL = 9cm and CM = 12 cm. Calculate the area of the quadrilateral.

The cost of fencing a square lawn at ₹ 14 per meter is ₹ 28000. Find the cost of mowing the lawn at ₹ 54per100 m2

The cost of harvesting a square field at ₹ 900 per hectare is ₹ 8100. Find the cost of putting a fence around it at ₹ 18 per meter.

The area of a square filed is 8 hectares. How long would a man take to cross it diagonally by walking at the rate of 4 km per hour?

Find the length of the diagonal of a square whose area is 128 cm2 . Also, find its perimeter.

Find the area and perimeter of a square plot of land whose diagonal is 24 m long.

The cost of painting the four walls of a room 12 m long at ₹ 30 per m2 is ₹ 7560 per m2 and the cost of covering the floor with the mat at ₹ dimensions of the room.

The dimensions of a room are 14 m x 10 m x 6.5 m There are two doors and 4 windows in the room. Each door measures 2.5 m x 1.2 m and each window measures 1.5 m x 1 m. Find the cost of painting the four walls of the room at ₹ 35 per m2 .

A 80 m by 64 m rectangular lawn has two roads, each 5 m wide, running through its middle, one parallel to its length and the other parallel to its breadth. Find the cost of gravelling the reads at ₹ 40 per m2 .

A carpet is laid on floor of a room 8 m by 5 m. There is border of constant width all around the carpet. If the area of the border is 12 m2

A room 4.9 m long and 3.5 m board is covered with carpet, leaving an uncovered margin of 25 cm all around the room. If the breadth of the carpet is 80 cm, find its cost at ₹ 80 per metre.

The length and breadth of a rectangular garden are in the ratio 9:5. A path 3.5 m wide, running all around inside it has an area of 1911m2 . Find the dimensions of the garden.

A footpath of uniform width runs all around the inside of a rectangular field 54m long and 35 m wide. If the area of the path is 420 m2 , find the width of the path.

A rectangular plot measure 125 m by 78 m. It has gravel path 3 m wide all around on the outside. Find the area of the path and the cost of gravelling it at ₹ 75 per m2

A rectangular park 358 m long and 18 m wide is to be covered with grass, leaving 2.5 m uncovered all around it. Find the area to be laid with grass.

The area of rectangle is 192cm2 and its perimeter is 56 cm. Find the dimensions of the rectangle.

A 36-m-long, 15-m-borad verandah is to be paved with stones, each measuring 6dm by 5 dm. How many stones will be required?

The floor of a rectangular hall is 24 m long and 18 m wide. How many carpets, each of length 2.5 m and breadth 80 cm, will be required to cover the floor of the hall?

A room is 16 m long and 13.5 m broad. Find the cost of covering its floor with 75-m-wide carpet at ₹ 60 per metre.

A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. The area of the lawn is 3375m2 . Find the cost of fencing the lawn at ₹ 65 per metre.

The area of a rectangular plot is 462m2is length is 28 m. Find its perimeter

One side of a rectangle is 12 cm long and its diagonal measure 37 cm. Find the other side and the area of the rectangle.

The length of a rectangular park is twice its breadth and its perimeter is 840 m. Find the area of the park.

The perimeter of a rectangular plot of land is 80 m and its breadth is 16 m. Find the length and area of the plot.

In the given figure, ABC is an equilateral triangle the length of whose side is equal to 10 cm, and ADC is right-angled at D and BD= 8cm. Find the area of the shaded region

Find the area and perimeter of an isosceles right angled triangle, each of whose equal sides measure 10cm.

Each of the equal sides of an isosceles triangle measure 2 cm more than its height, and the base of the triangle measure 12 cm. Find the area of the triangle.

The base of an isosceles triangle measures 80 cm and its area is 360 cm2. Find the perimeter of the triangle.

Find the length of the hypotenuse of an isosceles right-angled triangle whose area is 200cm2 . Also, find its perimeter

Find the area of a right – angled triangle, the radius of whose, circumference measures 8 cm and the altitude drawn to the hypotenuse measures 6 cm.

The base of a right – angled triangle measures 48 cm and its hypotenuse measures 50 cm. Find the area of the triangle.

If the area of an equilateral triangle is 81

11. If the area of an equilateral triangle is 36 Sol: cm2 , find its perimeter.

The height of an equilateral triangle is 6 cm. Find its area.

Each side of an equilateral triangle is 10 cm. Find (i) the area of the triangle and (ii) the height of the triangle.

The length of the two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 c m2 , find the perimeter of the triangle.

The difference between the sides at the right angles in a right-angled triangle is 7 cm. the area of the triangle is 60 c m2 . Find its perimeter.

The perimeter of a right triangle is 40 cm and its hypotenuse measure 17 cm. Find the area of the triangle.

The perimeter of a triangular field is 240m, and its sides are in the ratio 25:17:12. Find the area of the field. Also, find the cost of ploughing the field at ₹ 40 per m2