# CBSE CLASS 10 MATHS SAMPLE PAPER -3 2018 WITH ANSWERS

## Practice Paper – 3

Time Allowed : 3 Hours
Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) This model test paper contains 30 questions.
(iii) Questions 1 to 6 in section – A are very short answers type questions carrying mark each. (iv)Questions 7 to 12 in section – B are short answer type questions carrying 2 marks each.
(v) Questions 13 to 22 in section – C are long answer – I type questions carrying 3 marks each.
(vi) Questions 23 to 30 in section – D are long answer – II type questions carrying 4 marks each.

Section – A
1. Find the value of ‘a’ for which point $P\left(\frac{a}{3},2\right)$ is the mid point of the line segment joining the points Q( -5, 4) and P( -1, 0).
2. Write whether $\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt{5}}$ on simplification is rational or irrational number.
3. $\mathrm{tan}{1}^{0}\mathrm{tan}{2}^{0}\mathrm{tan}{3}^{0}.............\mathrm{tan}{89}^{0}$
4. Check if $\frac{x}{x-1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{x-1}{x}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{5}{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ is a quadratic equation.
5. ABC is an isosceles angled at C. Prove that $A{B}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2A{C}^{2}$
6. In the given figure , if PQ is a tangent then find the value of ∠POQ + ∠QPO.

Section - B
7. Solve : $\begin{array}{l}mx-ny={m}^{2}+{n}^{2},\\ x+y=2m\end{array}$
8. If the roots of the equation $\left(a-b\right){x}^{2}+\left(b-c\right)x+\left(c-a\right)=0$ are equal then prove that $2a=b+c$ .
9. The nth term of an A.P. is 5n + 2 . find the common difference.
10. If $\mathrm{tan}A=\sqrt{2}-1$ , Show that $\frac{\mathrm{tan}\text{\hspace{0.17em}}A}{1+{\mathrm{tan}}^{2}A}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\sqrt{2}}{4}$
11. Two right circular cones X and Y are made. X having three times the radius of Y and Y having half the volume of X. Calculate the ratio of height of X and Y.
12. Determine if the points (1, 5), (2, 3) and (-2, 1) are collinear.

Section – C

13. Show that $\left(1+\frac{1}{{\mathrm{tan}}^{2}\theta }\right)\left(1+\frac{1}{{\mathrm{cot}}^{2}\theta }\right)=\frac{1}{{\mathrm{sin}}^{2}\theta -{\mathrm{sin}}^{4}\theta }$
14. Find the mean age of 100 students of a town from the following data.

15. What is the area of a sector of a circle whose radius is ‘r’ and length of the arc is ‘l’ ?
16. The following figure represents a solid consisting of cylinder surmounted by a cone at one end and a hemisphere at the other end.

Given that the common radius is 3.5 cm , the length of cylinder is 6.5 cm and the total length is 12.8 cm , calculate the volume of the solid correct to the nearest integer.
17.Two coins are tossed simultaneously. Find the probability of getting –
18. Construct a triangle of side 5 cm , 6 cm and 7 cm and then a triangle similar to it whose sides are 4/5 of corresponding sides of the first triangle.
19. A man on the top of a 60m high building, observes angles of depression of two points on the same side of the building as 300 and 600, If one point is directly behind the other, Find the distance between the two points.
20. In the given figure, AB ꓕ BC and DE ꓕ AC. Prove that triangle ABC is similar to triangle AED.

21. A train covers a distance of 90km/s at a uniform speed if would have taken 30 minutes less if the speed had been 15km/hr more. Calculate the original duration of the journey.
22. The ninth term of A.P. is equal to seven times the second term and twelfth term exceeds five times the third term by 2. Find the first term and common difference.

Section – D

23. The points A(1, -2), B(2, 3), C(k, 2) and D(-4, -3) are the vertices of a parallelogram . Find the value of k and the altitude of the parallelogram corresponding to the base AB.
24. The radii of two concentric circles are 13cm and 8cm AB is a diameter of the bigger circle and BD is tangent to the smaller circle touching it at D and intersecting the larger circle at P, on producing , find the length of AP.
25. The table given below shows the percentage distribution of female technicians in the IT sector of various states and union territories (U. T.) of India. Find The mean percentage of female technicians by the step deviation method.

26. A round balloon of radius r subtends an angle α at the eye of the observer while the angle of elevation of its centre is β. Prove that the height of the centre of the balloon is $r\mathrm{sin}\beta .\mathrm{cos}ec\frac{\alpha }{2}$
27. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
28. A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16cm with radii of its lower and upper ends as 8cm and 20cm respectively find the cost of the bucket if the cost of metal sheet used is ₹15 per 100 square cm.
29. Prove that the perpendicular from origin to the line joining the points $\left(c\mathrm{cos}\alpha ,\text{\hspace{0.17em}}c\mathrm{sin}\alpha \right)\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\left(c\mathrm{cos}\beta ,\text{\hspace{0.17em}}c\mathrm{sin}\beta \right)$ bisects it as well.
30. A class of 20 boys and 15 girls is divided into n groups so that each group has x boys and y girls find x, y and n. What values are referred in class.

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