# BHARDWAJ CLASSES

## CBSE Class 10 Maths Sample Paper / Practice Paper

### Practice Paper – 4

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. Check whether the sequence defined by ${a}_{n}=2n-1$ is an A.P.
2. Evaluate: $\frac{{\mathrm{tan}}^{2}{60}^{0}+4{\mathrm{sin}}^{2}{45}^{0}+3{\mathrm{sec}}^{2}{30}^{0}+5{\mathrm{cos}}^{2}{90}^{0}}{\mathrm{cos}ec{30}^{0}+\mathrm{sec}{60}^{0}-{\mathrm{cot}}^{2}{30}^{0}}$
3. Find k for which ${x}^{2}+9k\text{\hspace{0.17em}}x+25=0$ has no real roots.
4. After how many decimal places will the decimal expansion of $\frac{{4}^{3}}{{2}^{4}\text{\hspace{0.17em}}{5}^{3}}$ terminate.
5. A letter of English alphabates is chosen at random.What is the probability that it is a letter of the word “MATHEMATICS” ?
6. A cuboidal matal plate of 1 cm thickness 9 cm breadth and 81 cm length is melted into a cube. Find the total surface area of cube.

Section – B

7. The side DC of square ABCD is produced to E, prove that $A{E}^{2}=C{E}^{2}+2DC×DE$
8.Find the value of $x$ if ,$\mathrm{tan}3x=\mathrm{sin}{45}^{0}.\mathrm{cos}{45}^{0}+\mathrm{sin}{30}^{0}$
9. Solve: $\frac{xy}{x+y}=\frac{1}{2},\text{\hspace{0.17em}}\frac{xy}{x-y}=\frac{1}{6},\text{\hspace{0.17em}}x+y\ne 0,\text{\hspace{0.17em}}x\ne y$
10. Can we have any natural no. ‘n’ for which 6n ends with the digit ‘0’ ?Justify your answer.
11. Distance between $A\left(x,y\right)$ and $B\left(-4,7\right)$ is $\sqrt{41}$ , Find $x\text{\hspace{0.17em}}and\text{\hspace{0.17em}}y$ . If $A\text{'}s$ ordinate is thrice of its abscissa.
12. The arithmetic mean of a set of 40 values is 65. If each of the 40 values is increased by 5, what will be the mean of the set of new values?

Section - C

13. If $a{\mathrm{cos}}^{3}\theta +3\text{\hspace{0.17em}}a\mathrm{cos}\theta .{\mathrm{sin}}^{2}\theta =m\text{\hspace{0.17em}}and\text{\hspace{0.17em}}a{\mathrm{sin}}^{3}\theta +3\text{\hspace{0.17em}}a{\mathrm{cos}}^{2}\theta .\mathrm{sin}\theta =n$ , Prove that ${\left(m+n\right)}^{\frac{2}{3}}+{\left(m-n\right)}^{\frac{2}{3}}=2{a}^{\frac{2}{3}}$
14. Prove that if the height of a tower and the distance of the point of observation from its foot, both are increased by 10 % , then the angle of elevation of its top remains unchanged.
15. Show that $\frac{ar\left(\Delta ABC\right)}{ar\left(\Delta DBC\right)}=\frac{AO}{DO}$ , if in the given figure, ABC and DBC are two triangles on the same base BC ,if AD intersects at O.
16. The dimensions of a metallic cuboid are 100 cm x 80 cm x 64 cm.It is melted and recast into a cube.Find the surface area of the cube.
17. Find the roots of $4{x}^{2}-16\left(p-q\right)x+\left(15{p}^{2}-34pq+15{q}^{2}\right)=0$
18. A bag contains 16 balls out of which $x$ are green. If 8 more green balls are put in the bag, the probability of drawing a green ball will be double than that $\frac{x}{16}$ , find $x$ .
19. If the points $\left(p,q\right),\left(m,n\right)\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\left(p-m,q-n\right)$ are collinear, show that $pn=qm$ .
20. The sum of numerator and denominator of a fraction is 12.If 1 is added to both the numerator and denominator , the fraction becomes $\frac{3}{4}$ .Find the fraction.
21. Construct tangents to a circle of radius 3 cm from a point on the concentric circle of radius 5 cm.
22. The length of the minute hand of a clock is 5 cm, find the area swept by the minute hand during 7:10 am to 7:45 am.

Section – D

23. Find the cubic polynomial with the sum, sum of product of its zeroes taken two at a time and the product of its zeroes are 0, -7, -6 respectively.
24. A train travels a distance of 300 km at a constant speed. If the speed of the train is increased by 5 km an hour, the journey would take 2 hours less. Find the speed of train.
25. The following table shows the ages of the patients admitted in a hospital during a year.
Find the mode and mean of the data. Compare and intercept the two measures of central tendency.
26. A box contains 40 balls of same shape and weight. Among the balls, 10 balls are white and 16 are red and rest are black. What is the probability that the ball drawn is not black.
27. A container open from the top and made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the milk which can completely fill the container at the rate of Rs. 20 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs. 8 per 100 cm2.
28. At ‘t’ minute past 2 pm, the time needed by the minute hand of a clock to show 3 pm was found to be 3 minutes less than $\frac{{t}^{2}}{4}$ minutes. Find ‘t’.
29. Prove that : $\frac{\mathrm{cot}A+\mathrm{cos}ecA-1}{\mathrm{cot}A-\mathrm{cos}ecA+1}=\frac{1+\mathrm{cos}A}{\mathrm{sin}A}$
30. Show that the sum of an A.P. whose first term is ‘a’ ,second term is ‘b’ , and the last term ‘c’ is equal to $\frac{\left(a+c\right)\left(b+c-2a\right)}{2\left(b-a\right)}$