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question_answer1) The ratio of the height of a tower and the length of its shadow on the ground is \[\sqrt{3}:1\]. What is the angle of elevation of the sun?
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question_answer2) Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?
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question_answer3)
| A number is chosen at random from the numbers\[3,\text{ }2,\text{ }1,\text{ }0,\text{ }1,\text{ }2,\text{ }3\]. |
| What will be the probability that square of this number is less then or equal to 1? |
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question_answer4) If the distance between the points \[(4,k)\] and \[(1,0)\] is 5, then what can be the possible values of k?
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question_answer5) Find the roots of the quadratic equation \[\sqrt{2}{{x}^{2}}+7x+5\sqrt{2}=0\].
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question_answer6) Find how many integers between 200 and 500 are divisible by 8?
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question_answer7) Prove that tangents drawn at the ends of a diameter of a circle are parallel to each other.
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question_answer8) Find the value of k for which the equation \[{{x}^{2}}+k(2x+k-1)+2=0\] has real and equal roots.
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question_answer9) Draw a line segment of length 8 cm and divide it internally in the ratio \[4:5\].
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question_answer10)
| In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If\[PA=12\text{ }cm,\text{ }QC=QD=3\text{ }cm\], then find \[PC+PD\]. |
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question_answer11) If \[{{m}^{th}}\] term of an A.P. is \[\frac{1}{n}\] and \[{{n}^{th}}\] term is \[\frac{1}{m}\], then find the sum of its first mn terms.
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question_answer12) Find the sum of n terms of the series \[\left( 4-\frac{1}{n} \right)+\left( 4-\frac{2}{n} \right)+\left( 4-\frac{3}{n} \right)+.......\]
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question_answer13) If the equation \[(1+{{m}^{2}}){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\] has equal roots then show that \[{{c}^{2}}={{a}^{2}}(1+{{m}^{2}})\].
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question_answer14) The \[\frac{3}{4}th\] part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.
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question_answer15)
| In the given figure, OACB is a quadrant of a circle with centre O and radius \[3.5\text{ }cm.\] If \[OD=2\text{ }cm\], find the area of the shaded region. |
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question_answer16) Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that \[\angle PTQ=2\angle OPQ.\]
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question_answer17) Show that \[\Delta \,ABC\], where \[A(-2,0),B(2,0),C(0,2)\] and \[\Delta \text{ }PQR\] where \[P(-4,0),Q(4,0),R(0,4)\] are similar triangles.
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question_answer18) The area of a triangle is 5 sq. units. Two of its vertices are (2, 1) and (\[3,\text{ }2\]). If the third vertex is \[\left( \frac{7}{2},y \right)\], find the value of y.
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question_answer19)
| Two different dice are thrown together. Find the probability that the numbers obtained |
| (i) have a sum less than 7 |
| (ii) have a product less than 16 |
| (iii) is a doublet of odd numbers. |
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question_answer20) A moving boat is observed from the top of a \[150\text{ }m\] high cliff moving away from the cliff. The angle of depression of the boat changes from \[60{}^\circ \] to \[45{}^\circ \] in 2 minutes. Find the speed of the boat in m/h.
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question_answer21) Construct an isosceles triangle with base \[8\text{ }cm\] and altitude\[4\text{ }cm\]. Construct another triangle whose sides are \[\frac{2}{3}\] times the corresponding sides of the isosceles triangle.
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question_answer22) Prove that the lengths of tangents drawn from an external point to a circle are equal.
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question_answer23)
| The ratio of the sums of first m and first n terms of an A. P. is \[{{m}^{2}}:{{n}^{2}}\]. |
| Show that the ratio of its mth and nth terms is \[2(m-1):(2n-1)\]. |
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question_answer24) Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream.
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question_answer25) If \[a\ne b\ne 0\], prove that the points \[(a,{{a}^{2}}),(b,{{b}^{2}})(0,0)\] will not be collinear.
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question_answer26) The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height Find the ratio of the volumes of the two parts.
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question_answer27)
| Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. |
| Who has the better chance to get the number 25. |
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question_answer28) A chord PQ of a circle of radius 10 cm subtends an angle of \[60{}^\circ \] at the centre of circle. Find the area of major and minor segments of the circle.
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question_answer29) The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is \[30{}^\circ \] and the angle of depression of its shadow in water of lake is \[60{}^\circ \]. Find the height of the cloud from the surface of water.
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question_answer30)
| In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square where O and O? are centres of the circles. Find the area of shaded region. |
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question_answer31) In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be height of standing water used for irrigating the park. Write your views on recycling of water.
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