General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections – A, B, C and D. (iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. (iv) Use of calculated is not permitted.
Question 1
A number is chosen at random from the number –3, –2, –1, 0, 1, 2, 3. What will be the probability that square of this number is less then or equal to 1? VIEW SOLUTION
Question 2
If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k? VIEW SOLUTION
Question 3
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? VIEW SOLUTION
Question 4
Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere? VIEW SOLUTION
Question 5
Prove that tangents drawn at the ends of a diameter of a circle are parallel to each other. VIEW SOLUTION
Question 6
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 cm, QC = QD = 3 cm, then find PC + PD. VIEW SOLUTION
Question 7
Find the roots of the quadratic equation $\sqrt{2}{x}^{2}+7x+5\sqrt{2}=0$. VIEW SOLUTION
Question 8
Find how many integers between 200 and 500 are divisible by 8. VIEW SOLUTION
Question 9
Find the value of k for which the equation x^{2} + k(2x + k − 1) + 2 = 0 has real and equal roots. VIEW SOLUTION
Question 10
Draw a line segment of length 7 cm and divide it internally in the ratio 2 : 3. VIEW SOLUTION
Question 11
The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is $\left(\frac{7}{2},y\right),$ find the value of y. VIEW SOLUTION
Question 12
Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles. VIEW SOLUTION
Question 13
Two different dice are thrown together. Find the probability that the numbers obtained
If the m^{th} term of an A. P. is $\frac{1}{n}$ and n^{th}^{ }term is $\frac{1}{m}$ then show that its (mn)^{th} term is 1. VIEW SOLUTION
Question 19
A metallic solid sphere of radius 10.5 cm is melted and recasted into smaller solid cones, each of radius 3.5 cm and height 3 cm. How many cones will be made? VIEW SOLUTION
Question 20
From the top of a 7 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of its foot is 45°. Find the height of the tower. VIEW SOLUTION
Question 21
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 the height of standing water used for irrigating the park. Write your views on recycling of water. VIEW SOLUTION
Question 22
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.
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. VIEW SOLUTION
Question 24
A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle. VIEW SOLUTION
Question 25
Prove that the lengths of tangents drawn from an external point to a circle are equal. VIEW SOLUTION
Question 26
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. VIEW SOLUTION
Question 27
If $a\ne b\ne 0$, prove that the points (a, a^{2}), (b, b^{2}) (0, 0) will not be collinear. VIEW SOLUTION
Question 28
Draw a right triangle in which the sides (other than the hypotenuse) are of lengths 4 cm and 3 cm. Now construct another triangle whose sides are $\frac{3}{5}$ times the corresponding sides of the given triangle. VIEW SOLUTION
Question 29
If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero. VIEW SOLUTION
Question 30
Two points A and B are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60° and 45° respectively. If the height of the tower is 15 m, then find the distance between the points. VIEW SOLUTION
Question 31
The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is $\frac{1}{27}$ of the given cone, then at what height it is cut from its base? VIEW SOLUTION