A quadrilateral in which a pair
Question: A quadrilateral in which a pair of opposite sides is parallel is-. Solution: trapezium We know that, in a trapezium, one pair of sides is parallel....
Read More →__________ is a regular quadrilateral.
Question: __________ is a regular quadrilateral. Solution: Square Since in square, all the sides are of equal length and all angles are equal....
Read More →If the solve the problem
Question: If $x=2 \cos t-\cos 2 t, y=2 \sin t-\sin 2 t$, find $\frac{d^{2} y}{d x^{2}}$ at $t=\frac{\pi}{2}$ Solution: Formula: (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{d}{d x} \cos x=\sin x$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \sin \mathrm{x}=-\cos \mathrm{x}$ (iv) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{\mathrm{n}}=\mathrm{nx}^{\mathrm{n}-1}$ (v) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\mathrm{wou})}{\mathrm{dt}} \cdot \frac{\mathrm...
Read More →The sum of all exterior angles
Question: The sum of all exterior angles of a polygon is. Solution: 360 As the sum of all exterior angles of a polygon is 360....
Read More →The sum of interior angles of
Question: The sum of interior angles of a polygon of n sides is __________right angles. Solution: The sum of interior angles of a polygon of n sides is 2n 4 right angles....
Read More →A regular polygon is a polygon
Question: A regular polygon is a polygon whose all sides are equal and allare equal. Solution: angles In a regular polygon, all sides are equal and all angles are equal....
Read More →A polygon is a simple closed
Question: A polygon is a simple closed curve made up of only. Solution: line segments , Since a simple closed curve made up of only line segments is called a polygon....
Read More →The number of diagonals
Question: The number of diagonals in a hexagon is __________. Solution: The number of diagonals in a hexagon is 9. We know that, The number of diagonals in a polygon of n sides is n(n 3)/2 Where n = 6 Then, = 6 (6 3)/2 = 6 3/2 = 18/2 = 9...
Read More →If the solve the problem
Question: If $y=500 e^{7 x}+600 e^{-7 x}$, show that $\frac{d^{2} y}{d x^{2}}=49 y$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{ax}}\right)}{\mathrm{dx}}=\mathrm{ae}^{\mathrm{ax}}$ (iii) $\frac{d}{d x} x^{n}=n x^{n-1}$ (iv) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\text { wou })}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}=\frac{\mathrm{dw}}{\mathrm{ds}} \cdot \frac{\mathrm{ds}}{\m...
Read More →The name of three-sided
Question: The name of three-sided regular polygon is-. Solution: equilateral triangle, as polygon is regular, i.e. length of each side is same....
Read More →The measure of each angle
Question: The measure of each angle of a regular pentagon is __________. Solution: The measure of each angle of a regular pentagon is 108. We know that, the sum of all the angles of a polygon is (n 2) 180o. Where n is the number of sides in the polygon, Then, pentagon has 5 sides, i.e. n = 5 So, (n 2) 180o (5 2) 180o 3 180o 540o Measure of each angle = 540o/5 = 108o...
Read More →A quadrilateral that is not a parallelogram
Question: A quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure is. Solution: kite By the property of a kite, we know that, it has two opposite angles of equal measure....
Read More →The number of sides of a regular polygon,
Question: The number of sides of a regular polygon, where each exterior angle has a measure of 36o, is __________. Solution: The number of sides of a regular polygon, where each exterior angle has a measure of 36o, is 10. We know that, the measure of each exterior angle of a regular polygon is 360o/n. Where n is the number of sides in the polygon, Then, exterior angle has a measure of 36o So, 36o= 360o/n n = 360o/36o n = 10...
Read More →The measure of each exterior angle
Question: The measure of each exterior angle of a regular polygon of 18 sides is __________. Solution: The measure of each exterior angle of a regular polygon of 18 sides is 20o. We know that, the measure of each exterior angle of a regular polygon is 360o/n. Where n is the number of sides in the polygon, Then, polygon has 18 sides, i.e. n = 18 So, 360o/18 = 20o...
Read More →Sum of the angles of a hexagon
Question: Sum of the angles of a hexagon is __________. Solution: Sum of the angles of a hexagon is 720o. We know that, the sum of all the angles of a polygon is (n 2) 180o. Where n is the number of sides in the polygon, Then, hexagon has 6 sides, i.e. n = 6 So, (n 2) 180o (6 2) 180o 4 180o 720o...
Read More →The measure of each exterior angle
Question: The measure of each exterior angle of a regular pentagon is __________. Solution: The measure of each exterior angle of a regular pentagon is 72o. We know that, the measure of each exterior angle of a regular pentagon is 360o/n. Where n is the number of sides in the polygon, Then, pentagon has 5 sides, i.e. n = 5 So, 360o/5 = 72o...
Read More →A manufacturer of TV sets produced 6000 units in the third year and 7000
Question: A manufacturer of TV sets produced 6000 units in the third year and 7000 units in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the production (i) in the first year, (ii) in the $10^{\text {th }}$ year, (iii) in 7 years. Solution: Hint: - In the question it is mentioned that the production increases by a fixed number every year. So it is an A.P. $\left(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots \ldots a_{n-1}, a_{n}\right)$. Given:...
Read More →If the solve the problem
Question: If $y=e^{a \cos ^{-1} x}$, prove that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}(\log \mathrm{x})}{\mathrm{dx}}=\frac{1}{\mathrm{x}}$ (ii) $\frac{\mathrm{d}\left(\cos ^{-1} \mathrm{x}\right)}{\mathrm{dx}}=\frac{-1}{\sqrt{1+\mathrm{x}^{2}}}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{\mathrm{n}}=\mathrm{nx}^{\mathrm{n}-1}$ (iv) chain rule $\f...
Read More →The sum of all __________
Question: The sum of all __________ of a quadrilateral is 360o. Solution: The sum of all angles of a quadrilateral is 360o....
Read More →The diagonals of the quadrilateral
Question: The diagonals of the quadrilateral DEFG are __________ and __________. Solution: The diagonals of the quadrilateral DEFG are DF and EG....
Read More →In quadrilateral WXYZ,
Question: In quadrilateral WXYZ, the pairs of opposite angles are __________. Solution: In quadrilateral WXYZ, the pairs of opposite angles are W and Y, X and Z....
Read More →In quadrilateral ROPE,
Question: In quadrilateral ROPE, the pairs of adjacent angles are __________. Solution: In quadrilateral ROPE, the pairs of adjacent angles are RO and OP, OP and PE, PE and ER, ER and RO....
Read More →If the solve the problem
Question: If $y=\cos ^{-1} x$, find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ alone. Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{d\left(\cos ^{-1} x\right)}{d x}=\frac{-1}{\sqrt{1+x^{2}}}$ (iii) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\text { wous })}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}=\frac{\mathrm{dw}}{\mathrm{ds}} \cdot \frac{\mathrm{ds}}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}$ Given: $-y=\co...
Read More →In quadrilateral HOPE,
Question: In quadrilateral HOPE, the pairs of opposite sides are __________. Solution: In quadrilateral HOPE, the pairs of opposite sides are HO and PE, HE and OP....
Read More →A man arranges to pay off a debt of ₹36000 by 40 annual instalments which
Question: A man arranges to pay off a debt of ₹36000 by 40 annual instalments which form an AP. When 30 of the instalments are paid, he dies, leaving one - third of the debt unpaid. Find the value of the first instalment. Solution: Given: - Total debt = Rs. 36000 A man pays this debt in 40 annual instalments that forms an A.P. After annual instalments, that man dies leaving one - third of the debt unpaid. So, Within 30 instalments he pays two - thirds of his debt. Sum of $n$ terms in an Arithmet...
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