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2. A particle moves along the curve \[ \begin{array}{l}x=2 t^{2}, \quad y=t^{2}-4 t \quad \text { and } \quad z=3 t-5 \\ \text { where } t \text { is the time. Find the components of its velocity and acceleration at time } t=1 \text {, in the } \\ \text { direction } \hat{i}-3 \hat{j}+2 \hat{k} \text {. } \\ \text { (Nagpur, Summer 200l) Ans. } \frac{8 \sqrt{14}}{7},-\frac{\sqrt{14}}{7}\end{array} \]
Calculus Jan 22, 2025
2. A particle moves along the curve \[ \begin{array}{l}x=2 t^{2}, \quad y=t^{2}-4 t \quad \text { and } \quad z=3 t-5 \\ \text { where } t \text { is the time. Find the components of its velocity and acceleration at time } t=1 \text {, in the } \\ \text { direction } \hat{i}-3 \hat{j}+2 \hat{k} \text {. } \\ \text { (Nagpur, Summer 200l) Ans. } \frac{8 \sqrt{14}}{7},-\frac{\sqrt{14}}{7}\end{array} \]
Calculus Jan 22, 2025
2. A particle moves along the curve \[ \begin{array}{l}x=2 t^{2}, \quad y=t^{2}-4 t \quad \text { and } \quad z=3 t-5 \\ \text { where } t \text { is the time. Find the components of its velocity and acceleration at time } t=1 \text {, in the } \\ \text { direction } \hat{i}-3 \hat{j}+2 \hat{k} \text {. } \\ \text { (Nagpur, Summer 200l) Ans. } \frac{8 \sqrt{14}}{7},-\frac{\sqrt{14}}{7}\end{array} \]
Calculus Jan 22, 2025
1 The coovdinate of the end point of a diancia of 1 circle zre \( \langle-5,3\rangle \) and \( \langle 3,-4\rangle \) Find 2. Equation of the curcle. 1) Lengtis of the diameter c Equation of the diameter
Geometry Jan 22, 2025
33. Solve \( \sin \theta=\tan \theta \). (A) \( 200^{\circ} \) (B) \( 90^{\circ} \) (C) \( 60^{\circ} \) (D.) \( 0^{\circ} \) 34. The sum of the first three terms of an Arithmetic progression (A.P.) is 18 . If the first term is 4 . Find their product. (A) 260 (B) 210 (C) 192 (D) 130 35. Evaluate \( \lim _{x \rightarrow 3}\left(\frac{x^{2}-2 x-3}{x-3}\right) \). (A) 4 (B) 3 (C) 2 (D) 0 36. Given that \( \overrightarrow{A B}=5 i+3 j \) and \( \overrightarrow{A C}=2 i+5 j \), find \( \overrightarrow{B C} \). (A) \( 3 i-2 j \) (B) \( -7 i-8 j \) (C) \( -3 i+2 j \) (D) \( 3 i+8 j \) 37. Find the derivative of \( 3 x^{2}+\frac{1}{x^{2}} \). (A) \( 6 x-\frac{1}{2 x^{2}} \) (B) \( 6 x-\frac{2}{x^{3}} \) (C) \( 6 x-\frac{1}{2 x} \) (D) \( 6 x+2 x^{2} \) 38. A body starts from rest and moves in a straight line with a uniform acceleration of \( 5 \mathrm{mls}{ }^{2} \). How far, in \( m \), does it go in 10 seconds? (A) 50 m (B) 250 m (C) 350 m (D) 500 m 39. If \( n \) items are arranged two at a time, the number obtained is 20 . Find the value of \( n \). (A) 40 (B) 15 (C) 10 (D) 5 40. The gradient of point \( P \) on the curve \( y=3 x^{2}-x+3 \) is 5 . Find the coordinates of \( P \). (A) \( (1,5) \) (B) \( (1,7) \) (C) \( (1,13) \) (D) \( (1,17) \) 41 Evaluate \( \frac{1}{1-\sin 60^{\circ}} \) leaving your answer in surd form. (A) \( 4-2 \sqrt{3} \) (B) \( 2-\sqrt{3} \) (C) \( 1-\sqrt{3} \) (D) \( 4+2 \sqrt{3} \) 42. Calculate, correct to one decimal place, the standard deviation of the numbers \( 85,84,83,86 \) and 87 . (A) 1.8 (B) 1.6 (C) \( \cdot 1.4 \) (D) 2.0 43. Find the value of \( x \) at the point of intersection of the curve \( y=x^{2}+2 x-3 \) and the line \( y+x=1 \). (A) \( (1,-2) \) (B) \( \left(0,,^{4}\right) \) (C) \( (1,-4) \) (D) \( (2,3) \) 44. Given that \( n=3 \), evaluate \( \frac{1}{(n-1)!}-\frac{1}{(n+1)!} \) (A) 12 (B) \( 2 \frac{1}{2} \) (C) \( \frac{11}{24} \) (D) 2
Calculus Jan 22, 2025
3. Choose \( y \) in order that the vectors \( \vec{a}=7 \hat{i}+y \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+\hat{k} \), \( \vec{c}=5 \hat{i}+3 \hat{j}+\hat{k} \) are linearly dependent.
Other Jan 22, 2025
3. Choose \( y \) in order that the vectors \( \vec{a}=7 \hat{i}+y \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+\hat{k} \), \( \vec{c}=5 \hat{i}+3 \hat{j}+\hat{k} \) are linearly dependent.
Other Jan 22, 2025
3 Write the equation at a line perpendicular to \( x-2 y+6=0 \) and Passing, through \( (3,-2) \)
Algebra Jan 22, 2025
23. Given that \( =x^{2}-3 x+9 \), find the value of \( \frac{d y}{d x} \) at \( (0,-3) \). (A) \( -\frac{1}{2} \) (B) \( -\frac{1}{4} \) (C) \( \frac{1}{4} \) (D) \( \frac{1}{2} \) 24. The mean of four numbers is 5 and the mean of another three numbers is 12 . Find the mean of seven numbers. (A) 10 (B) 9 (C) 8 (D) 7 25. Find the gradient of the normal to the curve \( =x^{3}-x^{2} \), at the point where \( x=2 \). (A) \( \frac{1}{8} \) (B) \( -\frac{1}{8} \) (C) 1 (D) \( -\frac{1}{24} \) 26. Which of the following is not an equation of a circle? (A) \( 3 x^{2}+3 y^{2}+5 x+7 y=5 \) (B) \( x^{2}+y^{2}- \) 27. Find the equation of the straight line that passes through \( (2,-5) \) and perpendicular to the line \( 3 x+2 y-4=0 \). (A) \( 3 y+2 x+5=0 \) (B) \( 3 y-2 x+15=0 \) (C) \( 3 y+2 x-19=0 \) 28. Calculate, correct to one decimal place, the length of the line joining points \( X(3,5) \) and \( (5,1) \). (A) 4.0 (D) \( 3 y-2 x+19=0 \) (B) 4.5 (C) 4.7 (D) 5.0 29. Find the unit vector in the direction of \( -5 i+12 j \). (A) \( \frac{1}{13}(-5 i-12 j) \) (B) \( \frac{1}{13}(5 i-12 j) \) (C) \( \frac{1}{13}(5 i+ \) 30. A curve is given by \( y=x^{2}-4 x-12 \). Find the axis of symmetry of the curve. (A) \( x=2 \) (B) \( y=2 \) (C) \( x=-2 \) (D) \( y=-2 \) 31. Find the coefficient of the 6 th term in the binomial expansion of \( \left(1-\frac{2 x}{3}\right)^{10} \) in ascending powers of \( x \) (A) \( -\frac{896 x^{6}}{9} \) (B) \( -\frac{896 x^{6}}{27} \) (C) \( -\frac{896 x^{5}}{27} \) (D) \( -\frac{896 x^{5}}{9} \) 32. The probabilities that Atta and Tunde will hit a target in a shooting contest are \( \frac{1}{6} \) and \( \frac{1}{9} \) respectively. Find the probability that only one of them will hit the target. (A) \( \frac{1}{54} \) (B) \( \frac{13}{54} \) (C) \( \frac{20}{27} \) (D) \( \frac{41}{54} \)
Calculus Jan 22, 2025
13. A body of mass 42 kg increases its speed from \( 15 \mathrm{~m} / \mathrm{s} \) to \( 43 \mathrm{~m} / \mathrm{s} \) in 12 seconds. Find the force acting c the body. (A) 52 N (B) 98 N (C) 150 N (D) 203 N 14. Find the minimum value of \( =x^{2}+6 x-12 \). (A) -12 (B) -6 (C) -3 (D) -21 15. Given that \( y=2 x-1 \) and \( \Delta x=0.1 \), find \( \Delta y \). (A) 0.20 (B) 0.15 (C) 0.10 (D) 0.05 16. An exponential sequence (G.P.) is given by \( \frac{9}{2}, \frac{3}{4}, \frac{1}{8}, \ldots \ldots \). Find its sum to infinity. (A) \( 13 \frac{1}{2} \) (B) \( 4 \frac{1}{5} \) (C) \( € \) (D) \( 5 \frac{2}{5} \) 17. If \( r=i+2 j \) and \( =-I+3 j \), find \( |2 n-r| \). (A) 8.5 (B) 5.0 (C) 4.0 (D) 3.6 18. Find the area of a circle whose equation is \( { }^{2}+y^{2}-4 \cdot x+8 y+11=0 \). (A) \( 3 \pi \) (B) \( 9 \pi \) (C) \( \epsilon \) 19. Find the fourth term in the binomial expansion of \( (3 x-y)^{6} \). (A) \( -540 x^{3} y^{3} \) (B) \( -27 x^{3} \) (D) \( 12 \pi \) (C) \( 540 x^{4} y^{2} \) (D) \( -540 x^{4} y^{2} \) 20. Given that \( f(x)=x^{2}+3 x+1 \), find the value of \( x \) at the turning point. (A) 2 (B) \( 1 \frac{1}{2}(C \) (C) \( -1 \frac{1}{2} \) 21. In how many ways can 12 people be seated on a bench if only 5 spaces are available? (A) 950 2 (B) 11880 (C) 792 (D) 495 22. If \( K x^{2}-12 x+K=0 \) has equal roots, find the values of \( K \). \( (A)+2(B)+3 \) \( (C)+.6 \) (D) +9
Physics Jan 22, 2025
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